# Millers Theorem

In general, the miller theorem is used for converting any circuit having the configuration of figure 2.48 (a) to another configuration shown in figure 2.48 (b)

If $Z$ is the impedance connected between two nodes, node 1 and node 2, it can be replaced by the two impedances $Z_1$ and $Z_2$ ; Where $Z_1$ is connected between node 1 and ground and $Z_2$ is connected between node 2 and ground.

Proof Of Miller’s Theorem

Miller’s theorem states that the effect of resistance $Z$ on the input circuit is a ratio of the input voltage $V_{i}$ to the current $I$ which flows from the input to the output.

$\displaystyle Z_1 = \frac {V_1}{I}$

$\displaystyle I = \frac {V_i - V_0}{Z} = V_i \biggl (\frac {1 - \frac {V_0}{V_i} }{Z} \biggr )$

$\displaystyle I = V_i \biggl ( \frac {1-A_v}{Z} \biggr )$

$\displaystyle Z_1 =\frac {Z}{1-k}$

$\displaystyle Z_2 = \frac {V_0}{I}$

$\displaystyle I = \frac {V_0 - V_i}{Z} = V_0 \biggl ( \frac {1 - \frac {V_i}{V_0}}{Z} \biggr )$

$\displaystyle I = V_0 \biggl( \frac {1 - \frac {1}{A_V}}{Z} \biggr )$

$\displaystyle Z_2 = \frac {Z}{1 - \frac {1}{k}}$