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-\subsection{Self-Resonant Frequency} +We measured inductance and Q-factor of each test coupon using a Keysight U1733C LCR meter at \qty{100}{\kilo\hertz}. We +measured DC resistance using a Keysight 34465A multimeter in four-wire resistance mode. 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_inductor_params}. + +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, while not particularly relevant for our application, we decided to evaluate the high-frequency performance of +twisted inductors. We found that going from a single-layer spiral inductor to a two-layer spiral inductor decreases the +self-resonant frequency, this effect being more pronounced with higher turn count. Intuitively, this makes sense if we +consider the mechanics 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. +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 +(usually around $4$). + +Interestingly, we observe that this decrease in high-frequency performance is counteracted by larger trace pair count +$k$. While our test samples focused on smaller turn counts, we observe an increase from an SRF of +\qty{8.9}{\mega\hertz} for a standard $n=25,k=1$ inductor to \qty{10.6}{\mega\hertz} for $n=25,k=13$. + +In conclusion, 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 on other +performance parameters. + +\begin{table} + \begin{tabular}{cc|ccc|} + Turn Count $n$& + Trace pair count $k$& + Inductance $L \left[\mu H\right]$& + Q-factor& + DC resistance $R_\text{ESR} \left[\Omega\right]$\\ + \end{tabular} + \caption{Inductor sample design parameters and measured characteristics. All inductors have outer diameter + \qty{35}{\milli\meter} and inner diameter \qty{15}{\milli\meter}.} +\end{table} \subsection{Coupling}