# transient analysis of linear system

Discussion in 'Electronic Design' started by wombat, Aug 17, 2006.

1. ### wombatGuest

Hi All,

First of all this is not homework it's just that linear systems and
transient circuit analysis hasn't been in the job description for a
while, actually ever.

R1 A R2 B R3
+---/\/\/\/\----+-----/\/\/\/\-----+-----/\/\/\/\-------+
| | | |
x(t) | | C1 | C2 | y(t)
===== ----- ----- =====
=== ----- ----- ===
| | | |
| | | |
+---------------+------------------+--------------------+
GND

Anyway the circuit is shown above. Clearly in steady state it's just a
voltage divider of the difference of Vx and Vy. The problem is that Vx
and Vy vary with time (out of my control). I need to report VA and VB to
the user but it must be the steady state result. In other words I must
filter out the transient effects caused by x and y. Please note that I
can't modify the circuit in any way. I know all the values for caps and
I can also measure *all* voltages. I even know the nominal values for
the resistors. The point of all this is to 'see' if the resistors change
through the "fog" caused the time varying sources.

My idea was to somehow use the system response [h(t)] to work out the
steady state result for A and B. Perhaps divide VA(t) by h(t) ????
eg in the case of VA:

x(t) --->| |
| h(t) |---> VA(t)
y(t) --->| |

I guess the first thing is, am I on the right track? Secondly I could do
with some tips on calculating h(t) at A and B.

I really appreciate any help.

2. ### John WoodgateGuest

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.

3. ### Stanislaw FlattoGuest

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

4. ### The PhantomGuest

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?

5. ### linnixGuest

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.

t -> infinite

6. ### wombatGuest

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.

7. ### wombatGuest

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.

8. ### The PhantomGuest

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?

9. ### Abstract DissonanceGuest

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

10. ### The PhantomGuest

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?

11. ### MarkGuest

Do you have PSPICE?

Mark

12. ### Jim ThompsonGuest

[snip]

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

13. ### Stanislaw FlattoGuest

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

Cheers

Stanislaw

14. ### John O'FlahertyGuest

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?

15. ### The PhantomGuest

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.

16. ### The PhantomGuest

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

17. ### wombatGuest

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

18. ### wombatGuest

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.

19. ### wombatGuest

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

20. ### wombatGuest

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