Download Complex Waveforms

Complex Waveforms
At the end of this section you should be able to:
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define a complex wave
recognise periodic functions
recognise the general equation of a complex waveform
use harmonic synthesis to build up a complex wave
recognise characteristics of waveforms containing odd, even or odd and even
harmonics, with or without phase change
calculate rms and mean values, and form factor of a complex wave
calculate power associated with complex waves
perform calculations on single-phase circuits containing harmonics
define and perform calculations on harmonic resonance
list and explain some sources of harmonics
In previous work, a.c. supplies have been assumed to be sinusoidal, this being a form of
alternating quantity commonly encountered in electrical engineering. However, many supply
waveforms are not sinusoidal. For example, sawtooth generators produce ramp waveforms,
and rectangular waveforms may be produced by multivibrators. A waveform that is not
sinusoidal is called a complex wave. Such a waveform may be shown to be composed of the
sum of a series of sinusoidal waves having various interrelated periodic times.
A function f(t) is said to be periodic if f(t + T), f(t) for all values of t, where T is the interval
between two successive repetitions and is called the period of the function f(t). A sine wave
having a period of 2/ is a familiar example of a periodic function.
A typical complex periodic-voltage waveform, shown in Figure 1, has period T seconds and
frequency f hertz. A complex wave such as this can be resolved into the sum of a number of
sinusoidal waveforms, and each of the sine waves can have a different frequency, amplitude
and phase.
The initial, major sine wave component has a frequency f equal to the frequency of the
complex wave and this frequency is called the fundamental frequency. The other sine wave
components are known as harmonics, these having frequencies which are integer multiples of
frequency f. Hence the second harmonic has a frequency of 2f, the third harmonic has a
frequency of 3f, and so on. Thus if the fundamental (i.e. ,supply) frequency of a complex
wave is 50 Hz, then the third harmonic frequency is 150 Hz, the fourth harmonic frequency is
200 Hz, and so on.
150
100
50
-100
-150
Figure 1
t
3t
/4
t/2
0
-50
t/4
0
The General Equation of a Complex Waveform.
The instantaneous value of a complex voltage wave v acting in a linear circuit may be
represented by the general equation
v = V1m sin(t + 1) + V2m sin(2t + 2)+….…+ Vnm sin(nt + n)volts
(1)
Here V1m sin(t + 1) represents the fundamental component of which V1m is the maximum
or peak value, frequency, f = /2 and 1 is the phase angle with respect to time, t = 0.
Similarly, V2m sin(2t + 2) represents the second harmonic component, and Vnm sin(nt +
n) represents the nth harmonic component of which Vnm is the peak value, frequency =
n/2 (= nf) and n is the phase angle.
In the same way, the instantaneous value of a complex current i may be represented by the
general equation
i = I1m sin(t + 1) + I2m sin(2t + 2)+….…+ Inm sin(nt + n)volts
(2)
Where equations (1) and (2) refer to the voltage across and the current flowing through a
given linear circuit, the phase angle between the fundamental voltage and current is 1 = (11), the phase angle between the second harmonic voltage and current is 2 = (2 - 2) and so
on.
It often occurs that not all harmonic components are present in a complex waveform.
Sometimes only the fundamental and odd harmonics are present, and in others only the
fundamental and even harmonics are present.
Harmonic synthesis
Harmonic analysis is the process of resolving a complex periodic waveform into a series of
sinusoidal components of ascending order of frequency. Many of the waveforms met in
practice can be represented by mathematical expressions similar to those of equations (1) and
(2), and the magnitude of their harmonic components together with their phase may be
calculated using Fourier series (see Higher Engineering Mathematics). Numerical methods
are used to analyse waveforms for which simple mathematical expressions cannot be
obtained. In a laboratory, waveform analysis may be performed using a waveform analyser
which produces a direct readout of the component waves present in a complex wave.
By adding the instantaneous values of the fundamental and progressive harmonics of a
complex wave for given instants in time, the shape of a complex waveform can be gradually
built up. This graphical procedure is known as harmonic synthesis (synthesis meaning the
putting together of parts or elements so as to make up a complex whole').
A number of examples of harmonic synthesis will now be considered.
Example 1
Consider the complex voltage expression given by
vm = 100 sin t + 30 sin 3t volts
The waveform is made up of a fundamental wave of maximum value 100V and frequency,
f = /2 hertz and a third harmonic component of maximum value 30V and frequency =
3/2(=3f), the fundamental and third harmonics being initially in phase with each other.
