Update paper for newer submission template

This reverts commit cb81979a7f.

coil_mag_solvers.yml

@@ -1,21 +1,3 @@
-Coil_Solver:
-  Equation: CoilSolver
-  Procedure: '"CoilSolver" "CoilSolver"'
-  Linear System Solver: Iterative
-  Linear System Preconditioning: ILU1
-  Linear System Max Iterations: 1000
-  Linear System Convergence Tolerance: 1e-10
-  Linear System Iterative Method: BiCGStab
-  Linear System Residual Output: 10
-  Steady State Convergence Tolerance: 1e-06
-  Normalize Coil Current: True
-  Nonlinear System Consistent Norm: True
-  Coil Closed: True
-  Narrow Interface: True
-  Save Coil Set: True
-  Save Coil Index: True
-  Calculate Elemental Fields: True
-
 Static_Current_Conduction:
   Equation: Static Current Conduction
   Variable: Potential

coil_parasitics.py

@@ -208,13 +208,13 @@ def inductance(mesh_file, sim_dir, solver_method):
     fr4 = elmer.load_material('fr4', sim, 'coil_mag_materials.yml')
     copper = elmer.load_material('copper', sim, 'coil_mag_materials.yml')
 
-    solver_coil = elmer.load_solver('Coil_Solver', sim, 'coil_mag_solvers.yml')
+    solver_current = elmer.load_solver('Static_Current_Conduction', sim, 'coil_mag_solvers.yml')
     solver_magdyn = elmer.load_solver('Magneto_Dynamics', sim, 'coil_mag_solvers.yml')
     if solver_method:
         solver_magdyn.data['Linear System Iterative Method'] = solver_method
     solver_magdyn_calc = elmer.load_solver('Magneto_Dynamics_Calculations', sim, 'coil_mag_solvers.yml')
 
-    copper_eqn = elmer.Equation(sim, 'copperEqn', [solver_coil, solver_magdyn, solver_magdyn_calc])
+    copper_eqn = elmer.Equation(sim, 'copperEqn', [solver_current, solver_magdyn, solver_magdyn_calc])
     air_eqn = elmer.Equation(sim, 'airEqn', [solver_magdyn, solver_magdyn_calc])
 
     bdy_trace = elmer.Body(sim, 'trace', [physical['trace'][1]])
@@ -237,20 +237,20 @@ def inductance(mesh_file, sim_dir, solver_method):
     bdy_if_bottom.material = copper
     bdy_if_bottom.equation = copper_eqn
 
-    comp_coil = elmer.Component(sim, 'Coil', [bdy_trace])
+    potential_force = elmer.BodyForce(sim, 'electric_potential', {'Electric Potential': 'Equals "Potential"'})
-    comp_coil.data['Desired Current Density'] = 'Real 1.0'
+    bdy_trace.body_force = potential_force
-
-    current_force = elmer.BodyForce(sim, 'Source', {
-        'Current Density 1': 'Equals "CoilCurrent e 1"',
-        'Current Density 2': 'Equals "CoilCurrent e 2"',
-        'Current Density 3': 'Equals "CoilCurrent e 3"',
-        })
-    bdy_trace.body_force = current_force
 
     # boundaries
     boundary_airbox = elmer.Boundary(sim, 'FarField', [physical['airbox_surface'][1]])
     boundary_airbox.data['Electric Infinity BC'] = 'True'
 
+    boundary_vplus = elmer.Boundary(sim, 'Vplus', [physical['interface_top'][1]])
+    boundary_vplus.data['Potential'] = 1.0
+    boundary_vplus.data['Save Scalars'] = True
+
+    boundary_vminus = elmer.Boundary(sim, 'Vminus', [physical['interface_bottom'][1]])
+    boundary_vminus.data['Potential'] = 0.0
+
     with tempfile.TemporaryDirectory() as tmpdir:
         tmpdir = sim_dir if sim_dir else Path(tmpdir)
 
@@ -261,8 +261,6 @@ def inductance(mesh_file, sim_dir, solver_method):
                    stdout_log=(tmpdir / 'ElmerGrid_stdout.log'),
                    stderr_log=(tmpdir / 'ElmerGrid_stderr.log'))
         solver_stdout, solver_stderr = (tmpdir / 'ElmerSolver_stdout.log'), (tmpdir / 'ElmerSolver_stderr.log')
-        print('Solver stdout:', solver_stdout)
-        print('Solver stderr:', solver_stderr)
         res = elmer_solver(tmpdir,
                    stdout_log=solver_stdout,
                    stderr_log=solver_stderr)

coil_test_board.kicad_prl

@@ -10,6 +10,7 @@
     "opacity": {
       "images": 0.6,
       "pads": 1.0,
+      "shapes": 1.0,
       "tracks": 1.0,
       "vias": 1.0,
       "zones": 0.6
@@ -62,9 +63,10 @@
       35,
       36,
       39,
-      40
+      40,
+      41
     ],
-    "visible_layers": "fffffff_ffffffff",
+    "visible_layers": "ffffffff_ffffffff",
     "zone_display_mode": 0
   },
   "git": {
@@ -75,9 +77,71 @@
   },
   "meta": {
     "filename": "coil_test_board.kicad_prl",
-    "version": 3
+    "version": 4
+  },
+  "net_inspector_panel": {
+    "col_hidden": [
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ],
+    "col_order": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8,
+      9
+    ],
+    "col_widths": [
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0
+    ],
+    "custom_group_rules": [],
+    "expanded_rows": [],
+    "filter_by_net_name": true,
+    "filter_by_netclass": true,
+    "filter_text": "",
+    "group_by_constraint": false,
+    "group_by_netclass": false,
+    "show_unconnected_nets": false,
+    "show_zero_pad_nets": false,
+    "sort_ascending": true,
+    "sorting_column": 0
   },
   "project": {
     "files": []
+  },
+  "schematic": {
+    "selection_filter": {
+      "graphics": true,
+      "images": true,
+      "labels": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pins": true,
+      "symbols": true,
+      "text": true,
+      "wires": true
+    }
   }
 }

coil_test_board.kicad_pro

@@ -6,14 +6,14 @@
         "apply_defaults_to_fp_fields": false,
         "apply_defaults_to_fp_shapes": false,
         "apply_defaults_to_fp_text": false,
-        "board_outline_line_width": 0.049999999999999996,
+        "board_outline_line_width": 0.05,
-        "copper_line_width": 0.19999999999999998,
+        "copper_line_width": 0.2,
         "copper_text_italic": false,
         "copper_text_size_h": 1.5,
         "copper_text_size_v": 1.5,
         "copper_text_thickness": 0.3,
         "copper_text_upright": false,
-        "courtyard_line_width": 0.049999999999999996,
+        "courtyard_line_width": 0.05,
         "dimension_precision": 4,
         "dimension_units": 3,
         "dimensions": {
@@ -24,13 +24,13 @@
           "text_position": 0,
           "units_format": 1
         },
-        "fab_line_width": 0.09999999999999999,
+        "fab_line_width": 0.1,
         "fab_text_italic": false,
         "fab_text_size_h": 1.0,
         "fab_text_size_v": 1.0,
         "fab_text_thickness": 0.15,
         "fab_text_upright": false,
-        "other_line_width": 0.09999999999999999,
+        "other_line_width": 0.1,
         "other_text_italic": false,
         "other_text_size_h": 1.0,
         "other_text_size_v": 1.0,
@@ -41,11 +41,11 @@
           "height": 1.524,
           "width": 1.524
         },
-        "silk_line_width": 0.09999999999999999,
+        "silk_line_width": 0.1,
         "silk_text_italic": false,
         "silk_text_size_h": 1.0,
         "silk_text_size_v": 1.0,
-        "silk_text_thickness": 0.09999999999999999,
+        "silk_text_thickness": 0.1,
         "silk_text_upright": false,
         "zones": {
           "min_clearance": 0.5
@@ -63,15 +63,20 @@
         "copper_edge_clearance": "error",
         "copper_sliver": "warning",
         "courtyards_overlap": "error",
+        "creepage": "error",
         "diff_pair_gap_out_of_range": "error",
         "diff_pair_uncoupled_length_too_long": "error",
         "drill_out_of_range": "error",
         "duplicate_footprints": "warning",
         "extra_footprint": "warning",
         "footprint": "error",
+        "footprint_filters_mismatch": "ignore",
+        "footprint_symbol_mismatch": "warning",
         "footprint_type_mismatch": "ignore",
         "hole_clearance": "error",
         "hole_near_hole": "error",
+        "hole_to_hole": "warning",
+        "holes_co_located": "warning",
         "invalid_outline": "error",
         "isolated_copper": "warning",
         "item_on_disabled_layer": "error",
@@ -98,7 +103,9 @@
         "text_thickness": "warning",
         "through_hole_pad_without_hole": "error",
         "too_many_vias": "error",
+        "track_angle": "error",
         "track_dangling": "warning",
+        "track_segment_length": "error",
         "track_width": "error",
         "tracks_crossing": "error",
         "unconnected_items": "error",
@@ -111,27 +118,29 @@
         "min_clearance": 0.0,
         "min_connection": 0.0,
         "min_copper_edge_clearance": 0.5,
+        "min_groove_width": 0.0,
         "min_hole_clearance": 0.25,
         "min_hole_to_hole": 0.25,
-        "min_microvia_diameter": 0.19999999999999998,
+        "min_microvia_diameter": 0.2,
-        "min_microvia_drill": 0.09999999999999999,
+        "min_microvia_drill": 0.1,
         "min_resolved_spokes": 2,
         "min_silk_clearance": 0.0,
-        "min_text_height": 0.7999999999999999,
+        "min_text_height": 0.8,
         "min_text_thickness": 0.08,
         "min_through_hole_diameter": 0.3,
         "min_track_width": 0.0,
-        "min_via_annular_width": 0.09999999999999999,
+        "min_via_annular_width": 0.1,
         "min_via_diameter": 0.5,
         "solder_mask_to_copper_clearance": 0.0,
         "use_height_for_length_calcs": true
       },
       "teardrop_options": [
         {
-          "td_onpadsmd": true,
+          "td_onpthpad": true,
           "td_onroundshapesonly": false,
+          "td_onsmdpad": true,
           "td_ontrackend": false,
-          "td_onviapad": true
+          "td_onvia": true
         }
       ],
       "teardrop_parameters": [
@@ -206,6 +215,7 @@
       "mfg": "",
       "mpn": ""
     },
+    "layer_pairs": [],
     "layer_presets": [],
     "viewports": []
   },
@@ -234,6 +244,7 @@
         "microvia_drill": 0.1,
         "name": "Default",
         "pcb_color": "rgba(0, 0, 0, 0.000)",
+        "priority": 2147483647,
         "schematic_color": "rgba(0, 0, 0, 0.000)",
         "track_width": 0.2,
         "via_diameter": 0.6,
@@ -242,7 +253,7 @@
       }
     ],
     "meta": {
-      "version": 3
+      "version": 4
     },
     "net_colors": null,
     "netclass_assignments": null,
@@ -257,7 +268,7 @@
       "pos_files": "",
       "specctra_dsn": "",
       "step": "",
-      "svg": "",
+      "svg": "svg_vis",
       "vrml": ""
     },
     "page_layout_descr_file": ""

make_wide_coils.py

@@ -0,0 +1,100 @@
+#!/bin/sh
+
+import math
+import subprocess
+import tempfile
+
+from gerbonara import utils
+from gerbonara.cad.kicad import pcb as kicad_pcb
+from gerbonara.cad.kicad import graphical_primitives as kicad_gr
+from gerbonara.cad.kicad import footprints as kicad_fp
+
+outer_dias = [40, 65, 90]
+
+nks = [[(1, 150),
+        (1, 300),
+        (1, 480)],
+       [(53, 50),
+        (53, 100),
+        (53, 160)],
+       [(53, 100),
+        (53, 200),
+        (53, 320)],
+       [(53, 150),
+        (53, 300),
+        (53, 480)],
+       [(53, 1),
+        (53, 1),
+        (53, 1)]]
+
+def make_pad(x, y, side, w, h, **kwargs):
+    f = kicad_fp.Footprint(name=f'conn_gen_01x{n}', _version=None, generator=None, at=kicad_fp.AtPos(x, y, 0), **kwargs)
+
+    for i in range(n):
+        f.pads.append(kicad_fp.Pad(
+            layers=['F.Mask', 'F.Cu'] if side == 'top' else ['B.Mask', 'B.Cu'],
+            number=str(i+1),
+            type=kicad_fp.Atom.smd,
+            shape=kicad_fp.Atom.rect,
+            at=kicad_fp.AtPos(0, 0, 0),
+            size=kicad_fp.XYCoord(w, h),
+            footprint=f))
+
+    return f
+
+for case_num, nk in enumerate(nks, start=1):
+    board_out = kicad_pcb.Board.empty_board()
+
+    for (i, d2), (n, k) in zip(enumerate(outer_dias), nk):
+        d1 = d2 - 15
+
+        with tempfile.NamedTemporaryFile(suffix='.kicad_pcb') as f:
+            args = 'python twisted_coil_gen_twolayer.py --circle-segments 331 --trace-width 0.13 --via-drill 0.3 --via-diameter 0.6 --via-offset 0.3'.split() + [
+                '--pcb', f.name,
+                '--turns', str(n), '--twists', str(k),
+                '--inner-diameter', str(d1), '--outer-diameter', str(d2),
+                ]
+
+            if k > 100:
+                args += ['--stagger-inner-vias', '--stagger-outer-vias', '--via-offset', '0.4']
+
+            subprocess.run(args, check=True)
+
+            for obj in kicad_pcb.Board.open(f.name).objects():
+                board_out.add(obj)
+
+        board_out.add(kicad_gr.Text(text=f'case {case_num} n={n} k={k}\\nd={d1:.1f}-{d2:.2f}mm\\nt=0.15mm',
+                                at=kicad_gr.AtPos(0, (d1+d2)/2/2), layer=kicad_gr.TextLayer('F.SilkS')))
+
+        # Slots
+        if i < len(outer_dias)-1:
+            na = 8
+            r2 = d2/2
+            r = r2 + 2.5
+            aw = 2*math.asin(3 / 2 / r)
+            for j in range(na):
+                start = utils.rotate_point(r, 0, (j+1)*2*math.pi/na - aw/2)
+                end = utils.rotate_point(r, 0, j*2*math.pi/na + aw/2)
+                arc = kicad_gr.Arc(end=end, center=(0, 0), start=start, stroke=kicad_gr.Stroke(width=1.2), layer='Edge.Cuts')
+                board_out.add(arc)
+
+        # Connecting pads
+        board_out.add(make_pad(d2/2+1, 0, 'top', 2, 0.7))
+        board_out.add(make_pad(d2/2+1, 0, 'bottom', 2, 0.7))
+        board_out.add(kicad_pcb.TrackSegment(start=kicad_pcb.XYCoord(d2/2, 0), end=kicad_pcb.XYCoord(d2/2+1, 0), width=0.15, layer='F.Cu'))
+        board_out.add(kicad_pcb.TrackSegment(start=kicad_pcb.XYCoord(d2/2, 0), end=kicad_pcb.XYCoord(d2/2+1, 0), width=0.15, layer='B.Cu'))
+
+    # Circular board outline
+    board_out.add(kicad_gr.Circle(center=kicad_gr.XYCoord(0, 0), end=kicad_gr.XYCoord(d2/2+5, 0), layer='Edge.Cuts'))
+
+    # Silk crosshairs
+    r = d2/2 * math.sin(math.pi/4)
+    board_out.add(kicad_gr.Line(start=kicad_gr.XYCoord(-r, -r), end=kicad_gr.XYCoord(r, r), layer='F.SilkS'))
+    board_out.add(kicad_gr.Line(start=kicad_gr.XYCoord(r, -r), end=kicad_gr.XYCoord(-r, r), layer='F.SilkS'))
+    board_out.add(kicad_gr.Line(start=kicad_gr.XYCoord(-r, -r), end=kicad_gr.XYCoord(r, r), layer='B.SilkS'))
+    board_out.add(kicad_gr.Line(start=kicad_gr.XYCoord(r, -r), end=kicad_gr.XYCoord(-r, r), layer='B.SilkS'))
+    r = d2/2
+    board_out.add(kicad_gr.Line(start=kicad_gr.XYCoord(0, r), end=kicad_gr.XYCoord(0, -r), layer='B.SilkS'))
+    board_out.add(kicad_gr.Line(start=kicad_gr.XYCoord(r, 0), end=kicad_gr.XYCoord(-r, 0), layer='B.SilkS'))
+
+    board_out.write(f'wide_coil_{case_num}.kicad_pcb')

paper/.gitignore

@@ -0,0 +1,7 @@
+*.bbl
+*.blg
+*.aux
+*.log
+*.bcf
+*.run.xml
+*.out

paper/Makefile

@@ -8,7 +8,7 @@ MAKEFLAGS += --no-builtin-rules
 
 main_tex ?= paper
 
-VERSION_STRING := $(shell git describe --tags --long --dirty)
+VERSION_STRING := $(shell git describe --tags --long)
 
 all: ${main_tex}.pdf
 
@@ -17,8 +17,26 @@ all: ${main_tex}.pdf
 	biber $* 
 	pdflatex -shell-escape $<
 
+paper_biber.bib:
+	biber --output-format=bibtex --output-resolve paper.bcf
+
+submission.zip:
+	rm -f submission.zip
+	zip -r submission.zip paper.tex paper.bib version.tex \
+		figures/klein-spulen-schwingkreise-korbspule.jpg\
+		figures/k_ripple_plot.pdf\
+		figures/nk_combined.pdf\
+		figures/nk_complex_illust.pdf\
+		figures/rms_ripple_double_rotation_n3_r4.pdf\
+		figures/saacke-radiotechnik-3-ledionspule.jpg\
+		figures/svg_vis_paper_plain.png\
+		figures/svg_vis_paper.png\
+		figures/symmetry_3turn_n_twist.pdf\
+		figures/test_schematic.pdf\
+		figures/setup_probe_small.jpg
+
 version.tex: ${main_tex}.tex paper.bib
-	echo "${VERSION_STRING}" > $@
+	echo -n "${VERSION_STRING}" > $@
 
