Mitch Richling: Measuring Impedance

If one knows the series resistance (\(R_L\) or \(R_{CS}\)), then one may directly compute the value of \(C\) or \(L\) from \(V_R\), \(V_S\), and \(f\):

\[ C = \frac{V_{R}}{2 \pi f \, \sqrt{R^{2} V_{S}^{2} - \left(R +R_{\mathit{CS}} \right)^{2} V_{R}^{2} }} \;\;\;\;\;\;\;\;\;\;\;\;\; L = \frac{\sqrt{R^{2} V_{S}^{2} - \left(R +R_{L} \right)^{2} V_{R}^{2}}}{2 \pi f V_{R}} \]

If the series resistance (\(R_L\) or \(R_{CS}\)) may be safely ignored, then the above equations may be replaced by less complex alternatives.

\[ C = \frac{V_R}{2 \pi f R \sqrt{V_S^2-V_R^2}} = \frac{\sqrt{V_S^2-V_C^2}}{2 \pi f R V_C} = \frac{V_R}{2 \pi f R V_C} \;\;\;\;\;\;\;\;\;\;\;\;\; L = \frac{R \sqrt{V_S^2-V_R^2}}{2 \pi f V_R} = \frac{R V_L}{2 \pi f \sqrt{V_S^2-V_L^2}} = \frac{R V_L}{2 \pi f V_R} \]

Back in the day, when signal sources were higher voltage and sliderules were king, the advice was to set the generator output voltage and frequency so that \(V_S=10\mathrm{V}\) and \(V_R=1\mathrm{V}\). In this case the above simplify to:

\[ C \approx 0.01599567362 \frac{1}{f R} \;\;\;\;\;\;\;\;\;\;\;\;\; L \approx 1.583571689 \frac{R}{f} \]

Note it is the ratio \(V_S\) to \(V_R\) that really matters in the formula above (i.e. so long as \(V_S=10\cdot V_R\), then the previous formulas work). Today a more useful rule is adjust the generator output voltage and frequency so that \(V_S=2\cdot V_R\). In this case we have the following:

\[ C = \frac{1}{2\pi f R \sqrt{3}} \approx 0.09188814925 \frac{1}{f R} \;\;\;\;\;\;\;\;\;\;\;\;\; L = \frac{R\sqrt{3}}{2\pi f} \approx 0.2756644477 \frac{R}{f} \]

Finally we note that when \(V_S=V_R\), we have the following:

\[ C=\frac{1}{2\pi f R} \;\;\;\;\;\;\;\;\;\;\;\;\; L=\frac{R}{2\pi f} \]

These formulas are for sinusoidal waveforms with voltage expressed as amplitude. Note any voltage measurement that is a constant multiple of amplitude, in the sinusoidal universe, will work just as well – ex: \(V_{pk}\), \(V_{p-p}\), \(V_{avg}\), and \(V_\mathrm{RMS}\).

The formulas above allow for many different options with regard to procedure and what must be measured. As such, we discuss more than one method below. The first method involves measuring various parameters (two voltages, one frequency, and one resistance), and directly computing the capacitance or inductance. This first method optionally allows us to take into consideration the series resistance of the impedance in question. The second method involves tuning the circuit to a particular frequency (rather like the resonant frequency of an RLC circuit), and using the resistance and frequency to compute the capacitance or inductance.

I normally measure the resister first. I ignore series resistance when I can, but for inductors I am generally forced to measure the DC series resistance so I do that next – be careful with delicate coils.

The next step is to stimulate the circuit with a sinusoidal signal and measure \(V_S\), \(V_R\), & \(f\). Best measurements are made when the voltage drop across \(R\) is 20%-75% of the output of the signal generator.

With an oscilloscope

• Wire up the test circuit
• Oscilloscope ground leads go to the bottom of the circuit, and probe tips go to \(V_S\) & \(V_R\)
• Setup the scope to take automatic measurements of: \(V_S\), \(V_R\), \(f\), and the phase difference between the voltages
• Select an \(R\) value and voltage output level that will produce a safe current through the unknown impedance
• Energize the circuit and adjust \(f\) & \(R\) so that \(V_R\) is 25% to 75% of \(V_S\)
  • Don't overload the DUT! Changes in both \(R\) and \(f\) change the current
• Compute the answer

With a DMM or VOM

• Wire up the test circuit
• Select an \(R\) value and voltage output level that will produce a safe current through the unknown impedance
• Set meter to AC voltage. If your DMM supports it, set a secondary measurement to frequency.
• Meter leads: black to ground and red to \(V_S\).
• Energize the circuit and measure \(V_S\)
• Deenergize the circuit, and move the red lead to \(V_R\)
• Energize the circuit
• Adjust \(f\) & \(R\) so that \(V_R\) is 25% to 75% of \(V_S\)
  • Don't overload the DUT! Changes in both \(R\) and \(f\) change the current
• Compute the answer

If doing the computation by hand, you can simplify things. Set \(V_S\) to 10V and adjust \(R\) & \(f\) till \(V_R\) is 1V.

If the series resistance (\(R_L\) or \(R_{CS}\)) may be safely ignored, then an alternative technique is to find an \(R\) and \(f\) for which the resistor and unknown impedances are of the same magnitude.

Appropriate values for \(R\) and \(f\) may be found by tuning the circuit under power.

With an oscilloscope

Set the oscilloscope to measure the phase difference between \(V_S\) & \(V_R\). Then tune the circuit until the phase difference is \(45^\circ\).

With a pair of DMMs or VOMs

Measure voltage \(V_C\) or \(V_L\) and \(V_R\). Then tune the circuit until they read equal. Then remeasure both voltages with the same meter.

For most coils you can just clamp on a DMM and use the ohm meter function; however, for smaller coils the DMM may provide too much current and damage the coil. To measure the resistance of delicate coils, use the same measuring circuit above with a larger resistor. Instead of stimulating with a sinusoidal signal, we use DC. So connect a bench power supply set to minimum voltage or connect your signal generator using only a DC offset voltage set to zero. Slowly ramp up the voltage until you see a tiny current flow, then use Ohm's law to compute the DC resistance.

The algebra can be a bit painful, so here is a bit of maple code that might help. In addition you will find R code for each equation. You can download the maple code and a PDF version of the maple code.