Miller theorem

The Miller theorem refers to the process of creating equivalent circuits. It asserts that a floating impedance element, supplied by two voltage sources connected in series, may be split into two grounded elements with corresponding impedances. There is also a dual Miller theorem with regards to impedance supplied by two current sources connected in parallel. The two versions are based on the two Kirchhoff's circuit laws.

Miller theorems are not only pure mathematical expressions. These arrangements explain important circuit phenomena about modifying impedance (Miller effect, virtual ground, bootstrapping, negative impedance, etc.) and help in designing and understanding various commonplace circuits (feedback amplifiers, resistive and time-dependent converters, negative impedance converters, etc.). The theorems are useful in 'circuit analysis' especially for analyzing circuits with feedback and certain transistor amplifiers at high frequencies.

There is a close relationship between Miller theorem and Miller effect: the theorem may be considered as a generalization of the effect and the effect may be thought as of a special case of the theorem.

Miller theorem (for voltages)

Definition

The Miller theorem establishes that in a linear circuit, if there exists a branch with impedance

Z

{\displaystyle Z}

, connecting two nodes with nodal voltages

V

1

{\displaystyle V_{1}}

and

V

2

{\displaystyle V_{2}}

, we can replace this branch by two branches connecting the corresponding nodes to ground by impedances respectively

Z

1

−

K

{\displaystyle {\frac {Z}{1-K}}}

and

K

Z

K

−

1

{\displaystyle {\frac {KZ}{K-1}}}

, where

K

=

V

2

V

1

{\displaystyle K={\frac {V_{2}}{V_{1}}}}

. The Miller theorem may be proved by using the equivalent two-port network technique to replace the two-port to its equivalent and by applying the source absorption theorem.