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I have a "circuit," that's not really an electrical circuit but rather a physical process that follows the same functional form as an RLC circuit with regards to how it responds to excitations--basically, it's a second-order linear differential system. However, I have no real way of measuring "R," "L," or "C" except by looking at how the "circuit" responds to stimulus. I can, however, modify R, L, or C in a relative way (i.e., make whatever C is 1.5x bigger) by making changes to the system design.

Every day, the system is excited by a step function. It responds in a manner very similar to an "under-damped" RLC circuit--that is, it greatly over-shoots, then oscillates around the step around once and hour, slowly settling down after about a day. I know this, because I measure the fluctuations in the levels of my system over time.

My question is: Given that it isn't possible or practical to measure R, L, or C (and hence I can't calculate damping factor), but it is possible to make modifications to the relative ratios R, L, and C to each other, is there any way for me to analytically determine how to modify my system to make it critically damped, based on the measurements I can make on the system as it exists now?

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    \$\begingroup\$ It is perfectly possible to deduce the values from the step response. Just write down the equation and put the measured values in. Or easier - simulate the equation adjusting the values until getting a waveform similar to the observed. \$\endgroup\$
    – Eugene Sh.
    Commented Mar 23, 2016 at 18:52
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    \$\begingroup\$ You can estimate the damping factor graphically. Any text covering 2nd order systems should be able to help. But having the damping factor will only tell you the relationship between whatever you are calling R, L, C -> z = 0.5*R*sqrt(C/L) \$\endgroup\$
    – Spehro 'speff' Pefhany
    Commented Mar 23, 2016 at 18:53
  • \$\begingroup\$ @EugeneSh. I'm really trying to keep away from "adjust it until it looks right." What I want is an analytical formula where I stick numbers in and get numbers out. That said, I'm not getting much in the way of answers through Google--what "measured values" would I put in? \$\endgroup\$
    – Frank
    Commented Mar 23, 2016 at 19:28
  • \$\begingroup\$ @SpehroPefhany I know whatever calculation I make won't give me exact values for R, L, or C. What I want is something like "current damping factor is 0.25, so I can multiply R by 4, C by 16, or L by 1/16 to achieve critical damping." Adjusting any of the parameters has cost associated with it, so the exact adjustment I make will also depend on minimizing those costs. \$\endgroup\$
    – Frank
    Commented Mar 23, 2016 at 19:31
  • \$\begingroup\$ Here you can get different measurable quantities and equations to plug them in. \$\endgroup\$
    – Eugene Sh.
    Commented Mar 23, 2016 at 19:33

2 Answers 2

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is there any way for me to analytically determine how to modify my system to make it critically damped, based on the measurements I can make on the system as it exists now?

enter image description here

You appear to be able to measure the damped resonant frequency (\$\omega_d\$ = red above). This is the imaginary part of the complex value for \$\omega_n\$ and you can ascertain zeta (\$\zeta\$) from how the ringing decays.

From this you can determine the natural resonant frequency using this formula: -

\$\omega_d = \omega_n\sqrt{1-\zeta^2}\$

I believe that is all you need to know.

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With your physical measurements, you have two knowns, the observed frequency, which will be close to the natural frequency, and the rate at which the oscillation decays, related to the Q. The easiest way to record the latter is the reduction in peak amplitude for each successive swing.

Although you think you have three unknowns, R, L and C, you only really need two at this stage, because the value of any one of these is unknown to a scaling factor, related to the impedance of the circuit. Your two unknowns can be any two ratios, conveniently L/R and C/R. You could invert the equations for frequency and damping and solve for the two ratios, or just plug those ratios into a simulator and adjust until a transient response reproduces the observed frequency and rate of decay. I know which I'd do!

If you want to nail the impedance as well, you can make a 2nd measurement adding (say) 1kg extra mass to the C, repeat the parameter extraction, and calibrate what C means in terms of mass (in the case it's a mass spring system, you don't tell us), and therefore the other parameters.

A version of this was very popular for audio hobbyists building their own loudspeaker enclosures. To extract the loudspeaker parameters, first measure their natural resonant frequency and the damping, then add the extra mass of a bit of plasticine or blu-tak to middle, and repeat.

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  • \$\begingroup\$ Technically the natural resonant frequency is not known because it is a damped system. It can be calculated once zeta is determined however. \$\endgroup\$
    – Andy aka
    Commented Mar 23, 2016 at 21:58
  • \$\begingroup\$ @Andyaka well spotted, I did write that wondering whether anybody would. With the level of damping the OP describes, the observed and natural are very close. The distinction is irrelevant if you fit via a simulator, and if you understand the equations well enough to invert them, then you're ahead of the game anyway. I wonder what the system is? An IN system is 42 minutes period IIRC? \$\endgroup\$
    – Neil_UK
    Commented Mar 24, 2016 at 4:54
  • \$\begingroup\$ Maybe @Frank can explain what the system is but probably the best solution is to find zeta via model tweaking an RLC circuit then deriving the natural frequency. \$\endgroup\$
    – Andy aka
    Commented Mar 24, 2016 at 8:11
  • \$\begingroup\$ I didn't want to get too far into describing the system (because it's pretty complicated and would seem to distract from my question), but it's a thermal transfer system where the heat flux entering the system is dependent on the integral of the temperature. The heat capacity of the system is the C component, while the time-varying heat transfer rate gives the R and the L components. The resulting damped resonant period is measured in hours, which makes it hard to "experiment with it until it looks right," especially considering I have a few dozen of these things. \$\endgroup\$
    – Frank
    Commented Mar 24, 2016 at 12:16
  • \$\begingroup\$ How does a thermal system oscillate, unless it's got a marginally stable control loop closed round it? If so, a better approach would be to open the loop and measure the transfer function, then you could design for closed loop stability/damping in one go. \$\endgroup\$
    – Neil_UK
    Commented Mar 24, 2016 at 12:22

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