Sunday 6 December 2015

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?



Answer



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\$;


schematic


simulate this circuit – Schematic created using CircuitLab


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



schematic


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.



schematic


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


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


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


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


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