IEEE Solid-State Circuits Magazine - Spring 2015 - 8

c ircu it intu itions

Ali Sheikholeslami

Bandwidth Extension

Welcome to "Circuit Intuitions!" This
is the fifth article of a column series
that appears regularly in this magazine. 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 send your e-mails to
ali@ece.utoronto.ca.
The past four articles in this series
were all related to the small-signal
operation of MOS circuits at low frequencies, and as such, we ignored all
the parasitic capacitances in the circuits. In this article, we look at circuit
behavior at higher frequencies, where
the effect of parasitic capacitances
cannot be ignored. We review a quick
way of determining the bandwidth
of single-pole circuits, and present
methods of extending bandwidth.
A small-signal model of a transistor
including its parasitic capacitances is
shown in Figure 1, where g m and ro
are the small-signal parameters at low
frequencies, as introduced in the first
article of this series, and C gs, C gd,
C db, and C sb are parasitic capacitances when the transistor is in saturation region. We ignore g mb in this
model so as to simplify the analysis.
We will now use this model to provide
intuition into the frequency behavior
of a common-source amplifier.
In the first article in this series, we
said looking into a node we see the
Digital Object Identifier 10.1109/MSSC.2015.2418154
Date of publication: 25 June 2015

s p r i n g 2 0 15

Norton (or Thevenin) equivalent circuit, which consists of a current source
in parallel with a resistor. Extending
this concept to higher frequencies,
we must replace the equivalent resistance by an equivalent impedance to
take into account the effects of parasitic capacitances. Determining the
frequency response of a circuit then
becomes equivalent to determining
the frequency response of RC circuits
(assuming we have no inductors for
now). Let us apply this idea to a simple common-source amplifier.
Figure 2 shows a common-source
circuit with a capacitive load C L
being driven by an ideal voltage
source Vin . We would like to determine the Norton equivalent of this
circuit looking into its output node.
The short circuit current (I sc) at
the output can be written as

I sc = (- g m + sC gd) Vin .
This current is only affected by
C gd as all other capacitors are either
shorted in this process (such as
C db, C L, C sb) or draw their own currents from the signal source but
have no influence on I sc .
Now let us determine the impedance looking into the output node.
For this, we zero the voltage source
and observe that the equivalent
impedance is
Z eq = R eq

1 ,
sC eq

where R eq = ro R L, and C eq = C L +
C dg + C db . Since Vo = I sc # Z eq, we can
write
(1 - sC gd /g m)
Vo
(s) = - g m R eq
.
Vin
(1 + sR eq C eq)

D
RL
Cgd

Cdb

Cgs

Csb

Cgd

Vin

Isc

S
(a)

(a)
D

Cgd

Cdb

G
Cgs

gmVgs gmbVbs

(b)

ro
Csb

Figure 1: (a) An MOS transistor and its
parasitic capacitances and (b) a MOS transistor small-signal model at high frequencies.

IEEE SOLID-STATE CIRCUITS MAGAZINE

Isc

Req

Ceq

(b)
Figure 2: (a) A common-source amplifier
with its load resistance and capacitance
and (b) the Norton equivalent circuit at the
output node.

Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Spring 2015