Reduce and Return Approach

The network of [Fig. 1] is redrawn as [Fig. 2(a)]. For this discussion, let us assume that voltage $V_4$ is desired. As described in the previous section, first combine the series resistors $R_3$ and $R_4$ to form an equivalent resistor $R'_T$ as shown in [Fig. 2(b)]. Resistors $R_2$ and $R'_T$ are then in parallel and can be combined to establish an equivalent resistor $R_T$ as shown in [Fig. 2(c)]. Resistors $R_1$ and $R_T$ are then in series and can be combined to establish the total resistance of the network as shown in [Fig. 2(d)]. The reduction phase of the analysis is now complete. The network cannot be put in a simpler form. We can now proceed with the return phase whereby we work our way back to the desired voltage $V_4$. Due to the resulting series configuration, the source current is also the current through $R_1$ and $R_T$. The voltage across $R_T$ (and therefore across $R_2$) can be determined using Ohm's law as shown in [Fig. 2(e)]. Finally, the desired voltage $V_4$ can be determined by an application of the voltage divider rule as shown in [Fig. 2(f)]. The reduce and return approach has now been introduced. This process enables you to reduce the network to its simplest form across the source and then determine the source current. In the return phase, you use the resulting source current to work back to the desired unknown. For most single-source series-parallel networks, the above approach provides a viable option toward the solution. In some cases, shortcuts can be applied that save some time and energy. Now for a few examples.