 .PHONY: clean
 clean:

paper/figures/test_schematic.svg

@@ -0,0 +1,825 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+   width="185.356mm"
+   height="93.594002mm"
+   viewBox="0 0 185.356 93.594002"
+   version="1.1"
+   id="svg1"
+   inkscape:version="1.4 (e7c3feb100, 2024-10-09)"
+   sodipodi:docname="test_schematic.svg"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:svg="http://www.w3.org/2000/svg">
+  <sodipodi:namedview
+     id="namedview1"
+     pagecolor="#ffffff"
+     bordercolor="#666666"
+     borderopacity="1.0"
+     inkscape:showpageshadow="2"
+     inkscape:pageopacity="0.0"
+     inkscape:pagecheckerboard="0"
+     inkscape:deskcolor="#d1d1d1"
+     inkscape:document-units="mm"
+     inkscape:zoom="1.6641134"
+     inkscape:cx="244.27422"
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+     inkscape:window-width="3183"
+     inkscape:window-height="1783"
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+       inkscape:collect="always"
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paper/paper.tex

@@ -1,4 +1,4 @@
-\documentclass[conference,compsoc]{IEEEtran}
+\documentclass[journal,10pt,a4paper]{IEEEtran}
 
 \usepackage[T1]{fontenc}
 \usepackage[
@@ -13,51 +13,959 @@
 \usepackage{amssymb,amsmath}
 \usepackage{eurosym}
 \usepackage{wasysym}
-
 \usepackage[binary-units]{siunitx}
-\DeclareSIUnit{\baud}{Bd}
-\DeclareSIUnit{\year}{a}
 \usepackage{commath}
 \usepackage{graphicx,color}
+\usepackage{colortbl}
 \usepackage{subcaption}
+\usepackage{placeins}
 \usepackage{array}
 \usepackage{censor}
 \usepackage{hyperref}
 \usepackage{makecell}
 
+\DeclareSIUnit{\baud}{Bd}
+\DeclareSIUnit{\year}{a}
+\DeclareSIUnit{\rpm}{rpm}
 \renewcommand{\floatpagefraction}{.8}
 \newcommand{\degree}{\ensuremath{^\circ}}
 \newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
 \newcommand{\partnum}[1]{\texttt{#1}}
+\newcommand{\todo}[1]{\textbf{TODO}\footnote{#1}}
+\newlength{\figurescale}
+\setlength{\figurescale}{\linewidth}
 
 \begin{document}
 
-\date{}
+\date{January 27 2025}
+\author{\IEEEauthorblockN{Jan Sebastian Götte},\thanks{Corresponding author. Jan Sebastian Götte is with the Technical
+University of Darmstadt, 64283 Darmstadt, Germany (e-mail: jan.goette@tu-darmstadt.de).}
+\and
+\IEEEauthorblockN{Björn Scheuermann}\thanks{Björn Scheuermann is with the Technical University of Darmstadt,
+64283 Darmstadt, Germany (e-mail: scheuermann@kom.tu-darmstadt.de).}
+\thanks{This work has been funded by the LOEWE initiative (Hesse, Germany) within the emergenCITY center
+    (LOEWE/1/12/519/03/05.001(0016)/72) as well as by Technical University of Darmstadt.}
+}
 \title{Wireless Power Transfer with a Twist:
-Achieving Rotation-Invariant Coupling using Multi-Layer PCB Inductors}
+Achieving Rotation-Invariant Coupling using Twisted Multi-Layer PCB Inductors}
 \maketitle
 
 \begin{abstract}
+    We present \emph{twisted inductors}, a planar inductor layout that interleaves multiple spiral traces across two
+    layers, increasing Self-Resonant Frequency (SRF), providing higher inductance, and improving rotational field
+    symmetry compared to conventional layouts. Twisted inductors generalize both conventional planar spiral inductors as
+    well as planar toroidal inductors. We experimentally show that in Wireless Power Transfer (WPT) through an axially
+    rotating joint in Inertial Hardware Security Modules (IHSMs), the improved symmetry of twisted inductors results in
+    decreased output ripple. We further provide measurements of 39 test coupons showing that twisted inductors improve
+    SRF by up to \qty{58}{\percent} and increase inductance by up to \qty{6.5}{\percent} compared to conventional planar
+    spiral inductors.
 \end{abstract}
 
 \section{Introduction}
 
+Inductive Wireless Power Transfer (WPT) is a widely used technology supported by a large corpus of research literature
+\cite{
+    awuahNovelCoilDesign2023,
+    batraEffectFerriteAddition2015,
+    curranModelingCharacterizationPCB2015,
+    fanSimultaneousWirelessPower2024,
+    leeSimpleWirelessPower2017,
+    liWirelessPowerTransfer2015,
+    maierContributionSystemDesign2019,
+    mooreApplicationsWirelessPower2019,
+    mouEnergyEfficientAdaptiveDesign2017,
+    mouWirelessPowerTransfer2015,
+    zhangWirelessPowerTransfer2019}.
+While working on an application of inductive WPT in a Inertial Hardware Security Module (IHSM) as previously
+published by \textcite{gotteCantTouchThis2022}, we found ourselves presented with an unusual set of constraints
+attempting WPT through a rotating joint using a planar inductor implemented in a Printed Circuit Board (PCB)---a set of
+constraints that does not seem to be addressed adequately in the existing literature on inductive WPT yet.
+
+Inertial Hardware Security Modules are a hardware security primitive that discourages tampering with a payload such as a
+single-board computer by rotating a tamper-sensing enclosure around the payload. The tamper-sensing enclosure
+continuously monitors itself for tampering using sensors such as tamper-sensing meshes~\cite{TamperResistance2020} and
+accelerometers. When the tamper-sensing enclosure signals a tamper alarm to the payload, the payload immediately
+destroys all sensitive data to prevent the attacker from gaining access to it. In principle, an IHSM is similar to an
+ATM that responds to attempts at opening its vault by dispensing dye over the bank notes within, rendering them
+unusable.
+
+In our IHSM implementation, the tamper-sensing enclosure rotates at speeds in the range from
+\qtyrange{1000}{3000}{\rpm}. The tamper sensing circuit on the rotating enclosure is powered through a pair of WPT
+inductors located on the IHSM's axis of rotation. The large centrifugal acceleration prohibits the use of batteries or
+liquid electrolyte capacitors on the rotating part, and makes heavy components such as large Multilayer Ceramic
+Capacitors (MLCCs) challenging to balance. Planar inductors that are patterned directly into a PCB provide a
+cost-effective and lightweight solution to this problem, but the coarse pattern resolution of PCBs results in a strict
+upper limit to the turn count that can be achieved in an inductor with a given area.
+
+Planar inductors are usually considered approximately axisymmetric. In our application, we found that the field
+asymmetry in feasible PCB inductors is large enough that axial rotation of two such inductors results in an oscillation
+of their coupling coefficient that leads to voltage ripple on the secondary side, especially when the coils are
+misaligned.
+
+In other inductive WPT systems, this issue is mitigated by several factors: First, for this effect to matter in the
+first place, the two coils have to be rotating with respect to one another. In ferrite core inductors, the core is the
+major factor shaping the magnetic field and evens out the small effect of winding asymmetry. In wire-wound inductors,
+the often higher turn count and the tightly packed, circular wires render this effect negligible. Finally, the output
+ripple caused by this oscillation can be filtered through a voltage regulator or by using a large decoupling capacitor
+on the secondary side if the application can accomodate such components on the rotating part.
+
+\begin{figure}
+    \begin{center}
+        \subcaptionbox{\raggedright A classic planar spiral inductor}{
+            \includegraphics[width=0.25\figurescale]{figures/svg_vis_paper_plain.png}}
+        \subcaptionbox{\raggedright Our proposed inductor layout}{
+            \includegraphics[width=0.25\figurescale]{figures/svg_vis_paper.png}}
+        \subcaptionbox{\raggedright A honeycomb coil in \textcite{saackeRadiotechnikIIIEmpfanger1926}}{
+            \includegraphics[width=0.15\figurescale]{figures/saacke-radiotechnik-3-ledionspule.jpg}}
+        \subcaptionbox{\raggedright A basket-woven coil in \textcite{kleinSpulenUndSchwingungskreise1941}}{
+            \includegraphics[width=0.15\figurescale]{figures/klein-spulen-schwingkreise-korbspule.jpg}}
+    \end{center}
+    \caption{Illustration of our proposed inductor layout compared to contemporary conventional planar inductors and
+        honeycomb as well as basket-woven coils from the early days of wireless radio.}
+    \label{fig_illust_honeycomb_basket}
+\end{figure}
+
+While there exist a corpus of prior work focusing on efficient power transfer between two coils whose position relative
+to one another cannot be precisely controlled as is the case in wireless phone charging systems as well as in proposed
+WPT electric vehicle chargers~\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017},
+it is generally assumed that the two coils remain quasi-stationary with respect to one another.
+
+There exists a body of work on inductive power transfer through rotating joints but here the focus often lies on higher
+power budgets than our application requires, which in practice requires more space and a ferrite or laminated iron
+core~\cite{
+    fanSimultaneousWirelessPower2024,
+    songRotationLightweightWirelessPower2019,
+    wangCoaxialNestedCouplersBased2020,
+    }.
+Often, these rotating joint WPT systems use coaxial structures, but segmented approaches exist, too~\cite{
+    wangNovelRotatingWireless2024,
+    yanFreeRotationWirelessPower2023,
+    xiaRotaryWirelessPower2024,
+    liWirelessPowerTransfer2021,
+}.
+In lower-power applications, segmented approaches are more common. A key challenge in segmented approaches is the
+reduction of secondary-side ripple induced when the segments' alignment changes throughout one revolution~\cite{
+    zhangWirelessSensorPower2024,
+}, which usually requires additional secondary-side circuitry.
+
+\subsection{Twisted inductors}
+
+In this paper, we propose a layout for circular PCB inductors that uses a number of series-connected interleaved spirals
+to achieve a topological equivalent to a torus knot from mathematical knot theory. Our layout twists the inductor's
+windings around one another by connecting the interleaved spiral segments with a ring of vias each on the inside and
+outside of the inductor's windings. Our approach improves rotation ripple in WPT through a rotating joint. Furthermore,
+it provides better performance beyond our particular use case, and improves upon conventional contemporary planar
+inductors by applying similar principles to those which inspired the polygonal basket-woven air coils used in early
+radio sets. We show that we can layout a twisted inductor for any number of layer inversions that is co-prime to the
+inductor's turn count. Our approach opens up a design space for inductor layouts that interpolate between planar spiral
+inductors on one end, and planar toroidal inductors on the other end, gerneralizing a super-set to a number of previous
+approaches to the design of planar inductors.
+
+We observe that in high-frequency applications, a moderate number of layer inversions increases the spacing between the
+beginning and end of the inductor's conductor, where the majority of the inductor's AC current flows. This decreases the
+parasitic capacitance of the inductor and increases its Self-Resonant Frequency (SRF), raising its maximum possible
+operating frequency and improving its efficiency at lower operating frequencies.
+
+\subsection{Contributions}
+Our contributions in this paper include:
+\begin{itemize}
+    \item We introduce twisted inductors, a planar inductor layout that improves rotational symmetry in WPT through
+        rotating joins, and promises improved high-frequency behavior in other applications.
+    \item We provide detailed instructions for the construction of such layouts, including a mathematical analysis of
+        the available parameter space.
+    \item We provide an analytical model of inductance and DC equivalent series resistance of our scheme.
+    \item We validate our scheme and provide laboratory measurements of the basic parameters of 39 test specimens
+        comparing our scheme to conventional layouts.
+    \item We further present the results of Finite Element Method (FEM) simulations to validate our inductance and ESR
+        approximations.
+    \item Finally, to analyze the degree of rotational symmetry in our proposed scheme, we provide the results of a
+        large number of automated measurements of coupling between pairs of inductors under various rotations, offsets,
+        distances and load conditions.
+\end{itemize}
+
 \section{Related Work}
-\subsection{Twisted Inductors in RFIC Design}
+
-\subsection{Basket-Woven Air Coils}
+\subsection{Inductive WPT in Practice}
+
+Inductive WPT has been proposed in a large number of
+scenarios~\cite{zhangWirelessPowerTransfer2019,mouWirelessPowerTransfer2015}, each of which comes with a set of unique
+constraints. When WPT is used to charge an electric toothbrush, the implementation cost of the system is critical, while
+efficiency and total power output are of little concern. Mechanically, in an electric toothbrush's charging system, the
+position and spacing of the transmitter and receiver coils can easily be controlled down to millimeter precision.
+
+In contrast to this, wireless smartphone charging is a much more demanding application. Here, the total cost of the
+system is only secondary, but the receiver's form factor is critical, and total power output as well as efficiency
+become major objectives. At the same time, in wireless smartphone charging, position tolerances are very coarse, and the
+two coils in the charging base and in the phone may be positioned more than a centimeter off-axis, with a gap of several
+millimeters and potentially not even in parallel planes.
+
+Power transfer across large distances is even more of a concern in implantable medical
+devices~\cite{mooreApplicationsWirelessPower2019}. Where a wireless phone charger must be able to bridge distances of a
+few millimeters, an implantable medical device might be situated underneath several centimeter of tissue and bones. At
+the same time, cost is of (almost) no concern in this medical application, which enables the use of complex
+manufacturing techniques, customized electronic components and exotic materials.
+
+While all of the aforementioned applications transfer somewhere between a few hundred milliwatts and several watts of
+power, at the other end of the spectrum there is a large body of research suggesting the use of inductive wireless power
+transfer for the charging of electric vehicles
+(EVs)~\cite{liWirelessPowerTransfer2015,mouEnergyEfficientAdaptiveDesign2017}. In this application, the wireless power
+transfer system usually replaces the conventional wired charging connector, which improves the systems' user experience
+given the strong force required to seat or unseat these rather large connectors, as well as the heft of the required
+water-cooled cables. In this application, size is of little concern, but at charging rates up to tens of kilowatt,
+efficiency becomes critical.
+%When charging an EV at a rate of \qty{10}{\kilo\watt}, an efficiency improvement of just
+%$0.1\%$ corresponds to a reduction in power dissipation of \qty{10}{\watt}. Besides the monetary cost of the power lost
+%this way, each small improvement enables a reduction in size of heat sinks and other cooling components, which directly
+%translates to a decrease in cost.
+
+\subsection{Core materials in WPT}
+
+Across application areas, air-core inductors are often used for WPT since in most applications, an air gap of several
+millimeters or more is expected~\cite{curranModelingCharacterizationPCB2015}. Especially in low-power application such
+as mobile device charging, the size and weight of ferrites is an obstacle to their use, and at lower power levels losses
+are less of a concern.
+
+A common way to use ferrites in WPT applications is by magnetically shielding the inductor's back side with a ferrite
+plate such that the field does not extend beyond the coil's back side, thereby increasing the intended mutual inductance
+while simultaneously reducing eddy current losses when the WPT coils are placed near metal
+objects~\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}. Similar to
+how the trace layouts of planar WPT coils are optimized to improve power transfer efficiency, the layout of ferrite
+components has been proposed for optimization~\cite{batraEffectFerriteAddition2015}.
+
+\subsection{PCB inductor design for wireless power transfer}
+
+Today, air-core inductors are the standard solution in inductive WPT links. Since in most WPT applications an air gap of
+several millimeters between the sending and receiving assemblies is expected, adding a ferrite core does not result in a
+large improvement in coupling. Instead, the impact of this distance as well as misalignment is reduced by maximizing the
+area of the air-core inductors used, or by tiling multiple
+inductors~\cite{curranModelingCharacterizationPCB2015,wangNovelRotatingWireless2024,zhangDynamicWirelessPower2025}.
+
+WPT inductors tend to be mostly planar coils with only a few layers, so implementing them in a PCB process seems
+natural. Using a PCB for the inductor has the potential to reduce implementation cost since PCBs are cheap, and they can
+also serve as structural support. However, implementing inductors in PCBs has several disadvantages. First, due to the
+limited layer count of common PCB processes and due to structure size limitations, the number of windings that can be
+fit into a given volume is much lower than in wire-wound inductors. Second, due to a PCB's copper layers being thin
+compared to its dielectric substrate\footnote{common values are \qtyrange{15}{30}{\micro\meter} copper thickness and
+\qtyrange{600}{1600}{\micro\meter} substrate thickness} PCB inductors tend to have poor DC resistance, albeit the thin
+copper layer decreases skin effect losses compared to a solid, round conductors of the same cross-sectional area.
+However, PCBs can still not approach the performance of litz wire used in high-frequency WPT coils, which commonly use
+wire diameters in the range of tens of micrometers~\cite{zhaoDesignOptimizationLitzWire2023}.
+\textcite{lopeFrequencyDependentResistancePlanar2014} and \textcite{nomotoSplittingConductorsCoils2024} propose a
+mitigation that aims to emulate a litz wire's structure in large, high-current PCB inductors, but their mitigation is
+heavily limited by the structure size achievable in common PCB manufacturing
+processes~\cite{nguyenReviewComparisonSolid2020}.
+
+A further factor that limits the high-frequency performance of PCB inductors is distributed capacitance. Not only does a
+large air coil exhibit more parasitic capacitance than an equivalent, smaller ferrite-core inductor simply due to its
+size, when implemented in a PCB process a large fraction of the electrical fields responsible for this capacitance pass
+through the PCB's substrate, not air. The relative permittivity $\epsilon_r$ of common PCB substrates typically lies in
+the range of $4$ to $5$~\cite{mumbyDielectricPropertiesFR41989}, which increases the distributed capacitance compared to
+a pure air-core inductor by approximately that same factor.
+
+\subsection{Planar Inductors in RFIC Design}
+
+Beyond WPT, planar inductors are commonly used in radio frequency integrated circuits (RFICs). In RFIC design, the major
+challenges are area optimization and precisely predicting the inductor's characteristics during the design phase. Common
+optimizations include applying a variable trace pitch~\cite{lopez-villegasImprovementQualityFactor2000} and variable
+trace width~\cite{hsuAnalyticalDesignAlgorithm2008}.
+
+In RFICs, inductors are commonly designed as \emph{balanced} inductors with a grounded central node. Such designs
+interleave two counter-wound planar spiral inductors on the same layer with the help of some jumper connections on a
+second layer~\cite{daneshDifferentiallyDrivenSymmetric2002,martinMultiturnTwistedInductor2016}. The use of such designs
+in RFIC design is mainly focused on their electrical symmetry, so that the two input ports can be fed with a fully
+differential signal, with the inductor loading both driver outputs equally across the inductor's frequency range.
+
+% Note: They note that the main point behind the design is electrical symmetry of the two ports to make driving the
+% thing differentially cleaner. We should adopt this observation for our inductors, which likewise are electrically
+% symmetric when compared to a single-layer spiral inductor.
+
+\subsection{A Brief Historical Diversion on Basket-Woven Air Coils}
+
+Since the early days of radio engineering, the parasitic capacitance of inductors has been a point of
+concern~\cite{nesperHandbuchDrahtlosenTelegraphie1921,flemingPrinciplesElectricWave1910}. Going back to the early days
+of wireless telegraphy after the turn of the twentieth century, coils with high inductance were needed for the
+construction of both transmitters and receivers, but the ferrites that would later permit their compact construction
+were still being developed. The ferromagnetic core material of choice back then was laminated iron, which was only
+useful at low frequencies due to eddy current losses. As a result, the inductors in radio circuits of the era were often
+constructed as air-core coils. While air-core inductors are immune to core saturation, the poor magnetic permeability of
+air necessitates a large number of wide turns of wire to reach useful inductance values, which for reasons of
+practicality or leakage inductance often could not be wound as a single layer cylindrical coil. This could be resolved
+by winding an inductor with many turns on multiple layers, which improves compactness and leakage inductance, but this
+in turn gives rise to increased distributed capacitance as now turns with a large voltage differential are layered right
+on top of each other.
+
+Before the invention of ferrites, a number of ways were devised to decrease distributed capacitance in multilayer
+inductors. These methods can be divided into two general categories: Optimizing the connecting order of turns and
+optimizing the winding schema of turns. Both aim at increasing spacing between parts of the coil that have a large
+voltage differential.
+
+The connecting order of turns was optimized at the assembly level by stacking coils in a particular
+way~\cite{flemingPrinciplesElectricWave1910} and at the component level by winding coils in a particular way to minimize
+the voltage differential between adjacent turns---a technique that is still used to this
+day~\cite{lopeFirstSelfresonantFrequency2021}. The main winding optimization in the first category concerns winding the
+turns of a cylindrical multilayer inductor not layer by layer, but instead layering them diagonally, effectively
+connecting adjacent turns in a diagonal zigzag pattern. Then as now, wound inductors applying this technique were not
+feasible to manufacture reliably by machine, but the technique can be closely replicated in PCB inductors as shown in
+\textcite{leePrintedSpiralWinding2011}. The main limiting factors in a PCB implementation are the requirement for a
+large number of vias inside the inductor's turns limiting the achievable turn count\footnote{In PCBs, as opposed to
+integrated circuits (ICs), vias limit the achievable turn count when they need to be placed in-line inside the turns as
+opposed to on the inside or outside because a PCB's minimum trace/space widths are usually much smaller than the
+smallest feasible via, consisting of a minimum-size drill surrounded by a minimum-size annular ring.} and increasing
+equivalent series resistance (ESR) through the thin trace sections that are necessary to accomodate the via structure,
+as well as the layer pairing limitations when blind vias are used in multilayer PCBs.
+
+This lack of a way to wind high frequency inductors with a machine led to the creation of a number of related winding
+schemes that include honeycomb and basket woven coils
+\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
+filbigLehrbuchHochfrequenztechnik1942,
+kleinSpulenUndSchwingungskreise1941,
+meinkeTaschenbuchHochfrequenztechnik1956,
+nottebrockSpulen1950,
+struttVerstarkerUndEmpfanger1951,
+wiggeRundfunktechnischesHandbuch1930,
+zicknerSpulen1927}. The simplest such winding technique is the universal winding as described in depth by
+\textcite{querfurthCoilWindingDescription1954}. In a simple, cylindrical wire-wound inductor, the windings are laid down
+one right next to the other, until the end of the winding area is met, where the winding direction is reversed. One
+layer of such windings forms a helix whose pitch is equal to the wire diameter. A universal winding uses the same
+helical scheme reversing at the coil ends, but uses a helical pitch larger than the wire diameter to form a structure
+similar to a spool of sewing thread.
+
+Other winding techniques include honeycomb and basket woven coils, some historic examples of which are shown in Fig.\
+\ref{fig_illust_honeycomb_basket}. In a honeycomb coil, like in an universal winding, subsequent winding layers are
+wound at a criss-cross pattern. The characteristic feature of honeycomb coils is that the winding machine is adjusted to
+produce large air gaps between adjacent windings, resulting in a three-dimensional rhomboid pattern that is vaguely
+reminiscent of a honeycomb's structure.
+
+In basket-woven coils, a mandrel consisting of an odd number of sticks pointing either radially or axially is used, and
+the wire is woven between adjacent sticks in an alternating direction. While visually similar to honeycomb coils, this
+winding technique is more suited to homebrew construction and less amenable to mass production by machine. In axially
+basket-woven coils, the mandrel can be pulled out after the coil is finished. Like honeycomb coils, the resulting
+structure can be made mechanically stable with some lacquer, with the turns carrying the layers where they cross.
+
+Both construction techniques apply similar principles to those leading to the improved high-frequency behavior of
+twisted inductors that we describe in this paper.\footnote{Interestingly, the winding schemes of both honeycomb and
+basket-woven coils are also governed by the same coprimality condition between the number of turns and the number of
+inversions within each turn that we describe for our twisted inductors below, although we could not find an example in
+historic literature where this condition was explicitly stated~\cite{eppenAnforderungenEinzelteileRundfunkempfanger1927,
+kleinSpulenUndSchwingungskreise1941, wiggeRundfunktechnischesHandbuch1930, querfurthCoilWindingDescription1954}.}
 
