Question

Find the equivalent resistances of the networks shown in the figure (32-E21) between the points a and b.

Answer 1

(a) The circuit can be simplified stepwise, as shown below.


The effective resistance between the points a and b,
${\mathrm{R}}_{\mathrm{eff}}=\frac{{\displaystyle \frac{5r}{3}}\times r}{{\displaystyle \frac{5r}{3}}+r}=\frac{5r}{8}$

(b) The circuit can be simplified, as shown below.

The effective resistance between the points a and b,
${\mathrm{R}}_{\mathrm{eff}}=\left(\frac{r}{3}\right)+r=\frac{4r}{3}$
(c)


From the figure, it can be seen that axbya is a balanced Wheatstone bridge. The resistors in branch xy will, thus, become ineffective. The circuit can be simplified as under

The effective resistance between the points a and b,
${\mathrm{R}}_{\mathrm{eff}}=\left(\frac{2r\times 2r}{2r+2r}\right)=r$

(d) The circuit can be simplified as shown below.

The effective resistance between the points a and b,
${\mathrm{R}}_{\mathrm{eff}}=\frac{r}{4}$

(e) The circuit can be redrawn as shown below.


Now, we can see that the circuit is a balanced Wheatstone bridge. So, the branch xy will become ineffective. Thus, the simplified circuit will become as shown below.

The effective resistance between the points a and b,
${\mathrm{R}}_{\mathrm{eff}}=\left(\frac{2r\times 2r}{2r+2r}\right)=r$