IEEE Solid-State Circuits Magazine - Fall 2015 - 7

c ircu it intu itions

Ali Sheikholeslami

Miller's Approximation

W

Welcome to the seventh article in
this column series. As the title suggests, each article provides insights
and intuitions into circuit design
and analysis. These articles are
aimed at undergraduate students
but may serve the interests of other
readers as well. If you read this article, I would appreciate your comments and feedback, as well as your
requests and suggestions for future
articles in this series. Please e-mail
your comments to me at: ali@ece.
utoronto.ca.
In the previous article, we presented an intuitive view of Miller's
theorem, especially as it applies to
resistors. In this article, we use Miller's theorem to estimate the bandwidth of an amplifier with a capacitor
between its input and output nodes.
Figure 1 shows an ideal voltage
amplifier (i.e., one with infinite input
impedance and zero output impedance) with a constant voltage gain of
" - A 0 " and a capacitor C 12 between
its input and output nodes. Miller's
theorem says that we can replace C 12
with two capacitors, C 1M and C 2M ,
connected from the input and output
node, respectively, to ground, where,
C 1M = C 12 (1 + A 0)
C 2M = C 12 (1 + 1/A 0) .

(1)
(2)

If we assume A 0 & 1, then Miller's
theorem tells us that the capacitor
between the two nodes appears as
much larger at the input node (by a factor of +A 0) but as the same capacitor
(by a factor of +1) at the output node.
This makes intuitive sense because a
Digital Object Identifier 10.1109/MSSC.2015.2475995
Date of publication: 2 December 2015

small voltage increment at the input
results in a much larger decrement
at the output, which in turn attracts a
large amount of charge on the capacitor plates, as if the capacitor were
much larger! From the perspective of
the output node, however, a change
in the output voltage corresponds to
a much smaller change at the input.
We can then simply assume the input
node is grounded. This is equivalent
to saying the capacitor seen from the
output is the same as C 12 .
When the amplifier is ideal but its
gain (A) is frequency dependent (i.e.,
not constant) or when the amplifier
is nonideal (e.g., has a finite output
impedance), there may be confusion
as how to apply Miller's theorem or
how useful it may be. We focus on
this in the remainder of this article.
First, let us examine a slightly
generalized case where the amplifier is ideal but it has a frequencydependent gain A v (j~), instead of
constant - A 0, with a single pole frequency, fp . In other words, assume
A v (j~) =

- A0
.
jf
1+
fp

Note that since this is an ideal
amplifier, adding C 12 across it will
not change the voltage transfer
function. We can therefore apply
Miller's theorem (1), to arrive at the
following equation:

1+
C 1M = C 12 (1 + A 0)

jf
fp (1 + A 0)
.
jf
1+
fp

This equation simply states that
C 1M is now frequency dependent. At

C12
I2

I1
-A0

V1

V2 = -A0V1

Ideal Voltage Amplifier
(a)
Ideal Voltage Amplifier
V1
I1

-A0

C1M

V2 = -A0V1
I2
C2M

C1M = C12 (1 + A0)
1
C2M = C12 (1 + )
A0
(b)
Figure 1: Miller's theorem: (a) capacitance
C 12 is connected between the input and
output nodes of an ideal voltage amplifier
with a constant gain of - A 0, and (b) Miller's
equivalent circuit.

low frequencies (below fp), C 12 is
simply multiplied by the dc gain of the
amplifier. At midfrequencies (between
fp and + A 0 fp), when the voltage
gain drops, C 1M drops also. Beyond
the amplifier's unity-gain frequency
(i.e., A 0 fp), C 1M approaches C 12 . This
makes intuitive sense because, at
very high frequencies, the gain of the
amplifier will approach zero and the
output node becomes grounded, producing C 12 at the input node.
We do not need to derive an equation for the output node capacitance
(C 2M ) in this case because C 2M will
have no impact on V2, given that the
amplifier is assumed to have zero
output impedance.

IEEE SOLID-STATE CIRCUITS MAGAZINE

fa l l 2 0 15

7
http://www.utoronto.ca
Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Fall 2015
