I need help on how to calculate resistances of this circuit

[Vex]

Hello Guys,

The circuit is shown in the attachment. The known values are resistances R1, R2, R3, R4, voltages U1, U3, U2, and current can be measured on A1 and A2.
The switches can be controlled (all closed, all open or one closed, one open). The resistances R5, R6 are unknown.

Can we somehow calculate these resistances with this known values and controlling the switches, and if we can, what are the equations for calculating it?

Thank you.

[ArthurDent]

Ah, looks like another homework assignment problem. First, with the switches open you can state a general equation for the current in the circuit circuit U1/R5/R6 as IR5=U1/(R5+R6). Then close SW1 and knowing the current added (A1) through R5 start getting values. Do the work.

[kulky64]

1. SW1=open, SW2=closed -> measure current I_A2
2. SW1=closed, SW2=open -> measure current I_A1
3. R6 = (U1*(U3-U2-U1+(R1+R3)*I_A1-(R2+R4)*I_A2)) / (U3*I_A2-(U1+U2+(R2+R4-R1-R3)*I_A2)*I_A1)
    R5 = -R6*(U1+U2+(R2+R4)*I_A2) / ((R2+R4+R6)*I_A2+U2)
Verify with simulation if it is correct.

[Vex]

Thank you, i did the simulation and the results are correct!
If you have time, can you please explain in a little more detail how did you get this equations?

Thank you.

[kulky64]

Let's consider first the case when SW1=open and SW2=closed. Leftmost branch of this circuit is completely dead in this case and A1 reads zero. Now you have effectively only two sources in the circuit and A2 shows some current. Derive the expression for this current using your preferred method (I used superposition principle):
I_A2 = f(R1, R2, R3, R4, R5, R6, U1, U2, U3)
Well, R1, R3 and U3 will not appear in this function for obvious reason, but I left it as most general case.
Then consider the case when SW1=closed and SW2=open. Now rightmost branch is completely dead and A2 reads zero. You have again effectively only two sources in this circuit and now derive equation for current through A1:
I_A1 = f(R1, R2, R3, R4, R5, R6, U1, U2, U3)
Now R2, R4 and U2 are irrelevant a will not appear in this equation.

You have now system of two equations with only two unknown values of R5 and R6, because you measure I_A2 and I_A1 and the rest is given. Solve this system of equations and done.