Finding missing current and voltage in a circuit involving superposition

Question

We are given a black box that contains only linear circuit elements and a pair of ports. We conduct the following two experiments with this black box.

1. With the right port open, we applied V1 = 2V to the left port and measured I1 = 4mA and V2 = 8V.

2. With the left port shorted, we applied I2 = -10 mA to the right port and measured I1 = 5mA and V2 = -10V.

Now we connect a current source to the right port and a voltage source to the left port. I1 = 8mA and V2 = 4V now. Find I2 and V1, in mA and V each.

I don't get this. Since we have analyzed the circuit separately for each source, adding them should give me the final values of I1, V2, I2, V1 etc. However, that's not quite true—from cases 1 and 2, I should have I1 = 4 + 5 = 9 mA, but it's 8mA. Hence, I thought that the black box consumes 1mA; hence I2 = 0 + (-10) - 1 = -11 mA. Similarly, V2 = 4V even though adding the V2 from first two cases gives -2; hence, the black box provides an additional 6V, and so V1 = 2 + 0 + 6 = 8V. However, this approach is wrong.

How might I solve this?

Answer 1

from cases 1 and 2, I should have I1 = 4 + 5 = 9 mA, but it's 8mA.

4 mA + 5 mA is what you would get if you applied 2 V on the left and -10 mA on the right.

But that isn't what you were asked. You were asked what happens if you apply 4 V on the left and 8 mA on the right.

So, to use superposition, first use test result 1 to find out what happens if you apply 4 V on the left (with the right port open). Then use test result 2 to find out what happens if you apply 8 mA on the left. Then sum those results to get your final answer.

Hence, I thought that the black box consumes 1mA;

The black box contains only linear elements. Therefore it can't "consume" any current.

Edit

OK, let me spell things out.

When they say the system is linear, it means you can write these equations for the unknown variables

$$I_1 = G_{11} V_1 + A_{12} I_2$$ $$V_2 = A_{21} V_1 + R_{22} I_2$$

where \$G_{11}\$, \$A_{12}\$, \$A_{21}\$ and \$R_{22}\$ are characteristic parameters of the black box that you need to discover. The names of the parameters are arbitrary, but you should think about why I might have named them what I did.

Your two test measurements give you enough information to determine the unknown parameters.

Then you can plug in any \$V_1\$ and \$I_2\$ you like (for example, the ones you were asked to solve for), and find the resulting \$I_1\$ and \$V_2\$ (for example, the result you were asked to find).