##### 1.8 MILLER'S THEOREM

Miller’s theorem states that if an impedance is connected between the input and output nodes in an amplifier, having a reference node *N*, then this impedance can be replaced by two impedances, one connected between the input and the reference node and the other connected between the output and the reference node.

Consider an amplifier in which *N* is the reference node and 1 and 2 are the input and output nodes. A resistance *R*^{′} is connected between nodes 1 and 2, as shown in Fig. 1.48(a).

Let *I*_{1} be the current leaving node1.

where *A _{V}* =

*V*

_{2}

*/V*

_{1}is the voltage gain and

Thus, if *R*^{′} is removed and a resistance *R*_{1} is connected between nodes 1 and *N*, the current drawn from node 1 is still *I*_{1}, as shown in Fig. 1.48(b).

Let us now calculate *I*_{2} using Fig. 1.48(c).

where

Then if *R*_{2} is connected between node 2 and *N* the current drawn from node 2 is still *I*_{2}, Fig. 1.48(d).

This suggests that we can replace *R*^{′} by *R*_{1} and *R*_{2}. Voltage shunt feedback amplifier can be analysed using Miller’s theorem. We can also see the advantage of op-amp integrator by using the Miller’s theorem, when we study low-pass circuits.

**FIGURE 1.48(a)** A multi-node network

**FIGURE 1.48(b)** Network when *R*_{1} is connected between 1 and *N*

**FIGURE 1.48(c)** Network to calculate *I*_{2}

**FIGURE 1.48(d)** Circuit when *R*_{2} is connected between 2 and *N*

#### 1.8.1 The Dual of Miller's Theorem

The dual of the Miller’s theorem states that if an impedance is connected between node 3 and the reference node *N*, in an amplifier then this impedance can be replaced by two impedances, one connected in series with the input and the other connected in series with the output.

Consider an amplifier in which a resistance *R*^{′} is connected between node 3 and reference node *N* as shown in Fig. 1.49(a). Then the dual of Miller’s theorem states that the resistance *R*^{′} now can be replaced by two resistances, *R*_{1} connected in series with the input node 1 and the *R*_{2} connected in series with the output node 2 as in Fig. 1.49(b) where:

and

where *A _{I}* is the current gain.

Current series feedback amplifier can be analysed using the dual of the Miller’s theorem. The advantage of using op-amp differentiator is seen when we study high-pass circuits.