Network analysis (electrical circuits)

Analysis of Resistive Circuits

Equivalence

Two electric circuits are said to be equivalent with respect to a pair of terminals if the voltages across the terminals and currents through the terminals are identical for both networks.

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If $V_{1}=V_{2}$ and $I_{1}=I_{2}$, then with respect to terminals ab and xy , circuit 1 and circuit 2 are equivalent.

Resistors in Series and in Parallel

Resistors in Series: $R_{e}q=R_{1}+R_{2}+...+R_{n}$

Resistors in Parallel: $\left({\frac {1}{R_{eq}}}\right)=\left({\frac {1}{R_{1}}}\right)+\left({\frac {1}{R_{2}}}\right)+...+\left({\frac {1}{R_{n}}}\right)$

Special Case: Two resistors in parallel: $R_{eq}=\left({\frac {R_{1}R_{2}}{R_{1}+R_{2}}}\right)$

Delta-Wye Transformation

The transformation is used to establish equivalence for networks with 3 terminals.

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For equivalence, the resistance between any pair of terminals must be the same for both networks.

Delta-to-Wye Transformation Equations

$R_{1}=\left({\frac {R_{a}R_{b}}{R_{a}+R_{b}+R_{c}}}\right)$

$R_{2}=\left({\frac {R_{b}R_{c}}{R_{a}+R_{b}+R_{c}}}\right)$

$R_{3}=\left({\frac {R_{c}R_{a}}{R_{a}+R_{b}+R_{c}}}\right)$

Wye-to-Delta Transformation Equations

$R_{a}=\left({\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2}}}\right)$

$R_{b}=\left({\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3}}}\right)$

$R_{c}=\left({\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}}}\right)$