 \section{Twisted Inductor Design} 
 
+In this section, we present a detailed derivation of the layout of twisted inductors. We approach this layout by
+construction. Let us first consider a simple, planar, circular spiral coil with a fixed pitch. We will ignore trace
+width for now, and consider the trace a thin wire. We will assume the inductor's ports are both located on the positive
+$x$-Axis on top of one another on different layers, which also helps to minimize the loop area of the inductor's
+connections.
+
+The trace trajectory of a standard planar spiral inductor can be parametrized in polar coordinates $r, \varphi$ based
+on an Archimedean spiral:
+
+\begin{equation}
+    r = a\cdot\varphi
+    \label{eqn_arch_spi_basic}
+\end{equation}
+
+An Archimedean spiral defined this way always starts at the origin, and it continues to infinity. Let us re-parametrize
+this spiral to a curve parameter $t$ with range $\left[0,1\right]$, such that $t=0$ corresponds to the start of the
+inductor and $t=1$ corresponds to its end. As is customary in PCB inductors, we place the inductor's start on its outer
+circumference.
+
+To improve layer utilization, a common technique in PCB inductor design is to use both layers of the PCB for the
+inductor's spiral trace, instead of only using the bottom layer for a straight jumper trace. Using both layers this way
+allows for wider traces, which lowers resistive losses. We can accomodate this optimization in our definition by
+re-defining our normalized radius to allow both positive and negative values, defining negative values to designate
+traces on the PCB's bottom layer as follows. Fig.\ \ref{fig_nk_combined} shows both a simple and a two-layer
+spiral inductor in the first two columns. 
+
+Let $n$ be the turn count of our inductor. The resulting parametrization is:
+
+\begin{align}
+    \varphi &= 2\pi n t\\\nonumber
+    r &= r_1 + \left|1 - 2 t\right| \left(r_2 - r_1\right)
+    \label{eqn_twolayer_spiral}
+\end{align}
+
+The resulting spiral trace starts at radius $r_2$ on the positive $x$ axis, and spirals inward until it meets $r_1$,
+where the sign indicates a layer change, and the trace reverses to continue back to $r_2$ on another layer. In its PCB
+realization, at $r_1$, a via would be placed to connect the end of the spiral trace to a jumper trace on the other layer
+of the PCB leading back to the start.
+
+\subsection{From Spiral to Twisted Inductor}
+
+Extending the above parametrization of a spiral inductor's layout, we propose planar \emph{twisted inductors} based on
+two core observations:
+
+\begin{description}[\IEEEsetlabelwidth{foo}]
+    \item[Observation 1.]\hfill\\When using an archimedean spiral, multiple such spirals using the same pitch can be
+        interleaved by spreading out their start and end points at regular angular intervals.
+    \item[Observation 2.]\hfill\\In a two-layer spiral inductor (Fig.\ \ref{fig_nk_combined}), we can adjust the turn
+        count of the pair of traces to move the end point of the bottom layer trace anywhere on the inductor's outer
+        radius.
+\end{description}
+
+Setting the inversion count to $k=1$ in our proposed scheme yields the conventional two-layer counterwound
+scheme~\cite{lopeFirstSelfresonantFrequency2021,sproHighVoltageInsulationDesign2021,leePrintedSpiralWinding2011}.
+
+Combining these two observations, we find that by choosing a number $k$ of inversions, i.e. layer jumps, that is coprime
+to the number of total turns of the inductor $n$, we achieve a layout where all $k$ pairs of top and bottom-layer traces
+naturally connect in series, with the resulting spirals on the top and bottom layers interleaving cleanly. Fig.\
+\ref{fig_nk_combined} shows a layout with $n=3$ turns with both a single inversion ($k=1$), which results in a
+conventional two-layer inductor, and with $k=2$ inversions, creating two interleaved spirals on both the top and the
+bottom layer of the PCB. Fig.\ \ref{fig_nk_complex_illust} in the Appendix of this paper shows
+additional layout examples for other values of $n$ and $k$. For $k=\frac{1}{2}$, we get a standard single-layer planar
+spiral inductor for any turn count $n$, and for $k=1$ we get a standard two-layer planar spiral inductor for any turn
+count $n$. In this paper, we will call all layouts with $k\ge 2$ \emph{Twisted Inductors}. The coordinate description of
+Equation\ \ref{eqn_twolayer_spiral} thus becomes:
+
+\begin{align}
+    \varphi &= 2\pi n t\\\nonumber
+    r &= r_1 + \left|1 - \left( 2 k t \mod 2 \right) \right| \left(r_2 - r_1\right)
+    \label{eqn_twisted_spiral}
+\end{align}
+
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=\figurescale]{figures/nk_interleave_illust.pdf}
+%    \end{center}
+%    \caption{single-layer spiral inductor's layout (left), a conventional two-layer planar inductor's layout (middle),
+%    and a twisted inductor with two inversions (right). All three inductors have $n=3$ turns. Traces on the PCB top
+%    side are shown in red, traces on the bottom side in blue. In the twisted inductor, each layer contains two
+%    archimedean spirals that interleave at a regular spacing. The four spirals of the inductor are connected in series
+%    such that they form three total turns.}
+%    \label{fig_nk_interleave_illust}
+%\end{figure}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=\figurescale]{figures/nk_combined.pdf}
+    \end{center}
+    \caption{Inductor layouts for several sets of turn count $n$ and inversion count $k$. The top row shows the actual
+    trace layout in cartesian coordinates, the bottom row visualizes the winding schema.
+    }
+    \label{fig_nk_combined}
+\end{figure}
+
+Topologically, the shape of our inductors can be described as a $(k, n)$-torus knot. From knot theory, we know that such
+a torus knot exists if and only if both $n$ and $k$ are co-prime. Fig.\ \ref{fig_nk_combined} illustrates a derivation
+of the coprimality requirement. If we plot the spiral in polar coordinates on a cartesian plot we observe that for a
+$n$-turn coil with $k$ inversions, the trace crosses the $\varphi$ axis once for each inversion, wrapping around $r$.
+Likewise, it crosses the $r$ axis once for each turn of the inductor, wrapping around $\varphi$. Based on this, we can
+re-label the angular axis in steps from $0$ to $k$, and re-label the radial axis in steps from $0$ to $n$. Labelling the
+new angular axis $i$ and the new radial axis $j$, in the resulting integer lattice, the trace has slope $1$. We can
+state the trace's trajectory as a function of a curve parameter $t \in [0, nk]$ as $f(t) = (i, j) = (t \mod n, t \mod
+k)$. To produce a valid inductor, the trace must not intersect anywhere. Thus, the system of congruences
+
+\begin{align}
+    t &\equiv i \mod n\\
+    t &\equiv j \mod k
+\end{align}
+
+must have a unique solution $t \in [0, nk]$ for all $(i, j)$. This statement corresponds exactly to the Chinese
+Remainder Theorem, which states that this solution is unique if and only if $k$ and $n$ are coprime.
+
+In the following paragraphs, we will derive analytical expressions for Ohmic resistance and inductance of inductors
+derived under this schema.
+
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=0.8\figurescale]{figures/nk_chinese_remainder_illust.pdf}
+%    \end{center}
+%    \caption{Illustration of the winding pattern of two twisted inductors. The upper plots show the inductor's actual
+%    layout with the traces on each side of the substrate colored in red (top) and blue (bottom), respectively. The lower
+%    plots show the same traces, but \emph{unwrap} the annulus by plotting the traces' polar coordinates on cartesian
+%    axes. The left axis labels show the normalized radius, with $0$ being at the inductor's inner diameter, $1$ being at
+%    its outer diameter on the top layer, and $-1$ being at its outer diameter on the bottom layer. The top and right
+%    axes labels show the axis scaled to match indices $i\in\left[0, n\right]$ and $j\in\left[0, k\right]$,
+%    respectively.}
+%    \label{fig_nk_chinese_remainder_illust}
+%\end{figure}
+
+\subsubsection{Ohmic Resistance}
+
+The arc length $l$ of a spiral can be calculated from its turn count $n$ and the average of its inner and outer diameter
+$\frac{2 r_1 + 2 r_2}{2}=r_1+r_2$ as $l = n\pi\left(r_1 + r_2\right)$. Since going from a standard inductor to a twisted
+inductor does not change its turn count nor its dimensions, the combined arc length of all traces of the twisted
+inductor does not change. Twisted inductors require two additional vias per inversion, which will increase DC resistance
+slightly, but the contribution of these vias will remain small in practical applications since the overall number of
+vias is still no more than a couple per turn, and since each via only bridges the short distance between the inductor's
+layers.
+
+As a general expression, for a standard or twisted inductor with turn count $n$ and twist count~$k$, given via
+resistance $R_\text{via}$ and trace cross-sectional area $A_\text{tr}$, we derive a first order approximation of the
+inductor's DC resistance as follows.
+
+\begin{equation}
+    R_L = n\pi\left(r_1 + r_2\right)\frac{\rho_\text{Cu}}{A_\text{tr}} + \left(2k-1\right)R_\text{via}
+\end{equation}
+
+\subsubsection{Inductance}
+
+Even for geometrically simple inductors, analytically calculating their inductance is a surprisingly hard problem whose
+complexity quickly escalates when geometrically complex inductors are analyzed, when realistic wire shapes as opposed to
+thin wire or current sheet approximations are used, and when taking into account differing magnetic permeabilities of
+air or dielectrics and core materials. Instead of precise analytical models, a number of approximations are commonly
+used. A commonly referenced approximation for the inductance of planar spiral inductors is given by
+\textcite{mohanSimpleAccurateExpressions1999}, whose current-sheet approximation for circular planar spiral inductors we
+will use here to estimate our inductor's inductance. The current-sheet approximation from
+\textcite{mohanSimpleAccurateExpressions1999} reads:
+
+\begin{equation}
+    \label{eqn_mohan_approx}
+    L = \frac{\mu n^2 d_\text{avg} c_1}{2}\left(\ln\left(c_2/\rho\right)+c_3\rho+c_4\rho^2\right)
+\end{equation}
+
+In this equation, $c_{1-4}$ denote four empirically determined coeficcients that are specific to the coil's shape. The
+values for circular coils are $c_{1-4}=(1.00, 2.46, 0.00, 0.20)$. $\mu$ is the magnetic permeability of air (for an
+air-core inductor), $n$ is the number of turns, $d_\text{avg}$ is the \emph{average} turn radius, i.e. $d_\text{avg} =
+2\frac{r_1 + r_2}{2} = r_1 + r_2$. $\rho = \frac{r_2-r_1}{r_2+r_1}$ is the planar spiral inductor's \emph{fill ratio}.
+The fill ratio encodes the fact that the inductor's turns have less flux linkage the closer to the inductor's center we
+get. While turns close to the outside have good flux linkage due to their inner area overlapping well with that of other
+turns, turns close to the center not only have a loop area that is only a fraction of that of turns further outwards,
+the closer we get to the center, the larger is also the fraction of the field lines returning as leakage flux on the
+outside of the inner turn that pass through the inner part of turns further outwards, flipping the sign and contributing
+\emph{negative} mutual inductance.
+
+As Equation\ \ref{eqn_mohan_approx} approximates the inductor's whole set of windings as a single, uniform current
+sheet, the turn count only appears as a single factor of $n^2$ in the equation, with $\rho$ and $c_{1-4}$ correcting for
+the inductor's geometry. To account for twisted inductors, we can separate the inductor into a set of $2k$ simple planar
+spiral inductor \emph{branches} that are connected in series by the twisted inductor's vias. Compared to a simple spiral
+inductor, for each branch, the inductance according to Equation\ \ref{eqn_mohan_approx} stays the same except that the
+factor $n^2$ drops to $\left(\frac{n}{2k}\right)^2$ because the $n$ windings are evenly distributed across the $2k$
+branches. Let us now make two assumptions. First, we will assume that the flux linkage between both sides of the
+inductor is approximately one. This assumption is grounded in the fact that for practical designs, the substrate
+thickness will be small compared to the inductor's diameter. Second, we will for now ignore the spiral inductor's field
+asymmetry and assume that the flux linkage between two intertwined branches on the same side of the substrate is
+approximately one. In our measurements below we show that for simple spiral inductors this asymmetry, while problematic
+in our application, is small in absolute terms, and grows smaller with increasing turn count.
+
+Based on these two assumptions, we can model the twisted inductor as a set of $2k$ series-connected spiral inductors
+that are perfectly coupled, with full flux linkage. This results in the total series inductance gaining back the factor
+$\frac{1}{2k^2}$ that each branch lost, resulting in identical inductances for a simple planar spiral inductor and a
+twisted inductor with the same size and turn count according to Equation\ \ref{eqn_mohan_approx}. This approximation
+introduces an error due to the imperfect flux linkage between the two sides of the substrate, and between two spiral
+branches located at an angular offset from each other. In our experiments, we found that for our test inductors,
+compared to inductances measured with an LCR meter, this error is below \qty{10}{\percent} for $n=5$ turns or more, and
+for our test samples matches the performance of Equation\ \ref{eqn_mohan_approx} for the simple planar spiral inductor
+case.
+
+\subsection{CAD Integration}
+
+To allow for easy design with twisted inductors and to speed up the laboratory prototyping we performed for this paper,
+we created a tool that generates arbitrary twisted inductor layouts, and that is able to output these layouts as PCB
+footprint files for the open source KiCad EDA CAD tool~\cite{KiCadEDA}. We integrated the ESR and inductance
+approximations as derived above with our tool, so that it provides immediate design feedback when generating inductors.
+In order to minimize ESR and maximize PCB area utilization, we made the tool automatically calculate the largest
+possible trace width when given a minimum clearance specification.
+
+To handle outputting PCB geometry in a format that can be read from KiCad, we utilized the open source EDA file format
+library \emph{gerbonara}. To support the FEM simulations that are described in the next section below, our tool contains
+functionality to map gerbonara's geometry representation into that of gmsh~\cite{geuzaineGmsh3DFinite2009}, the FEM
+mesher that we chose to interface with Elmer FEM~\cite{ruokolainenElmerCSCElmerfemElmer2023}.
+
+Our inductor design tool is available in this paper's supplementary material as well as at the git repository linked at
+the end of this paper.
+
+\section{FEM Simulation}
+
+To validate our analytical approximations, we performed a series of FEM simulations in Elmer FEM. For a number of
+inductor layouts, we performed simulations to determine ohmic resistance and inductance. Due to limitations in our
+gmsh/Elmer toolchain, we were unable to run simulations for parasitic capacitance and self-resonance, or for coupling
+behavior of coil pairs. We found that for these cases which require larger, more complex meshes, gmsh would frequently
+crash during meshing, and where we were able to produce meshes, Elmer would only converge for some of them. While these
+are problems that can be solved through either a more skillful description of the problem in gmsh and Elmer, or by using
+more robust software such as Simulia CST, we decided to instead experimentally measure these quantities instead (cf.\
+Section\ \ref{sec_experiments}). While our measurements only cover a small number of inductor samples, their results are
+more reliable than results from FEM and can serve as a baseline for future work on such simulations.
+
+We conducted our FEM simulations as follows:
+
+\paragraph{Ohmic Resistance}
+In Elmer FEM, we can use the built-in joint static current and joule heating solver to determine the ohmic resistance at
+a given current.
+
+\paragraph{Inductance}
+We let Elmer determine inductance by first using its coil solver to determine the volumetric current density in our mesh
+given a test current, then applying its magnetodynamics solver to solve the electromagnetic field. Elmer provides
+routines to derive the total magnetic field energy $U_\text{mag}$ from an EM field solution. Since we have only our
+inductor under test inside the simulation volume, with test current $I_\text{test}$, we can then derive the inductor's
+inductance according to the well-known relation~\cite{meeekerFiniteElementMethod2015}:
+
+\begin{equation}
+    L = \frac{2\cdot U_\text{mag}}{I_\text{test}^2}
+\end{equation}
+
+\section{Experimental Validation}
+\label{sec_experiments}
+
+To experimentally validate our design with real-world inductors, we produced 24 test coupons with a number of variations
+of twisted inductors with winding count $n$ between $1$ and $25$, and twist count ranging from $k=\frac{1}{2}$ (simple
+single-sided spiral inductor) to $k=37$. All test inductors had an inner diameter of \qty{15}{\milli\meter} and an outer
+diameter of \qty{35}{\milli\meter} corresponding to the space available in our IHSM implementation.
+
+\subsection{Inductance and DC resistance}
+
+We measured the inductance and DC resistance of each test coupon using a Keysight U1733C LCR meter at
+\qty{100}{\kilo\hertz} for inductance and a Keysight 34465A multimeter in four-wire configuration for DC resistance. We
+further determined the self-resonant frequency of each inductor using a LiteVNA64 handheld vector network analyzer. The
+results of our measurements are shown in Table\ \ref{tab_coupons}.
+
+We found our inductance approximation to be accurate within \qty{10}{\percent} and our ESR approximation to be accurate
+within \qty{20}{\percent} for inductors with three turns or more. For lower turn-count inductors, inductance
+measurements are difficult because the small absolute inductances involved are easily disturbed by stray inductances,
+and ESR measurements are affected by contact and trace resistance even when measurements are taken in four-wire mode.
+
+In accordance with our design intuition, we found that for high turn count inductors, the doubled trace width that is
+afforded by splitting a simple spiral inductor across two PCB layers in any two-layer configuration improves ESR by
