transient analysis of linear system

I don't think LPF's will help. The reason being is that when the resistors change (this was explained in a previous post 5560128 - not sure if that number is of any use, found it in the header) they change approx +/- 10% of their value for only 10 secs. I think the kind of filtering you are suggesting would attenuate it too much and I would miss the event.

However recent postings (cheers John O'Flaherty) have made me think of the whole thing in a slightly different way... If I can calculate what A and B should be by using the nominal values in a mathematical model together with the measurements from the sources then I can just compare the expected A or B with the measured A or B.

Ok I've just simulated it with SPICE. Two identical circuits driven by the same sources except the 'real-world' circuit has an altered resistor. Obviously the waveforms vary. The hard bit for me (being mathematically challenged) is to run the model in real-time on a device. Any suggestions? I guess I could use SPICE algorithms that use the measured source voltages as a pseudo piece-wise source???

wombat

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.

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?"

Try the physical component simulation. You can claim you are using a RISC analog computer to generate a comparison standard.

-- John

Ok, the expected resistance change could be 100%. The resistance is unlikely increase much (unless wire is cut) but could drop close to 0. However as I said I need to detect when in drops by %10 of it's value. Usually this 'event' where the it drops by 10% only occurs for 5-10secs. I need to detect this event, that is the purpose of the gadget.

Vx and Vy are not random noise as such but I can't predict them, I can only measure them. Vx should always be at least 20% greater than Vy to ensure there is current flow through the system. For arguments sake let's say that Vx is 100kV which ramps up to 150kV and back down again over the course of 2 mins (not necessarily a nice linear ramp though). Vy tries to keep itself around 20kV below Vx.

wombat

RE: "Try the physical component simulation"

Very funny. I wish it was that easy. A few 0402 parts hidden under a black plastic square and I make the claim... problem is that this isn't even an electrical system, just one that can be modelled by one.

I think the solution will be something like: A numerical SPICE style solver that digest the measurements of Vx and Vy and spits out the expected voltage of VA and VB in real-time. That result is compared with the measured values of VA and VB. I probably only need to calculate every half a second to successfully catch the event so it's probably doable with a decent micro. Thoughts?

wombat

they

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You might set it up as a state variable system. The state variables are the variables that represent the current state of the system, that is, the capacitor voltages and inductor currents. Since the form of your system is fixed, you only need to figure out the form of its representation once. The current state of the system, the inputs, and the system form are represented as matrices, and you can calculate the next state of the system by fixed procedures. This is a time-domain procedure. There is a chapter of a textbook available free on the net that describes the procedure. I can't be of further help because I haven't studied the chapter yet!

Engineering Circuit Analysis | Chapter 19 State-Variable Analysis

-- John

Not even close.

Your system is undetermined. The problem statement is to predict a new steady state for Va and Vb as a function of R1,2,3. These resistors are on the order of 2e15 ohms and the capacitors are on the order of 2e-12 for a time constant of 4e3, or thousands of seconds, and this holds for relatively minor +/-10% change in R. Then Vx and Vy exhibit a drift characteristic on the order of hundreds of seconds. You can get an idea of what happens by thinking of C1 and C2 as DC sources, batteries, of magnitude steady state Va and Vb. As the resistor fluctuate at a rate nearly instantaneous relative to the circuit time constants, all voltages remain unchanged, and charge will be circulated through the resistors to maintain those node voltages constant. Looks like you have everything wrong, attempting to measuring a circuit parameter that nature is forcing to be constant, meaning you have to measure *current* to detect the resistor changes, the voltage measurements will barely move by ppm and be undiscernible from drift. And what does this have to do with your original ill-posed resistor network that was another failed identification problem? You're a starting to look like a big waste of time.

If R1, R2 and R3 are *unknown* functions of time, then how can you solve it period?

Unless you're just asking for an generalized differential equation?

If R1, R2 and R3 are *known* functions of time then R1(S), R2(S) and R3(S) exist.

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC\'s and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
|       http://www.analog-innovations.com           |    1962     |
             
I love to cook with wine.      Sometimes I even put it in the food.

they

of

Thanks for that John. Ok I've had a read. Lets assume I could get my head around all that and develop a state-space model for the system. The problem as I see it that I can't describe the inputs with a mathematical equation. They are measured values that change due to the influence of the quasi-random environment. This is why I'm thinking that some sort of iterative numerical method (like SPICE) might be the way to go.

You mention determining the next state of the system by a fixed procedure. I didn't see anything on that. The inputs were all 'cos' this or 'sin' that...

[snip]

Is it the value of the resistors that you are trying to determine, since you said "I can also measure *all* voltages"?

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC\'s and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
|       http://www.analog-innovations.com           |    1962     |
             
I love to cook with wine.      Sometimes I even put it in the food.

At 1000G-ohm source impedance and pf shunt? I don't think so-do you believe this clueless ME has a hint in hell of what he's trying to do or knows what is even close to realizable? Bwahahaha- as you say.

My original post was probably a little premature and therefore misleading with regard to the problem statement so I'll clarify.

"I need to know if the resistors change by more than 10% while having to contend with the sources of x and y moving up and down."

Regarding the time constant, I have modelled the circuit. As an example: When R2 decreases it's resistance by 10% (to 1800G) point B changes to it's maximum voltage (however only 0.4% change) in under 8 secs.

