# Non-trivial equivalent resistance. At school we have been learning the rules for equivalent resistance in series and parallel (which I covered at least two years ago, so I'm bored) and I asked my teacher about the circuit shown, which cannot be decomposed into a collection of series and parallel circuits.

I spent most of a lesson just bashing it with algebra, and in the end I found a three-line fraction for $R_{eq}$ in terms of $R_{1}$, $R_{2}$, $R_{3}$, $R_{4}$ and $R_{5}$.

So I was wondering: is there a nice method? Note by Sophie Crane
5 years, 2 months ago

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I think delta - star method can help .

- 5 years, 2 months ago

Can you give me an explanation/link to an explanation? Thanks

- 5 years, 2 months ago

@Sophie Crane It's trivial using Y-$\Delta$ transformation. Just apply this on any of the ends of $\text{R}_{5}$.

- 5 years, 1 month ago

Thank you very much. This is an excellent technique and I had never heard of it before. Awesome! :D

- 5 years, 1 month ago

If u use loop rule instead of junction rule, the expression would have been simpler. For making it even simpler u can assume the circuit to be connected across a good emf(I mean u can take different emfs if values of the resistances are known)

- 5 years, 2 months ago

I used both. The emf is an arbitrary value. All resistances are unknown, and are to be treated as algebraic variables. I am trying to find a general expression for the equivalent resistance in terms of these variables.

- 5 years, 2 months ago

That would be a very big formula

- 5 years, 2 months ago

It is. That's why I posted this.

- 5 years, 2 months ago

But u can still assume any value for emf as that wouldn't change the equivalent resistance.

- 5 years, 2 months ago

Correct me if I'm wrong, but isn't this the outline of a Wheatstone bridge? If $\dfrac{R_2}{R_4}=\dfrac{R_1}{R_3}$, then the equivalent can be easily calculated since the resistance $R_5$ will be ineffective. Is that condition given, or the variables $R_1,R_2,R_3,R_4,R_5$ can have any positive real value?

- 5 years, 1 month ago

You are correct on every count. The condition is that the resistors can take on any positive real value.

- 5 years, 1 month ago

I got a fraction of 8 terms over 4 terms

- 5 years, 1 month ago

Did you use delta-wye or something else?

- 5 years, 1 month ago