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Since we know what the current/voltage output of a RLC circuit is as a function of $R,L,C$ hooking up an $RLC$ circuit is one way to determine an unknown capacitance or inductance?

How did people measure capacitances and inductances before the invention of the oscilloscope?

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  • $\begingroup$ You've asked a question moments ago about a method where there was no need of an oscilloscope ;-) $\endgroup$ – Massimo Ortolano Jan 11 '17 at 21:42
  • $\begingroup$ Indeed I did. And I find that method to be very awkward and I'm hoping there's an easier method... $\endgroup$ – Joshua Benabou Jan 11 '17 at 21:45
  • $\begingroup$ It depends on what is your target accuracy: at the order of 10-20%, you can use an oscilloscope: at the $10^{-3}$ level you cannot. $\endgroup$ – Massimo Ortolano Jan 11 '17 at 21:50
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Resonant methods for the measurement of electrical impedance were quite common in the past.

A once common instrument that allows resonant impedance measurements is the $Q$-meter. A $Q$-meter is an instrument capable of measuring the quality factor $Q$ of a resonant circuit.

The basic schematic of a $Q$-meter is the following (from [1]):

enter image description here

The instrument is composed of: i) a voltage source with adjustable frequency and very low output resistance $R$; ii) a calibrated variable capacitor $C_v$; and iii) a voltmeter with with very high input impedance whose scale is calibrated in terms of $Q$.

The measurement is carried out by substitution. Suppose, for instance, that you want to measure an unknown capacitance $C_x$ at a certain frequency $f$. You proceed as follows:

  1. Adjust the source frequency to $f$.
  2. Connect an inductor with suitable value to the instrument so that, together with the variable capacitor, it forms a resonant circuit.
  3. Adjust $C_v$ until the voltmeter reads the maximum value (resonance). Record the value of the calibrated capacitance ($C_1$).
  4. Connect the unknown capacitor $C_x$ to the instrument in parallel to the variable capacitance (parallel insertion).
  5. Adjust $C_v$ until you attain resonance again. Record the value of the calibrated capacitance ($C_2$).

Since you didn't change the source frequency and the inductance, the total capacitance is not changed between points 3 and 5. Thus, the unknown capacitance is given by

$$C_x = C_1-C_2.$$

Furthermore, from the $Q$ readings, you can obtain the capacitor's loss.

A dual procedure, with series insertion, can be used for inductance measurements.

An example of $Q$-meter: Boonton Q-Meter Type 160-A.

[1] E. Rubiola, Laboratorio di misure elettroniche, CLUT, Torino, 1993.

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