+approximately a factor of two. Going from a simple single-layer spiral inductor to a simple two-layer spiral inductor
+($k=1$), we observe that the resulting inductance decreases by up to \qty{15}{\percent}. We suspect that the main factor
+leading to this decrease is radial magnetic flux leakage through the PCB material between the inductor's layers.
+Comparing simple two-layer inductors with $k=1$ to the twisted inductors with larger $k$ values that we propose in this
+paper, we observe almost identical performance for $k>1$ with decreases of less than \qty{0.5}{\percent} going from
+$k=1$ to $k=3$ irrespective of turn count. From these measurements we can conclude that the flux linkage of twisted
+inductors almost perfectly matches that of simple two-layer inductors.
+
+Finally, we decided to evaluate the high-frequency performance of twisted inductors. It is well-known that self-resonant
+frequency decreases when going from a single-layer spiral inductor to a two-layer spiral inductor while keeping
+inductance and dimensions constant~\cite{zhangImprovedCompensationMethod2025}. Our measurements show this effect, with
+it being more pronounced with higher turn count. Intuitively, this makes sense if we consider the mechanism of inductor
+self-resonance. The primary contributor to self resonance, particularly in higher turn count inductors, is capacitive
+coupling between the inductor's windings. In a single-layer spiral inductor, this effect gets partially mitigated since
+the strongest coupling exists between adjacent windings, which here have only a small voltage differential as only a
+fraction of the inductor's total voltage appears across each winding. Compared to this, when the inductor is constructed
+as a simple two-layer inductor with $k=1$, now the start and end windings of the inductor, which have the highest
+voltage differential, are located right on top of each other with the substrate in between. Making things worse, common
+PCB substrates have a relative permittivity much larger than air (usually around $4$).
+
+We observe that this decrease in high-frequency performance is eventually counteracted by increasing inversion count
+$k$. While our test samples focused on smaller turn counts, we observe a notable increase from a self-resonant frequency
+of \qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$. Prompted by
+this observation, we produced another set of 15 samples focusing on this aspect. We report our results of this
+investigation in the following section.
+
+In conclusion to the above measurement results, we observe that twisted inductors \emph{improve} high-frequency
+performance compared to simple two-layer inductors while closely matching them in ESR and inductance. While they peform
+worse than simple single-layer inductors in high-frequency performance, the increased trace width that two-layer
+inductors allow for lowers resistive losses by approximately a factor of four. In applications where resistive losses
+lead to the choice of a two-layer inductor, twisted inductors provide improved high-frequency performance at no
+additional cost and without compromising other performance parameters.
+
+\setlength{\tabcolsep}{4pt}
+\begin{table}
+    \caption{Inductor sample design parameters and measured characteristics. All inductors have outer diameter
+        \qty{35}{\milli\meter} and inner diameter \qty{15}{\milli\meter}. The missing values in the simulation results
+        columns result from the solver failing to converge.}
+    \begin{tabular}{cc|cc|cc|ccc}
+        \multicolumn{2}{c|}{}&
+        \multicolumn{2}{c|}{\textbf{Design}}&
+        \multicolumn{2}{c|}{\textbf{Simulation}}&
+        \multicolumn{3}{c}{\textbf{Measured}}\\
+
+        $n$&
+        $k$&
+        $L$&
+        $R$&
+        $L$&
+        $R$&
+        $L$&
+        $f_\text{res}$&
+        $R$\\
+
+        &
+        &
+        $\left[\unit{\micro\henry}\right]$&
+        $\left[\unit{\ohm}\right]$&
+        $\left[\unit{\micro\henry}\right]$&
+        $\left[\unit{\ohm}\right]$&
+        $\left[\unit{\micro\henry}\right]$&
+        $\left[\unit{\mega\hertz}\right]$&
+        $\left[\unit{\ohm}\right]$\\\hline
+
+        \rowcolor[gray]{0.9}
+        $1$&  $3$&  $0.03$&   $0.0095$& $0.039$&  $0.008$&  $0.056$&   ${465}$& ${0.0143}$\\
+        $1$&  $4$&  $0.03$&   $0.0108$& $0.040$&  $0.008$&  ${0.059}$&   $460$& $0.0150$\\
+        $1$&  $5$&  $0.03$&   $0.0123$& $0.041$&  $0.009$&  $0.055$&   $460$& $0.0166$\\
+        \hline\rowcolor[gray]{0.9}
+        $2$&  $1$&  $0.12$&   $0.0253$& $0.127$&  $0.024$&  $0.149$&   ${246}$& ${0.0284}$\\
+        $2$&  $3$&  $0.12$&   $0.0270$& $0.128$&  $0.025$&  ${0.152}$&   $241$& $0.0291$\\
+        $2$&  $5$&  $0.12$&   $0.0299$& $0.130$&  $0.027$&  $0.147$&   $226$& $0.0300$\\
+        \hline\rowcolor[gray]{0.9}
+        $3$&  $1$&  $0.26$&   $0.0454$& $0.262$&  $0.044$&  ${0.287}$&   ${146}$& $0.0461$\\
+        $3$&  $4$&  $0.26$&   $0.0479$& $0.265$&  $0.046$&  ${0.286}$&   ${146}$& ${0.0455}$\\
+        \hline\rowcolor[gray]{0.9}
+        $5$&  $1$&  $0.73$&   $0.0755$& $0.670$&  $0.074$&  ${0.693}$&   $61.3$& $0.0778$\\
+        $5$&  $3$&  $0.73$&   $0.0763$& $0.671$&  $0.074$&  ${0.694}$&   ${70.3}$& $0.0727$\\
+        $5$&  $7$&  $0.73$&   $0.0802$& $0.675$&  $0.077$&  ${0.694}$&   $68.0$&  ${0.0672}$\\
+        \hline\rowcolor[gray]{0.9}
+        $10$& $1$&  $2.90$&   $0.2513$& $2.700$&  $0.250$&  ${2.718}$&   $24.1$& $0.2322$\\
+        $10$& $3$&  $2.90$&   $0.2520$& $2.700$&  $0.250$&  $2.714$&   ${28.6}$& $0.2255$\\
+        $10$& $7$&  $2.90$&   $0.2554$& $2.700$&  $0.252$&  $2.713$&   $28.1$& ${0.2122}$\\
+        \hline\rowcolor[gray]{0.9}
+        $25$& $1$&  $18.15$&  $1.8843$& $16.900$& $1.900$&  $16.938$&  $8.84$&   $1.7024$\\
+        $25$& $3$&  $18.15$&  $1.8851$& N/A&      N/A&      $16.919$&  $8.60$& $1.6360$\\
+        $25$& $13$& $18.15$&  $1.9016$& $16.900$& $1.900$&  $16.931$&  ${10.56}$& ${1.5429}$\\
+        $25$& $37$& $18.15$&  $2.0197$& $17.100$& $2.000$&  ${17.066}$& $10.31$&  $1.6980$\\
+
+    \end{tabular}
+    \label{tab_coupons}
+\end{table}
+
+\subsection{Inductance and Frequency Behavior of Larger Coils}
+
+To investigate the high-frequency behavior of twisted inductors further, we produced and measured 15 additional sample
+inductors that were larger (up to \qty{90}{\milli\meter} outer diameter) and that had a higher turn count (up to 53)
+compared to our initial set of samples. The parameters of these new samples and our measurement results are shown in
+Table\ \ref{tab_wide_coils}. In these results, we can identify three clear trends. First, the ESR of twisted inductors
+is generally poorer when compared to two-layer spiral inductors. This increase in ESR is due to the large number of vias
+used in these sample inductors. It should be noted that while twisted inductors have worse ESR compared to conventional
+two-layer inductors, in our first set of test coupons we saw that their ESR is still better than that of a single-layer
+inductor because the traces can be made wider. Our second observation is that in every set of samples from this second
+run of physically larger inductors, twisted inductors outperform conventional planar inductors in self-resonant
+frequency by a considerable margin with an increase in SRF of up to \qty{58}{\percent} from our
+$d_2=\qty{65}{\milli\meter}$ sample going from $k=1$ to $k=100$.
+
+Our third observation is that unlike in the smaller inductors from Table\ \ref{tab_coupons}, in these larger instances,
+twisted inductors show increased inductance by approximately \qty{3.7}{\percent} for our smallest samples, and
+\qty{6.5}{\percent} for our largest samples. This behavior indicates that large twisted inductors indeed behave like a
+combination between a conventional planar spiral inductor and a conventional planar toroidal inductor. Comparing the
+magnitude of this increase with the measurements listed in Table\ \ref{tab_wide_coils} for planar toroidal inductors, we
+see that this effect exceeds what one would reach by a simple series configuration of both styles of inductor,
+indicating a contribution from flux linkage.
+
+\begin{table}
+    \caption{Parameters and measurement results of a set of larger sample inductors. Bold values indicate best
+        performance at a given size. Shaded rows indicate conventional planar toroidal ($n=1$) or two-layer planar
+        spiral inductors ($k=1$).}
+    \begin{tabular}{cc|cc|ccc|c}
+        $d_1$&
+        $d_2$&
+        $n$&
+        $k$&
+        $L$&
+        $R_\text{ESR}$&
+        $f_\text{Res}$&
+        $C_\text{p}$\\
+        $\left[\unit{\milli\meter}\right]$&
+        $\left[\unit{\milli\meter}\right]$&
+        &
+        &
+        $\left[\unit{\micro\henry}\right]$&
+        $\left[\unit{\ohm}\right]$&
+        $\left[\unit{\mega\hertz}\right]$&
+        $\left[\unit{\pico\farad}\right]$\\\hline
+        \rowcolor[gray]{0.9}
+        $25$&$40$&$1$   &$150$& $5.00$&         $11.0$&          N/A&              N/A\\
+        \rowcolor[gray]{0.9}
+        $25$&$40$&$53$  &$1$&   $120$&          $\mathbf{19.6}$& $18.0$&           $0.65$\\
+        $25$&$40$&$53$  &$50$&  $121$&          $22.6$&          $\mathbf{27.5}$&  $\mathbf{0.28}$\\
+        $25$&$40$&$53$  &$100$& $123$&          $26.9$&          $26.5$&           $0.29$\\
+        $25$&$40$&$53$  &$150$& $\mathbf{125}$& $33.2$&          $24.0$&           $0.35$\\\hline
+        \rowcolor[gray]{0.9}
+        $50$&$65$&$1$   &$300$& $10.2$&         $21.9$&          N/A&              N/A\\
+        \rowcolor[gray]{0.9}
+        $50$&$65$&$53$  &$1$&   $270$&          $\mathbf{35.7}$& $10.0$&           $0.94$\\
+        $50$&$65$&$53$  &$100$& $272$&          $41.9$&          $\mathbf{15.8}$&  $\mathbf{0.37}$\\
+        $50$&$65$&$53$  &$200$& $277$&          $50.1$&          $13.3$&           $0.52$\\
+        $50$&$65$&$53$  &$300$& $\mathbf{280}$& $65.0$&          $13.8$&           $0.48$\\\hline
+        \rowcolor[gray]{0.9}
+        $75$&$90$&$1$   &$480$& $17.3$&         $35.5$&          N/A&              N/A\\
+        \rowcolor[gray]{0.9}
+        $75$&$90$&$53$  &$1$&   $441$&          $\mathbf{50.7}$& $7.00$&           $1.17$\\
+        $75$&$90$&$53$  &$160$& $444$&          $60.8$&          $\mathbf{10.0}$&  $\mathbf{0.57}$\\
+        $75$&$90$&$53$  &$320$& $461$&          $76.2$&          $8.75$&           $0.72$\\
+        $75$&$90$&$53$  &$480$& $\mathbf{470}$& $92.9$&          $8.00$&           $0.84$\\
+    \end{tabular}
+    \label{tab_wide_coils}
+\end{table}
+
+
+\subsection{Coupling and its Sensitivity to Radial Offset}
+
+To evaluate twisted inductors in our WPT application, we measured the variation of the coupling between a pair of
+inductors using an automated measurement setup consisting of a 3D gantry built from an old 3D printer, with a fourth
+rotation axis provided by a small servo that allows us to position two inductor test coupons at arbitrary offsets and
+angles to one another (cf.\ Fig.\ \ref{fig_setup_probe}).
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=.85\figurescale]{figures/test_schematic.pdf}
+    \end{center}
+    \caption{The test schematic used in all measurements. For direct coupling factor measurements, the load resistor was
+    disconnected. We measure voltage at the output of the function generator to account for drop in its internal output
+    resistance.}
+    \label{fig_test_schematic}
+\end{figure}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=.5\figurescale]{figures/setup_probe_small.jpg}
+    \end{center}
+    \caption{The automated measurement setup we used to measure coupling while sweeping through linear and angular
+    displacements.}
+    \label{fig_setup_probe}
+\end{figure}
+
+To approximate our application, we loaded the secondary inductor with a \qty{10}{\ohm} resistor while providing a signal
+at a \qty{300}{\kilo\hertz} carrier frequency to the primary inductor from a Siglent SDG6022X function generator as
+shown in Fig.\ \ref{fig_test_schematic}. We measured both the input and output voltages of the coupled inductor pair
+using Keysight 34465A multimeters in AC Root Mean Square (RMS) mode.
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=\figurescale]{figures/symmetry_3turn_n_twist.pdf}
+    \end{center}
+    \caption{RMS output voltage of the test circuit from Fig.\ \ref{fig_test_schematic} for three pairs of matching
+    inductors with one inductor rotating w.r.t.\ the other. The inductors have $n=3$ turns each and $k=\frac{1}{2}$,
+    $k=1$, and $k=3$, respectively. For each $k$, voltage curves are plotted for a number of different radial offsets
+    between the two inductor's centers.}
+    \label{fig_symmetry_3turn_n_twist}
+\end{figure}
+
+Fig.\ \ref{fig_symmetry_3turn_n_twist} shows the ratio between input and output voltage of our test link for a set of
+three-turn inductors with multiple inversion numbers $k$ when one inductor is rotated. In practical WPT setups, the
+transmitter and receiver coils are rarely aligned perfectly, so we show measurements across a range of radial offsets.
+In line with our inductance measurements, coupling is lower at $k>0$ compared to a single-layer spiral inductor. Across
+one revolution, we find that the single-layer spiral inductor exhibits the most voltage ripple, with simple two-layer
+inductors with $k=1$ already improving ripple. For $k$ above $1$, ripple amplitude stay sconstant, but energy is shifted
+into higher frequencies that are easier to passively filter on the WPT link's secondary side in our application.
+
+Expanding our measurements in the previous section, we performed a series of measurements rotating both inductors. In
+these measurements, the coils' distance is fixed \qty{1}{\milli\meter} and the radial offset is set to a worst-case
+value of \qty{4}{\milli\meter}. Fig.\ \ref{fig_rms_ripple_n3} shows the normalized output voltage of a WPT link made
+from three-turn inductors with rotation of one inductor shown on the horizontal axis, and the rotation of the other 
+shown on the vertical axis.
+
+We performed similar measurements on 24 of our test coupons at \qty{1}{\milli\meter} and \qty{4}{\milli\meter} radial
+offsets. Fig.\ \ref{fig_k_ripple_plot} shows the combined results of these measurements, with worst-case voltage
+variation plotted across inversion count $k$ for multiple turn counts $n$ and radial offsets $r$. In this graph, we see
+that twisted inductors improve ripple compared to conventional designs, even at a low inversion count such as $k=3$.
+
+Concluding our measurements, we achieved our primary objective of reducing coupling variation under rotation, with
+twisted inductors ($k>1$) improving over conventional two-layer spiral inductors, which perform better than simple
+single-layer spiral inductors. This improvement is greatest for inductors with low turn count and consequentially coarse
+pitch, as their turns deviate the furthest from a set of ideal, concentric circles.
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=.85\figurescale]{figures/k_ripple_plot.pdf}
+    \end{center}
+    \caption{RMS Voltage ripple as a percentage of total RMS
+    output voltage in a rotating WPT setup with $R_L=\qty{10}{\ohm}$, coil distance $d=\qty{1}{\milli\meter}$ plotted
+    w.r.t. inductor inversion count $k$. Measurements were taken at two radial offsets of $r=\qty{1}{\milli\meter}$ and
+    $\qty{4}{\milli\meter}$. The shaded area indicates conventional coil layouts, with the remainder of the plot showing
+    twisted inductors.}
+    \label{fig_k_ripple_plot}
+\end{figure}
+
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=.6\figurescale]{figures/field_plot_3d_n5_k0.pdf}
+%    \end{center}
+%    \caption{The coupling between a pair of identical coils (here two simple spiral inductors with $n=5$ and
+%    $k=\frac{1}{2}$)
+%    visualized in three dimensions. The $x$ and $y$ axis show in-plane displacement, and the $z$ axis shows output
+%    amplitude in arbitrary units. Height and rotation are fixed to \qty{1}{\milli\meter} and \qty{0}{\degree},
+%    respectively. The most prominent aspects of this plot are that coupling falls off steeply with distance, and that
+%    the rotation-dependent variation is small in comparison. The circular valley around the central peak is the region
+%    where one inductor is mostly outside the other inductors, and intersects the field lines returning from the other
+%    inductor's back, leading to a negative coupling coefficient.}
+%    \label{fig_field_plot_3d}
+%\end{figure}
+
+\begin{figure}
+    \begin{center}
+        \includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n3_r4.pdf}
+    \end{center}
+    \caption{RMS ripple magnitude as a percentage of mean RMS output voltage, plotted against the rotation of each of
+    the two inductors for three variants of three-turn coils. The two coils were kept at a constant
+    \qty{4}{\milli\meter} radial offset, and the output coil was loaded with a \qty{10}{\ohm} load. The plots share the
+    same color scale to allow for visual comparison.
+    % , plots for $n=5$ can be found in Figure\ \ref{fig_rms_ripple_n5} and plots for $n=\{10,25\}$ in Figures
+    % \ref{fig_rms_ripple_n10} and \ref{fig_rms_ripple_n25} in the Appendix.
+    }
+    \label{fig_rms_ripple_n3}
+\end{figure}
+
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n5_r4.pdf}
+%    \end{center}
+%    \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 5-turn coils.}
+%    \label{fig_rms_ripple_n5}
+%\end{figure}
+
+\section{Future Work}
+
+Our derivation of twisted inductors opens up a space for future research. On the practical side, as part of our inductor
+design tool, we extended the EDA file format library gerbonara with code to automatically map gerbonara's geometry
+description to the gmsh FEM mesher. This code may be of independent interest since it allows for the extraction of FEM
+meshes from not just individual planar components, but PCBs in any file format supported by gerbonara such as KiCad's
+native file format, as well as the Gerber file format supported by the majority of EDA tools.
+
+On the theoretical side, the fact that our twisted inductor model generalizes both one- or two-layer planar spiral
+inductors as well as planar toroidal inductors would make the deduction of key parameters such as inductance and
+distributed capacitance by mathematical analysis or by finite element methods interesting. Furthermore, the precise
+contribution of vias to the twisted inductor's parasitics is interesting, especially for layouts with large values of
+inversion count $k$. We suspect that via influence will be frequency dependant as vias and traces have distinct DC
+resistances, and skin effect will affect both to a differring extent.
+
 \section{Conclusion}
 
