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A series-parallel circuit is formed by connecting a set of parallel resistors in series.

- Connect \(R_1\) and \(R_2\) in parallel to obtain an effective resistance of \(R_{P1}\).
- Similarly, connect \(R_3\) and \(R_4\) in parallel to get an effective resistance of \(R_{P2}\).
- These parallel segments of resistors are then joined in series.

*Series-parallel combination of resistors*

The formula for the effective resistance of the parallel combination of resistors is

$\frac{1}{{R}_{P}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{2}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{3}}$

For two resistors in the circuit, the effective resistance is given as

$\begin{array}{l}\frac{1}{{R}_{\mathit{P1}}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{2}}\\ \\ \frac{1}{{R}_{\mathit{P2}}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{2}}\end{array}$

Using the effective resistance of the series circuit, ${R}_{S}=\phantom{\rule{0.147em}{0ex}}{R}_{1}+{R}_{2}+{R}_{3}$, the net effective resistance of the series-parallel combination of resistors is,

${R}_{\mathit{Total}}=\phantom{\rule{0.147em}{0ex}}{R}_{\mathit{P1}}+{R}_{\mathit{P2}}$

Parallel connection of series resistors:

A parallel-series circuit is formed by connecting a set of series resistors in parallel.

- Connect \(R_1\) and \(R_2\) in series to get an effective resistance of \(R_{S1}\).
- Similarly, connect \(R_3\) and \(R_4\) in series to get an effective resistance of \(R_{S2}\).
- These series segments of resistors are then joined in parallel.

*Parallel-series combination of resistors*

Using the effective resistance of the series circuit, ${R}_{S}=\phantom{\rule{0.147em}{0ex}}{R}_{1}+{R}_{2}+{R}_{3}$, we get

$\begin{array}{l}{R}_{\mathit{S1}}=\phantom{\rule{0.147em}{0ex}}{R}_{1}+{R}_{2}\\ \\ {R}_{\mathit{S2}}=\phantom{\rule{0.147em}{0ex}}{R}_{3}+{R}_{4}\end{array}$

Using the formula for the effective resistance of the parallel combination of resistors $\frac{1}{{R}_{P}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{2}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{3}}$, the net effective resistance of parallel-series combination of resistors

*is*$\frac{1}{{R}_{\mathit{Total}}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{S1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{S2}}$.