Equivalent Capacitance and ESR of Paralleled Capacitors

22 POWER CAPACITORS www.murata-ps.com Equivalent Capacitance and ESR of Paralleled Capacitors

Parallel connection of capacitors is widely used in power electronics to decrease high frequency ripples and current stress, to decrease power dissipation and operating temperature, to shape frequency response, and to boost reliability. Alexander Asinovski, Principal Engineer, Murata Power Solutions, Mansfield, USA

Parallel connection of capacitors is admittance Ysk of the series Csk , Rsk capacitor Cpe (5). widely used in power electronics to connections equal to each other: Ypk = Ysk ,

decrease high frequency ripples and Re( Ypk ) = RE( Ysk ) and Im( Ypk ) = Im( Ysk ). It Based on the analysis presented above, current stress, to decrease power follows: calculation procedure for equivalent series dissipation and operating temperature, to capacitance, ESR, voltage ripples , and RMS shape frequency response, and to boost (2) currents in the capacitors is as follows: reliability. Main questions a designer faces 1. Calculate reactances of individual with regard to the parallel connection of (3) capacitances according to formula (4). capacitors are: What are equivalent 2. Determine equivalent parallel

capacitance and ESR (electric series where parameters Cpk , Rpk of the capacitors resistance) values? What is high frequency Xsk =1/2(2 ␲ fc sk ) (4) based on equations (2) and (3). ripple voltage? What are individual RMS is the reactance of the individual capacitor. 3. Calculate equivalent parallel capacitance

currents? Cpe of the structure, its reactance Xpe ,

If all capacitors in the parallel connection After individual parallel capacitance Cpk and equivalent parallel resistance Rpe

are identical with equal capacitance values, and resistance Rpk values are calculated according to formulas (5), (9), and (6).

Csk = C , k = 1,2... N and equal ESR values according to (2) and (3), equivalent 4. Calculate equivalent series capacitance

Rsk = Rs the answers are obvious: Cse is parallel capacitance Cpe can be easily found Cse and ESR Rse of the structure according

directly proportional to the number of as the sum of Cpk to formulas (7) and (8).

capacitors N, Cse = NC , Rse is inversely 5. Obtain RMS ripple voltage V using

proportional to N: Rse = Rs / N , ripple (5) equation (1).

voltage V (RMS value) for a sinusoidal 6. Calculate RMS currents Ik in the current excitation i (t) = I͌2 sin ( 2␲ft ) and real part of equivalent admittance can capacitors based on the formula with frequency f, and RMS value I is be found as the sum of admittances

1/ Rpk . Rpe can be obtained as a reverse (10) , (1) value of that sum: It is worthwhile to note that ESR values are where Xse =1/(2 ␲ fC se )is the reactance of (6) strong functions of frequency. A designer the equivalent capacitor Cse and individual should use ESR data specified by capacitor

RMS currents in the capacitors are Equivalent series capacitance Cse and ESR manufacturers at a given frequency of

identical: Ik = I / N . Rse of the system can be found by operation. An example of a comprehensive

In case capacitors in the parallel conversion of the parallel Cpe , Rpe source of data for ceramic and polymer connection are not identical, with different connection to the equivalent series aluminum electrolytic capacitors is found

capacitance Csk and ESR Rsk values, the connection Cse , Rse . To obtain relationships on the Murata Manufacturing Co., Ltd.

solution to the problem is not trivial. The between Cse and Cpe and also between Rse (MMC) website http://ds.murata.co.jp/

direct approach would be obtaining an and Rpe , set impedance Zpe of the parallel software/simsurfing/en-us/index.html.

analytical expression for the input Cpe , Rpe and impedance Zse of the series To illustrate the calculation procedure

impedance of the parallel connection in Cse , Rse connections equal to each other: let’s determine equivalent parameters,

the algebraic form Z = Re Z – j Im Z and Zpe = Zse , Re Zpe = Re Zse and Im Zpe = voltage ripple, and current distribution for a

using the formulas Rse = Re Z , Xse = Im Z Im Zse . It follows: parallel connection at of three ceramic and Cse =1/(2 ␲ fX se ). A less complicated (7) capacitors GRM21BR60J226ME39L and approach taken below is based on the one polymer capacitor

conversion of series, Csk , Rsk connections to (8) where ESASD40J107M015K00 from MMC.

equivalent parallel Cpk , Rpk connections. To Assuming the following input data: obtain relationships between Rpk and Rsk , (9) f =200 kHz , Cs1 = Cs2 = Cs3 =22 ␮F, Rs1 = and also between Cpk and Csk , set Rs2 = Rs3 = 4m ⍀ Cs4 =100 ␮F, Rs4 = 8m ⍀, admittance Ypk of the parallel Cpk , Rpk and is reactance of the equivalent parallel I = 2A .

Issue 8 2013 Power Electronics Europe www.power-mag.com www.murata-ps.com POWER CAPACITORS 23

1. For reactance of each individual ␮F, Rp4 = 16m ⍀. ESR Rse according to formulas (7) and capacitance according to formula (4) we 3. For equivalent parallel capacitance Cpe , (8) are: Cse = 143.4 ␮F, Rse = 2.76m Ω. have: Xs1 = Xs2 = Xs3 = 3.6 m ⍀, Xs4 = its reactance Xpe , and equivalent parallel 5. For RMS ripple voltage based on 0.8m ⍀. resistance Rpe of the structure according equation (1) we obtain: V = 12.4 mV. 2. Equivalent parallel parameters Cpk , Rpk of to formulas (5), (9), and (6) we 6. RMS currents according to formula (10) the capacitors based on formulas (2) calculate: Cpe = 115 ␮F, Xpe = 6.9 m ⍀, in ceramic and polymer capacitors are and (3) are: Cp1 = Cp2 = Cp3 = 21.7 ␮F, Rpe = 13.9 m ⍀. respectively: I1 = I2 = I3 = 341mA, I4 = Rp1 = Rp2 = Rp3 = 331 m ⍀, Cp4 = 49.7 4. Equivalent series capacitance Cse and 1.1 A.

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