Thevenin Equivalent Resistance of complex circuit

I'm trying to find the Thevenin equivalent of a circuit of which I have already found the Thevenin voltage. Unfortunately I can't figure out how these series parallel combinations work to find the Thevenin resistance. Any help is appreciated. A numerical solution is not required so the values of the resistors are not given.

• Pedantic, but you probably don't mean numerical. Numerical solutions are (and I'm drastically oversimplifying here) guess-and-check schemes to solve a problem with numbers, such as Newton-Raphson iteration. – esilk Jul 11 '18 at 20:57

This one is a good exercise for the extra-element theorem or EET which is part of the fast analytical circuits techniques or FACTs. The cool thing with this approach is that you can break a quite complex arrangement such as yours in a succession of small individual sketches all independent from each other. Should you make a mistake, it is easy to solve the guilty drawing and fix the whole thing

With the EET, you identify an element in the circuit which bothers you when determining the resistance $R_{th}$ (or any other transfer function - yes, a resistance or an impedance is a transfer function) in your network. Here, $R_4$ is causing problems and I will set it to infinity (remove it) for the first approach. In this mode, I will calculate the reference resistance or $R_{ref}$ when $R_4$ is gone. Then, I will determine the resistance offered by $R_4$ terminals when the excitation $I_T$ is 0 A or the response $V_T$ is 0 V. When you want to determine a resistance or an impedance across some connections, you connect a test current source $I_T$ (the stimulus) which produces a response $V_T$ across the terminals. The resistance is simply $R=\frac{V_T}{I_T}$.

I have arranged all the steps in the below drawings with a small intermediate EET in between. Nothing insurmountable, I did not write a single line of algebra, just did inspect the schematics:

Then, a quick dc point analysis with SPICE tells you if what you computed with Mathcad is ok or not:

The results are gathered here and confirm the dc points analysis. The expression describing $R_{th}$ is quite ugly : )

• Thank you so much, this really helped a lot! I'm actually creating a lab manual with one of my professors so this was a circuit I had made myself, so when I couldn't solve it myself I was very confused. Clearly I made it way to complicated without even realizing! – Scott Sutherland Jul 12 '18 at 13:42
• No problem, do not forget to validate the answer then : ) These FACTs are extremely powerful and should be introduced to students once they master classical KVL-KCL analyses. Good luck with this manual! – Verbal Kint Jul 12 '18 at 14:09

There are no series or parallel resistors in this circuit. You need to find the short circuit current at the output, then calculate $$R_{TH} = \frac{V_{OC}}{I_{SC}}$$

• Or use Y-delta transformations. – John D Jul 11 '18 at 20:52

For your given circuit (I'm assuming you've properly removed all independent sources), apply a test voltage Vt at the input indicated by Rth. Solve symbolically, with a method such as nodal analysis, to determine the current flowing out of your test voltage (It). Your Thevenin resistance will be given by Vt/It.

Alternatively, use a test current and solve for the voltage it develops across its nodes.