# Capacitor calculation with real and imaginary parts for a resonant converter

#### SiCEngineer

Joined May 22, 2019
444
Hi All,

I have designed a resonant LCC converter for my thesis. It has been some time since I visited the theory and believe I used the following document by Robert Steigerwald to select the resonant components: IEEE Xplore Full-Text PDF:

I was attempting to rearrange this equation for the parallel resonant capacitor, CP, so that I could calculate what value it needs to be at my switching frequency, resonant frequency, Q factor, input/output voltage requirements, and I get the following equation when simplified:

CP = 8.56nF + j22.72nF.

My question relates to the fact that this number involves a complex number. Now, it's been many years since I worked with a complex number, so this is a pretty basic question - but what exactly does this part of the equation relate to? Obviously, the parallel capacitance cannot have any reactive parts, so I placed a capacitor of 9nF into my simulation, and I happily get the 3,000V output voltage from my 243V input, as theory would suggest. I made it slightly larger to account for circuit voltage drops and other inefficiencies. So it seems that it would be appropriate to take only the real part of the above equation into the calculation for the required parallel capacitor - but the "imaginary" part must have some kind of function that possibly "changes" the effective voltage gain - I assume, since this is a variable frequency controller, is demonstrating how the gain changes as the operating switching frequency deviates higher and further away from the series resonant frequency.

So my question is, is it appropriate to ignore the effect of the gain attenuating frequency term when calculating the parallel resonant capacitance? Is there any effect of this gain variation on the selection of the parallel resonant capacitance that I should concern myself with?

I have done a lot of searching today regarding this and haven't came across anything of use. It seems so obvious to just ignore the complex term but I'd like to be able to fully understand its effect on the gain with respect to the capacitance values!

Best regards,
SIC

I have attached an image here for those who may not have access to IEEExplore.

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#### Papabravo

Joined Feb 24, 2006
21,264
Hi All,

I have designed a resonant LCC converter for my thesis. It has been some time since I visited the theory and believe I used the following document by Robert Steigerwald to select the resonant components: IEEE Xplore Full-Text PDF:

I was attempting to rearrange this equation for the parallel resonant capacitor, CP, so that I could calculate what value it needs to be at my switching frequency, resonant frequency, Q factor, input/output voltage requirements, and I get the following equation when simplified:

CP = 8.56nF + j22.72nF.

My question relates to the fact that this number involves a complex number. Now, it's been many years since I worked with a complex number, so this is a pretty basic question - but what exactly does this part of the equation relate to? Obviously, the parallel capacitance cannot have any reactive parts, so I placed a capacitor of 9nF into my simulation, and I happily get the 3,000V output voltage from my 243V input, as theory would suggest. I made it slightly larger to account for circuit voltage drops and other inefficiencies. So it seems that it would be appropriate to take only the real part of the above equation into the calculation for the required parallel capacitor - but the "imaginary" part must have some kind of function that possibly "changes" the effective voltage gain - I assume, since this is a variable frequency controller, is demonstrating how the gain changes as the operating switching frequency deviates higher and further away from the series resonant frequency.

So my question is, is it appropriate to ignore the effect of the gain attenuating frequency term when calculating the parallel resonant capacitance? Is there any effect of this gain variation on the selection of the parallel resonant capacitance that I should concern myself with?

I have done a lot of searching today regarding this and haven't came across anything of use. It seems so obvious to just ignore the complex term but I'd like to be able to fully understand its effect on the gain with respect to the capacitance values!

Best regards,
SIC

I have attached an image here for those who may not have access to IEEExplore.
A couple of things.
1. You cannot use units of Farads for the real part of a capacitive impedance.
2. A complex impedance has a real part with units of Ω's that is not frequency dependent
3. A complex impedance has an imaginary part, the capacitive reactance, which is frequency dependent and has units of Ω's
4. The magnitude of a complex impedance is real and has units of Ω's
5. The phase of a complex impedance, in radians, is the arctan(Im/Re)
Sometimes the real part of a capacitor is referred to as the ESR (Equivalent Series Resistance). The following expression may be illustrative:

$$Z_{C}\;=\;R\;+\;\cfrac{1}{j\omega C}\;=\;R\;-\;\cfrac{j}{\omega C}$$

Your expression is hard to interpret without knowing the frequency for which it is valid.

