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transient analysis of linear system

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

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  1. wombat

    wombat Guest

    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. 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. 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.
     
  4. The Phantom

    The Phantom Guest

    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. linnix

    linnix Guest

    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
     
  6. wombat

    wombat Guest

    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.
     
  7. wombat

    wombat Guest

    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 Phantom

    The Phantom Guest

    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. 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 Phantom

    The Phantom Guest

    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. Mark

    Mark Guest

    Do you have PSPICE?

    Mark
     
  12. Jim Thompson

    Jim Thompson Guest

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

    Cheers

    Stanislaw
    Slack user from Ulladulla.
     
  14. 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 Phantom

    The Phantom Guest

    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 Phantom

    The Phantom Guest

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

    wombat Guest

    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. wombat

    wombat Guest

    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. wombat

    wombat Guest

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

    wombat Guest


    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
     
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