Two Capacitor Problem

[tinska.h]

Hey Everyone,

Imagine A circuit of C1 -> R1 -> C2
And C1 is also connected to R2 and then to GND,

Where C1 initial voltage is lets say 60V, and C2 initial voltage is 10V. C1=100µF, C2=5µF, R1 = 100Ohm, R2 = 500Ohm. How would one solve C2 voltage in respect of time? Any steady state equation tricks? If going through calculus / Laplace is the only way, how would I arrange initial problem, if e.g. charging capacitor equation is CR*dV/dT+Vc=Vsource?

Thank you in advance!

 

[BradtheRad]

Lacking a schematic, we can only speak in generalities.

From your description it sounds as though two neighboring resistors form a voltage divider. Therefore it might help to think in terms of what is the voltage at the middle node? How is it influenced by the weighted voltage being sent through each resistor? (Does a capacitor send voltage at one end? Is 0V ground at the other end?)

In addition sometimes you can predict the final situation of the capacitors, that is, the final charge on them, or the proportion between one and the other.

 

[c_mitra]

You need to set up the equations and solve - you may or may not get a solution in a closed form but a sketch is the first thing that is needed before we can make a comment...

 

[tinska.h]

Hey, here is the schematic. PR1 is the node I'm interested in time domain. I'm trying to create a basic equation for PR1's V(t), but feeling bit confused now.

Initial conditions for C1 is 60V and C2 is 5V.

 

[BradtheRad]

Immediately after Time=0, C1 discharges partly toward 0V ground, and partly to charge C2.

Calculate RC time constants for each capacitor. This tells you how quickly the capacitor charges and discharges. Look at a small time increment which is much smaller than both time constants. Figure how much change occurs on each capacitor during that increment.

Repeat many times.

Eventually the capacitors have equal charge. Eventually they both discharge to 0V.

 

[albbg]

If you are still interested in the solution, here is my calculations:

 

[zorro]

Exact calculation gives the solution for the general case.
But in many cases, like this one, we can get an approximate solution (but quite accurate) by inspection.

Initial values are:

V₁₀ = 60 V
V₂₀ = 5 V

Let's proceed ad follows:

The circut has two time constants, one fast (tau1) and one slow (tau2).
Let's assume that tau1 << tau2.
During the first part, the response due to tau2 is almost constant and tau1 dominates.
During the second part, the response due to tau1 has practically vanished and tau2 dominates.

Qualitatively:

1) As:
1a) C2*R1 << C1*R2 (the time constant at which charges or discharges the smaller capacitor is much faster that the time constant at which the larger capacitor can be discharged or charged), and
1b) C1*V10 >> C2*V20 (the initial charge in C1 is much larger than in C2),
Then:
The voltage on C2 raises in a "short time" up to a value Vm, at which both capacitors have the same voltage. This rise has a time constant tau1.

2) After that, both capacitors continue to have approximately the same voltage and are discharged across R2 with a time constant tau2.

We have:
tau1 ≈ R1*C2 = 500 us
tau2 ≈ R2*(C1+C2) = 52.5 ms
So we verify the assumption tau1 << tau2.

Now we must find Vm. The total initial charge contained in C1 and C2 is redistributed in their parallel during the first part of the transient (only a small part is lost in that time):

C1*V₁₀ + C2*V₂₀ ≈ (C1+C2)*Vm

Vm ≈ (C1*V₁₀ + C2*V₂₀) / (C1+C2)

Replacing:
Vm ≈ 57.4 V
tau1 ≈ 500 us
tau2 ≈ 52.5 ms

Fron albbg's solution (taking his rounded final values):
Vm ≈ 57.9 V
tau1 ≈ 476 us
tau2 ≈ 52.6 ms

Regards

Z