Unfortunately the current through the resistors can't be measured so I have to rely on voltage measurement. It sounds pretty extreme, 0.4% accuracy is hard to come by but if I measure differentially across the resistor it equates to ~5% change - definitely achievable.

The previous problem is related but my methodology changed when I realised I couldn't do it that way. It wasn't solvable.

wombat

That's correct I am trying to determine the R's. My original post was misleading because I was thinking that if I could somehow 'convert' the measured voltages at A,B using some sort of inverse system I would arrive at a steady state version of A,B. With that knowledge I could calculate the R's. My mistake.

I haven't been able to make much sense as to what he's trying to accomplish, though I've re-read the whole thread :-(

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC\'s and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
|       http://www.analog-innovations.com           |    1962     |
             
I love to cook with wine.      Sometimes I even put it in the food.

[snip]

Are the R values very high as Bloggs implied?

I haven't found where you stated the R and C ranges. Can you re-state?

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC\'s and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
|       http://www.analog-innovations.com           |    1962     |
             
I love to cook with wine.      Sometimes I even put it in the food.

That is exactly the question the OP would like to have answered.

The OP has clearly said that they vary with time and he would like to detect when they vary and by how much. This means that transform methods aren't applicable, or at best, only approximately applicable.

I'm still not sure what your after but if I interpret what you said above correctly then all you want to do is determine the change in R?

This is quite easy if you can measure the sources and have some standard value of R to compare to. All you have to do is measure the voltage across R and the voltage sources and compare it to what would theoretically be expected with the standard value that R is suppose to be.

i.e., You use the "theoretical" values of the resistanaces and capacitances and the experimental voltages sources and then the experimental values for all and then compare for differences.

If your circuit is fixed and you can solve it algebraically then all you have to do is "plug in" the measured values and the theoretical values and compare. If you can't solve the circuit algebraically then you will have to do numerics on it to get the results.

You could do more advanced mathematics(stochastic DE's and such) but I think for your problem it would be quite easy since you can measure the voltage of the voltage source. (else it would be impossible for an arbitrary voltage source because you can't tell if the extra voltage drop on the resistors are coming from the change in resistance or from the voltage source)

for example, R +---/\\/\\/\\/\\----+ | |

If, say, you are trying to figure out the change in R then its quite easy.

Vx - I*R - Q/C = 0

==>

I = dQ/dt

so

Vx - dQ/dt*R - Q/C = 0

and Q(t) = exp(-t/RC)*(Q0 + int(Vx(s)*exp(s/RC)/R,s=0..t))

but VR(t) = dQ/dt*R =

so the voltage drop across the resistor is given by

VR(t) = Vx(t) - exp(-t/RC)/RC*int(Vx(s)*exp(s/RC),s=0..t) - VC0*exp(-t/RC)

But we can measure VR(t), Vx(t), and VC0 and hopefully we know the theoretical value for C(else its more complicated and probably impossible to measure) then we can calculate the value R for this(numerically).

i.e., say the theoretical value for R is 10ohms. Then we can measure and plug into the equation above and test for different R's until we get a true statement. This R then can be compared with 10 ohms to see how much it varied to a "hidden" variable.

If, say, C = 1, VC0 = 0, and Vx(t) = t when we start measuring(So we will have to sample the voltage sourc) and suppose that we measure the voltage across the resistor at the end of 1 sec to be 1/2V

then

1/2 = 1 - exp(-1/R)*int(s*exp(s/R),s=0..1)

we get the equation

1/2 = 1 - R^2*exp(-1/R) - R^2 + R

We can solve this numerically to find out what R has to be to produce those measurements: This equation has solutions at about 1.065 ohms.

So one would have a change in about 9 ohms.

Anyways, Thats just an example and the measurements are made up for the purpose of demonstrating what you could do. It might not be the best way to do this though depending on the circuit and such. If you could measure the current then it would be hell of a lot easier as you would only need to measure the voltage and current of the resistor you want to measure and then compute its value using ohms law... and it would be correct regardless of the circuit topology(for the most part).

The above method may not work well though since its possible to have multiple solutions and there might be stability issues in trying to implement it.

Ultimately it would probably be much easier to just measure the current along with the voltage. You can also do the same to measure the capacitance since V = Q/C and I = dQ/dt but you will need to sample the current enough to build a history for the integration.

Ofcourse if this was some type of research then you would be using much more complication mathematics and physics to get the answers but I doubt you want to spend the next 10 years on that.

Hopefully I'm on the right track with what you want to do. If so then one thing you have to realize is that there are probably thousands of factors that can change the resistance and can complicate matters. Even the simple RC circuit above is quite complicated in this aspect if both the resistance and capacitance can change. If you don't know what is making them change and cannot model that sufficiently then it can be very difficult to measure there change by limiting yourself to measuring only one aspect of them(such as there voltage). The reason is simple. What we measure as a change of resistance my actually be from a change of capacitance and vice versa. So you would have to be vary careful not to allow this type of situation to occur in your method. Also, say, assuming the capacitance is constant when it is not could contribute. So, in this problem is not only what you measure but what you don't. This is not to say that there might be simplifications and approximations that could make the problem much easier to deal with. I'm just not completely clear on what you are trying to do.

Anyways,

Jon

He said the resistors are on the order of 2000 G ohms, Fred. That's 2e12 ohms, giving a time constant of 4 seconds, not 4000 seconds.

Yes they are. The R's are ~2000G and the C's are ~2pF. They're not really electrical bits so I know I can't by them at Digikey.