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Series connection of parallel resistors:
A series-parallel circuit is formed by connecting a set of parallel resistors in series.
1. Connect $$R_1$$ and $$R_2$$ in parallel to obtain an effective resistance of $$R_{P1}$$.
2. Similarly, connect $$R_3$$ and $$R_4$$ in parallel to get an effective resistance of $$R_{P2}$$.
3. 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.
1. Connect $$R_1$$ and $$R_2$$ in series to get an effective resistance of $$R_{S1}$$.
2. Similarly, connect $$R_3$$ and $$R_4$$ in series to get an effective resistance of $$R_{S2}$$.
3. 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

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