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WAVESHAPING
INTRODUCTION
This section investigates the effect of C-R and L-R circuits on a rectangular waveform, and
relates it to the time constant of the circuit.
OBJECTIVES
when you have completed this section, you will be able to:

explain the effect of circuit time constants on rectangular waveforms, as demonstrated
using an oscilloscope

relate these waveforms to simple concepts of integration and differentiation.
GENERATION OF RECTANGULAR WAVEFORMS
Before we can see the effects of simple series circuits with various time constants on a
rectangular waveform, we first have to generate the waveform, and have something to
measure the waveform after it has passed through the circuit.
Although waveforms are now generated electronically, it is easier to understand the effect of
time-constants on the waveforms by thinking of them as being generated by the switching in
and out of d.c. sources. Figure 25 illustrates how a square-wave can be generated by manual
switching.
1
S
2
V
Volts
VS
Time
t
t
Figure 25
Switch S is placed in position 1 for a certain time and then moved over 'instantaneously' to
position 2 and held there for a certain time. The switch is then moved back to position 1, and
held there for an equal period of time. This movement of the switch between positions 1 and
2 creates the square-wave. Obviously manual switching is very limited in speed and also the
waveforms have uneven periods, hence the need for an electronic generator. Electronic
switching can be performed virtually instantaneously and the periodic time of the waveform
can be varied over a wide range, from seconds to nanoseconds.
If a square-wave is required with both positive and negative values, then two separate power
sources are connected alternately by a switch S as shown in Figure 26.
27
S
2
VS
V
Volt
s
VS
Time mS
Figure 26
DISPLAYING AND MEASURING WAVEFORMS
Waveforms are displayed and measured by means of a cathode ray oscilloscope.
RECTANGULAR WAVEFORMS CONSIDERED AS TRANSIENTS
The waveforms are generated by an electronic pulse generator, which we saw above could be
thought of as switching between various voltage levels. By considering rectangular
waveforms in this way, as changes in d.c. levels, we can apply our previous work on d.c.
transients to explain the effect of the time-constants in series C-R and L-R circuits.
At this point, you might be thinking that the square wave in Figure 26 is really a.c. - and you
would be right! We'll leave the explanation of why we look at it as d.c. until the end of this
section.
The pulse generator applies a square-wave to the input of the C-R or circuit under test, as
shown in Figure 27.
Figure 27
28
The output of the circuit is measured by connecting an oscilloscope as shown in Figure 27.
The output voltage VR generated across the resistor is fed to the Y-plates of the oscilloscope.
The time-base frequency is adjusted to display a number of cycles and the Y-amplitude
control is set to give the required vertical amplification of the voltage waveform.
If the input square-wave is positive-going, with a peak value of 1 volt and a period of 1 ms
(0.5 ms on and 0.5 ms off), its waveform is as shown in Figure 28.
C-R CIRCUIT: OUTPUT TAKEN FROM RESISTOR
lV
Voltage
VR
0
Time (mS)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-l V
Figure 28
If the square wave of Figure 28 is applied to a C-R circuit as shown in Figure 27, with a timeconstant considerably smaller than the period, the resultant waveform of the output voltage
VR is as shown in Figure 29.
lV
Voltage
VR
0
Time (mS)
1.0
2.0
3.0
-l V
Figure29
when the supply is switched to +1 volt, the potential on both sides of the capacitor becomes
+1 volt, and the voltage VR across the resistor is 1 volt. Hence the instant rise of VR.
The capacitor now charges at a rate determined by the time-constant. This results in a voltage
drop VR across the resistor, until the capacitor has fully charged, when the voltage across the
resistor will have fallen to zero volts; as the time constant is much smaller than the period,
this will happen well before the voltage from the pulse generator drops to zero. The curve of
the first 0.5 ms period, measured by the oscilloscope will be one of exponential decay as
shown in Figure 29.
