Extend related work section

paper/paper.tex

@@ -162,6 +162,36 @@ inductors below, although in contemporary literature, this condition is never ex
 \cite{eppenAnforderungenEinzelteileRundfunkempfanger1927, kleinSpulenUndSchwingungskreise1941,
 wiggeRundfunktechnischesHandbuch1930}.
 
+\subsection{PCB inductor design for wireless power transfer}
+
+For wireless power transfer, air-core inductors with or without ferrite magnetic shielding are the standard solution.
+Since in most 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. Meanwhile, in many WPT applications,
+especially for charging portable devices or medical implants, some misalignment between the sending and receiving coils
+is expected. Using the available space with an air-core inductor that has a large cross-sectional area reduces the
+impact of this misalignment.
+
+Looking at such WPT inductors, they 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.
+
+Implementing inductors in PCBs has a number of 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, PCB inductors tend to have poor DC resistance. A PCBs' thin but wide trace cross-section helps with
+skin effect compared to a solid conductor. However, PCBs can still not approach the performance of litz wire used in
+high-frequency WPT coils, which commonly use wire diameters in the tens of micrometer
+range\cite{zhaoDesignOptimizationLitzWire2023}. \textcite{lopeFrequencyDependentResistancePlanar2014} 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.
+
+A further factor that limits the high-frequency performance of PCB inductors is distributed capacitance. Not only do
+large air coils exhibit more parasitic capacitance than much smaller ferrite-core inductors simply due to their 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.
+
 \subsection{Twisted Inductors in RFIC Design}
 
 The simplest twisted inductor as shown below with $k=1$ inversion corresponds to the counterwound scheme that is
@@ -200,7 +230,14 @@ Besides the monetary cost of the power lost this way, each small improvement ena
 and other cooling components, which directly translates to a decrease in cost.
 
 \subsection{Air-Core Inductors for Inductive Power Transfer}
-\subsection{Ferrite or Iron-Core Inductors for Inductive Power Transfer}
+
+In inductive wireless power transfer, air-core inductors are often used since in most applications, an air gap of
+several millimeters or more is expected, and adding a ferrite core would not change the system's performance by much in
+these circumstances. A common way to use ferrites in WPT applications is magnetically shielding the inductor's back side
+with a ferrite plate such that the field does not extend beyond the coil's back side, and to reduce eddy current losses
+when the WPT coils are placed near metal
+objects\cite{batraEffectFerriteAddition2015,leeSimpleWirelessPower2017,muehlmannMutualCouplingModeling2012}.
+
 
 \section{Twisted Inductor Design} 
 
@@ -462,7 +499,7 @@ effect gets partially mitigated since the strongest coupling exists between adja
 the SRF have 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 dielectric constant much larger than air
+the substrate in between. Making things worse, common PCB substrates have a relative permittivity much larger than air
 (usually around $4$).
 
 Interestingly, we observe that this decrease in high-frequency performance is counteracted by larger inversion count