How can I solve for the Thevenin resistance in this circuit?

[Novark]

I'm doing some practice problems, and I can't seem to figure out where to even begin with this one.

I've tried solving for R_{T} (Thevenin resistance) by removing the 9V source and adding a 1V test source across the load, and I've also tried solving for R{_T} by leaving the 9V source in the circuit.

If someone could walk me through the steps, I would greatly appreciate it!

 

[berkeman]

I'm doing some practice problems, and I can't seem to figure out where to even begin with this one.

I've tried solving for R_{T} (Thevenin resistance) by removing the 9V source and adding a 1V test source across the load, and I've also tried solving for R{_T} by leaving the 9V source in the circuit.

If someone could walk me through the steps, I would greatly appreciate it!

Start with either KVL or KCL equations. What is the power in RL? How does varying the value of RL affect the power equation? What's the power for a small value of RL like 1 Ohm? What about 1k Ohm?

What technique in calculus is used to maximize something?

 

[willem2]

I've tried solving for R_{T} (Thevenin resistance) by removing the 9V source and adding a 1V test source across the load, and I've also tried solving for R{_T} by leaving the 9V source in the circuit.

You can find the thevenin resistance by finding the open circuit voltage and the short circuit current.
The tevenin resistance is (open circuit voltage)/(short circuit current)

Finding these can IMO most easily be done by finding an equation for the potential of the point between the 3 resistances with KCL.

 

[hadiffina]

To find Rth :

short out 9v voltage supply and open RL resistor
the independent 3Vx source remains.
Introduce voltage source Vth from RL terminal
and Isc flowing into the 4ohm resistor from Vth

using KVL with mesh currents both are clockwise,
The below equation is derived :

<< complete solution deleted by berkeman >>

 

[berkeman]

To find Rth :

short out 9v voltage supply and open RL resistor
the independent 3Vx source remains.
Introduce voltage source Vth from RL terminal
and Isc flowing into the 4ohm resistor from Vth

using KVL with mesh currents both are clockwise,
The below equation is derived :

<< complete solution deleted by berkeman >>

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