Finding the Thevenin equivalent Resistance for circuit with voltage-dependent sources

Question

A linear circuit containing two voltage dependent sources, as shown. The elements in this circuit have the following values: \$R_1 = 7.2k\$ Ohms, \$R_2 = 12k\$ Ohms, \$R_3 = 3.3k\$ Ohms, \$\alpha = 0.025\$ A/V and \$\beta = 5\$.

Calculate the numerical value for the Thevenin equivalent resistance for terminal A-B.

I am having a hard time solving this. Letting \$i\$ be the current flowing into \$R_3\$, \$i_1\$ the current flowing through \$R_1\$ and \$i_2\$ the current through \$R_1\$—and the node between R3 and R1 be e_1;

Using KVL and KCL, I've managed to find that

\$i + I = i_1 + i_2\$ Hence \$\frac{e_1-\beta v_2}{R_3} + \alpha v_1 = \frac{v_2}{R_2} + \frac{v_1}{R_1}\$

also from one of the loops around V, R3 and R2, I have \$e_1 - 2\beta v_2 + v_2 = 0\$.

I have 3 unknowns so far, so I'm trying to find another equation, but I'm not sure what to do from here.

Answer 1

Homework - no ready to copy solutions will be written, but some guidance.

Connect an ideal voltage source Us to AB and calculate how much current your circuit takes from it. You'll see that the current taken from Us is (QCF) x Us, where (QCF) is a quite complex formula which contains alpha, beta and the resistances R1, R2, R3.

This means that your circuit behaves like a resistor 1/(QCF). That's the Thevenin equivalent resistance. The Thevenin equivalent voltage of your circuit is zero.

This circuit has controlled sources. You cannot use solution methods which remove any of the controlled sources or any of the branches which control the sources. Make straight Kirchoff's current law equation where v2 is the unknown voltage. You can express v1 with Us and v2.

Answer 2

To find thevenin equivalent resistance just short the voltage source and open the current source. You will Get Rth=R3||R2+R1