# 1.8 Miller’s Theorem – Pulse and Digital Circuits

##### 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 I1 be the current leaving node1.

where AV = V2/V1 is the voltage gain and

Thus, if R is removed and a resistance R1 is connected between nodes 1 and N, the current drawn from node 1 is still I1, as shown in Fig. 1.48(b).

Let us now calculate I2 using Fig. 1.48(c).

where

Then if R2 is connected between node 2 and N the current drawn from node 2 is still I2, Fig. 1.48(d).

This suggests that we can replace R by R1 and R2. 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 R1 is connected between 1 and N

FIGURE 1.48(c) Network to calculate I2

FIGURE 1.48(d) Circuit when R2 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, R1 connected in series with the input node 1 and the R2 connected in series with the output node 2 as in Fig. 1.49(b) where:

and

where AI 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.