Update experimental section with results

paper/paper.tex

@@ -264,9 +264,62 @@ To experimentally validate our design with real-world inductors, we produced tes
 twisted inductors with winding count $n$ between $1$ and $25$, and twist count ranging from $k=0$ (simple single-sided
 spiral inductor) to $k=37$.
 
-\subsection{Inductance and Parasitic Capacitance}
+\subsection{Inductance, Q-factor and DC resistance}
 
-\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}