After 0.5 ms the supply is switched off, zero volts appears on one side of the capacitor, but
because the capacitor is fully charged, the other side of the capacitor will be at -1 volt, hence
the fall of VR. The capacitor discharges, again the rate being determined by the timeconstant, until there is again zero volts registering across the resistor
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If the time-constant T is altered to one half of the period then the resulting curve is shown in
Figure 30. Obviously the time-constant is greater, therefore the capacitor will take longer to
charge, hence the steady slope of the curve.
lV
Voltage
VR
Time (mS)
-l V
Figure 30
As before, the moment the input rises to 1 volt, the output voltage VR rises to 1 volt. The
curve between points A and B on the trace represents the capacitor charging. Because of the
longer time-constant, the capacitor does not now become fully charged before the supply
drops to zero volts. when the supply drops to zero, the output also drops by 1 volt, as is
shown between points B and C. The point C is higher than -1 volt. The capacitor now
discharges, and the curve is shown between points C and D. Again the supply rises to +1 volt,
resulting in the curve rising to point E. This point E will be slightly lower than +1 volt. This
situation will continue until the positive and negative output voltage traces will be mirror
images of one another, and the trace level will appear equidistant, above and below the axis.
If the time-constant is much greater than the period, then the resulting waveform is as shown
in Figure 31.
lV
A
B
E
Voltage
VR
F
0.5 V
Time
Voltage
VR
C D
Time
-0.5 V
-l V
Figure 31
As before, the time-constant determines the charging and discharging rate of the capacitor;
therefore with a time-constant greater than the period, the capacitor will be only slightly
charged during the time for which the supply first remains positive. The curve is shown
between points A and B in Figure 31a. when the supply drops to zero, the trace immediately
drops by 1 volt, to a point C, just below the axis. The discharge of the capacitor will be
proportional to the charge, therefore there will only be a slight curve, as shown between
points C and D.
When the supply again goes positive, there is a 1 volt increase in the output voltage, to point
E, which is a value less than +1 volt but more than point B. when the supply is switched off,
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there will be a sharp fall of 1 volt to point G on the curve, which will be more negative than
the point C. This situation continues until the trace of the output voltage, shows not quite a
square-wave, averaging between +0.5 V and –0.5 V as shown in Figure 3lb.
L-R CIRCUIT: OUTPUT TAKEN FROM RESISTOR
Having considered C-R circuits, we will now consider the effect which an circuit has on a
square-wave.
If the square-wave shown in Figure 28 is fed into the L-R circuit shown in Figure 27, with a
time-constant equal to one half of the period (0.5 ms), then the output waveform will be as
shown in Figure 32.
lV
Voltage
VR
Time (mS)
-l V
Figure 32
The output voltage VR rises gradually to the peak voltage. This is due to the back e.m.f.
produced by the inductance, opposing the current in the circuit, ultimately affecting the
output voltage across the resistance; the time-constant determines the rate of rise of the
current.
If the time-constant was reduced to 0.1 ms, then the output voltage VR would rise
exponentially, and sufficiently rapidly to reach almost the input voltage of 1 volt. The trace
on the oscilloscope would be as shown in Figure 33.
lV
Voltage
VR
Time (mS)
Figure 33
Finally if the time-constant is considerably smaller than the period, then the rise of the output
voltage would be considerably faster. The trace on the oscilloscope would nearly resemble
the input square-wave, as shown in Figure 34.
31
lV
Voltage
VR
Time (mS)
-lV
Figure 34
C-R CIRCUIT: OUTPUT TAKEN FROM CAPACITOR
L
R
output
input
input
R
output
C
Figure 35
The C-R circuit we looked at above had the output taken from the resistor. If the output is
taken from the capacitor instead, as shown in Figure 35, it will have a different waveform.
We can take a short cut to finding out what this is by writing down the expression for the
capacitor voltage following switch-on that we saw in a previous handout.
Vc =V( l – e-t/T),where T = CR
This is the same as the resistor voltage in the L-R circuit we looked at above, shown in Figure
35 b.
VR = V(1 – e-t/T), where we now use T = L/R
So the output voltage from the capacitor of a C-R circuit will be the same as the output
voltage from the resistor of an WR circuit with the same time constant. But the current drawn
from the square-wave source at the input is not the same; and in a practical application, the
voltage will only be the same if the inevitable internal resistance of the actual inductor is
much less than R.
