Re: transient analysis of linear system

On Fri, 18 Aug 2006 09:17:47 +0200, wombat
<wombat@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

The Phantom wrote:

On Fri, 18 Aug 2006 01:39:10 +0200, wombat
<wombat@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:


The Phantom wrote:

On Thu, 17 Aug 2006 21:50:18 +0200, wombat
<wombat@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:



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.


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?


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).


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).


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.


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?


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.



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)?

That's why I asked you what the maximum expected change in the resistors
is. You said you needed to "...pick up variance in resistance of +/-10%
when the variance occurs for more than 5 secs.", but you haven't yet told
me what the max expected change is.

I'm thinking that if the change isn't too much and doesn't happen too
rapidly, you may be able to treat the system as though it *were* linear
without too much error.


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.

In another post, I asked you:

" Are you saying that they (Vx and Vy) 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?"


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

wombat

.

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