# What is the correct voltage current eq. for a variable capacitance capacitor?

#### simplex

What is the correct voltage - current equation for a variable capacitance capacitor?

If the capacitor has a fix value, C, then the following expression holds:

i(t)=C*dv/dt,

where i is the current that charges the capacitor and v the voltage across it.

If C=C(t) then i(t)=C(t)*dv/dt ???!!.
Something tells me that this formula is wrong. I am not sure 100% but it looks wrong.

What I need is the correct relation between i(t) and v(t) when C=C(t).
Can I get some help from you?
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Description of the problem in detail:

I have a capacitor whose capacitance, C is a function of time, t. More precisely, the distance between its plates is varied in time, by a mechanical device, according to a known law d=d(t) where d(t) is a function of t only and does not depend of the voltage, current, etc.

As d=d(t) and C=eps*S/d, also C=C(t). So, I can say that I have a capacitor whose capacitance changes in time according to a known, given function, independent of the voltage, current and other electrical parameters and dependent only of the time, t.

This variable capacitor is part of a circuit and as long as I do not know the current - voltage equation of it, I can not go further to analyze the behavior of the circuit which I need for an experiment.

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#### wywong

If the capacitor has a fix value, C, then the following expression holds:

i(t)=C*dv/dt,

where i is the current that charges the capacitor and v the voltage across it.

If C=C(t) then i(t)=C(t)*dv/dt ???!!.
Something tells me that this formula is wrong. I am not sure 100% but it looks wrong.

What I need is the correct relation between i(t) and v(t) when C=C(t).
The important equation here is:

Q(t) = C(t)V(t)

So,

$$I = \frac{dQ(t)}{dt} = V(t)\frac{dC(t)}{dt}+C(t)\frac{dV(t)}{dt}$$

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