How do i find Ro when Ro is adjusted for maxmimum power transfer?

Question

How and what is Ro in this circuit when it is adjusted for maximum power? and what is its corresponding power?

The circuit: https://imgur.com/a/NvWeAZM

Answer 1

Make a thevin equivalent of the circuit including everything to the left of Ro and terminals a & b become the connection points for Ro. You will need only to find R_thevinin in order to find Ro for maximum power. Find R_thevinin by redrawing the circuit with the independent voltage sources shorted. Leave the dependent source in place but convert it to a voltage source. Apply a voltage across a&b of Vab and calculate the current in the thevinin loop. Solve the resulting algebraic equation for Vab/I and that will be the thevin resistance. Note that this value will be a function of the admittance gain of the independent current source (.5). I found the thevinin resistance to be 4.49 ohms.

Now, draw a thevinin loop containing a voltage source in series with 4.49 ohms and Ro.

Write an equation ... P=(Vth/(4.49+Ro))². Find the derivative dp/dRo and set = 0 to find the maximum of P. you will find that the maximum power occurs when Ro =4.49 ohms. This is the explanation of why the load impedance is matched to the source impedance for maximum power transfer.

To find the value of P when 4.49 ohms is connected, find the thevin voltage of the circuit without Ro connected using superposition and then calculate P. You will find the use of a wye-delta conversion to be helpful to find the thevin voltage.