transient analysis of linear system

In message , dated Thu,

Can you not connect low-pass filters to points A and B and measure the outputs of the filters? The filters could be RC, LC or op-amp.

How slowly do Vx and Vy vary? There are digital methods that will average Va and Vb over hours, if you need to.

It would help enormously if you _DON'T_ define perfect components. NO R1;R2;R3 but Z1;Z2;Z3 and so on which are all functions of time. Now kick this mess with unit transient but don't expect steady state as result.

Have fun

Stanislaw Slack user from Ulladulla.

You need to give as much detail as you can for problems like this. For example, detail that you didn't give that would be helpful:

Tell us more about the nature of Vx and Vy. Are they sine waves (what frequency) that vary only slowly? Or, do they vary quickly? What varies, the frequency? The amplitude? If they aren't sine waves, what are they? What are their nominal characteristics and how do they vary? Are they steady most of the time with some variation only occasionally? Or, do they vary constantly?

How much are the resistors likely to vary? One percent? Ten percent? Quickly, or slowly? How quickly or slowly? What are the nominal values of the resistors and capacitors?

To what accuracy do you need to determine the change in the resistors? How quickly must you report the change in the resistors when they do change?

It could very well be a home work problem. Or perhaps a test for new job candidates.

You have 7 unknowns: Ir1, Ir2, Ir3, Ic1, Ic2, Va, Vb

And 7 Equations: Ir1 = (Vx - Va) / R1 Ir2 = (Va - Vb) / R2 Ir3 = (Vb - Vy) / R3 Ic1 = C1 (d Va / dt ) Ic2 = C2 (d Vb / dt ) Ir1 = Ir2 + Ic1 Ir2 = Ir3 + Ic2

Give me X(t) and Y(t) and solve them.

Steady state: t -> infinite

Thanks for your interest. I'll try to answer your questions but as far as I see it it's a mathematical issue so the nature of waveforms is largely irrelevant (within reason).

x and y vary but there is no waveform that can be associated with them. They are at the whim of nature. However they move relatively slowly, perhaps 50% of their nominal value in one minute. Sometimes they are essentially constant but I don't have the luxury of measuring only when they are constant. I must monitor them all, all of the time for a discrepancy event.

I need to pick up variance in resistance of +/-10% when the variance occurs for more than 5 secs. The resistors are ~2000G ohms (that's right giga) and the caps 2.2pF.

An accuracy of +/-2% of the nominal would be great. I can report the change up to 1 minute after the event.

Now that I look at it I guess one could consider it analogous to a series of strain gauges where the overall excitation voltage floats around and the caps represent stray capacitance to ground.

Rightyo R's to Z's. When you say unit transient do you mean a delta function? If so what do I do next? I have seen that theory applied mathematically using convolution but I need to convert the system (as seen from A and B) into the s domain for convolution, correct?

Cheers.

Are you saying that they are essentially random noise? Are they bipolar? That is, do they present both positive and negative polarities, with an average of zero? What is the maximum voltage they attain?

This is a system of differential equations. Now setting them up may or may not be easy depending on the method you choose. What you should know or realize is that the fourier or laplace transform will transform a linear system of DE's into a system of equations. By doing this you get away from having to solve the DE's but you'll probably have to look up the inversions in some big book.

The idea is very simple though. You just use the standard methods to write your equations down. Kirchoff laws or thevinen equivilents, etc... Then you write in the equivilent time dependent quantities for your components.

use the fact that the current through the caps is I = dQ/dt and the voltage across them is Q/C. By doing this and noting that Vx and Vy are independent functions you should arrive at a system of DE's in terms of two dependent quantities and they should be symmetrical(because your problem is symmetrical. Its very similar to the standard method of finding the currents in a steady state problem except now one treats each quantity as time dependent(except you probably will want to assume the resistances and capacitances are constant) and then solve your system by standard method. Its not hard but there is some algebra involved.

Again, the idea is to do the standard "math" on this type of problem and try to get down to a system that just involves two unknowns. You won't be able to do this without considering the fact that I = dQ/dt for each of the caps(so you have two extra equations to use to help simplify) and that Vx and Vy are independent.

i.e., you'll have, say, for the first loop,

Vx - I1*R1 - Qa/Ca = 0

but I1 = I2 + Ia where Ia is the current through the cap.

apon substitution we have

Vx - I2*R1 - Ia*R1 - Qa/Ca = 0

but we know Ia = dQa/dt so it is depend on Qa and we can consider it reduced. I2 though is not and we will need to use other equations on it.

so Vx - I2*R1 - dQa/dt*R1 - Qa/Ca = 0

by symmetry one has

Vy - I2*R3 - dQb/dt*R3 - Qb/Cb = 0

So in both of these equations we need some form of I2 that is simplified(only involving the constants, Vx, Vy, Qa, Qb, dQa/dt, and dQb/dt)

Once you find that then you have your two equations DE equations that can be solved for a solution.

i.e., all you have to do know is find an equation for I2 that you can plug into the above equations. I'll leave that to you though.

Jon

You should understand that since the resistors are time-varying, this is not a linear system in the traditional sense, and traditional methods of solving linear systems are not applicable (except for short times while the resistors are nearly constant).

Are Vx and Vy low impedance so that you might load them with another network without upsetting your existing circuit?

Are the voltages at points A and B converted (A-D, probably) so that they are available as numbers for number crunching, and, if so, how much computer power do you have available?

Do you have PSPICE?

Mark

Change to Laplace notation such that Xc = 1/(C*S)

Calculate y(S)

Partial fraction expand y(S)

Convert y(S) expansion terms to y(t)

Trivial ;-)

...Jim Thompson

Correct. In changing conditions time is honorable guest at the table. Don't insult him!

Cheers

Stanislaw Slack user from Ulladulla.

What if you could take the measurements of the voltage sources, and apply them either to a simulation program or to a circuit set up to represent the nominal values of your subject circuit, but with handier values- 2 Mohm and 2.2 nF, say, and then see at some interval how much the subject circuit deviates from the simulation/test circuit?

-- John

Is +-10% the maximum change in the resistors that will ever occur? And how fast will the change occur? If it's slow enough you may be able to treat the circuit as though its differential equations had constant coefficients for a few seconds at a time.

How would you calculate y(S), since R1, R2 and R3 are unknown functions of time?

Thanks again for the input Phantom. I appreciate the resistors are time varying but given that is what I am trying to detect can't we just work out how we _expect_ the circuit to behave (when R's aren't time varying) and then compare that to what is measured (when R's might be time varying)?

Vx and Vy are low impedance and can be loaded with external circuitry. In fact that was my first solution. I added a 'T' (R-C-R) circuits in parallel with R1 and R3 and then tuned them to compensate for the changes. This worked extremely well but I prefer the mathematical method as it simplifies fabrication of the system.

All voltages are available for number crunching. Computing power is relatively decent. I have DSPs, FPGAs and microcontrollers at my disposal.

wombat

That's a possibility (I can measure all voltages) but I want to avoid requiring a full blown PC for the calcs if I can. DSPs, FPGAs and microcontrollers are preferable for the computations due to size limitations.

Out of interest has anyone ever done that? Calculated a SPICE model out in real-time using real time-varying sources. I guess it's like having a piecewise source with the 'pieces' coming from real world measurements. Interesting.

Of course, I've been using it solidly for 2 weeks on this problem. How can it help me?

I can connect LPF's to A and B however I don't see the benefit. A and B are already 'filtered' by the nature of the arrangement of R's and C's. That's the problem! If there was no capacitance in the system it would be dead easy, just a voltage divider of the two sources at any instant. Needless to say the capacitance won't go away.

wombat