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CAPACITOR ESR MEASUREMENT TECHNIQUEJ.E. Dolan and H.R. BoltonDepartment of Electrical, Electronic, and SystemsEngineering, UWCC, P.O. Box 904, Cardiff, CF1 3YH, U.K.

Abstract

Capacitor losses were determined for a particulartype of pulse capacitor. The technique used involvedmeasurement of terminal voltage and current waveformsduring charge and discharge phases; comparison of thederived terminal energy flows and the correspondingchanges in the stored energy gives the internalcapacitor loss. The measured l o s s figures were furtherused to calculate effective values of capacitor ESR fort,hr pulse conditions; these would be directlyapplicable in numerical circuit modelling of pulsedcircuits. The values obtained under both unipolar andvoltage reversal conditions appeared to validate theuse of a single appropriate value of ESR under pulsedconditions.

IntroductionOur interest in capacitor losses arose from work

on bench-top proof-of-principle experiments on energyrecovery with Melville line magnetic pulse compressioncircuits [l], from which it became clear that the lossper stage can be considerable. In the literature,attention has tended to focus on magnetic switchlosses, but it began to seem as though other componentsin the system might be absorbing at least as large aproportion of the pulse energy as the magnetic cores. Aparticular concern in repetitive feeds is capacitorheating and the need to derate capacitors accordingly,particularly i f capacitor voltages are reversed. Thefollowing work was therefore intended to give some ideaof the proportion of pulse energy loss to be expectedi n capacitors.

Principle of Capacitor loss-measurementsA 0.15pF, 40nH. 65KV Maxwell type S S pulsecapacitor was chosen for the tests. The manufacturersquoted capacitance value of 0.149pP was approximatelychecked using an LCR meter, and also by measuring CR

time-constants with known resistances.In an effort to minimise added circuit inductanceand resistance, the connections to the capacitorterminals were made through parallel copper plates.Initially, measurements of resonant frequency anddecrement per cycle during oscillatory discharge weremade: however, as discussed by Nunnally et a1 [ 2 ] . areasonably accurate assessment of capacitor loss canonly be made if the loss in external components is low,and deductions from these measurements foundered on thehigh external circuit loss experienced in the small-scale experimental rig. This led to the idea ofobtaining the loss through direct measurement of theinstantaneous power flow at the capacitor terminals.

The procedure is as follows; waveforms forcapacitor terminal voltage and current are obtained,and the terminal power waveform is then given by theproduct of terminal voltage and current waveforms. Thetime-integral of the power waveform gives the netterminal energy flow. Because the capacitor internalinductance is very low, the terminal voltage at zerocurrent is a close approximation to the nominalcapacitor voltage, and so the change in stored energybetween maximum and mimimum voltage points can beaccurately estimated. The loss is then given by

loss = net terminal energy flow- change in 1/2CV energy

Principle of current measurementBecause the capacitor internal resistive voltagedrop is significant, it was found that deduction of thecircuit current from the rate of change of capacitorterminal voltage was unsatisfactory. A particularfeature of the experiment was therefore themeasurement of current by use of a 30-turn Rogowski-like inductive pick-up coil with turn diameter of aboutlcm, and axial length of about 3cm, located between thecapacitor terminal plates. The pick-up coil generates avoltage given by

(1 )p = M di/dtwhere M is a constant of proportionality. Integrating(1) w.r.t. time gives the capacitor current:

In order to calibrate M, carrying out the doubleintegral of (1) w.r.t. time gives the change incapacitor charge, &Q, multiplied by M;

Vp dt dt = M 11:i dt

= M AQ ( 3 )1s::

M can then be found by substituting for / \ Q , which isgiven byAQ c (v2 - vl) (4)

The method obviously relies on having accurate valuesfor both the capacitance, C, and the initial and finalcapacitor voltages.

Experimental measurementsThe low energy test circuit of figure 1 was used.C1, a 0.1pF capacitor, is d.c. charged, and is switchedto ring across to Cz, the 0.15pP test capacitor.Switching is performed by the mercury-wetted reedrelay, in series with a magnetic switch, MS1, formedfrom a ferrite bead with dimensions- i.d. 1.2mm- 0.d. 3.4mm- w 5.0mmand wound with 27 turns. With the second magnetic

switch, MS2, disconnected from the circuit, C2 chargesup, and then discharges back into C1 after a furtherdelay due to MS1. The rationale for the use of magneticswitches is that their high blocking state inductancecauses the already low internal inductive voltage dropin the test capacitor to become negligible at zerocurrent, allowing more accurate estimatibn of theinternal capacitor voltage.

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/i n du ct. v e p i cK- p

p i c k - u p voltage(volts)

Figure 1inductive pick-up for current measurementCircuit for capacitor loss measurements, with

3 5 v ~terminal voltage(volts)

(a) CharainR efficiency

2 p s

Figure 2 shows the measured terminal and pick-upvoltage waveforms for capacitor C2, and also thecomputed current and power waveforms. C2 was chargedfrom OV to 23.25V, giving a change in stored energy of40.5~5, hile the terminal input energy was computed as47.7pJ. This gives the charging efficiency, heredefined as the ratio of energy in to energy stored, as40.5/47.7 = 8 5% .5 h Ijb) Discharnina efficiency I

Figure 3 shows the discharging voltage and currentwaveforms for capacitor C2, with C2 ringing back intoC?. The incomplete discharge of C2 is due to theC2 were 23V and 7.W respectively, giving a change instored energy of -35.4pJ. The terminal energy flow wascomputed as -32.35. This gives the dischargingefficiency, here defined as the ratio of energy out tochange in stored energy, as 32.3/35.4 91%.