Since the maximum value of the third harmonic is 30 V and that of the fundamental is 100 V,
the resultant waveform va is said to contain 30/100, i.e., ‘30% third harmonic’. In Figure 2,
the fundamental waveform is shown by the broken line plotted over one cycle, the periodic
time being 2/ seconds. On the same axis is plotted 30sin3t, shown by the dotted line,
having a maximum value of 30 V and for which three cycles are completed in time T
seconds. At zero time, 30sin3wt is in phase with 100 sint.
150
100
50
t
3t
/4
t/2
-50
t/4
0
0
t/1
2
Figure 2
-100
-150
The fundamental and third harmonic are combined by adding ordinates at intervals to
produce the waveform for va as shown. For example, at time T/12 seconds, the fundamental
has a value of 50 V and the third harmonic a value of 30 V. Adding gives a value of 80 V for
waveform va at time T/12 seconds. Similarly, at time T/4 seconds, the fundamental has a
value of 100 V and the third harmonic a value of -30 V. After addition, the resultant
waveform va is 70 V at time T/4. The procedure is continued between t 0 and r T to produce
the complex waveform for va. The negative half-cycle of waveform va is seen to be identical
in shape to the positive half-cycle.
Example 2
Consider the addition of a fifth harmonic component to the complex waveform of Figure.2,
giving a resultant waveform expression
vb = 100sin t + 30sin3t + 20 sin 5t volts
150
100
50
-100
-150
Figure 3
t
3t
/4
t/2
-50
t/4
0
0
Figure 3 shows the effect of adding (100sint +
30sin3t) obtained from Figure 2 to 20sint volts.
The shapes of the negative and positive half-cycles
are still identical If further odd harmonics of the
appropriate amplitude and phase were added to vb a
good approximation to a square wave would result.
Example 3
Consider the complex voltage expression given by
=
l00sint + 30sin (3t + /2)volts
This expression is similar to voltage va in that the peak value of the fundamental and third
harmonic are the same. However the third harmonic has a phase displacement of /2 radians
leading (i.e., leading 30 sin 3wt by /2 radians). Note that. since the periodic time of the
fundamental is T seconds, the periodic time of the third harmonic is T/3 seconds, and a phase
150
100
50
t
3t
/4
t/2
-50
t/4
0
t/1
2
0
-100
-150
displacement of /2 radian or 1/4 cycle of the third harmonic represents a time interval of
(T/3)  1/4, i.e., T/12 seconds.
Figure 4 shows graphs of 100 sint and 30sin(3t + /2) over the time for one cycle of the
fundamental. When ordinates of the two graphs are added at intervals, the resultant waveform
vc is as shown. The shape of the waveform vc is quite different from that of waveform va
shown in Figure 2, even though the percentage third harmonic is the same. If the negative
half-cycle in Figure 4 is reversed it can be seen that the shape of the positive and negative
half-cycles are identical.
Example 4
Consider the complex voltage expression given by vd = 100 sin t + 30 sin (3t - /2) volts
150
100
50
-100
-150
Figure 5
t
3t
/4
t/2
0
-50
t/4
0
The fundamental, 100 sin t and the third harmonic component 30 sin (3t - /2) are plotted
in Figure 5, the latter lagging 30 sin (3t - /2) by /2 radian or T/12 seconds. Adding
ordinates at inter-gives the resultant waveform vd as shown. The negative half-cycle vd is
identical in shape to the positive half-cycle.
Example 5
Consider the complex voltage expression given by ve = 100 sin t + 30 sin (3t + ) volts
150
100
50
t
3t/4
t/2
-50
t/4
0
0
-100
-150
Figure 6
The fundamental, 100 sin t and the third harmonic component + 30 sin (3t + ), are
plotted as shown in Figure 6, the latter leading + 30sin(3t + ) by  radian or T/6 seconds.
Adding ordinates at inter gives the resultant waveform ve as shown. The negative half-cycle o
is identical in shape to the positive half-cycle.
Example 6
Consider the complex voltage expression given by 100 sin t - 30 sin (3t + /2) volts figure
7c
The phasor representing 30sin(3t + /2) is shown in Figure 7(a) at time t = 0. The phasor
representing -30sin(3t + ) is shown in Figure 7(b) where it is seen to be in the opposite
direction to that shown in Figure 7(a).
