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)

bjt27

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.

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\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} 

bjt29

\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}} 

 

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