mismatch of C 1 and C2. The initial and final charges on 2.:

0 t_-

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i 5 v ~I

terminal

01-0 1 2PS

Figure3 Test capacitor discharging waveforms

The experimentally measured capacitor loss is thedifference between the terminal input energy and theincrease in stored energy: equating this with theright-hand term in (5) gives a value for ReSr. Thecapacitor voltage was measured at the start and end ofthe nominal half-sine current waveforms, at whichpoints the actual current is practically zero. Thevalues obtained for tests (a),(b), and (c) above wereas follows:

- 0.54 ohms (a)- 0.55 ohms (b)- 0.48 ohms (c)The values obtained for (a) and (b) are remarkablyclose. The value for (c) is lower, which could bebecause the pulse length on ring-round is longer:1.45~s against 1.1 ~s. ultiplying 0.48 ohms by thefactor J(1.45/1.1) gives 0.55 ohms, equivalent to thevalues for (a) and (b), which suggests that theexplanation for the lower value in (c) may be a Sfdependence of the ESR.

Discussion of resultsTests (a) and (b) indicate capacitor losses under

conditions of unipolar charging and discharging of 10-15% of the change in stored energy. However, when thecapacitor voltage was reversed, in test (c), around 27%of the initial stored energy was lost in the capacitor.Qualitatively, the higher loss measured on voltagereversal ties in with the fact that pulsed capacitorshave to be derated according to the degree of reversal.Calculation of the pulse ESR shows that the capacitorcontinues to behave in a linear manner on voltagereversal; the larger proportion of loss on voltagereversal is simply due to the larger pulse current.

Possible limitations of the experimentalmethodology used are, firstly, the need for an integralcalculation, which is prone to drift, as can be seenfrom the waveforms for current, power, and energy ontest (a). However, this was probably due mainly toinaccuracies in the oscilloscope trace readings, whichwere performed manually. Secondly, use of the inductivepick-up for current measurement relies on theassumption that the current distribution pattern acrossthe parallel plate capacitor feed remains uniformduring the pulse, which was not actually verified.Third, the proportion of energy dissipation in thecapacitor must be large enough to be measurable;otherwise a calorimetric method is indicated, asdiscussed by Rust and McDuff [ 3 ] for very low-loss rep-rate designs.

25V ,

terminal vultsi

-15v I . .0 1 2PS

f c

terminal current(ampsiI

Figure 4 Test capacitor ring-round waveforms

High current testThe low voltage at which the above tests werecarried out could mean that the ESR figures obtainedare not valid for full-voltage operation of the Maxwellcapacitor, although one would probably expect thecapacitor to continue to behave linearly at highervoltage levels. A brief experiment was carried out inwhich the test capacitor was d.c. charged to around4kV, and then switched onto a second, identical,capacitor through a self-breaking spark-gap, as shown

in figure 5. Heavy copper wire was used for theconnections. The oscillatory charging waveform for C2is also shown in figure 5; the half-cycle time was1.2511s, corresponding closely to the above tests. Thepeak current was around 680Amps. The effective circuitresistance was calculated from the oscillationdecrement as 2 x 0.55ohms (*SI). This result suggeststhat the capacitor equivalent series resistance formsthe dominant circuit loss mechanism, and that the valueof ESR determined at low (1-5A) current l eve l s remainedvalid at the higher power levels.

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"d0- .

1'-rI

I

-breaking g a p

Figure 5 High voltage oscillatory charging waveform

ConclusionsThe experimental method allows precise assessmentof internal capacitor losses under fast pulseconditions, when the effect of the external circuitcannot be neglected. Proportions of initial energy lostin the test capacitor during charge transfer with pulselengths of 1-1.5~s ere found to be between 10-30%. Theindividual contributions of conduction, dielectric, andcontact losses to the ESR cannot, of course, bedistinguished. The measurements carried out appeared toindicate a constant ESR with pulse amplitude, with aputative Jf frequency dependence, but further workwould obviously be necessary to establish the form of

frequency dependence, whether the ESR remains constantat up to the full charge level, and its variation underrep-rate conditions.The ESR of the Maxwell pulse capacitor under testwas estimated as around 0.50hms at the pulse lengthused. The manufacturers quoted values for thecapacitance and series inductance of the capacitor were0.15pF and 40nH respectively, giving values ofcharacteristic impedance of only $(0.04/0.15)= 0.520hms, and nominal resonant half-cycle time ofapproximately $0.15~0.04 = 243ns. For efficient energyextraction, the capacitor would have to be dischargedwith a pulse length much longer than the capacitorself-resonant half-cycle time of 24011s. As the Maxwellcapacitor is specifically designed for pulsedapplications, this suggests that fast pulses and highefficiency operation of capacitors may be ratherincompatible objectives.

References[11 Power, A . J . , Bolton, H.R., Bessel, W. (1987).Simulation of a pulsed power scheme feeding a denseZ-pinch plasma load, 6TH IEEE P.P. Conf., 696-699.[ 2 ] Nunnally, W.C., Kristiansen, M . , Hagler, M.O .(1975), Differential measurement of fast energydischarge capacitor inductance and resistance, IEEETrans. Instrumentation and Measurement, Vol. 24, No.2 , 112-114.[3] Rust, K., McDuff, G. (1986), 'Calorimetricmeasurements of the 'equivalent series resistance' oflow-loss, high repetition rate discharge capacitors',IEEE 17th Power Mod. Symp.

This work was supported by the U.K. Science andEngineering Research Council.

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