-\section*{Availability}
+In this paper, we introduced a novel layout approach for planar, multi-layer inductors. Our \emph{twisted} inductors
-This is version \texttt{\input{version.tex}\unskip} of this paper, generated on \today.
+generalize several types of conventional planar inductors including conventional single- or two-layer planar spiral
+inductors as well as planar toroidal inductors. For inversion count parameter $k\ge 2$, twisted inductors produce
+magnetic field distributions that have better rotational symmetry along the inductor's main axis compared to either
+conventional single- or two-layer planar spiral inductors, which yields lower output ripple in WPT through rotating
+joints and enables the use of smaller and lighter secondary-side circuitry, improving efficiency.
 
-% The git repository with the
+Furthermore, besides the advantages twisted inductors show in our particular application, we found that our sample
-% LaTeX source for this paper as well as our data analysis and demo code can be found at:
+twisted inductors have up to \qty{50}{\percent} improved self-resonant frequency as well as up to \qty{6.5}{\percent}
+increased inductance compared to conventional two-layer planar spiral inductors.
 
-% \center{\url{https://git.jaseg.de/nice-coils.git}}
+We base our evaluation on laboratory measurements on a set of 39 sample inductors in total, including an automated,
+four-dimensional mapping of the coupling between a pair of identical inductors. We provide both an analytical
+description of twisted inductor construction as well as a set of Open-Source tools for their design, available at the
+link at the end of this paper.
 
 \printbibliography[heading=bibintoc]
 
+%\section{Supplemental plots}
+
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=\figurescale]{figures/symmetry_10turn_n_twist.pdf}
+%    \end{center}
+%    \caption{Coupled RMS output voltage of three pairs of matching inductors with $n=10$ turns each and
+%    $k=\frac{1}{2}$, $k=1$, and $k=3$, respectively, shown as in Figure\ \ref{fig_symmetry_3turn_n_twist}}
+%    \label{fig_symmetry_10turn_n_twist}
+%\end{figure}
+
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n10_r4.pdf}
+%    \end{center}
+%    \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 10-turn coils.}
+%    \label{fig_rms_ripple_n10}
+%\end{figure}
+
+%\begin{figure}
+%    \begin{center}
+%        \includegraphics[width=.75\figurescale]{figures/rms_ripple_double_rotation_n25_r4.pdf}
+%    \end{center}
+%    \caption{RMS ripple magnitude as shown in Figure\ \ref{fig_rms_ripple_n3} for four different 25-turn coils.}
+%    \label{fig_rms_ripple_n25}
+%\end{figure}
+
+\FloatBarrier
+%\section{Layout examples}
+%\label{sec_appendix_layout_examples}
+
+\begin{figure*}
+    \appendix
+    \begin{center}
+        \includegraphics[width=.75\textwidth]{figures/nk_complex_illust.pdf}
+    \end{center}
+    \caption{Layout examples for a number of combinations of turn count $n$ and inversion count $k$. Note that in this
+    illustration we chose values for $n$ and $k$ such that all pairs are coprime.}
+    \label{fig_nk_complex_illust}
+\end{figure*}
 \end{document}

paper/version.tex

@@ -1 +1 @@
-
+final-tpel-submission-2025-01-27-0-g99d4905

sim_runner.py

@@ -14,6 +14,7 @@ import time
 import math
 import json
 import subprocess
+import re
 
 import tqdm
 import click
@@ -25,7 +26,7 @@ def mesh_args(db, coil_id, mesh_type, mesh_file, outfile, **kwargs):
     rows = dict(db.execute('SELECT key, value FROM results WHERE coil_id=?', (coil_id,)).fetchall())
     rows.update(kwargs)
     args = ['python', '-m', 'twisted_coil_gen_twolayer', mesh_type, mesh_file, '--pcb']
-    for k, v in rows:
+    for k, v in rows.items():
         prefix, _, k = k.partition('.')
         if v != 'False' and prefix == 'gen':
             args.append('--' + k.replace('_', '-'))
@@ -104,9 +105,10 @@ def list_runs(ctx):
 
 @cli.command()
 @click.option('-r', '--run-id')
+@click.option('-l', '--log-dir', default='logs', type=click.Path(dir_okay=True, file_okay=False, path_type=pathlib.Path))
 @click.option('-m', '--mesh-dir', default='meshes')
 @click.pass_context
-def list_coils(ctx, run_id, mesh_dir):
+def list_coils(ctx, run_id, log_dir, mesh_dir):
     db = ctx.obj['db_connect']()
     if run_id is None:
         run_id, = db.execute('SELECT run_id FROM runs ORDER BY timestamp DESC LIMIT 1').fetchone()
@@ -122,9 +124,11 @@ def list_coils(ctx, run_id, mesh_dir):
             'gen.inner_diameter': 'ID[mm]',
             'gen.outer_diameter': 'OD[mm]',
             'calculated_fill_factor': 'Fill factor',
-            'calculated_approximate_inductance': 'L [µH]',
+            'calculated_approximate_inductance': 'L [µH] (design)',
             'calculated_trace_length': 'track len [mm]',
-            'calculated_approximate_resistance': 'R [mΩ]'}
+            'calculated_approximate_resistance': 'R [mΩ] (design)',
+            'sim_inductance': 'L [µH] (sim)',
+            'sim_resistance': 'R [mΩ] (sim)'}
     out = []
     for row in db.execute('SELECT *, MAX(meshes.timestamp) FROM coils LEFT JOIN meshes ON coils.coil_id=meshes.coil_id WHERE run_id=? GROUP BY coils.coil_id, mesh_type ORDER BY meshes.timestamp', (run_id,)).fetchall():
         if row['timestamp']:
@@ -151,6 +155,19 @@ def list_coils(ctx, run_id, mesh_dir):
         if 'calculated_fill_factor' in params:
             params['calculated_fill_factor'] = f'{float(params["calculated_fill_factor"]):.03f}'
 
+        log_file = log_dir / (pathlib.Path(row['filename']).stem + '_elmer_self_inductance.log')
+        if log_file.is_file():
+            log = log_file.read_text()
+            resistance = re.search(r'Coil resistance calculated by solver: ([0-9.e+-]*) (\w?)Ω', log)
+            inductance = re.search(r'Inductance calucated from field: ([0-9.e+-]*) (\w?)H', log)
+            si_prefix = {'': 1, 'm': 1e-3, 'µ': 1e-6, 'n': 1e-9, 'p': 1e-12}
+            if resistance:
+                resistance = float(resistance.group(1)) * si_prefix[resistance.group(2)]
+                params['sim_resistance'] = format(resistance*1e3, '.3f')
+            if inductance:
+                inductance = float(inductance.group(1)) * si_prefix[inductance.group(2)]
+                params['sim_inductance'] = format(inductance*1e6, '.3f')
+
         out.append([row['coil_id'], row['mesh_type'], state, row['timestamp']] + [params.get(key, '-') for key in keys])
 
     print(tabulate(out, headers=['coil', 'mesh', 'state', 'time'] + list(keys.values()), disable_numparse=True, stralign='right'))
@@ -180,13 +197,13 @@ def run(ctx, run_id, log_dir, mesh_dir):
 
 @run.command()
 @click.option('-j', '--num-jobs', type=int, default=1, help='Number of jobs to run in parallel')
+@click.option('-t', '--mesh-type', type=click.Choice(['split', 'normal', 'mutual']), default=['split', 'normal', 'mutual'], multiple=True)
 @click.pass_context
-def generate_meshes(ctx, num_jobs):
+def generate_meshes(ctx, num_jobs, mesh_type):
     db = ctx.obj['db_connect']()
     rows = [row['coil_id'] for row in db.execute('SELECT coil_id FROM coils WHERE run_id=?', (ctx.obj['run_id'],)).fetchall()]
-    mesh_types = ['split', 'normal', 'mutual']
 
-    params = list(itertools.product(rows, mesh_types))
+    params = list(itertools.product(rows, mesh_type))
     all_files = {get_mesh_file(db, ctx.obj['mesh_dir'], ctx.obj['run_id'], coil_id, mesh_type): (coil_id, mesh_type) for coil_id, mesh_type in params}
     todo = [(coil_id, mesh_type) for f, (coil_id, mesh_type) in all_files.items() if not f.is_file()]
 
@@ -232,7 +249,7 @@ def self_inductance(ctx, num_jobs):
                 with tempfile.TemporaryDirectory() as tmpdir:
                     try:
                         tqdm.tqdm.write(f'Processing {mesh_file}')
-                        res = subprocess.run(['python', '-m', 'coil_parasitics', 'inductance', '--sim-dir', tmpdir, mesh_file], check=True, capture_output=True)
+                        res = subprocess.run(['python', '-m', 'coil_parasitics', 'inductance', '--sim-dir', tmpdir, mesh_file], check=True, capture_output=True, text=True)
                         logfile.write_text(res.stdout+res.stderr)
                     except subprocess.CalledProcessError as e:
                         print(f'Error running simulation, rc={e.returncode}')
@@ -266,8 +283,8 @@ def self_inductance(ctx, num_jobs):
     q.join()
 
 @run.command()
-@click.option('target_hosts', type=click.Path(exists=True, dir_okay=False, path_type=pathlib.Path), help='File with one SSH target host name per line')
+@click.argument('target_hosts', type=click.Path(exists=True, dir_okay=False, path_type=pathlib.Path))
-@click.option('job_file', type=click.Path(exists=True, dir_okay=False, path_type=pathlib.Path), help='JSON job description file')
+@click.argument('job_file', type=click.Path(exists=True, dir_okay=False, path_type=pathlib.Path))
 @click.pass_context
 def run_mutual_inductance(ctx, target_hosts, job_file):
     db = ctx.obj['db_connect']()
@@ -301,7 +318,7 @@ def run_mutual_inductance(ctx, target_hosts, job_file):
                                        'ElmerGrid_stderr.log',
                                        'ElmerSolver_stdout.log',
                                        'ElmerSolver_stderr.log']:
-                            subprocess.run(['scp', f'{host}:{mesh_abs}/{workdir.name}/{fn}', str(host_log(Path(fn).stem))], check=True, capture_output=True, text=True)
+                                subprocess.run(['scp', f'{host}:{mesh_abs}/{workdir.name}/{fn}', str(host_log(Path(fn).stem))], check=True, capture_output=True, text=True)
 
                         except subprocess.CalledProcessError as e:
                             print(f'Error copying simulation logs, rc={e.returncode}', file=sys.stderr)

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+</svg>

svg-vis-paper-plain-vias.svg

@@ -0,0 +1,159 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+   width="45.349998mm"
+   height="42.200001mm"
+   viewBox="0 0 45.349998 42.200001"
+   version="1.1"
+   id="svg1"
+   inkscape:version="1.4 (e7c3feb100, 2024-10-09)"
+   sodipodi:docname="svg-vis-paper-plain-vias.svg"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:svg="http://www.w3.org/2000/svg">
+  <sodipodi:namedview
+     id="namedview1"
+     pagecolor="#ffffff"
+     bordercolor="#666666"
+     borderopacity="1.0"
+     inkscape:showpageshadow="2"
+     inkscape:pageopacity="0.0"
+     inkscape:pagecheckerboard="0"
+     inkscape:deskcolor="#d1d1d1"
+     inkscape:document-units="mm"
+     inkscape:zoom="6.6386364"
+     inkscape:cx="60.102704"
+     inkscape:cy="110.94146"
+     inkscape:window-width="3840"
+     inkscape:window-height="2091"
+     inkscape:window-x="0"
+     inkscape:window-y="0"
+     inkscape:window-maximized="1"
+     inkscape:current-layer="layer1" />
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+     id="defs1" />
+  <g
+     inkscape:label="Layer 1"
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+     id="layer1">
+    <g
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+       transform="translate(-349.4,-134.36458)" />
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+</svg>

twisted_coil_gen_twolayer.py

@@ -1,5 +1,6 @@
 #!/usr/bin/env python3
 
+import warnings
 import subprocess
 import sys
 import math
@@ -605,6 +606,8 @@ def print_valid_twists(ctx, param, value):
 @click.option('--outer-diameter', type=float, default=50, help='Outer diameter [mm]')
 @click.option('--svg-out', type=click.Path(writable=True, dir_okay=False, path_type=Path), help='Filename to output SVG illustration of the coil to')
 @click.option('--inner-diameter', type=float, default=25, help='Inner diameter [mm]')
+@click.option('--stagger-inner-vias/--no-stagger-inner-vias', default=False, help='Stagger inner via ring')
+@click.option('--stagger-outer-vias/--no-stagger-outer-vias', default=False, help='Stagger outer via ring')
 @click.option('--trace-width', type=float, default=None)
 @click.option('--via-diameter', type=float, default=0.6)
 @click.option('--two-layer/--single-layer', default=True)
@@ -617,7 +620,6 @@ def print_valid_twists(ctx, param, value):
 @click.option('--twists', type=int, default=1, help='Number of twists per revolution. Note that this number must be co-prime to the number of turns. Run with --show-twists to list valid values. (default: 1)')
 @click.option('--circle-segments', type=int, default=64, help='When not using arcs, the number of points to use for arc interpolation per 360 degrees.')
 @click.option('--show-twists', callback=print_valid_twists, expose_value=False, type=int, is_eager=True, help='Calculate and show valid --twists counts for the given number of turns. Takes the number of turns as a value.')
-@click.option('--close-loop', is_flag=True, help='Close coil loop for simulation meshes')
 @click.option('--clearance', type=float, default=None)
 @click.option('--arc-tolerance', type=float, default=0.02)
 @click.option('--mesh-split-out', type=click.Path(writable=True, dir_okay=False, path_type=Path))
@@ -635,7 +637,7 @@ def generate(outfile, turns, outer_diameter, inner_diameter, via_diameter, via_d
              footprint_name, layer_pair, twists, clipboard, counter_clockwise, keepout_zone, keepout_margin,
              arc_tolerance, pcb, mesh_out, magneticalc_out, circle_segments, mesh_split_out, copper_thickness,
              board_thickness, mesh_mutual_out, mutual_offset_x, mutual_offset_y, mutual_offset_z, mutual_rotation_z,
-             two_layer, svg_out, close_loop):
+             two_layer, svg_out, stagger_inner_vias, stagger_outer_vias):
 
     if 'WAYLAND_DISPLAY' in os.environ:
         copy, paste, cliputil = ['wl-copy'], ['wl-paste'], 'xclip'
@@ -643,7 +645,10 @@ def generate(outfile, turns, outer_diameter, inner_diameter, via_diameter, via_d
         copy, paste, cliputil = ['xclip', '-i', '-sel', 'clipboard'], ['xclip', '-o', '-sel' 'clipboard'], 'wl-clipboard'
 
     if gcd(twists, turns) != 1:
-        raise click.ClickException('For the geometry to work out, the --twists parameter must be co-prime to --turns, i.e. the two must have 1 as their greatest common divisor. You can print valid values for --twists by running this command with --show-twists [turns number].')
+        raise click.ClickException(f'For the geometry to work out, the --twists parameter must be co-prime to --turns, i.e. the two must have 1 as their greatest common divisor. You can print valid values for --twists by running this command with --show-twists [turns number]. Right now, both are divisible by {gcd(twists, turns)}.')
+
+    if (stagger_inner_vias or stagger_outer_vias) and twists%2 != 0:
+        raise click.ClickException('For --stagger-inner/outer-vias to work, --twists must be even and --turns must be odd.')
 