3 rads-1
150
30 V
/2 rads
Figure 7a
100
50
Figure 7b
-100
-150
Figure 7c
30 V
3 rads-1
t
3t
/4
t/2
0
-50
 rads
t/4
0
-30sin(3t + /2) is the same as 30sin(3t - /2). Thus
vf = 100sint - 30 sin (3t + /2) = 100sint + 30 sin (3t + /2)
The waveform representing this expression has already been plotted in Figure 5 and is
repeated in figure 7c.
General conclusions on examples 1 to 6
Whenever odd harmonics are added to a fundamental waveform, whether initially in phase
with each other or not, the positive and negative half-cycles of the resultant complex wave
are identical in shape (i.e., in Figures 2 to 7, the values of voltage in the third quadrant between T/2 seconds and 3T/4 seconds - are identical to the voltage values in the first
quadrant - between 0 and T/4 seconds, except that they are negative, and the values of voltage
in the second and fourth quadrants are identical, except for the sign change). This is a feature
of waveforms containing a fundamental and odd harmonics and is true whether harmonics are
added or subtracted from the fundamental.
From Figures 2 to 6, it is seen that a waveform can change its shape considerably as a result
of changes in both phase and magnitude of the harmonics.
Example 7
Consider the complex current expression given by ia =10 sin t + 4 sin 2t amperes
Current ia consists of a fundamental component, 10sint, and a second harmonic component,
4 sin 2t, the components being initially in phase with each other. Current ia contains 40%
second harmonic. The fundamental and second harmonic are shown plotted separately in
Figure 8. By adding ordinates at intervals, the complex waveform representing i a is produced
as shown. It is noted that if all the values in the negative half-cycle were reversed then this
half-cycle would appear as a mirror image of the positive half-cycle about a vertical line
drawn through time, t = T/2.
15
10
5
-10
-15
Figure 8
t
3t
/4
t/2
-5
t/4
0
0
Example 8
Consider the complex current expression given by :
ib =10 sin t + 4 sin 2t + 3sin4t amperes
The waveforms representing (10 sin t + 4 sin 2t) and the fourth harmonic component,
3sin 4t are each shown separately in Figure 33.% the former waveform having been
produced in Figure 8. By adding ordinates at intervals, the complex waveform for ib is
produced as shown in Figure 9. If the half-cycle between times T/2 and T is reversed then it is
seen to be a mirror image of the half-cycle lying between 0 and T/2 about a vertical line
drawn through the time, t = T/2.
15
10
5
t
3t
/4
t/2
-5
t/4
0
0
-10
-15
Figure 9
Example 9
Consider the complex current expressions given by iC = 10 sin t + 4 sin(2t + /2) amperes
The fundamental component,10 sin t, and the second harmonic component, having an
amplitude of 4A and a phase displacement of /2 radian leading (i.e., leading 4 sin2t by
15
10
5
t
3t
/4
t/2
-5
t/4
0
0
-10
-15
-20
Figure 10
/2 radian or T/8 seconds), are shown plotted separately in Figure 10. By adding ordinates at
intervals, the complex waveform for iC is produced as shown. The positive and negative halfcycles of the resultant waveform iC are seen to be quite dissimilar.
Example 10
Consider the complex current expression given by id = 10 sin t + 4 sin(2t + ) amperes.
The fundamental, 10 sin t, and the second harmonic component which leads 4 sin2t by 
radians are shown separately in Figure11. By adding ordinates at intervals, the resultant
waveform id is produced as shown. If the negative half-cycle is reversed, it is seen to be a
15
10
5
t
/4
3t
t/2
-5
t/4
0
0
-10
-15
Figure 11
mirror image of the positive half-cycle about a line drawn vertically through time, t = T/2.
General conclusions on examples 7 to 10
Whenever even harmonics are added to a fundamental component
(a)
if the harmonics are initially in phase or if there is a phase-shift of  radians, the
negative half-cycle, when reversed, is a mirror image of the positive half-cycle about
a vertical line drawn through time,t T1
(b)
if the harmonics are initially out of phase with each other (i.e., other than  radians),
the positive and negative half-cycles are dissimilar.
These are features of waveforms containing the fundamental and even harmonics.
Example 11
Consider the complex voltage expression given by vg = 50sint +25sin2t + 15sin3t volts
The fundamental and the second and third harmonics are each shown separately in Figure 12.
By adding ordinates at intervals, the resultant waveform vg produced as shown. If the
negative half-cycle is reversed, it appears as a mirror image of the positive half-cycle about a
vertical line drawn through time = T/2.
