# System Identification (Linear Passive Circuit)

I'm need to find what's inside the following black box but I'm not sure how to exactly do this. simulate this circuit – Schematic created using CircuitLab

I'm thinking that I might use the circuit's bode plot as well as the step and impulse response to find it's transfer function. Does this sound right?

• Under the assumption that the black box contains an LTI system, only one of the three is required since the step response is the time integral of the impulse response and the transfer function is the Fourier transform of the impulse response. – Alfred Centauri Feb 16 '14 at 2:11

Assuming the Black Box contains a model that is: 1. time invariant (i.e. does not change over time); 2. is Linear (i.e superposition applies).

Then the Black Box can be modelled as an AR or ARMA model.

System ID techniques can be applied to this. Matlab is an excellent choice for this.

Theoretically Autocorrelation with a white noise input excitation is a popular technique. But you don't need to know the theory to get good results from the Matlab System ID Toolbox.

A Bode Plot would also be helpful. Technically, that would be a non-parametric System ID method. The AR/ARMA model method would be a Parametric System ID method.

UPDATE: As you've identified it as a linear passive circuit, then the following procedure might help:

1. Apply a DC input voltage. Measure the output voltage (open circuit) to confirm there are no sources inside.
2. Apply a DC input voltage. Measure the short circuit output current. You can now calculate the DC source resistance or Thevenin resistance. You should know about that from your course work.
3. Apply an AC source to the input and sweep the frequency. Measure the output AC magnitude (rms) and phase with respect to the input signal. Plot output/input to produce a Bode Plot (Magnitude and Phase).
4. Identify the poles and zeroes inside; i.e. the combined impedance of the source.

This is essentially what the ARMA model would have produced automatically.

• I'm a sophomore electrical engineer at the moment and I don't believe we've covered arma models or that autocorrelation thing. Is there some simpler way to do it? This is for some prelab work so we should be able to do it ourselves with our previous experience (AC theory and the beginnings of signals/systems). – codedude Feb 15 '14 at 21:16
• Oh wow...I should have thought of that. haha thanks for your answer. :) – codedude Feb 16 '14 at 14:51
• I just noticed it said to do this in the time and frequency domain. What you described seemed to the frequency domain method. What about the time domain? – codedude Feb 16 '14 at 22:05

IF the system is linear, then you are right, it can be characterized solely from its impulse response. It can be thought of as convolving the input by the impulse response.

Often the impulse response is difficult to meaure. One way around this is to measure the step response, which is usually easier, then differentiate that. That may have a lot of noise, but after some low pass filtering you can often get a reasonable impluse response that is impossible to measure directly. I go into a lot more detail, including a real world example, here: https://electronics.stackexchange.com/a/27861/4512