     if (mesh_out or mesh_split_out or mesh_mutual_out) and not pcb:
         raise click.ClickException('--pcb is required when --mesh-out, --mesh-mutual-out or --mesh-split-out are used.')
@@ -708,9 +713,9 @@ def generate(outfile, turns, outer_diameter, inner_diameter, via_diameter, via_d
     print(f'    {degrees(outer_via_angle):.1f} deg / via', file=sys.stderr)
 
     # Check if the vias of the inner ring are so large that they would overlap
-    if inner_via_angle*twists > 2*pi:
+    if inner_via_angle*twists > (4*pi if stagger_inner_vias else 2*pi):
-        min_dia = 2*((via_diameter + clearance) / (2*sin(pi / twists)) + via_offset)
+        min_dia = 2*((via_diameter + clearance) / (2*sin(pi / twists * (2 if stagger_inner_vias else 1))) + via_offset)
-        raise click.ClickException(f'Error: Overlapping vias in inner via ring. Calculated minimum inner diameter is {min_dia:.2f} mm.')
+        warnings.warn(f'Overlapping vias in inner via ring. Calculated minimum inner diameter is {min_dia:.2f} mm.')
 
     pitch = clearance + trace_width
     t, _, b = layer_pair.partition(',')
@@ -862,7 +867,7 @@ def generate(outfile, turns, outer_diameter, inner_diameter, via_diameter, via_d
         x = inverse[i]*floor(2*sweeping_angle / (2*pi)) * 2*pi
         (x0, y0), (xn, yn), clen = do_spiral(0, outer_radius, inner_radius, start_angle, fold_angle, (x + start_angle)/total_angle, (x + fold_angle)/total_angle, circle_segments)
         if two_layer:
-            do_spiral(1, inner_radius, outer_radius, fold_angle, end_angle, (x + fold_angle)/total_angle, (x + end_angle)/total_angle)
+            do_spiral(1, inner_radius, outer_radius, fold_angle, end_angle, (x + fold_angle)/total_angle, (x + end_angle)/total_angle, circle_segments)
         else:
             dr = outer_radius - inner_radius
             xq = xn + cos(fold_angle) * dr
@@ -875,7 +880,11 @@ def generate(outfile, turns, outer_diameter, inner_diameter, via_diameter, via_d
             path.move(xn, yn)
             path.line(xq, yq)
 
-        xv, yv = inner_via_ring_radius*cos(fold_angle), inner_via_ring_radius*sin(fold_angle)
+        r = inner_via_ring_radius
+        if stagger_inner_vias:
+            if i%2 != 0:
+                r -= 2*via_offset
+        xv, yv = r*cos(fold_angle), r*sin(fold_angle)
         pads.append(make_via(xv, yv, layer_pair))
         if not isclose(via_offset, 0, abs_tol=1e-6):
             lines.append(make_line(xn, yn, xv, yv, layer_pair[0]))
@@ -883,8 +892,12 @@ def generate(outfile, turns, outer_diameter, inner_diameter, via_diameter, via_d
         svg_vias.append(Tag('circle', cx=xv, cy=yv, r=via_diameter/2, stroke='none', fill='white'))
         svg_vias.append(Tag('circle', cx=xv, cy=yv, r=via_drill/2, stroke='none', fill='black'))
 
-        if i > 0 or close_loop:
+        if i > 0:
-            xv, yv = outer_via_ring_radius*cos(start_angle), outer_via_ring_radius*sin(start_angle)
+            r = outer_via_ring_radius
+            if stagger_outer_vias:
+                if i%2 != 0:
+                    r += 2*via_offset
+            xv, yv = r*cos(start_angle), r*sin(start_angle)
             pads.append(make_via(xv, yv, layer_pair))
             if not isclose(via_offset, 0, abs_tol=1e-6):
                 lines.append(make_line(x0, y0, xv, yv, layer_pair[0]))
@@ -892,7 +905,7 @@ def generate(outfile, turns, outer_diameter, inner_diameter, via_diameter, via_d
             svg_vias.append(Tag('circle', cx=xv, cy=yv, r=via_diameter/2, stroke='none', fill='white'))
             svg_vias.append(Tag('circle', cx=xv, cy=yv, r=via_drill/2, stroke='none', fill='black'))
 
-    l_total = clen*twists*2
+    l_total = clen*twists*(2 if two_layer else 1)
     print(f'Approximate track length: {l_total:.2f} mm', file=sys.stderr)
     A = copper_thickness/1e3 * trace_width/1e3
     rho = 1.68e-8

wide_coil_1.kicad_prl

@@ -0,0 +1,136 @@
+{
+  "board": {
+    "active_layer": 0,
+    "active_layer_preset": "All Layers",
+    "auto_track_width": true,
+    "hidden_netclasses": [],
+    "hidden_nets": [],
+    "high_contrast_mode": 0,
+    "net_color_mode": 1,
+    "opacity": {
+      "images": 0.6,
+      "pads": 1.0,
+      "shapes": 1.0,
+      "tracks": 1.0,
+      "vias": 1.0,
+      "zones": 0.6
+    },
+    "selection_filter": {
+      "dimensions": true,
+      "footprints": true,
+      "graphics": true,
+      "keepouts": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pads": true,
+      "text": true,
+      "tracks": true,
+      "vias": true,
+      "zones": true
+    },
+    "visible_items": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      8,
+      9,
+      10,
+      11,
+      12,
+      13,
+      15,
+      16,
+      17,
+      18,
+      19,
+      20,
+      21,
+      22,
+      23,
+      24,
+      25,
+      26,
+      27,
+      28,
+      29,
+      30,
+      32,
+      33,
+      34,
+      35,
+      36,
+      39,
+      40,
+      41
+    ],
+    "visible_layers": "ffffffff_ffffffff",
+    "zone_display_mode": 0
+  },
+  "git": {
+    "repo_password": "",
+    "repo_type": "",
+    "repo_username": "",
+    "ssh_key": ""
+  },
+  "meta": {
+    "filename": "wide_coil_1.kicad_prl",
+    "version": 4
+  },
+  "net_inspector_panel": {
+    "col_hidden": [
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ],
+    "col_order": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8,
+      9
+    ],
+    "col_widths": [],
+    "custom_group_rules": [],
+    "expanded_rows": [],
+    "filter_by_net_name": true,
+    "filter_by_netclass": true,
+    "filter_text": "",
+    "group_by_constraint": false,
+    "group_by_netclass": false,
+    "show_unconnected_nets": false,
+    "show_zero_pad_nets": false,
+    "sort_ascending": true,
+    "sorting_column": -1
+  },
+  "project": {
+    "files": []
+  },
+  "schematic": {
+    "selection_filter": {
+      "graphics": true,
+      "images": true,
+      "labels": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pins": true,
+      "symbols": true,
+      "text": true,
+      "wires": true
+    }
+  }
+}

wide_coil_1.kicad_pro

@@ -0,0 +1,276 @@
+{
+  "board": {
+    "3dviewports": [],
+    "design_settings": {
+      "defaults": {
+        "apply_defaults_to_fp_fields": false,
+        "apply_defaults_to_fp_shapes": false,
+        "apply_defaults_to_fp_text": false,
+        "board_outline_line_width": 0.05,
+        "copper_line_width": 0.2,
+        "copper_text_italic": false,
+        "copper_text_size_h": 1.5,
+        "copper_text_size_v": 1.5,
+        "copper_text_thickness": 0.3,
+        "copper_text_upright": false,
+        "courtyard_line_width": 0.05,
+        "dimension_precision": 4,
+        "dimension_units": 3,
+        "dimensions": {
+          "arrow_length": 1270000,
+          "extension_offset": 500000,
+          "keep_text_aligned": true,
+          "suppress_zeroes": false,
+          "text_position": 0,
+          "units_format": 1
+        },
+        "fab_line_width": 0.1,
+        "fab_text_italic": false,
+        "fab_text_size_h": 1.0,
+        "fab_text_size_v": 1.0,
+        "fab_text_thickness": 0.15,
+        "fab_text_upright": false,
+        "other_line_width": 0.1,
+        "other_text_italic": false,
+        "other_text_size_h": 1.0,
+        "other_text_size_v": 1.0,
+        "other_text_thickness": 0.15,
+        "other_text_upright": false,
+        "pads": {
+          "drill": 0.8,
+          "height": 1.27,
+          "width": 2.54
+        },
+        "silk_line_width": 0.1,
+        "silk_text_italic": false,
+        "silk_text_size_h": 1.0,
+        "silk_text_size_v": 1.0,
+        "silk_text_thickness": 0.1,
+        "silk_text_upright": false,
+        "zones": {
+          "min_clearance": 0.5
+        }
+      },
+      "diff_pair_dimensions": [],
+      "drc_exclusions": [],
+      "meta": {
+        "version": 2
+      },
+      "rule_severities": {
+        "annular_width": "error",
+        "clearance": "error",
+        "connection_width": "warning",
+        "copper_edge_clearance": "error",
+        "copper_sliver": "warning",
+        "courtyards_overlap": "error",
+        "diff_pair_gap_out_of_range": "error",
+        "diff_pair_uncoupled_length_too_long": "error",
+        "drill_out_of_range": "error",
+        "duplicate_footprints": "warning",
+        "extra_footprint": "warning",
+        "footprint": "error",
+        "footprint_filters_mismatch": "ignore",
+        "footprint_symbol_mismatch": "warning",
+        "footprint_type_mismatch": "ignore",
+        "hole_clearance": "error",
+        "hole_to_hole": "warning",
+        "holes_co_located": "warning",
+        "invalid_outline": "error",
+        "isolated_copper": "warning",
+        "item_on_disabled_layer": "error",
+        "items_not_allowed": "error",
+        "length_out_of_range": "error",
+        "lib_footprint_issues": "warning",
+        "lib_footprint_mismatch": "warning",
+        "malformed_courtyard": "error",
+        "microvia_drill_out_of_range": "error",
+        "missing_courtyard": "ignore",
+        "missing_footprint": "warning",
+        "net_conflict": "warning",
+        "npth_inside_courtyard": "ignore",
+        "padstack": "warning",
+        "pth_inside_courtyard": "ignore",
+        "shorting_items": "error",
+        "silk_edge_clearance": "warning",
+        "silk_over_copper": "warning",
+        "silk_overlap": "warning",
+        "skew_out_of_range": "error",
+        "solder_mask_bridge": "error",
+        "starved_thermal": "error",
+        "text_height": "warning",
+        "text_thickness": "warning",
+        "through_hole_pad_without_hole": "error",
+        "too_many_vias": "error",
+        "track_dangling": "warning",
+        "track_width": "error",
+        "tracks_crossing": "error",
+        "unconnected_items": "error",
+        "unresolved_variable": "error",
+        "via_dangling": "warning",
+        "zones_intersect": "error"
+      },
+      "rules": {
+        "max_error": 0.005,
+        "min_clearance": 0.0,
+        "min_connection": 0.0,
+        "min_copper_edge_clearance": 0.5,
+        "min_hole_clearance": 0.25,
+        "min_hole_to_hole": 0.25,
+        "min_microvia_diameter": 0.2,
+        "min_microvia_drill": 0.1,
+        "min_resolved_spokes": 2,
+        "min_silk_clearance": 0.0,
+        "min_text_height": 0.8,
+        "min_text_thickness": 0.08,
+        "min_through_hole_diameter": 0.3,
+        "min_track_width": 0.0,
+        "min_via_annular_width": 0.1,
+        "min_via_diameter": 0.5,
+        "solder_mask_to_copper_clearance": 0.0,
+        "use_height_for_length_calcs": true
+      },
+      "teardrop_options": [
+        {
+          "td_onpadsmd": true,
+          "td_onroundshapesonly": false,
+          "td_ontrackend": false,
+          "td_onviapad": true
+        }
+      ],
+      "teardrop_parameters": [
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_round_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_rect_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_track_end",
+          "td_width_to_size_filter_ratio": 0.9
+        }
+      ],
+      "track_widths": [],
+      "tuning_pattern_settings": {
+        "diff_pair_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 1.0
+        },
+        "diff_pair_skew_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        },
+        "single_track_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        }
+      },
+      "via_dimensions": [],
+      "zones_allow_external_fillets": false
+    },
+    "ipc2581": {
+      "dist": "",
+      "distpn": "",
+      "internal_id": "",
+      "mfg": "",
+      "mpn": ""
+    },
+    "layer_pairs": [],
+    "layer_presets": [],
+    "viewports": []
+  },
+  "boards": [],
+  "cvpcb": {
+    "equivalence_files": []
+  },
+  "libraries": {
+    "pinned_footprint_libs": [],
+    "pinned_symbol_libs": []
+  },
+  "meta": {
+    "filename": "wide_coil_1.kicad_pro",
+    "version": 1
+  },
+  "net_settings": {
+    "classes": [
+      {
+        "bus_width": 12,
+        "clearance": 0.2,
+        "diff_pair_gap": 0.25,
+        "diff_pair_via_gap": 0.25,
+        "diff_pair_width": 0.2,
+        "line_style": 0,
+        "microvia_diameter": 0.3,
+        "microvia_drill": 0.1,
+        "name": "Default",
+        "pcb_color": "rgba(0, 0, 0, 0.000)",
+        "priority": -1,
+        "schematic_color": "rgba(0, 0, 0, 0.000)",
+        "track_width": 0.2,
+        "via_diameter": 0.6,
+        "via_drill": 0.3,
+        "wire_width": 6
+      }
+    ],
+    "meta": {
+      "version": 4
+    },
+    "net_colors": null,
+    "netclass_assignments": null,
+    "netclass_patterns": []
+  },
+  "pcbnew": {
+    "last_paths": {
+      "gencad": "",
+      "idf": "",
+      "netlist": "",
+      "plot": "gerber_wide_1",
+      "pos_files": "",
+      "specctra_dsn": "",
+      "step": "",
+      "svg": "",
+      "vrml": ""
+    },
+    "page_layout_descr_file": ""
+  },
+  "schematic": {
+    "legacy_lib_dir": "",
+    "legacy_lib_list": []
+  },
+  "sheets": [],
+  "text_variables": {}
+}

wide_coil_2.kicad_prl

@@ -0,0 +1,136 @@
+{
+  "board": {
+    "active_layer": 0,
+    "active_layer_preset": "All Layers",
+    "auto_track_width": true,
+    "hidden_netclasses": [],
+    "hidden_nets": [],
+    "high_contrast_mode": 0,
+    "net_color_mode": 1,
+    "opacity": {
+      "images": 0.6,
+      "pads": 1.0,
+      "shapes": 1.0,
+      "tracks": 1.0,
+      "vias": 1.0,
+      "zones": 0.6
+    },
+    "selection_filter": {
+      "dimensions": true,
+      "footprints": true,
+      "graphics": true,
+      "keepouts": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pads": true,
+      "text": true,
+      "tracks": true,
+      "vias": true,
+      "zones": true
+    },
+    "visible_items": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      8,
+      9,
+      10,
+      11,
+      12,
+      13,
+      15,
+      16,
+      17,
+      18,
+      19,
+      20,
+      21,
+      22,
+      23,
+      24,
+      25,
+      26,
+      27,
+      28,
+      29,
+      30,
+      32,
+      33,
+      34,
+      35,
+      36,
+      39,
+      40,
+      41
+    ],
+    "visible_layers": "ffffffff_ffffffff",
+    "zone_display_mode": 0
+  },
+  "git": {
+    "repo_password": "",
+    "repo_type": "",
+    "repo_username": "",
+    "ssh_key": ""
+  },
+  "meta": {
+    "filename": "wide_coil_2.kicad_prl",
+    "version": 4
+  },
+  "net_inspector_panel": {
+    "col_hidden": [
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ],
+    "col_order": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8,
+      9
+    ],
+    "col_widths": [],
+    "custom_group_rules": [],
+    "expanded_rows": [],
+    "filter_by_net_name": true,
+    "filter_by_netclass": true,
+    "filter_text": "",
+    "group_by_constraint": false,
+    "group_by_netclass": false,
+    "show_unconnected_nets": false,
+    "show_zero_pad_nets": false,
+    "sort_ascending": true,
+    "sorting_column": -1
+  },
+  "project": {
+    "files": []
+  },
+  "schematic": {
+    "selection_filter": {
+      "graphics": true,
+      "images": true,
+      "labels": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pins": true,
+      "symbols": true,
+      "text": true,
+      "wires": true
+    }
+  }
+}