80
60
40
20
t
t/2
0
t/4
-20
3t
/4
0
-40
-60
-80
Figure 12
Example 12
Consider the complex voltage expression given by
vh = 50sint +25sin(2t - ) + 15sin (3t + /2)volts
The fundamental, the second harmonic lagging by  radian and the third harmonic leading by
/2 radian are initially plotted separately, as shown in Figure13. Adding ordinates at intervals
gives the resultant form vh as shown. The positive and negative half-cycles are see to be quite
dissimilar.
100
80
60
40
20
t
3t
/4
t/2
-20
t/4
0
0
-40
-60
General conclusions on examples 11 and 12
Whenever a waveform contains both odd and even harmonics:
(a)
if the harmonics are initially in phase with each other, the negative cycle, when
reversed, is a mirror image of the positive half about a vertical line drawn through
time, t = T/2;
(b)
if the harmonics are initially out of phase with each other, the positive and negative
half-cycles are dissimilar.
Example 13
Consider the complex current expression given by
i = 32 + 50sint + 20sin (2t - /2) mA
The current i comprises three components - a 32 mA d.c. component, a fundamental of
amplitude 50 mA and a second harmonic of amplitude 20 mA, lagging by /2 radian. The
fundamental and second harmonic are shown separately in Figure 14. Adding ordinates at
interval the complex waveform 50 sin t + 20 sin(2t - (/2)).
t
3t
/4
t/2
0
120
100
80
60
40
20
0
-20
-40
-60
t/4
This waveform is then added to the 32 mA d.c. component to produce the waveform i as
shown. The effect of the d.c. component is seen to be to shift the whole wave 32 mA upward.
The waveform approaches that expected from a half-wave rectifier .
Problem 1. A complex waveform v comprises a fundamental voltage of 240 V rms and
frequency 50 Hz, together with a 20% third harmonic which has a phase angle lagging by
34 rad at time 0. (a) Write down an expression to represent voltage V. (b) Use harmonic
synthesis to sketch the complex waveform representing voltage V over one cycle of the
fundamental component.
(a)
A fundamental voltage having an rms value of 240 V has a maximum value, or
amplitude of (V2)(240), i.e., 339.4 V.
If the fundamental frequency is 50 Hz then angular velocity,  = 2f = 2(50) = 100 rad/s.
Hence the fundamental voltage is represented by 339.4 sin l00t volts. Since the fundamental
frequency is 50 Hz, the time for one cycle of the fundamental is given by T = 1/f = 1/50 s or
20 ms.
The third harmonic has an amplitude equal to 20% of 339.4 V, i.e., 67.9 V. The frequency of
the third harmonic component is 3 x 50 150 Hz, thus the angular velocity is 2 (150), i.e.,
300 rad/s. Hence the third harmonic voltage is represented by 67.9 sin(300t - (3/4)) volts.
Thus
voltage, V = 339.4 sin 100t + 67.9 sin (300t - 3/4) volts
(b)
One cycle of the fundamental, 339.4 sin l00t, is shown sketched in Figure15,
together with three cycles of the third harmonic component, 67.9 sin(300t (3/4))
initially lagging by 3t/4 rad. By adding ordinates at intervals, the complex waveform
representing voltage is produced as shown. If the negative half-cycle is reversed, it is
seen to be identical to the positive half-cycle, which is a feature of waveforms
containing the fundamental and odd harmonics.
500
400
300
200
100
t
3t
/4
t/2
0
t/4
0
-100
-200
-300
-400
-500
Problem 2. For each of the periodic complex waveforms shown in Figure 16, suggest
whether odd or even harmonics (or both) are likely to be present.
(a)
If in Figure16(a) the negative half-cycle is reversed, it is seen to be identical to the
positive half-cycle. This feature indicates that the complex current waveform is composed of
a fundamental and odd harmonics only (see examples 1 to 6).
(b)
In Figure16(b) the negative half-cycle is quite dissimilar to the positive half-cycle.
This indicates that the complex voltage waveform comprises either
(i)
a fundamental and even harmonics, initially out of phase with each other (see
example 9), or
a fundamental and odd and even harmonics, one or more of the harmonics being
initially out of phase (see example 12)
(ii)
(c)
If in figure 16c the negative half-cycle is reversed, it is seen to be a mirror image of
the positive half-cycle about a vertical line drawn through time T/2. This feature
indicates that the complex e.m.f. waveform comprises either:
(i)
a fundamental and even harmonics initially in phase with each other (see examples 7
and 8), or
a fundamental and odd and even harmonics, each initially in phase with each other.
(see example 11).
(ii)