IEEE Solid-States Circuits Magazine - Fall 2019 - 15

Cc

Vo

+
Vi
-

Ri

Ci

FIGURE 2: The input signal is ac-coupled to
the rest of the circuit. All of the signal current will travel through the coupling capacitor, creating a zero in the transfer function.

Cgd
RS
+
Vi
-

M

Vo
RL

ISC

1

FIGURE 3: The signal from node 1 travels
through both the transconductance of the
transistor and the parasitic capacitance between the transistor's gate and drain. This
dual-path signal from node 1 to the output
node creates a zero in the transfer function.

and a lower impedance at higher frequencies. As a result, the final voltage
signal takes shape considering all the
high-frequency diversions, amplifications, and attenuations along the signal path.
Looking back, the overall transfer function of this circuit, Vo /Vi ^s h,
can be written as
Vo ^ h V1 ^ h V2 ^ h Vo ^ h
s =
s #
s #
s,
Vi
Vi
V1
V2
where s is the complex frequency in
the Laplace transform. This equation
states that the overall transfer function of the circuit is the product of
three other transfer functions, each
providing a frequency-dependent gain
or attenuation. We now focus on the
frequency-dependent treatment of the
signal on its journey.
We noted that, every time the signal arrives at a node with a capacitance, a part of it is diverted to the
ground instead of continuing along
the path to the output node. Since the
amount of the signal that is diverted
is frequency dependent, the ultimate gain of this circuit is frequency
dependent. We characterize the gain
of the amplifier as a function of fre-

quency with a transfer function having multiple poles and zeros.
The poles appear to form whenever
there is a capacitance to the ground
from any node along the signal path.
The zeros, which we have ignored so
far, appear when there is a capacitance directly on the signal path, for
example, when all or part of the signal
current travels through the capacitor
to arrive at the output. In the example
that we have covered thus far, this was
not the case: the signal never traveled
through a capacitor to arrive at the output; it was only wasted to the ground
along the way. We will demonstrate the
concept of zero in Figures 2 and 3.
Figure 2 displays a case where
the input voltage is ac-coupled to the
circuit through coupling capacitor
C c . In that case, the signal current
must travel through C c to reach the
output. The signal is impeded by the
high impedance of the capacitor at
low frequencies but admitted by the
high admittance of the capacitance at
high frequencies. As a result, the flow
of the signal is controlled with frequency, providing a high-pass filter
with a zero (at zero frequency) in the
transfer function.

Every time the
signal arrives
at a node with
a capacitance,
a part of it is
diverted to the
ground instead
of continuing
along the path to
the output node.

Figure 3 offers an example where
part of the signal travels through capacitor C gd to arrive at the output node. If
we assume node 1 has voltage V1, the
short circuit current at the output node
will be the sum of two currents:
I sc = - g m V1 + sC gd V1 .
In other words, the signal current
travels through both the transistor
(characterized by g m) and the capacitor (characterized by sC gd) . Since
the sum of those two currents can be

+
Vi
-

Vo

R2
Probe's
Resistance

R1

C1

Input Circuit of
an Oscilloscope
(a)
C2

+
Vi
-

Vo

R2
Probe's
Resistance

R1

C1

Input Circuit of
an Oscilloscope
(b)

FIGURE 4: (a) The oscilloscope's input
circuit low-pass filters the voltage of interest (Vi ) due to its input capacitance, C1. (b)
C2 creates a zero to cancel the pole of the
new circuit.

zero at s Z = g m /C gd , there is a zero
in the transfer function (Vo /Vi). If we
evaluate the magnitude of the transfer function at an angular frequency
equal to s Z (also known as the zero
frequency), we find a value that is
2 times the magnitude of the transfer function at lower frequencies.
There is a significant design effort
to compensate for the effect of parasitic capacitances that limit the circuit
bandwidth, which involves, essentially, altering or augmenting the signal
path. Figure 4(a) provides an example
where the input signal is low-pass filtered by the parasitic capacitance of
the measurement equipment (such as
an oscilloscope). The transfer function
of this circuit has a single pole at an
angular frequency of 1/ (R 1 R 2) C 1 .
To extend the bandwidth, we provide a
parallel path for the signal to reach the
output through an additional capacitor
in parallel with R 2, as shown in Figure 4(b). With C 2 in place, the short
circuit current at the output node can
be provided through R 2 or C 2, the sum
of which can produce a zero at
s Z = - 1/R 2 C 2 .

IEEE SOLID-STATE CIRCUITS MAGAZINE

(continued on p. 95)

FA L L 2 0 19

15
IEEE Solid-States Circuits Magazine - Fall 2019

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