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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.

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  • \$\begingroup\$ 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. \$\endgroup\$ – esilk Jul 11 '18 at 20:57
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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:

enter image description here

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

enter image description here

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

enter image description here

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  • \$\begingroup\$ 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! \$\endgroup\$ – Scott Sutherland Jul 12 '18 at 13:42
  • \$\begingroup\$ 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! \$\endgroup\$ – Verbal Kint Jul 12 '18 at 14:09
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There are no series or parallel resistors in this circuit. You need to find the short circuit current at the output, then calculate $$R_{TH} = \frac{V_{OC}}{I_{SC}}$$

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

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