wide_coil_2.kicad_pro

@@ -0,0 +1,276 @@
+{
+  "board": {
+    "3dviewports": [],
+    "design_settings": {
+      "defaults": {
+        "apply_defaults_to_fp_fields": false,
+        "apply_defaults_to_fp_shapes": false,
+        "apply_defaults_to_fp_text": false,
+        "board_outline_line_width": 0.05,
+        "copper_line_width": 0.2,
+        "copper_text_italic": false,
+        "copper_text_size_h": 1.5,
+        "copper_text_size_v": 1.5,
+        "copper_text_thickness": 0.3,
+        "copper_text_upright": false,
+        "courtyard_line_width": 0.05,
+        "dimension_precision": 4,
+        "dimension_units": 3,
+        "dimensions": {
+          "arrow_length": 1270000,
+          "extension_offset": 500000,
+          "keep_text_aligned": true,
+          "suppress_zeroes": false,
+          "text_position": 0,
+          "units_format": 1
+        },
+        "fab_line_width": 0.1,
+        "fab_text_italic": false,
+        "fab_text_size_h": 1.0,
+        "fab_text_size_v": 1.0,
+        "fab_text_thickness": 0.15,
+        "fab_text_upright": false,
+        "other_line_width": 0.1,
+        "other_text_italic": false,
+        "other_text_size_h": 1.0,
+        "other_text_size_v": 1.0,
+        "other_text_thickness": 0.15,
+        "other_text_upright": false,
+        "pads": {
+          "drill": 0.8,
+          "height": 1.27,
+          "width": 2.54
+        },
+        "silk_line_width": 0.1,
+        "silk_text_italic": false,
+        "silk_text_size_h": 1.0,
+        "silk_text_size_v": 1.0,
+        "silk_text_thickness": 0.1,
+        "silk_text_upright": false,
+        "zones": {
+          "min_clearance": 0.5
+        }
+      },
+      "diff_pair_dimensions": [],
+      "drc_exclusions": [],
+      "meta": {
+        "version": 2
+      },
+      "rule_severities": {
+        "annular_width": "error",
+        "clearance": "error",
+        "connection_width": "warning",
+        "copper_edge_clearance": "error",
+        "copper_sliver": "warning",
+        "courtyards_overlap": "error",
+        "diff_pair_gap_out_of_range": "error",
+        "diff_pair_uncoupled_length_too_long": "error",
+        "drill_out_of_range": "error",
+        "duplicate_footprints": "warning",
+        "extra_footprint": "warning",
+        "footprint": "error",
+        "footprint_filters_mismatch": "ignore",
+        "footprint_symbol_mismatch": "warning",
+        "footprint_type_mismatch": "ignore",
+        "hole_clearance": "error",
+        "hole_to_hole": "warning",
+        "holes_co_located": "warning",
+        "invalid_outline": "error",
+        "isolated_copper": "warning",
+        "item_on_disabled_layer": "error",
+        "items_not_allowed": "error",
+        "length_out_of_range": "error",
+        "lib_footprint_issues": "warning",
+        "lib_footprint_mismatch": "warning",
+        "malformed_courtyard": "error",
+        "microvia_drill_out_of_range": "error",
+        "missing_courtyard": "ignore",
+        "missing_footprint": "warning",
+        "net_conflict": "warning",
+        "npth_inside_courtyard": "ignore",
+        "padstack": "warning",
+        "pth_inside_courtyard": "ignore",
+        "shorting_items": "error",
+        "silk_edge_clearance": "warning",
+        "silk_over_copper": "warning",
+        "silk_overlap": "warning",
+        "skew_out_of_range": "error",
+        "solder_mask_bridge": "error",
+        "starved_thermal": "error",
+        "text_height": "warning",
+        "text_thickness": "warning",
+        "through_hole_pad_without_hole": "error",
+        "too_many_vias": "error",
+        "track_dangling": "warning",
+        "track_width": "error",
+        "tracks_crossing": "error",
+        "unconnected_items": "error",
+        "unresolved_variable": "error",
+        "via_dangling": "warning",
+        "zones_intersect": "error"
+      },
+      "rules": {
+        "max_error": 0.005,
+        "min_clearance": 0.0,
+        "min_connection": 0.0,
+        "min_copper_edge_clearance": 0.5,
+        "min_hole_clearance": 0.25,
+        "min_hole_to_hole": 0.25,
+        "min_microvia_diameter": 0.2,
+        "min_microvia_drill": 0.1,
+        "min_resolved_spokes": 2,
+        "min_silk_clearance": 0.0,
+        "min_text_height": 0.8,
+        "min_text_thickness": 0.08,
+        "min_through_hole_diameter": 0.3,
+        "min_track_width": 0.0,
+        "min_via_annular_width": 0.1,
+        "min_via_diameter": 0.5,
+        "solder_mask_to_copper_clearance": 0.0,
+        "use_height_for_length_calcs": true
+      },
+      "teardrop_options": [
+        {
+          "td_onpadsmd": true,
+          "td_onroundshapesonly": false,
+          "td_ontrackend": false,
+          "td_onviapad": true
+        }
+      ],
+      "teardrop_parameters": [
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_round_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_rect_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_track_end",
+          "td_width_to_size_filter_ratio": 0.9
+        }
+      ],
+      "track_widths": [],
+      "tuning_pattern_settings": {
+        "diff_pair_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 1.0
+        },
+        "diff_pair_skew_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        },
+        "single_track_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        }
+      },
+      "via_dimensions": [],
+      "zones_allow_external_fillets": false
+    },
+    "ipc2581": {
+      "dist": "",
+      "distpn": "",
+      "internal_id": "",
+      "mfg": "",
+      "mpn": ""
+    },
+    "layer_pairs": [],
+    "layer_presets": [],
+    "viewports": []
+  },
+  "boards": [],
+  "cvpcb": {
+    "equivalence_files": []
+  },
+  "libraries": {
+    "pinned_footprint_libs": [],
+    "pinned_symbol_libs": []
+  },
+  "meta": {
+    "filename": "wide_coil_2.kicad_pro",
+    "version": 1
+  },
+  "net_settings": {
+    "classes": [
+      {
+        "bus_width": 12,
+        "clearance": 0.2,
+        "diff_pair_gap": 0.25,
+        "diff_pair_via_gap": 0.25,
+        "diff_pair_width": 0.2,
+        "line_style": 0,
+        "microvia_diameter": 0.3,
+        "microvia_drill": 0.1,
+        "name": "Default",
+        "pcb_color": "rgba(0, 0, 0, 0.000)",
+        "priority": -1,
+        "schematic_color": "rgba(0, 0, 0, 0.000)",
+        "track_width": 0.2,
+        "via_diameter": 0.6,
+        "via_drill": 0.3,
+        "wire_width": 6
+      }
+    ],
+    "meta": {
+      "version": 4
+    },
+    "net_colors": null,
+    "netclass_assignments": null,
+    "netclass_patterns": []
+  },
+  "pcbnew": {
+    "last_paths": {
+      "gencad": "",
+      "idf": "",
+      "netlist": "",
+      "plot": "gerber_wide_2",
+      "pos_files": "",
+      "specctra_dsn": "",
+      "step": "",
+      "svg": "",
+      "vrml": ""
+    },
+    "page_layout_descr_file": ""
+  },
+  "schematic": {
+    "legacy_lib_dir": "",
+    "legacy_lib_list": []
+  },
+  "sheets": [],
+  "text_variables": {}
+}

wide_coil_3.kicad_prl

@@ -0,0 +1,136 @@
+{
+  "board": {
+    "active_layer": 0,
+    "active_layer_preset": "All Layers",
+    "auto_track_width": true,
+    "hidden_netclasses": [],
+    "hidden_nets": [],
+    "high_contrast_mode": 0,
+    "net_color_mode": 1,
+    "opacity": {
+      "images": 0.6,
+      "pads": 1.0,
+      "shapes": 1.0,
+      "tracks": 1.0,
+      "vias": 1.0,
+      "zones": 0.6
+    },
+    "selection_filter": {
+      "dimensions": true,
+      "footprints": true,
+      "graphics": true,
+      "keepouts": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pads": true,
+      "text": true,
+      "tracks": true,
+      "vias": true,
+      "zones": true
+    },
+    "visible_items": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      8,
+      9,
+      10,
+      11,
+      12,
+      13,
+      15,
+      16,
+      17,
+      18,
+      19,
+      20,
+      21,
+      22,
+      23,
+      24,
+      25,
+      26,
+      27,
+      28,
+      29,
+      30,
+      32,
+      33,
+      34,
+      35,
+      36,
+      39,
+      40,
+      41
+    ],
+    "visible_layers": "ffffffff_ffffffff",
+    "zone_display_mode": 0
+  },
+  "git": {
+    "repo_password": "",
+    "repo_type": "",
+    "repo_username": "",
+    "ssh_key": ""
+  },
+  "meta": {
+    "filename": "wide_coil_3.kicad_prl",
+    "version": 4
+  },
+  "net_inspector_panel": {
+    "col_hidden": [
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ],
+    "col_order": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8,
+      9
+    ],
+    "col_widths": [],
+    "custom_group_rules": [],
+    "expanded_rows": [],
+    "filter_by_net_name": true,
+    "filter_by_netclass": true,
+    "filter_text": "",
+    "group_by_constraint": false,
+    "group_by_netclass": false,
+    "show_unconnected_nets": false,
+    "show_zero_pad_nets": false,
+    "sort_ascending": true,
+    "sorting_column": -1
+  },
+  "project": {
+    "files": []
+  },
+  "schematic": {
+    "selection_filter": {
+      "graphics": true,
+      "images": true,
+      "labels": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pins": true,
+      "symbols": true,
+      "text": true,
+      "wires": true
+    }
+  }
+}

wide_coil_3.kicad_pro

@@ -0,0 +1,276 @@
+{
+  "board": {
+    "3dviewports": [],
+    "design_settings": {
+      "defaults": {
+        "apply_defaults_to_fp_fields": false,
+        "apply_defaults_to_fp_shapes": false,
+        "apply_defaults_to_fp_text": false,
+        "board_outline_line_width": 0.05,
+        "copper_line_width": 0.2,
+        "copper_text_italic": false,
+        "copper_text_size_h": 1.5,
+        "copper_text_size_v": 1.5,
+        "copper_text_thickness": 0.3,
+        "copper_text_upright": false,
+        "courtyard_line_width": 0.05,
+        "dimension_precision": 4,
+        "dimension_units": 3,
+        "dimensions": {
+          "arrow_length": 1270000,
+          "extension_offset": 500000,
+          "keep_text_aligned": true,
+          "suppress_zeroes": false,
+          "text_position": 0,
+          "units_format": 1
+        },
+        "fab_line_width": 0.1,
+        "fab_text_italic": false,
+        "fab_text_size_h": 1.0,
+        "fab_text_size_v": 1.0,
+        "fab_text_thickness": 0.15,
+        "fab_text_upright": false,
+        "other_line_width": 0.1,
+        "other_text_italic": false,
+        "other_text_size_h": 1.0,
+        "other_text_size_v": 1.0,
+        "other_text_thickness": 0.15,
+        "other_text_upright": false,
+        "pads": {
+          "drill": 0.8,
+          "height": 1.27,
+          "width": 2.54
+        },
+        "silk_line_width": 0.1,
+        "silk_text_italic": false,
+        "silk_text_size_h": 1.0,
+        "silk_text_size_v": 1.0,
+        "silk_text_thickness": 0.1,
+        "silk_text_upright": false,
+        "zones": {
+          "min_clearance": 0.5
+        }
+      },
+      "diff_pair_dimensions": [],
+      "drc_exclusions": [],
+      "meta": {
+        "version": 2
+      },
+      "rule_severities": {
+        "annular_width": "error",
+        "clearance": "error",
+        "connection_width": "warning",
+        "copper_edge_clearance": "error",
+        "copper_sliver": "warning",
+        "courtyards_overlap": "error",
+        "diff_pair_gap_out_of_range": "error",
+        "diff_pair_uncoupled_length_too_long": "error",
+        "drill_out_of_range": "error",
+        "duplicate_footprints": "warning",
+        "extra_footprint": "warning",
+        "footprint": "error",
+        "footprint_filters_mismatch": "ignore",
+        "footprint_symbol_mismatch": "warning",
+        "footprint_type_mismatch": "ignore",
+        "hole_clearance": "error",
+        "hole_to_hole": "warning",
+        "holes_co_located": "warning",
+        "invalid_outline": "error",
+        "isolated_copper": "warning",
+        "item_on_disabled_layer": "error",
+        "items_not_allowed": "error",
+        "length_out_of_range": "error",
+        "lib_footprint_issues": "warning",
+        "lib_footprint_mismatch": "warning",
+        "malformed_courtyard": "error",
+        "microvia_drill_out_of_range": "error",
+        "missing_courtyard": "ignore",
+        "missing_footprint": "warning",
+        "net_conflict": "warning",
+        "npth_inside_courtyard": "ignore",
+        "padstack": "warning",
+        "pth_inside_courtyard": "ignore",
+        "shorting_items": "error",
+        "silk_edge_clearance": "warning",
+        "silk_over_copper": "warning",
+        "silk_overlap": "warning",
+        "skew_out_of_range": "error",
+        "solder_mask_bridge": "error",
+        "starved_thermal": "error",
+        "text_height": "warning",
+        "text_thickness": "warning",
+        "through_hole_pad_without_hole": "error",
+        "too_many_vias": "error",
+        "track_dangling": "warning",
+        "track_width": "error",
+        "tracks_crossing": "error",
+        "unconnected_items": "error",
+        "unresolved_variable": "error",
+        "via_dangling": "warning",
+        "zones_intersect": "error"
+      },
+      "rules": {
+        "max_error": 0.005,
+        "min_clearance": 0.0,
+        "min_connection": 0.0,
+        "min_copper_edge_clearance": 0.5,
+        "min_hole_clearance": 0.25,
+        "min_hole_to_hole": 0.25,
+        "min_microvia_diameter": 0.2,
+        "min_microvia_drill": 0.1,
+        "min_resolved_spokes": 2,
+        "min_silk_clearance": 0.0,
+        "min_text_height": 0.8,
+        "min_text_thickness": 0.08,
+        "min_through_hole_diameter": 0.3,
+        "min_track_width": 0.0,
+        "min_via_annular_width": 0.1,
+        "min_via_diameter": 0.5,
+        "solder_mask_to_copper_clearance": 0.0,
+        "use_height_for_length_calcs": true
+      },
+      "teardrop_options": [
+        {
+          "td_onpadsmd": true,
+          "td_onroundshapesonly": false,
+          "td_ontrackend": false,
+          "td_onviapad": true
+        }
+      ],
+      "teardrop_parameters": [
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_round_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_rect_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_track_end",
+          "td_width_to_size_filter_ratio": 0.9
+        }
+      ],
+      "track_widths": [],
+      "tuning_pattern_settings": {
+        "diff_pair_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 1.0
+        },
+        "diff_pair_skew_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        },
+        "single_track_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        }
+      },
+      "via_dimensions": [],
+      "zones_allow_external_fillets": false
+    },
+    "ipc2581": {
+      "dist": "",
+      "distpn": "",
+      "internal_id": "",
+      "mfg": "",
+      "mpn": ""
+    },
+    "layer_pairs": [],
+    "layer_presets": [],
+    "viewports": []
+  },
+  "boards": [],
+  "cvpcb": {
+    "equivalence_files": []
+  },
+  "libraries": {
+    "pinned_footprint_libs": [],
+    "pinned_symbol_libs": []
+  },
+  "meta": {
+    "filename": "wide_coil_3.kicad_pro",
+    "version": 1
+  },
+  "net_settings": {
+    "classes": [
+      {
+        "bus_width": 12,
+        "clearance": 0.2,
+        "diff_pair_gap": 0.25,
+        "diff_pair_via_gap": 0.25,
+        "diff_pair_width": 0.2,
+        "line_style": 0,
+        "microvia_diameter": 0.3,
+        "microvia_drill": 0.1,
+        "name": "Default",
+        "pcb_color": "rgba(0, 0, 0, 0.000)",
+        "priority": -1,
+        "schematic_color": "rgba(0, 0, 0, 0.000)",
+        "track_width": 0.2,
+        "via_diameter": 0.6,
+        "via_drill": 0.3,
+        "wire_width": 6
+      }
+    ],
+    "meta": {
+      "version": 4
+    },
+    "net_colors": null,
+    "netclass_assignments": null,
+    "netclass_patterns": []
+  },
+  "pcbnew": {
+    "last_paths": {
+      "gencad": "",
+      "idf": "",
+      "netlist": "",
+      "plot": "gerber_wide_3",
+      "pos_files": "",
+      "specctra_dsn": "",
+      "step": "",
+      "svg": "",
+      "vrml": ""
+    },
+    "page_layout_descr_file": ""
+  },
+  "schematic": {
+    "legacy_lib_dir": "",
+    "legacy_lib_list": []
+  },
+  "sheets": [],
+  "text_variables": {}
+}

wide_coil_4.kicad_prl

@@ -0,0 +1,136 @@
+{
+  "board": {
+    "active_layer": 0,
+    "active_layer_preset": "All Layers",
+    "auto_track_width": true,
+    "hidden_netclasses": [],
+    "hidden_nets": [],
+    "high_contrast_mode": 0,
+    "net_color_mode": 1,
+    "opacity": {
+      "images": 0.6,
+      "pads": 1.0,
+      "shapes": 1.0,
+      "tracks": 1.0,
+      "vias": 1.0,
+      "zones": 0.6
+    },
+    "selection_filter": {
+      "dimensions": true,
+      "footprints": true,
+      "graphics": true,
+      "keepouts": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pads": true,
+      "text": true,
+      "tracks": true,
+      "vias": true,
+      "zones": true
+    },
+    "visible_items": [
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+      1,
+      2,
+      3,
+      4,
+      5,
+      8,
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+      10,
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+      12,
+      13,
+      15,
+      16,
+      17,
+      18,
+      19,
+      20,
+      21,
+      22,
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+      24,
+      25,
+      26,
+      27,
+      28,
+      29,
+      30,
+      32,
+      33,
+      34,
+      35,
+      36,
+      39,
+      40,
+      41
+    ],
+    "visible_layers": "ffffffff_ffffffff",
+    "zone_display_mode": 0
+  },
+  "git": {
+    "repo_password": "",
+    "repo_type": "",
+    "repo_username": "",
+    "ssh_key": ""
+  },
+  "meta": {
+    "filename": "wide_coil_4.kicad_prl",
+    "version": 4
+  },
+  "net_inspector_panel": {
+    "col_hidden": [
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ],
+    "col_order": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8,
+      9
+    ],
+    "col_widths": [],
+    "custom_group_rules": [],
+    "expanded_rows": [],
+    "filter_by_net_name": true,
+    "filter_by_netclass": true,
+    "filter_text": "",
+    "group_by_constraint": false,
+    "group_by_netclass": false,
+    "show_unconnected_nets": false,
+    "show_zero_pad_nets": false,
+    "sort_ascending": true,
+    "sorting_column": -1
+  },
+  "project": {
+    "files": []
+  },
+  "schematic": {
+    "selection_filter": {
+      "graphics": true,
+      "images": true,
+      "labels": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pins": true,
+      "symbols": true,
+      "text": true,
+      "wires": true
+    }
+  }
+}

