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Question
Find the equivalent resistances of the networks shown in the figure between the points a and b.
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Solution
(a) The circuit can be simplified stepwise, as shown below.
The effective resistance between the points a and b,
\[R_{eff} = \frac{\frac{5r}{3} \times r}{\frac{5r}{3} + r} = \frac{5r}{8}\]
(b) The circuit can be simplified, as shown below.
The effective resistance between the points a and b,
\[R_{eff} = \left( \frac{r}{3} \right) + r = \frac{4r}{3}\]
(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,
\[R_{eff} = \left( \frac{2r \times 2r}{2r + 2r} \right) = r\]
(d) The circuit can be simplified as shown below.
The effective resistance between the points a and b,
\[R_{eff} = \frac{r}{4}\]
(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,
\[R_{eff} = \left( \frac{2r \times 2r}{2r + 2r} \right) = r\]
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