Capacitor discharge circuits are used extensively in fast pulse generation. In such applications, the capacitor's stray inductance (or equivalent series inductance) and series resistance play an important role in determining performance of the pulse-generator circuits.

In practice, the equivalent series inductance (ESL) and the equivalent series resistance (ESR) values of the capacitors are typically unavailable. If they are supplied, they're not specified at frequencies of interest for fast-pulse power generation by capacitor discharge. In these applications, the ESL and ESR values of interest are those near the self-resonant frequency of the capacitor. While spark gap switches are used in conventional circuits to measure high-voltage capacitors, their variable resistance during the measurement period negatively affects the accuracy of the measurement. The circuit shown here tackles ESL/ESR measurement of capacitors by using a mercury-wetted relay *(Fig. 1)*.

In this circuit, the relay is employed as a fast closing switch. This bounceless switch has fast rise times (typically 10 ns or less) and low, stable on-state resistance. A variable voltage supply through R1 charges the capacitor under test (C1). The supply effectively isolates the source when the C1 discharge circuit is set into oscillation. Then the charged capacitor is connected through the mercury-wetted relay—once to a similar uncharged capacitor (C2), then short-circuited via the same connecting leads.

Switch SW1 turns on the mercury-wetted relay, and the C1 discharge circuit is set into oscillation. The oscillating current waveforms are stored on the digital storage oscilloscope, and basic circuit analysis determines the component values. Because identical circuit interconnections are maintained in both cases, the effects of stray inductance (L_{S}) and stray resistance (R_{S}) associated with the leads and mercury-wetted relay are eliminated. As a result, the ESL/ESR of the capacitor under test can be determined.

From basic circuit analysis, the current of an oscillating R-L-C series circuit is :

where

and

Taking the ratio of two successive current peaks of the oscillating waveform:

where

and

In each case, effective L, R, and C components for ESL/ESR measurement *(Fig. 2)* are:

- when C2 is not connected (short circuit):
- when C2, similar to C1, is connected:

L1 = ESL1+L_{S}, R1 = ESR1+R_{S}C1 = C1

L2 = 2ESL1 + Ls R2 = 2ESR1+R_{S}C = (C1)/2

Using basic circuit equations:

Similarly,

are determined from the monitored waveform in the respective cases.

and

ESL1=L2 - L1

Similarly,

and

ESR1=R2 - R1

Thus, the ESL and ESR values of the capacitor under test can be determined simply by measuring the waveforms *(Fig. 3)* and using the above formulas. The initial charging voltage of the capacitor must be adjusted to accommodate the peak current rating of the mercury-wetted relay. Implementing readily available relays, this circuit targets capacitor values up to about 10 nF. Because the measured current is always used in ratio, absolute calibration of the current probe isn't necessary. Generally, ESL values up to tens of nanohenries and ESR values up to hundreds of milliohms can be measured precisely with this simple setup.