IEEE Solid-States Circuits Magazine - Winter 2021 - 14

signify the filtering properties of the
RC circuit, that is, how the RC circuit
discriminates among input signals
of different frequencies or how it
filters the input signal according to
its frequency content. To see the filtering behavior of the RC circuit, we
distinguish three frequency regions
in this plot: 1) the low-frequency
region, where ~ % ~ p; 2) the highfrequency region, where ~ & ~ p; and
3) frequencies around ~ p. In each
region, the circuit exhibits a different behavior. At low frequencies, by
our definition, R % 1/~C, and hence
all the input voltage appears across
the capacitor. In other words, the circuit voltage gain is one. Also, in this
region, the phase difference between
the output and the input is close to
zero, as detailed in Figure 2(b). In
the time domain, one can arg ue
that, since the input voltage changes
slowly (i.e., it does not change much
in one time-constant RC), the capacitor voltage is able to track the input
voltage and not fall behind. That is,
v o ^ t h , v i ^ t h, and the current flowing
through the resistor and the capacitor i R ^ t h = i C ^ t h = (v i ^ t h - v o ^ t h /R is
close to zero.
At high frequencies, where ~ & ~ p,
we can simplify the transfer function equation to ~ p /j~. The reader
may recognize this as the transfer
function of an integrator, where the

gain magnitude is inversely proportional to ~ and the phase shift
from the input to the output is 90º.
Simply put, in region 2, the transfer function of the RC circuit is
close to that of an integrator. Let us
now see this in the time domain. At
high frequencies, the input voltage
changes so quickly that the capacitor cannot keep up and hence maintains a voltage close to zero. As a
result, we can assume all the input
voltage appears across the resistor,
creating a current proportional to
the input voltage. This current is
then fed directly to the capacitor,
where it is integrated, producing a
voltage proportional to the integral
of the input voltage, with a scaling factor of 1/RC . Note that there
is no contradiction in the output
voltage being close to zero and its
value (albeit close to zero) being
proportional to the integral of
the input voltage. For example, an
input signal Vi sin ^~t h with ~ & ~ p
results in an output voltage that
i s ^~ p /~h Vi sin ^~t - r/2h, w h i c h
i s b o t h c los e to ze r o ( b e c aus e
~ p /~ % 1) and proportional to the
integral of the input voltage.
So far, we have said the RC circuit
tracks the input voltage in region 1
and integrates it in region 2. How
about in region 3? It turns out that in
this region, the circuit neither tracks

ωp /ω
1

0 dB

2

1
(a)

Vo
( jw)
Vi

ω << ωp

ω /ωp ≅
0.1 - 10

ω >> ωp

3
-40 dB

(b)

Vo
( jw)
Vi

ωp

ω (rad/s)

0°
-45°
-90°

FIGURE 2: The (a) magnitude and (b) phase plots of the transfer function of a series RC circuit. The output tracks the input signal for frequencies far below ~ p but integrates the input
signal for frequencies far above ~ p . For frequencies between the two extremes, the output
neither tracks nor integrates the input signal.

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W I N T E R 2 0 2 1	

IEEE SOLID-STATE CIRCUITS MAGAZINE	

nor integrates the input but, rather,
does something in between. At ~ p,
for example, the circuit attenuates
the signal amplitude by 1/ 2 (which
is less than the value of one that is
required for tracking) and shifts its
phase by 45º (which is less than the
90º required by an integrator). One
can also say that, in this region, the
circuit partially tracks and integrates
the input.
With this basic understanding, let
us now examine the following three
statements about RC circuits and
determine if each is true or false:
1)	The RC circuit produces the time
average of its input signal at the
output.
2)	The RC circuit takes the running
average (also called the moving
window average) of the input signal, where the time window is related to the RC time constant.
3)	The RC circuit is continuously
interpolating between the input
and output voltages as time progresses.
Readers are encouraged to reflect
on these assertions before proceeding further.
Statement 1 is correct only for
dc inputs. As discussed earlier, a dc
input will appear intact at the output,
and, since the average of a dc voltage
is equal to the dc voltage itself, this
statement is true for dc inputs. However, any input signal that is changing with time will result in an output
signal that is also changing with
time, making statement 1 invalid, as
the average of a signal must be constant by definition. Statement 2 is
also correct only in special cases. It
is certainly correct if the input is a
pure dc voltage, as argued in relation
to statement 1. In addition, this statement is correct for low-frequency
signals, i.e., when ~ % ~ p, because,
similar to dc, the running average
of a low-frequency signal is identical to the signal itself. And, since the
output is equal to the input at  low
frequencies, this statement is true.
How about a signal with content
only at very low and very high frequencies (i.e., with content in regions
1 and 2 in Figure 2 but not in region 3)?
IEEE Solid-States Circuits Magazine - Winter 2021

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