# Finding the schematic of a black box with unknown impedances

I'm struggling with trying to understand a piece of homework in Electrical Engineering that I'm really trying to solve on my own but now I'm basically stuck.

The assignment is to find the schematic for the black box given below:

The black box can contain either 2 or 3 components according to the assignment. All impedances are unknown. The assignment is to explain how one would measure and analyze the circuit in order to understand which components are present and what their values are.

It's also given in the assignment that the system is a first order system which consists of RLC components in some combination (as I interpret it this would mean that it's an RL or RC circuit, so only 2 components then I suppose?). But I have no idea where to go from here... The main problem is that I'm lacking resources that I can further study for this also. Any help is much appreciated.

How do you know the system is first order, was this given in the assignment? I ask because you said the box contained impedances, not any energy storage elements (capacitors or inductors). If you can confirm that it does not have any energy storage elements you can have a looked at the theory for two-port networks. The two-port theory can still apply if the circuit contains capacitors and inductors but the source would then have to be AC.

Basic two-port network is defined as:

The two port method to determine the impedance parameters are based on the following two equations:

We analyse one port at a time, if we open circuit the output port then I2 will be zero and we have solutions for Z11 and Z22.

Similarly if we open circuit the input port I1 will be zero and we have solutions for Z22 and Z12:

Whether the impedances are due to capacitors, inductors or just plain resistors can be determined from the phase difference between the resultant currents and the applied voltage. Capacitors will make the current lag behind the voltage, inductors will make the current lead the voltage and resistors will create no phase shift.

• Indeed, it was given in the assignment that the system is first order. Updated the question to clarify this. It was also mentioned that the system consists of RLC components. Added this to my question too... Sorry about the confusion. Apr 6 '15 at 12:05
• I flagged this as "not an answer" because it looks more like a comment than an answer, but I think you can be forgiven since you don't have enough rep to comment :) Apr 6 '15 at 12:11
• Lol, Yeah I can't comment, I'll try to update the answer when I have enough info about the problem Apr 6 '15 at 12:13
• @dwightreid "Impedance" suggests Resistances AND Reactances, and Reactance is from capacitors/inductors. Impedance is the combination of a resistance and a reactance, like R+jXl or R-jXc Apr 6 '15 at 12:46
• Holy crap - I came back, and the comment had bloomed into an answer! +1 from me. Apr 6 '15 at 14:53

I can give you a possible approach. For any two-port containing passive elements you can measure its admittance matrix Y, at a given frequency.

simulate this circuit – Schematic created using CircuitLab

This matrix Y is such that: $$\begin{bmatrix} i_1 \\ i_2 \end{bmatrix}= \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \cdot \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$$

Actually $Y_{12}=Y_{21}$ for two-port containing linear passive elements, and therefore you only need to perform 3 measurements to your "black box". You would then have 3 complex numbers.

Then you need to calculate the admittance matrix of your "target" two-port:

Its admittance matrix is rather easy to find out in terms of the admittances of the components, and you'll end up with: $$\begin{bmatrix} Y_{1}+Y_{2} & -Y_{2} \\ -Y_{2} & Y_{3}+Y_{2} \end{bmatrix}$$ where $Y_{1}$, $Y_{2}$ and $Y_3$ are the admittances of the components $Y_1=\frac 1 {Z_1}$, $Y_2=\frac 1 {Z_2}$, etc.

You now simply match the coefficients of this matrix with the values measured from the "black box" and solve for $Y_{1}$, $Y_{2}$ and $Y_{3}$. Finally $Z_1$, $Z_2$ and $Z_3$ are obtained doing the inverse.

The admittance matrix was chosen because of your particular "target" two port topology.