Strange-Looking Circuit Diagrams

Let’s redraw this circuit diagram to make it clearer. Then we’ll apply the Strategy outlined above to calculate the equivalent resistance.

To redraw the diagram, consider the figure below. In the upper circuit, the blue resistors constitute a path from the positive terminal of the battery to the negative terminal. In parallel with this circuit are the red resistors, which constitute another path from the positive to negative terminal of the battery. The blue and red paths are shown more cleanly drawn in the lower circuit diagram. Note that, in both the upper and lower circuit diagrams, the blue and red paths connect the positive terminal of the battery to the negative terminal of the battery.

Now it is easier to see that $R_{\text{1}}\text{ and }{R}_2$ are in parallel, and the parallel combination is in series with $R_4$ . This combination in turn is in parallel with the series combination of $R_{\text{3}}\text{ and }{R}_5$ . First, we calculate the blue branch, which contains $R_1,R_{\text{2}}\text{, and }{R}_4$ . The equivalent resistance is

where we show the contribution from the parallel combination of resistors and from the series combination of resistors. We now calculate the equivalent resistance of the red branch, which is

Inserting these equivalent resistors into the circuit gives the circuit below.

These two resistors are in parallel, so they can be replaced by a single equivalent resistor with a resistance of

The final equivalent circuit is show below.