Calculate Equivalent Esr, Ripple Voltage, and Currents for Inequality Capacitors in Parallel Combination
In the second instalment of our Ideas for Design series, we delve into the fascinating world of parallel capacitor connections in power electronics. Alexander Asinovski, the principal engineer at Murata Power Solutions Inc., Mansfield, Mass., shares his insights on this topic.
The Impact of Parallel Capacitor Connections
Capacitors in parallel connections play a crucial role in power electronics, as they help decrease high-frequency ripples, current stress, power dissipation, operating temperature, and boost reliability. They also shape the frequency response and are essential in power distribution and decoupling applications.
The Analytical Expression for Input Impedance
The analytical expression for the input impedance (Z_{in}) of a parallel connection of capacitors, each with different capacitance (C_i) and equivalent series resistance (ESR) (R_i), is as follows:
[ Z_{in} = \frac{1}{\sum_i \frac{j \omega C_i}{1 + j \omega C_i R_i}} ]
This expression combines the frequency-dependent reactive impedance of each capacitor with its ESR and sums their parallel contributions. It captures how different capacitors and their associated ESRs affect the overall impedance.
Calculation Procedure
The calculation procedure for equivalent series capacitance (C), ESR (R), voltage ripples (V) (RMS value), and RMS currents (I) in the capacitors involves several steps, including Equations 2, 3, 5, 9, 6, 7, 8, and 10.
In the case of series (C,R) connections being converted to equivalent parallel (C,R) connections, the real part of equivalent admittance can be found as the sum of admittances (1/R), and (R) can be obtained as a reverse value of that sum.
Special Cases
When capacitors in a parallel connection aren't identical, the solution to find the equivalent values of capacitance and ESR isn't trivial and requires either an analytical expression for the input impedance or the conversion of series (C,R) connections to equivalent parallel (C,R) connections.
However, if all capacitors in a parallel connection are identical, the equivalent values of capacitance and ESR can be calculated as (C = NC) and (R = R/N), respectively.
Practical Considerations
When ESR values (R_i) are very small (ideal capacitors), the expression reduces to the parallel capacitance. In practical electronic designs, the equivalent series inductance (ESL) of capacitors can also be significant, especially at high frequencies, and would modify each (Z_i).
Careful layout and selection of capacitor ESRs and ESLs are essential to avoid undesirable effects such as anti-resonance peaks in impedance.
Example Calculation
For a given example with three ceramic capacitors (GRM21BR60J226ME39L) and one polymer capacitor (ESASD40J107M015K00) from Murata Manufacturing Co. Ltd. (MMC), the equivalent parallel capacitance (C) is 115 μF, its reactance (X) is 6.9 mΩ, and equivalent parallel resistance (R) is 13.9 mΩ.
The reactance of the equivalent parallel capacitor can be calculated using Equation 5. Individual RMS currents in the capacitors are identical and equal to (I/N). The RMS currents (I) in the ceramic and polymer capacitors for the given example are respectively: (I = 341) mA, (I = 1.1) A.
The equivalent series capacitance (C) for the given example is 143.4 μF, and the ESR (R) is 2.76 mΩ.
Conclusion
Understanding the behaviour of parallel capacitor connections in power electronics is essential for designing efficient and reliable power systems. The analytical expression for the input impedance of a parallel network of capacitors with distinct capacitances and ESRs provides the fundamental theoretical basis to model and compute the input impedance in circuit simulations or analytical studies.
This article includes high-resolution graphics and schematics, and it can be downloaded in .PDF format. We encourage designers to 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).
[References]
[5] W. S. Kao, M. K. Y. Wong, and W. K. L. Chan, "Capacitor Selection and Layout Techniques for Power Distribution," IEEE Transactions on Power Electronics, vol. 28, no. 6, pp. 1351-1362, June 2013.
Power electronics technology benefits significantly from the use of parallel capacitor connections, as they help improve high-frequency ripples, manage current stress, reduce power dissipation, lower operating temperatures, and enhance reliability. Furthermore, capacitors in a parallel arrangement shape the frequency response and are crucial in power distribution and decoupling applications.
