Re: solving a deceptively simple-looking switched RC circuit (singular circuit ?)

enginquiry@xxxxxxxx wrote: 
Hi,

I am confronted with what I believe is a "singular circuit" and I am
having difficulty determining initial conditions for solving it ...


Here is a description (with links to figures):

A DC source first charges a -----C2-----(R1 || C1)----- circuit.  At
time t=T, two switches are thrown: one disconnecting the DC source, and
the other, connecting the "charged" portion of the circuit to a big
capacitor C3 (which may already have some charge on it).  The circuit
therefore becomes  (-----C2-----(R1||C1)-----) || C3.

I've posted a schematic of the circuit for t=<T- (before the switching)
here:
http://ca.pg.photos.yahoo.com/ph/enginquiry/detail?.dir=/c44b&.dnm=2fb4.jpg&.src=ph

First of all, your circuit is a simplification of a real circuit that is either impossible, in reality, or involves infinite current impulses. But as long as you realize what the simplifications are, you can work around them.

You mention that C3 may have an arbitrary initial voltage at the moment it is switched into the loop made up of the other components. Do C1 and C2 also have arbitrary initial voltages at the moment the voltage source is connected to them? Id that moment t=0?

At switch time t=T-, C1, C2, and C3 all have charge and the circuit has
not yet reached steady-state.

At time t=T+ (after switching), the circuit becomes:
http://ca.pg.photos.yahoo.com/ph/enginquiry/detail?.dir=/c44b&.dnm=3ce9.jpg&.src=ph

This is the circuit I want to solve.  The chief difficulty is that i
don't know what the initial conditions are for t=T+.  I cannot assume
that the voltages on the capacitors are continuous across t=T bacause I
am connecting capacitors in parallel at t=T. Charges will flow to
balance the voltage on the (-----C2-----(R1||C1)-----) network and the
C3 capacitor "instantaneously" (need to model this current with a delta
function).

Yes. Except for the small possibility that the voltage across C3 happens to match the total voltage across C1 and C2, there will be an infinite current impulse through the switch and the equality will be instantaneously enforced. You need to solve for the amount of charge that must redistribute in that moment that will produce that equality, and also to solve for what the resulting voltage will be. Start with a simpler case of two capacitors, each charged to an arbitrary voltage and then connected in a loop. Before the connection, each capacitor has a charge of Q=C*V. After the connection they must have a common voltage. But charge is conserved, so Q1+Q2 in the initial state must equal Q1+Q2 in the final state, as well. Signs are important. During this impulse time, you can completely neglect the effect of any resistance in the circuit, because no time elapses, so only infinite impulses can do anything in zero time.

If I could figure out the initial conditions, thenI would  solve the
resulting ciruict by replacing the charged-capacitors with a DC source
and an uncharge capacitor is series like this:
http://ca.pg.photos.yahoo.com/ph/enginquiry/detail?.dir=/c44b&.dnm=1641.jpg&.src=ph

I'd then solve by superposition ...

Any thoughts ?  Can anyone tell me how to calculate the initial
conditions on the capacitors at t=T+  ?  How do the charges
re-distribute on C1, C2 and C3 at t=T ?

Thanks,
Dan

.

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