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L-R CIRCUIT: OUTPUT TAKEN FROM INDUCTOR
R
C
input
L
output
input
R
output
Figure 36
Similarly, when a voltage is applied to an L-R circuit like that in Figure 36, the voltage across
the inductor was shown, in a previous handout ,to be given by the following expression.
VL = V e-t/T
This will give the same voltage as that from the resistor of a C-R circuit like that in Figure
36b with the same time constant, shown previously to be as follows.
VR = V e-t/T
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PROGRESS QUESTION 14
Sketch the trace obtained of the output voltage across the resistor in a series C-R circuit,
when the circuit is connected to a square-wave supply whose period is very much longer
than the time-constant. Assume a period of 1 ms and a peak -to-peak value of 10 volts.
Check on page
PROGRESS QUESTION 15
Sketch the waveform of the output voltage obtained across a 1 kilohm resistor in series with a
1 H inductor, to which a 5 volt 100 kHz square wave is applied.
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SQUARE WAVES AS TRANSIENTS
We have looked at square waves as a series of transients. But they are really alternating
current with a particular waveform.
We could think of the square wave as a.c., and work out the effects of the various circuits as
follows:
1
Break down the square wave into sine waves of fundamental and odd harmonic
frequencies.
2
Work out the output voltage and phase from the circuit for each frequency.
3
Add together the individual frequencies to find the actual output waveform.
If we had done this, we would have ended up with the same answers after a lot more work
than looking at square waves as a series of transients.
R
C
R
L
R
R
C
L
Figure 37
If we had considered a sine waveform input, we would have taken the other view of these
circuits, and described them as simple filters, where Figure 37a and d are high-pass types,
and Figure 37b and c are low-pass types.
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DIFFERENTIATION AND INTEGRATION
Figure 38
Figure 38 shows a square wave input to a C-R circuit like that in Figure 37a with a short time
constant, and the corresponding output. The output is zero except when the input changes.
when the input increases, the output is positive; and when the input decreases, the output is
negative. This is similar to the mathematical operation of differentiation, that is finding the
rate of change, so this type of circuit, or the L-R circuit of Figure 37d with the same effect, is
often called a differentiator.
Average
voltage
Input
voltage
Time
Output
voltage
Time
Figure 39
Figure 39 shows a square wave input, and the corresponding output from a C-R circuit like
Figure 37c with a long time constant. The output corresponds to the long term trend of the
input, like mathematical integration, so this circuit is often called an integrator. The WR
circuit of Figure 37b has a similar effect in theory, but the resistance of a practical inductor
means that it is used less.
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SUMMARY
Series C-R circuits: output from resistor
In series C-R circuits with a square wave input, the output voltage across the resistor takes
the following forms:
1
With time constants much less than the period of the input, the output is a series of
positive and negative spikes.
2
As the time-constant increases, the spikes on the traces extend into sloping waveforms
which do not reach zero before the next transition of the input.
3
when the time constant is much greater than the period of the input, the output voltage
trace very nearly resembles the input square-wave.
Series L-R circuits: output from resistor
In series C-R circuits with a square wave input, the output voltage across the resistor takes
the following forms:
1
With time constants more than the period of the input, the output is in the form of a
triangular wave of lower amplitude than the input.
2
As the time-constant decreases, then the output rises and falls more rapidly until the
output voltage trace nearly resembles the input square-wave in amplitude, but with
rounded leading edges.
3
When the time constant is much less than the period of the input, the output voltage
trace very nearly resembles the input square wave.
Series C-R circuits: output from capacitor
In series C-R circuits with a square wave input, the output voltage across the capacitor is
similar to that from the resistor of an C-R circuit with the same time constant.
Series L-R circuits: output from Inductor
In series L-R circuits with a square wave input, the output voltage across the inductor is
similar to that from the resistor of an C-R circuit with the same time constant.
Differentiation and integration
A series circuit with a short time constant which responds to changes in the input is called a
differentiator. A series circuit with a long time constant which responds to the long term
trend of the input is called an integrator.
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