transient analysis of linear system

to

change

do

varies,

they

values of

Ok. I had intended to study this over the summer in any case. Using the reference I gave you , I found the form of the solution for Vc1 and Vc2. It remains constant for varying input voltage, but the coefficients of the pieces varies. This is a way of solving for Vc1 and Vc2 for a given input voltage, at a given time after the application of a step. This isn't designed to accommodate varying input, but it might still be usable. Suppose that you have a history of input voltages over 5 seconds. Average those inputs, and take that value as the impressed input for time zero. Then solve the circuit for the coefficients of the output equation, and then solve the output equation for t=2.5. If there is a difference between the capacitor voltages and the circuit solution, then the resistances didn't have the assumed values during that period (assuming nothing else can change). The method would depend on the inputs not changing so quickly that the average wasn't a valid approximation. The form of the equations is Vcx = a*exp(0.2273*t) + b*exp(0.6818*t) + c

The values of the coefficient of t in the exponentials depends on the form of the circuit and the values of R and C. The coefficients a,b,c depend on the applied voltage and are different for Vc1 and Vc2.

It's a little involved to show the derivation of the equations, and to recalculate a,b, and c, so I won't type that in unless you think it might be useful.

-- John

to

change

do

varies,

they

values of

Hi John. Thanks for that. I can't evaluate your work this weekend for fear of divorce... Anyway I'll definitely check it out next week. I can compare the output of SPICE with the output from the equation. If you can be bothered I would be really interested in your derivation. As I say I struggle a bit with maths. Feel free to cc to my mailbox direct.

Thanks again.

{snipped}

Wombat, It was too hard to try to type it all in, so I posted a bmp file of a handwritten derivation on alt.binaries.schematics.electronic. I also posted an MSword file of Mupad CAS code used to come up with values. There's still a slight problem, in that it's set up to find the voltage at time t, starting from zero on the caps. It would need redoing to get it to start from an arbitrary point; I won't be able to look at it any more because of other things, but, FWIW.

-- John

As a final afterthough, I tried verifying the results with an LTspice simulation, and the graph looked the same as the CAS output. Here is the LTspice file:

Version 4 SHEET 1 880 680 WIRE -48 160 -80 160 WIRE 64 160 32 160 WIRE 80 160 64 160 WIRE 192 160 160 160 WIRE 208 160 192 160 WIRE 336 160 288 160 WIRE 192 208 192 160 WIRE 336 208 336 160 WIRE -80 224 -80 160 WIRE 64 224 64 160 WIRE -80 336 -80 304 WIRE 64 336 64 288 WIRE 64 336 -80 336 WIRE 192 336 192 272 WIRE 192 336 64 336 WIRE 336 336 336 288 WIRE 336 336 192 336 WIRE 336 352 336 336 FLAG 336 352 0 SYMBOL voltage -80 208 R0 SYMATTR InstName V1 SYMATTR Value -.7 SYMBOL voltage 336 192 R0 SYMATTR InstName V2 SYMATTR Value 3 SYMBOL res 48 144 R90 WINDOW 0 0 56 VBottom 0 WINDOW 3 32 56 VTop 0 SYMATTR InstName R1 SYMATTR Value 2e12 SYMBOL res 176 144 R90 WINDOW 0 0 56 VBottom 0 WINDOW 3 32 56 VTop 0 SYMATTR InstName R2 SYMATTR Value 2e12 SYMBOL res 304 144 R90 WINDOW 0 0 56 VBottom 0 WINDOW 3 32 56 VTop 0 SYMATTR InstName R3 SYMATTR Value 2e12 SYMBOL cap 48 224 R0 SYMATTR InstName C1 SYMATTR Value 2.2e-12 SYMBOL cap 176 208 R0 SYMATTR InstName C2 SYMATTR Value 2.2e-12 TEXT -96 56 Left 0 !.tran 20 uic

-- John

If all three resistors are variable there's no way to reliably detect a change unless you can measure the current through one of them,

Bye. Jasen

you can't do it this way either.

measuring the rate of change of voltage on each capacitor will give you the current flowing through that capacitor

but without also knowing the current flowing through one of the resistors there can be no solution of the system.

Hi John,

Thanks again for your efforts. I had a look at alt.binaries.schematics.electronic but I couldn't find anything attached. Please repost or mail me direct.

Cheers, wombat

They're mailed.

-- John