# Determine Equivalent ESR, Ripple Voltage, And Currents For Unequal Capacitors In Parallel

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Capacitors often are connected in parallel in power electronics to decrease high-frequency ripples, current stress, decrease power dissipation, and operating temperature, as well as to shape frequency response and boost reliability. Yet designers have three critical questions about this technique:

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• What are the equivalent values of capacitance C_{se }and equivalent series resistance (ESR) R_{se}?

• What is the high-frequency ripple voltage?

• What are the individual RMS currents?

If all N capacitors in the parallel connection are identical *(Fig. 1)*, with equal capacitance values C_{sk} = C and equal ESR values R_{sk} = R_{s}, then for k = 1, 2, . . . N the answers are clear:

• C_{se} is directly proportional to the number of capacitors N: C_{se} = NC, and R_{se }is inversely proportional to N: R_{se }=R_{s}/N.

• Ripple voltage V (RMS value) is:

for a sinusoidal current excitation i(t) = I √2 sin (2πft) with frequency f, where X_{se} = 1/(2πfC_{se}) is the reactance of the equivalent capacitor C_{se }and RMS value I, and individual RMS currents in the capacitors are identical: I_{k} = I/N.

When the capacitors in the parallel connection aren’t identical, with different capacitance C_{sk} and ESR R_{sk} values, the solution to the problem isn’t trivial. The direct approach is to obtain an analytical expression for the input impedance of the parallel connection in the algebraic form Z = Re Z – j ImZ= Z_{se} Z and use the formulas R_{se }= Re Z, X_{se} = Im Z, and C_{se }= 1/(2πf X_{se}).

A less complicated approach is based on the conversion of series C_{sk},R_{sk} connections to equivalent parallel C_{pk}, R_{pk }connections. To obtain relationships between R_{pk} and R_{sk}, and also between C_{pk} and C_{sk}, set the admittance Y_{pk} of the parallel C_{pk}, R_{pk} pair and admittance Y_{sk} of the series C_{sk}, R_{sk} pair connections equal to each other: Y_{pk} = Y_{sk}, Re (Y_{pk}) = Re (Y_{sk}), and Im (Y_{pk}) = Im (Y_{sk}). Then:

where:

is the reactance of the individual capacitor.

After individual parallel capacitances C_{pk} and resistances R_{pk} are calculated according to Equations 2 and 3, equivalent parallel capacitance C_{pe }can be easily found as the sum of C_{pk}:

The real part of equivalent admittance can be found as the sum of admittances 1/R_{pk}. R_{pe} can be obtained as a reverse value of that sum:

The system’s equivalent series capacitance C_{se }and ESR R_{se } can be found by conversion of the parallel C_{pe}, R_{pe} connection to the equivalent series connection C_{se}, R_{se}. To obtain relationships between C_{se} and C_{pe} and also between R_{se} and R_{pe}, set impedance Z_{pe }of the parallel C_{pe}, R_{pe} and impedance Z_{se} of the series C_{se}, R_{se} connections equal to each other: Z_{pe }= Z_{se}, Re Z_{pe }=Re Z_{se}, Im Z_{pe }= Im Z_{se}. Then:

where:

is the reactance of the equivalent parallel capacitor C_{pe}*(Equation 5)*.

Based on this analysis, the calculation procedure for equivalent series capacitance C_{se}, ESR R_{se}, voltage ripples V, and RMS currents I_{k} in the capacitors is:

• Calculate reactances of individual capacitances according to Equation 4.

• Determine equivalent parallel parameters C_{pk}, R_{pk} of the capacitors based on Equations 2 and 3.

• Calculate equivalent parallel capacitance C_{pe} of the structure, its reactance X_{pe}, and equivalent parallel resistance R_{pe} according to Equations 5, 9, and 6.

• Calculate equivalent series capacitance C_{se} and ESR R_{se} of the structure according to Equations 7 and 8.

• Obtain RMS ripple voltage V using Equation 1.

• Calculate RMS currents I_{k} in the capacitors based on:

Note that ESR values R_{sk} are strong functions of frequency. A designer should use ESR data specified by capacitor manufacturers at a given frequency of operation, such as the data for ceramic and polymer aluminum electrolytic capacitors from Murata Manufacturing Co. Ltd. (MMC) *(http://ds.murata.co.jp/software/simsurfing/en-us/index.html)*.

To illustrate the calculation procedure, let’s determine equivalent parameters, voltage ripple, and current distribution for a parallel connection of three ceramic capacitors (GRM21BR60J226ME39L) and one polymer capacitor (ESASD40J107M015K00) from MMC *(Fig. 2)*.

Using the data f = 200 kHz, C_{s1} = C_{s2} = C_{s3} = 22 μF, R_{s1} = R_{s2} = R_{s3} = 4 mΩ, C_{s4} = 100 μF, R_{s4} =8 mΩ, I = 2 A, then:

• For reactance of each individual capacitance according to Equation 4, we have X_{si} = X_{s2} = X_{s3} = 3.6 mΩ, X_{s4} = 0.8 mΩ.

• Equivalent parallel parameters C_{pk}, R_{pk} of the capacitors based on Equations 2 and 3 are C_{p1} =C_{p2 }= C_{p3} = 21.7 μF, R_{p1} =R_{p2 }= R_{p3} = 331 mΩ, C_{p4} = 49.7 μF, R_{p4} = 16 mΩ.

• For equivalent parallel capacitance C_{pe}, its reactance X_{pe} and equivalent parallel resistance R_{pe} of the structure according to Equations 5, 9, and 6, we calculate C_{pe} = 115 μF, X_{pe} = 6.9 mΩ, R_{pe} = 13.9 mΩ.

• According to Equations 7 and 8, the equivalent series capacitance C_{se} and ESR R_{se} are C_{se} = 143.4 μF, R_{se} = 2.76 mΩ.

• For RMS ripple voltage V based on Equation 1, we obtain V = 12.4 mV.

• RMS currents according to Equation 10 in ceramic and polymer capacitors are respectively: I_{1} = I_{2} = I_{3} = 341 mA, I_{4} = 1.1 A.

This shows the technique can easily determine the parameter values in each of the capacitors.

*Alexander Asinovski **is principal engineer at Murata Power Solutions Inc., Mansfield, Mass. He holds BSEE and MSEE degrees from State Technical University, St. Petersburg, Russia, and a PhD from the University of Telecommunications, St. Petersburg. He can be reached at **[email protected]**.*

This file type includes high resolution graphics and schematics.