Harmonic Sources
3.10 Modulated Phase Control
3.11.2 Integral Cycle Control
va
vo
vo
Figure 3.75 Waveforms of the single-phase regulator of Figure 3.74
3.11.2 Integral Cycle Control
Instead of point on wave switching selection, this type of control is based on the switching of entire voltage half-cycles using a pair of back-to-back thyristors, as shown in Figure 3.77. It is often called ‘burst-firing’ and has found application in long time constant loads (e.g. temperature control in electric ovens).
The fundamental supply frequency cannot be used as a basis for the Fourier analysis in this case, because the period of repetition, and thus the lowest frequency produced, is now a variable subharmonic frequency.
If the number of ON cycles is N and the number of cycles over which the pattern is repeated is M the period of repetition is M/f , where f is the supply frequency.
The lowest frequency, which now becomes the fundamental frequency, is f/M Hz.
With reference to this lowest frequency, the current being analysed can be expressed as
i= Imaxsin(Mωt) (3.106)
Using the time reference indicated in Figure 3.77(b), the A0 and An Fourier coeffi-cients are zero, i.e. there are only sine terms, their general expression being
Bn= 2 π
2πf/M 0
[−Imaxsin(Mωt) sin(nωt)] d(ωt)
= −Imax·2M
π · sin((N/M)nπ )
M2− n2 (3.107)
Consider as an example the case where M = 5 and N = 1.
1.0
V0
V1
V3
V5
V7 V9 0.8
0.6
0.4
0.2
00 30 60 90
Angle of delay, a (degrees)
V Va
120 150 180
Figure 3.76 Harmonic content of the a.c. regulator of Figure 3.74
vs vL
vT
(a)
(b) iL
R
Imax
0 2p/10 2p x
Time = 5 × 1 s 50
Figure 3.77 Integral cycle control: (a) basic circuit; (b) voltage waveform
140 HARMONIC SOURCES
The lowest repetition frequency for a 50 Hz supply is therefore
f1 = f M = 50
5 = 10, (3.108)
i.e. n= 1 corresponds with 10 Hz.
The per unit levels of the various frequencies present, obtained from equation (3.107), are
f1(10 Hz)= 0.078 f8(80 Hz)= 0.078 f2(20 Hz)= 0.14 f9(90 Hz)= 0.033 f3(30 Hz)= 0.189 f10(100 Hz)= 0 f4(40 Hz)= 0.208 f11(110 Hz)= 0.019 f5(50 Hz)= 0.2 f12(120 Hz)= 0.025 f6(60 Hz)= 0.17 f13(130 Hz)= 0.021 f7(70 Hz)= 0.126 f14(140 Hz)= 0.01
When n is multiple of M the coefficients are zero, i.e. for 100 Hz, 150 Hz, etc.
This clearly shows that integral cycle control produces no harmonic frequencies. It does, however, produce inter-harmonic and subharmonic frequencies.
3.12 Discussion
Within the normal operating range the harmonic content of the transformer magnetising current is not significant. It is only during energisation and when operating above their normal voltage that transformers can considerably increase their harmonic contribution.
Similarly, the harmonic content of the internal e.m.f. of well-designed synchronous machines is also small. Rotor saliency in the presence of transmission system unbal-ance and/or load-injected harmonic currents is likely to be the main source of generator e.m.f. distortion. Induction motors produce time harmonics as a result of the har-monic content of the m.m.f. distribution, and these are speed dependent. However, the impact on the power system is small, partly due to operational diversity and partly to the smaller rating of these machines, when compared with the synchronous generators.
Power electronic devices constitute the main sources of harmonic current distortion.
The characteristics of the main power electronic devices have been described in this chapter with reference to their harmonic contribution under the assumption of specified terminal conditions. For most applications this is an acceptable approximation. How-ever the input terminal conditions (normally the voltage waveform) can change as a result of the interaction that exists between the nonlinear power electronic component and the rest of the system. A more rigorous analysis of the harmonic sources taking this effect into account is described in Chapter 8.
3.13 References
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142 HARMONIC SOURCES
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