wide_coil_4.kicad_pro

@@ -0,0 +1,276 @@
+{
+  "board": {
+    "3dviewports": [],
+    "design_settings": {
+      "defaults": {
+        "apply_defaults_to_fp_fields": false,
+        "apply_defaults_to_fp_shapes": false,
+        "apply_defaults_to_fp_text": false,
+        "board_outline_line_width": 0.05,
+        "copper_line_width": 0.2,
+        "copper_text_italic": false,
+        "copper_text_size_h": 1.5,
+        "copper_text_size_v": 1.5,
+        "copper_text_thickness": 0.3,
+        "copper_text_upright": false,
+        "courtyard_line_width": 0.05,
+        "dimension_precision": 4,
+        "dimension_units": 3,
+        "dimensions": {
+          "arrow_length": 1270000,
+          "extension_offset": 500000,
+          "keep_text_aligned": true,
+          "suppress_zeroes": false,
+          "text_position": 0,
+          "units_format": 1
+        },
+        "fab_line_width": 0.1,
+        "fab_text_italic": false,
+        "fab_text_size_h": 1.0,
+        "fab_text_size_v": 1.0,
+        "fab_text_thickness": 0.15,
+        "fab_text_upright": false,
+        "other_line_width": 0.1,
+        "other_text_italic": false,
+        "other_text_size_h": 1.0,
+        "other_text_size_v": 1.0,
+        "other_text_thickness": 0.15,
+        "other_text_upright": false,
+        "pads": {
+          "drill": 0.8,
+          "height": 1.27,
+          "width": 2.54
+        },
+        "silk_line_width": 0.1,
+        "silk_text_italic": false,
+        "silk_text_size_h": 1.0,
+        "silk_text_size_v": 1.0,
+        "silk_text_thickness": 0.1,
+        "silk_text_upright": false,
+        "zones": {
+          "min_clearance": 0.5
+        }
+      },
+      "diff_pair_dimensions": [],
+      "drc_exclusions": [],
+      "meta": {
+        "version": 2
+      },
+      "rule_severities": {
+        "annular_width": "error",
+        "clearance": "error",
+        "connection_width": "warning",
+        "copper_edge_clearance": "error",
+        "copper_sliver": "warning",
+        "courtyards_overlap": "error",
+        "diff_pair_gap_out_of_range": "error",
+        "diff_pair_uncoupled_length_too_long": "error",
+        "drill_out_of_range": "error",
+        "duplicate_footprints": "warning",
+        "extra_footprint": "warning",
+        "footprint": "error",
+        "footprint_filters_mismatch": "ignore",
+        "footprint_symbol_mismatch": "warning",
+        "footprint_type_mismatch": "ignore",
+        "hole_clearance": "error",
+        "hole_to_hole": "warning",
+        "holes_co_located": "warning",
+        "invalid_outline": "error",
+        "isolated_copper": "warning",
+        "item_on_disabled_layer": "error",
+        "items_not_allowed": "error",
+        "length_out_of_range": "error",
+        "lib_footprint_issues": "warning",
+        "lib_footprint_mismatch": "warning",
+        "malformed_courtyard": "error",
+        "microvia_drill_out_of_range": "error",
+        "missing_courtyard": "ignore",
+        "missing_footprint": "warning",
+        "net_conflict": "warning",
+        "npth_inside_courtyard": "ignore",
+        "padstack": "warning",
+        "pth_inside_courtyard": "ignore",
+        "shorting_items": "error",
+        "silk_edge_clearance": "warning",
+        "silk_over_copper": "warning",
+        "silk_overlap": "warning",
+        "skew_out_of_range": "error",
+        "solder_mask_bridge": "error",
+        "starved_thermal": "error",
+        "text_height": "warning",
+        "text_thickness": "warning",
+        "through_hole_pad_without_hole": "error",
+        "too_many_vias": "error",
+        "track_dangling": "warning",
+        "track_width": "error",
+        "tracks_crossing": "error",
+        "unconnected_items": "error",
+        "unresolved_variable": "error",
+        "via_dangling": "warning",
+        "zones_intersect": "error"
+      },
+      "rules": {
+        "max_error": 0.005,
+        "min_clearance": 0.0,
+        "min_connection": 0.0,
+        "min_copper_edge_clearance": 0.5,
+        "min_hole_clearance": 0.25,
+        "min_hole_to_hole": 0.25,
+        "min_microvia_diameter": 0.2,
+        "min_microvia_drill": 0.1,
+        "min_resolved_spokes": 2,
+        "min_silk_clearance": 0.0,
+        "min_text_height": 0.8,
+        "min_text_thickness": 0.08,
+        "min_through_hole_diameter": 0.3,
+        "min_track_width": 0.0,
+        "min_via_annular_width": 0.1,
+        "min_via_diameter": 0.5,
+        "solder_mask_to_copper_clearance": 0.0,
+        "use_height_for_length_calcs": true
+      },
+      "teardrop_options": [
+        {
+          "td_onpadsmd": true,
+          "td_onroundshapesonly": false,
+          "td_ontrackend": false,
+          "td_onviapad": true
+        }
+      ],
+      "teardrop_parameters": [
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_round_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_rect_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_track_end",
+          "td_width_to_size_filter_ratio": 0.9
+        }
+      ],
+      "track_widths": [],
+      "tuning_pattern_settings": {
+        "diff_pair_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 1.0
+        },
+        "diff_pair_skew_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        },
+        "single_track_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        }
+      },
+      "via_dimensions": [],
+      "zones_allow_external_fillets": false
+    },
+    "ipc2581": {
+      "dist": "",
+      "distpn": "",
+      "internal_id": "",
+      "mfg": "",
+      "mpn": ""
+    },
+    "layer_pairs": [],
+    "layer_presets": [],
+    "viewports": []
+  },
+  "boards": [],
+  "cvpcb": {
+    "equivalence_files": []
+  },
+  "libraries": {
+    "pinned_footprint_libs": [],
+    "pinned_symbol_libs": []
+  },
+  "meta": {
+    "filename": "wide_coil_4.kicad_pro",
+    "version": 1
+  },
+  "net_settings": {
+    "classes": [
+      {
+        "bus_width": 12,
+        "clearance": 0.2,
+        "diff_pair_gap": 0.25,
+        "diff_pair_via_gap": 0.25,
+        "diff_pair_width": 0.2,
+        "line_style": 0,
+        "microvia_diameter": 0.3,
+        "microvia_drill": 0.1,
+        "name": "Default",
+        "pcb_color": "rgba(0, 0, 0, 0.000)",
+        "priority": -1,
+        "schematic_color": "rgba(0, 0, 0, 0.000)",
+        "track_width": 0.2,
+        "via_diameter": 0.6,
+        "via_drill": 0.3,
+        "wire_width": 6
+      }
+    ],
+    "meta": {
+      "version": 4
+    },
+    "net_colors": null,
+    "netclass_assignments": null,
+    "netclass_patterns": []
+  },
+  "pcbnew": {
+    "last_paths": {
+      "gencad": "",
+      "idf": "",
+      "netlist": "",
+      "plot": "gerber_wide_4",
+      "pos_files": "",
+      "specctra_dsn": "",
+      "step": "",
+      "svg": "",
+      "vrml": ""
+    },
+    "page_layout_descr_file": ""
+  },
+  "schematic": {
+    "legacy_lib_dir": "",
+    "legacy_lib_list": []
+  },
+  "sheets": [],
+  "text_variables": {}
+}

wide_coil_5.kicad_prl

@@ -0,0 +1,136 @@
+{
+  "board": {
+    "active_layer": 0,
+    "active_layer_preset": "All Layers",
+    "auto_track_width": true,
+    "hidden_netclasses": [],
+    "hidden_nets": [],
+    "high_contrast_mode": 0,
+    "net_color_mode": 1,
+    "opacity": {
+      "images": 0.6,
+      "pads": 1.0,
+      "shapes": 1.0,
+      "tracks": 1.0,
+      "vias": 1.0,
+      "zones": 0.6
+    },
+    "selection_filter": {
+      "dimensions": true,
+      "footprints": true,
+      "graphics": true,
+      "keepouts": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pads": true,
+      "text": true,
+      "tracks": true,
+      "vias": true,
+      "zones": true
+    },
+    "visible_items": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      8,
+      9,
+      10,
+      11,
+      12,
+      13,
+      15,
+      16,
+      17,
+      18,
+      19,
+      20,
+      21,
+      22,
+      23,
+      24,
+      25,
+      26,
+      27,
+      28,
+      29,
+      30,
+      32,
+      33,
+      34,
+      35,
+      36,
+      39,
+      40,
+      41
+    ],
+    "visible_layers": "ffffffff_ffffffff",
+    "zone_display_mode": 0
+  },
+  "git": {
+    "repo_password": "",
+    "repo_type": "",
+    "repo_username": "",
+    "ssh_key": ""
+  },
+  "meta": {
+    "filename": "wide_coil_5.kicad_prl",
+    "version": 4
+  },
+  "net_inspector_panel": {
+    "col_hidden": [
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ],
+    "col_order": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7,
+      8,
+      9
+    ],
+    "col_widths": [],
+    "custom_group_rules": [],
+    "expanded_rows": [],
+    "filter_by_net_name": true,
+    "filter_by_netclass": true,
+    "filter_text": "",
+    "group_by_constraint": false,
+    "group_by_netclass": false,
+    "show_unconnected_nets": false,
+    "show_zero_pad_nets": false,
+    "sort_ascending": true,
+    "sorting_column": -1
+  },
+  "project": {
+    "files": []
+  },
+  "schematic": {
+    "selection_filter": {
+      "graphics": true,
+      "images": true,
+      "labels": true,
+      "lockedItems": false,
+      "otherItems": true,
+      "pins": true,
+      "symbols": true,
+      "text": true,
+      "wires": true
+    }
+  }
+}

wide_coil_5.kicad_pro

@@ -0,0 +1,276 @@
+{
+  "board": {
+    "3dviewports": [],
+    "design_settings": {
+      "defaults": {
+        "apply_defaults_to_fp_fields": false,
+        "apply_defaults_to_fp_shapes": false,
+        "apply_defaults_to_fp_text": false,
+        "board_outline_line_width": 0.05,
+        "copper_line_width": 0.2,
+        "copper_text_italic": false,
+        "copper_text_size_h": 1.5,
+        "copper_text_size_v": 1.5,
+        "copper_text_thickness": 0.3,
+        "copper_text_upright": false,
+        "courtyard_line_width": 0.05,
+        "dimension_precision": 4,
+        "dimension_units": 3,
+        "dimensions": {
+          "arrow_length": 1270000,
+          "extension_offset": 500000,
+          "keep_text_aligned": true,
+          "suppress_zeroes": false,
+          "text_position": 0,
+          "units_format": 1
+        },
+        "fab_line_width": 0.1,
+        "fab_text_italic": false,
+        "fab_text_size_h": 1.0,
+        "fab_text_size_v": 1.0,
+        "fab_text_thickness": 0.15,
+        "fab_text_upright": false,
+        "other_line_width": 0.1,
+        "other_text_italic": false,
+        "other_text_size_h": 1.0,
+        "other_text_size_v": 1.0,
+        "other_text_thickness": 0.15,
+        "other_text_upright": false,
+        "pads": {
+          "drill": 0.8,
+          "height": 1.27,
+          "width": 2.54
+        },
+        "silk_line_width": 0.1,
+        "silk_text_italic": false,
+        "silk_text_size_h": 1.0,
+        "silk_text_size_v": 1.0,
+        "silk_text_thickness": 0.1,
+        "silk_text_upright": false,
+        "zones": {
+          "min_clearance": 0.5
+        }
+      },
+      "diff_pair_dimensions": [],
+      "drc_exclusions": [],
+      "meta": {
+        "version": 2
+      },
+      "rule_severities": {
+        "annular_width": "error",
+        "clearance": "error",
+        "connection_width": "warning",
+        "copper_edge_clearance": "error",
+        "copper_sliver": "warning",
+        "courtyards_overlap": "error",
+        "diff_pair_gap_out_of_range": "error",
+        "diff_pair_uncoupled_length_too_long": "error",
+        "drill_out_of_range": "error",
+        "duplicate_footprints": "warning",
+        "extra_footprint": "warning",
+        "footprint": "error",
+        "footprint_filters_mismatch": "ignore",
+        "footprint_symbol_mismatch": "warning",
+        "footprint_type_mismatch": "ignore",
+        "hole_clearance": "error",
+        "hole_to_hole": "warning",
+        "holes_co_located": "warning",
+        "invalid_outline": "error",
+        "isolated_copper": "warning",
+        "item_on_disabled_layer": "error",
+        "items_not_allowed": "error",
+        "length_out_of_range": "error",
+        "lib_footprint_issues": "warning",
+        "lib_footprint_mismatch": "warning",
+        "malformed_courtyard": "error",
+        "microvia_drill_out_of_range": "error",
+        "missing_courtyard": "ignore",
+        "missing_footprint": "warning",
+        "net_conflict": "warning",
+        "npth_inside_courtyard": "ignore",
+        "padstack": "warning",
+        "pth_inside_courtyard": "ignore",
+        "shorting_items": "error",
+        "silk_edge_clearance": "warning",
+        "silk_over_copper": "warning",
+        "silk_overlap": "warning",
+        "skew_out_of_range": "error",
+        "solder_mask_bridge": "error",
+        "starved_thermal": "error",
+        "text_height": "warning",
+        "text_thickness": "warning",
+        "through_hole_pad_without_hole": "error",
+        "too_many_vias": "error",
+        "track_dangling": "warning",
+        "track_width": "error",
+        "tracks_crossing": "error",
+        "unconnected_items": "error",
+        "unresolved_variable": "error",
+        "via_dangling": "warning",
+        "zones_intersect": "error"
+      },
+      "rules": {
+        "max_error": 0.005,
+        "min_clearance": 0.0,
+        "min_connection": 0.0,
+        "min_copper_edge_clearance": 0.5,
+        "min_hole_clearance": 0.25,
+        "min_hole_to_hole": 0.25,
+        "min_microvia_diameter": 0.2,
+        "min_microvia_drill": 0.1,
+        "min_resolved_spokes": 2,
+        "min_silk_clearance": 0.0,
+        "min_text_height": 0.8,
+        "min_text_thickness": 0.08,
+        "min_through_hole_diameter": 0.3,
+        "min_track_width": 0.0,
+        "min_via_annular_width": 0.1,
+        "min_via_diameter": 0.5,
+        "solder_mask_to_copper_clearance": 0.0,
+        "use_height_for_length_calcs": true
+      },
+      "teardrop_options": [
+        {
+          "td_onpadsmd": true,
+          "td_onroundshapesonly": false,
+          "td_ontrackend": false,
+          "td_onviapad": true
+        }
+      ],
+      "teardrop_parameters": [
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_round_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_rect_shape",
+          "td_width_to_size_filter_ratio": 0.9
+        },
+        {
+          "td_allow_use_two_tracks": true,
+          "td_curve_segcount": 0,
+          "td_height_ratio": 1.0,
+          "td_length_ratio": 0.5,
+          "td_maxheight": 2.0,
+          "td_maxlen": 1.0,
+          "td_on_pad_in_zone": false,
+          "td_target_name": "td_track_end",
+          "td_width_to_size_filter_ratio": 0.9
+        }
+      ],
+      "track_widths": [],
+      "tuning_pattern_settings": {
+        "diff_pair_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 1.0
+        },
+        "diff_pair_skew_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        },
+        "single_track_defaults": {
+          "corner_radius_percentage": 80,
+          "corner_style": 1,
+          "max_amplitude": 1.0,
+          "min_amplitude": 0.2,
+          "single_sided": false,
+          "spacing": 0.6
+        }
+      },
+      "via_dimensions": [],
+      "zones_allow_external_fillets": false
+    },
+    "ipc2581": {
+      "dist": "",
+      "distpn": "",
+      "internal_id": "",
+      "mfg": "",
+      "mpn": ""
+    },
+    "layer_pairs": [],
+    "layer_presets": [],
+    "viewports": []
+  },
+  "boards": [],
+  "cvpcb": {
+    "equivalence_files": []
+  },
+  "libraries": {
+    "pinned_footprint_libs": [],
+    "pinned_symbol_libs": []
+  },
+  "meta": {
+    "filename": "wide_coil_5.kicad_pro",
+    "version": 1
+  },
+  "net_settings": {
+    "classes": [
+      {
+        "bus_width": 12,
+        "clearance": 0.2,
+        "diff_pair_gap": 0.25,
+        "diff_pair_via_gap": 0.25,
+        "diff_pair_width": 0.2,
+        "line_style": 0,
+        "microvia_diameter": 0.3,
+        "microvia_drill": 0.1,
+        "name": "Default",
+        "pcb_color": "rgba(0, 0, 0, 0.000)",
+        "priority": -1,
+        "schematic_color": "rgba(0, 0, 0, 0.000)",
+        "track_width": 0.2,
+        "via_diameter": 0.6,
+        "via_drill": 0.3,
+        "wire_width": 6
+      }
+    ],
+    "meta": {
+      "version": 4
+    },
+    "net_colors": null,
+    "netclass_assignments": null,
+    "netclass_patterns": []
+  },
+  "pcbnew": {
+    "last_paths": {
+      "gencad": "",
+      "idf": "",
+      "netlist": "",
+      "plot": "gerber_wide_5",
+      "pos_files": "",
+      "specctra_dsn": "",
+      "step": "",
+      "svg": "",
+      "vrml": ""
+    },
+    "page_layout_descr_file": ""
+  },
+  "schematic": {
+    "legacy_lib_dir": "",
+    "legacy_lib_list": []
+  },
+  "sheets": [],
+  "text_variables": {}
+}