Real coils are not purely imaginary

Abstract

Real coils deviate from idealized models due to finite wire resistance, skin depth effects, and parasitic capacitance. We analyze manufacturer data to extract key parameters and develop predictive models for practical coil behavior. This analysis is essential for accurate circuit optimization, particularly in applications such as LIGO RF filter design.

Keywords

LIGO, coils

Practical coils

In order to build a predictive model, it is essential to include various aspects of real coils which deviate from idealized coils. First of all, it is wound with wires of finite resistance, and therefore there is a resistive component to it. Wires stacked together also cause a capacitive effect. Overall, a practical coil can be modeled as shown in Figure 1.

Figure 1: The equivalent circuit for a real coil. In addition to the inductance, we have serial resistance originating from finite wire conductivity, skin depth effect and parallel capacitance.

\(R_\text{VAR}\) in Figure 1 represents the skin effect, and it is given by

\[\begin{eqnarray} R_\text{VAR}=k \sqrt{f} \label{eq:skindepth}, \end{eqnarray}\]

where \(k\) is a constant we will extract from manufacturer data. We also analyze CoilCraft’s product data sheet, remove suspected data points and use regression to estimate the parameters of the practical coils that we use in the optimization process. The most significant deviation from an ideal coil is the serial resistance, \(R_2\), and we can set our expectations about it without even looking at the data. The inductance of a coil is proportional to the square of the number of turns. This simply follows from the fact that the field created by a each turn passes through every other turn as well, i.e., the interaction is \(\propto N(N-1)\sim N^2\). The length of the wire used in the coil is proportional to the number of turns at the first order (a small quadratic order term appears as the turn diameter increases with more turns.) Overall we can expect: \[\begin{eqnarray} R_2\propto N \propto \sqrt{L}\label{eq:r2prop}. \end{eqnarray}\] We can also argue that \(k\) is linear function of the magnetic field, which inturn proportional to the inductance. Therefore, if I was forced to make a prediction, I would predict the following relation:

\[\begin{eqnarray} k \propto L\label{eq:kprop}. \end{eqnarray}\]

Manufacturers tabulate and publish these parameters. We will use the data from CoilCraft, as shown in Figure 2.

Figure 2: A leading coil manufacturer has a document on their coil model parameters.

Table 1: Coilcraft inductor parameters for various products.

Let’s look for some trends and outliers.

Figure 3: The parameter \(k\) vs \(L\).

Figure 4: \(C\) vs \(L\).

After removing the outlier points, we can check if the predictions in Eqs. \(\ref{eq:r2prop}\) and \(\ref{eq:kprop}\) hold by doing a scatter plot and a linear fit as seen in Figure 5.

Figure 5: Linear fit to the parameter \(k\) vs \(L\).

Figure 6: Linear fit to the parameter \(R_2\) vs \(\sqrt{L}\).

Figure 7: Linear fit to the parameter \(k\) vs \(L\).

Figure 8: Linear fit to the parameter \(k\) vs \(L\).

Practical capacitors

In the zoo of capacitors, things look similar. Figure Figure 9 shows the equivalent circuit for a practical capacitor.

Figure 9: Equivalent circuit for a real capacitor.

TDK has a note on ceramic capacitors as shown in Figure 10.

Figure 10: TDK notes on ESR in ceramic capacitors.