Consider an infinite ladder of network shown in Figure. A voltage is applied between points $A$ and $B$ . If the voltage is halved after each section, Find the ratio of $R_{1} / R_{2}$ .
Suggest a method to terminate it after a few sections without introducing much error in its attenuation.

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Solution

Voltage across $A B$ is $V$ , voltage across A'B' is V/2 i.e., voltage across $R_{2}$ is V/2.
Now from Kirchhoff's law, it is obvious that voltage across $R_{1}$ is
$V - \frac{V}{2} = \frac{V}{2}$
$R_{1} i = R_{2} \frac{i}{2}$ or $\frac{R_{1}}{R_{2}} = \frac{1}{2}$
Resistance will not be affected if equivalent resistance $R$ becomes independent of number of sections in the circuit. This is only possible if the terminating resistance $R_{0}$ is itself equal to equivalent resistance
(Fig. S5.34). The equivalent resistance of $R_{0}$ and $R_{2}$ is
$R^{'} = R_{0} = R_{1} + \frac{R_{0} R_{2}}{R_{0} + R_{2}}$
$R_{1}$ is in series with it, so equivalent resistance between $A$ and $B$ is
$R_{0} = R_{1} + \frac{R_{0} R_{2}}{R_{0} + R_{2}}$
According to proposition $R_{0} = R_{1} + \frac{R_{0} R_{2}}{R_{0} + R_{2}}$
Solving for $R_{0}$ , we get $R_{0} = \frac{R_{1}}{2} [1 + \sqrt{[1 + \frac{4 R_{2}}{R_{1}}]}]$
Thus, the circuit may be terminated after a few sections if resistance $R_{0}$ is connected in parallel as shown in Fig. S5.34.

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