# Basic rules to calculate the equivalent resistance of a resistor circuit

I have a certain circuit only containing resistors of different values. There is one 'input' and one 'output' for the current. How do I calculate the equivalent resistance of the circuit? Are there any basic rules to follow?

If determining replacement value is the only goal then I can think of the following steps:

1) Analyse the circuit into the smallest solvable sub-circuits possible (series and parallel);

2) Calculate series resistors $R_S = R_1 + R_2$;

simulate this circuit – Schematic created using CircuitLab

3) Calculate parallel resistors: $R_P = \frac{1}{\frac{1}{R_3}+\frac{1}{R_4}}$

simulate this circuit

4) Apply wye-delta (Y-Δ) transform or reverse

5) Repeat until solved or run the circuit through a circuit simulator like SPICE.

# Wye-delta (Y-Δ) transform

simulate this circuit

## Y→Δ

$$R_{ab} = R_{an} + R_{bn} + \frac{ R_{an} \cdot R_{bn} }{ R_{cn} }$$

$$R_{ac} = R_{an} + R_{cn} + \frac{ R_{an} \cdot R_{cn} }{ R_{an} }$$

$$R_{bc} = R_{bn} + R_{cn} + \frac{ R_{bn} \cdot R_{cn} }{ R_{an} }$$

## Δ→Y

$$R_{an} = \frac{ R_{ab} \cdot R_{ac} }{ R_{ab} + R_{ac} + R_{bc} }$$

$$R_{bn} = \frac{ R_{ab} \cdot R_{bc} }{ R_{ab} + R_{ac} + R_{bc} }$$

$$R_{cn} = \frac{ R_{ac} \cdot R_{bc} }{ R_{ab} + R_{ac} + R_{bc} }$$

• This is a good answer, straight to the point. Perhaps clarifying with examples and an explanation of wye-delta would make it even better? :-) (I meant this to be easily understandable for beginners, and think they will yike seeing that wiki :-)) – Keelan Mar 14 '13 at 19:51
• Ooh, cool. Wish I could upvote you some more for that edit! – Keelan Mar 14 '13 at 20:54
• In Dutch it is called "ster-driehoek transformatie" (star-triangle transform). – jippie Mar 14 '13 at 22:15
• driehoek always makes me smile. Three-Corner! – stanri Mar 15 '13 at 0:41
• @StaceyAnne Oh, it's a horror, really. For every mathematical term we have a different one. Fault of this guy. Driehoek isn't that bad, triangular is basically three-corner as well. But why do we have to call mathematics wiskunde (science of that what's true)!? – Keelan Mar 15 '13 at 8:05