11.22
First-Order
RC
Circuits
Averaging is a signal processing application that appears often in
analog and digital signal
processing. The waveform to be averaged is typically a rectangular pulse waveform with a slowly
varying DC content. An
averaging circuit produces an output that is the DC content of input signal.
Consider the rectangular pulse waveform shown in Fig. 11.5-5. It has an amplitude of 1 V, a
frequency of 1kHz and a duty ratio of 0.2. This waveform can be thought of as the sum of a DC voltage
of value 0.2 V and a pure alternating waveform with zero full-cycle area (
i.e., zero DC content). These
two components are also shown in Fig. 11.5-5.
1
0.2 ms
(a)
Time in ms
1 ms
v
s
(
t
)
0.8 V
0.2 V
–0.2 V
0.2
1
(b)
Time in ms
Time in ms
Fig. 11.5-5
A rectangular pulse waveform and its components
Assume that this waveform is applied to a Series
RC Circuit with
R
=
10 k
W
and
C
=
1
m
F from
t
=
0 .
The time constant is 10 ms. Output is taken across the capacitor. The total zero-state response can be
obtained by using superposition principle. Therefore, we expect the standard step response scaled by
0.2 to be present along with other components in the output. This step response component will reach
steady-state in about 5
t
, i.e., in 50ms and contribute 0.2 V steady component to the output after that.
The pure alternating component of applied voltage also will reach a periodic steady-state at the
output after about 50ms. We remember that this alternating component can be thought of as the sum
of infinitely many sinusoids of frequencies that are integer multiples of 1kHz. The lowest frequency
component will be 1kHz. The phasor impedance of capacitor and the resistor share the sinusoidal
voltage under steady-state and we are interested in the voltage absorbed by the capacitor. Let us
calculate the phasor impedance of capacitor at 1kHz. It is –
j159.2
W
. We see that this is only about
1.6% of the resistor value (10 k
W
) and hence we expect the capacitor to absorb only a very small
percentage of sinusoidal voltage at 1kHz – most of it will appear across the resistor. This will be more
so for other sinusoidal components with frequencies higher than 1kHz since phasor impedance of
capacitor goes down with frequency.
As a first approximation, we assume that the alternating component that appears across capacitor is
negligible when we calculate the alternating component of current in the circuit. Thus, the alternating
component of applied voltage is assumed to appear across 10k
W
almost entirely, thereby resulting in
a current whose waveshape will be the same as that of alternating component of voltage. This current
will vary between 0.08 mA and –0.02 mA. Now we work out the small voltage that appears across
capacitor due to this alternating current flow.
The half-cycle area of this current is 0.08 mA
×
0.2 ms (or 0.02mA
×
0.8 ms)
=
0.016
m
C. Hence,
the peak-to-peak voltage across the capacitor due to the alternating current flow will be 0.016
m
C/
1
m
F
=
0.016 V. Hence the total capacitor voltage will vary in the range 0.2
±
0.008 V. The variation is
±
4% of the desired average value of 0.2 V.
This approximate solution is confirmed by the accurate solution worked out using the method to
solve for periodic steady-state explained earlier in this stion. This is shown in Fig. 11.5-6. The output
FrequencyResponseofFirstOrderRCCircuits
11.23
waveform segments are actually exponential; but they appear nearly straight-line segments confirming
the validity of assumption employed in our approximate reasoning.
1 ms
1 ms
0.2 ms
0.192 V
0.20
0.15
0.5
1
0.208 V
Time in ms
0.2 ms
Time in ms
v
s
(
t
)
v
o
(
t
)
Fig. 11.5-6
Inputandoutputwaveformsofseries
RC
averagingcircuit
We should not ever forget that a circuit reaches steady-state only after covering the transient period.
This reasonably clean average value appears only after 50 ms of applying the input. We can increase
the time constant of the circuit to higher levels in order to make the output appear cleaner; but there is
a price to pay.
The cleaner output will take longer to establish. The average value of input is not likely
to remain constant forever in a practical application of averaging circuit. In fact, this value may be
used to code some information and will consequently change slowly. Typically, the duty ratio of the
input wave changes slowly while its frequency is kept constant. And the averaging circuit is expected
to track the change in average value faithfully. Obviously there is a conflict between the requirement
of a clean output and the requirement of a fast response to changing DC content at input. The time
constant of the circuit must be selected in such a way that it has enough speed to catch up with the
average value variation. And, if the ripple in the output is excessive with that value of time constant,
we better look for some other better technique to do averaging!
Parallel
RC Circuit can be employed for averaging current signals subject to similar constraints.
Essentially,
averaging is only a special case of low-pass filtering.
Good averaging performance
requires that
t
>>
T, where
T is the period of the input signal or its characteristic time of variation if
a regular period cannot be identified.
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