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Complex Waveforms At the end of this section you should be able to: 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(2t + 2)+….…+ Vnm sin(nt + 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(2t + 2) represents the second harmonic component, and Vnm sin(nt + 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(2t + 2)+….…+ Inm sin(nt + 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 = (11), 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 3t 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 30sin3t, 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 sint. 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 + 30sin3t + 20 sin 5t 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 (100sint + 30sin3t) obtained from Figure 2 to 20sint 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 = l00sint + 30sin (3t + /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 sint and 30sin(3t + /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 (3t - /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 (3t - /2) are plotted in Figure 5, the latter lagging 30 sin (3t - /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 (3t + ) 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 (3t + ), are plotted as shown in Figure 6, the latter leading + 30sin(3t + ) 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 (3t + /2) volts figure 7c The phasor representing 30sin(3t + /2) is shown in Figure 7(a) at time t = 0. The phasor representing -30sin(3t + ) 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(3t + /2) is the same as 30sin(3t - /2). Thus vf = 100sint - 30 sin (3t + /2) = 100sint + 30 sin (3t + /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 2t amperes Current ia consists of a fundamental component, 10sint, and a second harmonic component, 4 sin 2t, 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 2t + 3sin4t amperes The waveforms representing (10 sin t + 4 sin 2t) and the fourth harmonic component, 3sin 4t 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(2t + /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 sin2t 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(2t + ) amperes. The fundamental, 10 sin t, and the second harmonic component which leads 4 sin2t 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 = 50sint +25sin2t + 15sin3t 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 = 50sint +25sin(2t - ) + 15sin (3t + /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 + 50sint + 20sin (2t - /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(2t - (/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 34 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, = 2f = 2(50) = 100 rad/s. Hence the fundamental voltage is represented by 339.4 sin l00t 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(300t - (3/4)) volts. Thus voltage, V = 339.4 sin 100t + 67.9 sin (300t - 3/4) volts (b) One cycle of the fundamental, 339.4 sin l00t, is shown sketched in Figure15, together with three cycles of the third harmonic component, 67.9 sin(300t (3/4)) initially lagging by 3t/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)