That said if we take a complex impedance like Z = 8.56 + j22.72, we can compute

$$|Z|\;=\;\sqrt{(8.56)^2+(22.72)^2}\;=\;24.28\; \text{ Ω}$$

$$arg(Z)\;=\;arctan(22.72/8.56)\;=\;1.21 \text{ radians}\;=\;69.36 \text{ }^{\circ}$$

Last edited:

#### SiCEngineer

Joined May 22, 2019
444
A couple of things.
1. You cannot use units of Farads for the real part of a capacitive impedance.
2. A complex impedance has a real part with units of Ω's that is not frequency dependent
3. A complex impedance has an imaginary part, the capacitive reactance, which is frequency dependent and has units of Ω's
4. The magnitude of a complex impedance is real and has units of Ω's
5. The phase of a complex impedance, in radians, is the arctan(Im/Re)
Sometimes the real part of a capacitor is referred to as the ESR (Equivalent Series Resistance). The following expression may be illustrative:

$$Z_{C}\;=\;R\;+\;\cfrac{1}{j\omega C}\;=\;R\;-\;\cfrac{j}{\omega C}$$

Your expression is hard to interpret without knowing the frequency for which it is valid.

That said if we take a complex impedance like Z = 8.56 + j22.72, we can compute

$$|Z|\;=\;\sqrt{(8.56)^2+(22.72)^2}\;=\;24.28\; \text{ Ω}$$

$$arg(Z)\;=\;arctan(22.72/8.56)\;=\;1.21 \text{ radians}\;=\;69.36 \text{ }^{\circ}$$
Dear Papabravo,

Thank you very much for this response. It has helped a lot. I am not sure though whether it is an impedance. The equation shown is for a circuit gain, of which is a function of the parallel resonant capacitance, CP.

I rearranged that equation, with a Q factor of 2, Eo/Ed (intended circuit gain) = 1, w = 2*pi*210kHz, and ws = 2*pi*182kHz. Where w is the switching frequency and ws is the resonant frequency of the series circuit.

From there, I found an equation for CP - but it has a real part, 8.56, and an imaginary part: j22.72. That has where the confusion has came in, because this is calculation of a capacitance, and not an impedance. The gain is frequency dependent on the switching frequency and how far it deviates from the circuit resonant frequency - and the quality factor. So I guess that I will have to calculate a capacitance that gives me the intended minimum and maximum gains required in the circuit, as we go from the maximum to the minimum frequencies of the switching - respectively.

I'll give that a go today and report back.

Best regards,
SIC

#### Papabravo

Joined Feb 24, 2006
21,264
Dear Papabravo,

Thank you very much for this response. It has helped a lot. I am not sure though whether it is an impedance. The equation shown is for a circuit gain, of which is a function of the parallel resonant capacitance, CP.

I rearranged that equation, with a Q factor of 2, Eo/Ed (intended circuit gain) = 1, w = 2*pi*210kHz, and ws = 2*pi*182kHz. Where w is the switching frequency and ws is the resonant frequency of the series circuit.

From there, I found an equation for CP - but it has a real part, 8.56, and an imaginary part: j22.72. That has where the confusion has came in, because this is calculation of a capacitance, and not an impedance. The gain is frequency dependent on the switching frequency and how far it deviates from the circuit resonant frequency - and the quality factor. So I guess that I will have to calculate a capacitance that gives me the intended minimum and maximum gains required in the circuit, as we go from the maximum to the minimum frequencies of the switching - respectively.

I'll give that a go today and report back.

Best regards,
SIC
I don't think you can have a capacitance with a complex value. That makes absolutely no sense.

#### Janis59

Joined Aug 21, 2017
1,856
The real problem is that lossy capacitor is trying to wildly heat-up. Thus, when ceramic cap are cracking, some 9 grams weighty particles flies on You with some 1 km/sec, its indeed painful. Just take the good FLIR camera and inspect when it will be made ready.
But about the prime question - any parallel RC one may re-calculate to the serial R-C and them are habiting ABSOLUTE equal. Thus, any R parallel to C means the bit other value of R in series to that C. Aware it!
And last but not least, the 60 deg angle is sth madness-ful. Even tan(fi)=0.01 ir criminally much when target is 0.001 or 0.0001 or even 0.00001. But 60 - its walking catastrophy.

#### Janis59

Joined Aug 21, 2017
1,856
Papabravo: thanks for the link. This Your suggested doccument have indeed very intriguing name (sci-hub.se/10.1109/63.4347) however reading it returned more than disappointment. Firstly, all three ciruits are in no yotta different one from another. Suspect the mistake in illustrations. And from everyday used at least 10 topologies to show only 3 - thats very poor performance indeed. Sorry for criticism, but we are waiting from articles to acknow sth new, in most cases.