# State space modeling of parallel capacitors

1. Apr 9, 2012

### SunnyBoyNY

Hi there,

I have been thinking about how to actually describe a system input filter consisting of several caps and an inductance in the state-space representation.

The topology follows: a DC source to inductor to two parallel caps that are referenced to the same potential as the voltage source. One cap is film and the other one is electrolytic. Hence, the latter cap has significant resistance.

I cannot write the governing equations in a form where a time derivative of energy storage element could be written as a linear sum of other state space variables. This makes me think that the system cannot be described in the present form.

On the other hand, each cap can have different voltage so the two voltages are not slaved one to the other. Circulating current between the two (provided zero inductor current) is determined by the voltage difference and ESR (resistance).

Any thought?

SunnyBoy

2. Apr 10, 2012

### SunnyBoyNY

Okay, so it took me one more day to solve the problem. Of course, the system can be described via a system of first order DEs. Solution follows:

$\begin{pmatrix} \frac{d}{dt} i_L \\ \frac{d}{dt} v_{C1} \\ \frac{d}{dt} v_{C2} \\ \end{pmatrix} = \begin{pmatrix} -\frac{\text{Rl}}{L} & -\frac{1}{L} & 0 \\ \frac{1}{\text{C1}} & -\frac{1}{\text{C1} \text{Rc}} & \frac{1}{\text{C1} \text{Rc}} \\ 0 & \frac{1}{\text{C2} \text{Rc}} & -\frac{1}{\text{C2} \text{Rc}} \end{pmatrix} . \begin{pmatrix} i_L \\ v_{C1} \\ v_{C2} \\ \end{pmatrix}$

If the filter connects to a stiff voltage source, the output voltage ramp will ramp up to Vin according to the system eigenvalues.

I.e. for Vin = 40 V, C1 = 50 uF, C2 = 11 mF, Rc = 18 mOhms, L = 10 nH, Rl = 10 mOhms the eigenvalues are

-1056456+1411806 i
-1056456-1411806 i
-3249