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 Fig. 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 Fig. 1 represents the skin effect, and it is given by \[\begin{eqnarray} R_\text{VAR}=k \sqrt{f} \tag{1}, \end{eqnarray}\] 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: \[\begin{eqnarray} R_2\propto N \propto \sqrt{L}\tag{2}. \end{eqnarray}\] 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:

\[\begin{eqnarray} k \propto L\tag{3}. \end{eqnarray}\]

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

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

One may be inclined the data as is and plug these coils into LTspice simulations in a programmatic way [which is something we will do later] and pick the best combination of filters. However, note that these data points are coming from manufacturers tests and they might include noise and outliers. We have to scrutinize the data and reduce the jitter in the data. We can extract the table from the pdf and take a closer look.

// Jan, 2023
// extract a table from a pdf file
//https://www.coilcraft.com/getmedia/8b8dc5bb-0e94-4194-b919-ac2586274005/spice_1008cs.pdf
// download the pdf and save it under the same folder.
thefile="spice_1008cs";
theTable=Open(
    ".\"||thefile||".pdf",
    PDF Tables(
        Table(
            table name( thefile ),
            add rows( page( 2 ), Rect( 0.75, 1.3736, 4.0817, 4.7197 ) ),
            add rows( page( 2 ), Rect( 4.375, 1.3752, 7.7067, 4.5747 ) )
        )
    )
)[1];
colnames=theTable<<get column names();current data table(theTable);
column(colnames[4])<<set name("R1(ohms)");
column(colnames[5])<<set name("R2(ohms)");
theTable<< save( thefile||".csv" );
theTable<< close window();

Table 1 shows the parameters for various products from CoilCraft. See the pdf above for the detailed description of the test conditions.

Let’s look for some trends and outliers.

Figure 3: (Left) The parameter \(k\) trends linearly with \(L\), but the three data points shown as red stars fall of the regression. (Right) They also show strange capacitance values. We need to drop them from the data since they are questionable. … [+] These product might be using a different core. Fortunately, they are on the high end tail of the inductance, and hopefully we won’t need to go that high.

After removing the outlier points, we can check if the predictions in Eqs. (2) and (3) hold by doing a scatter plot and a linear fit as seen in Fig. 4.

Figure 4: (Left) The parameter \(k\) trends linearly with \(L\). (Right) \(R_2\) trends linearly with \(\sqrt{L}\). It has a discontinuity at around \(L=100\)nH. We should take a closer look at the model if the optimization results in values with \(L<100\)nH. … [+] These relations will help us exclude noise from the optimization process.

Figure 5: (Left) The parameter \(k\) trends linearly with \(L\). (Right) \(R_2\) trends linearly with \(\sqrt{L}\). It has a discontinuity at around \(L=100\)nH. We should take a closer look at the model if the optimization results in values with \(L<100\)nH. … [+] These relations will help us exclude noise from the optimization process.

In the dropdown below, you can select coils and see how the impedances of individual elements change with frequency. The coils in use at the current circuit now, 1008CS-101,1008CS-391,1008CS-181,1008CS-471, and 1008CS-182, are marked with the orange background.

Figure 6: Impedances of components for the selected coils. Red dashed lines show the harmonics to be notched out, and the green lines show the frequencies to be notched in, i.e., the signal outputs.

Practical capacitors

In the zoo of capacitors, things look similar. Figure 7 shows the equivalent circuit for a practical capacitor.

Figure 7: Equivalent circuit for a real capacitor. Hover over the orange-colored elements to see what their functions are. … [+] More stuff here.

TDK has a note on ceramic capacitors as shown in Fig. 8.

Figure 8: TDK notes on ESR in ceramic capacitors.

Fortunately, typically, \(R_\text{ESR}\) is very low; \(R_\text{isol}\) is very high, and finally the reactance due to \(L_\text{ESL}\) is also very low compared to the impedance of \(C\) at the frequecies we will be interested. It can be assumed that \(Z_c>\sim \text{several }\, \Omega's\), therefore, we can take the capacitor as a pure capacitive element, and take a note to remind ourselves to confirm this assumption is not violated in the final proposal.