Harmonic Sources

...

. ...

Transformer dc winding voltages

Current in dc winding 5

FIGURE 2. Voltage and current waveforms for Figure l a

The harmonic "spectra" represented by Equations and 2 are identical except that alternate harmonic pairs and 17th and are negative in Equation 1 while all are positive in Equation 2. This suggests that if the two types of rectifier are paralleled, with equal line currents, these alternate pairs of harmonics will cancel in the common source path.

When two equally rated rectifiers are combined, one with a transformer phase shift from the other. the resulting system is said to operate "12-pulse," or with a pulse number of 12. Additional rectifier units equally phase shifted from one another create other pulse numbers. Three units phase shifted from one another by 20' constitute an 18-pulse system; four units separated by result in 24-pulse operation. When the pulse number q is used in Equation 4, the orders of the harmonics may be determined. Table 1 gives the harmonics for q = 6, 18, and 24. Note also that the magnitude of the harmonic current is

I 3

6

Current in rectifytng elements

Harmonics

1

Characteristic AC Supply Line Harmonic Currents in 6-, 12-, 18-, and 24-Pulse Rectifiers

6-Pulse Rectifier Harmonic Current

and that only the following "characteristic" harmonics appear under ideal conditions:

= kq 1

Here, k is any integer, is the amplitude of the fundamental com- ponent, and q is the pulse number.

The square-cornered wave shapes of Figure 2 are possible only with idealized zero commutating reactance. The current waveforms will be modified by finite amounts of commutating reactance and phase retard. These two factors will affect the magnitude and phase angle of each har- monic but the same harmonic orders will prevail. For circuits such as those in Figure Figure 30 shows how finite commutating reactance results in a gradual rise time and fall time represented by the angle Controlling the dc voltage by thyristor phase retard delays the time of commutation by the angle a and reduces the commutation angle

(Figure The reduction in further modifies the harmonic ampli- tudes.

- - - -

Harmonic Sources ₃₃₅

I I _{I I}

60 120 240 300 FIGURE 3. of phase

degrees and overlap.

Harmonic current magnitudes in percent of the fundamental are tabu- lated in Table 1. The column headed "Theoretical" follows Equation 3, The other columns list calculated magnitudes for an assumed com- mutating reactance of 0.15 per unit one for phase retard angle of a

and the other for a Magnitudes for other values of and can be calculated using the classical equations of Fourier analysis.

With finite "real commutating reactance harmonic currents are reduced. presents curves of harmonic current amplitude versus for various angles of phase retard for harmonics up to the 25th. The last two columns of Table were taken from the printout of a com- puter calculation based on the classical equations relating these quantities.

It is not to be inferred from Table 1 that complete harmonic

occurs for the harmonics "eliminated" by higher than 6-pulse opera- tion. There will always be residual harmonics due to imperfect transformer phase shift, impedance unbalances, or unequal loadings or phase retard angles in the rectifiers. Under reasonably balanced condi- tions, residual harmonics are often assumed to be 10 to 20% of the value present in a 6-pulse rectifier of the same total rating. Under unbalanced conditions, greater amounts of residual harmonics are produced. For e r - ample, if one unit of a 4-unit balanced 24-pulse rectifier system is re- moved for service, the harmonics will be those of a 12-pulse rectifier a 6-pulse rectifier. Aluminum rectifier systems are always rated to deliver full power with one unit out of service. The harmonic duty of capacitor banks should always be calculated for projected condi- tions with the highest degree of unbalance.

As well as solid state rectifiers, there are other examples of switching devices that control the power in different types of loads. The ac

Harmonics

336

tor controller described in Chapters 5 and 6 is used for heating control in furnaces and other of resistive loads. We have already seen in Chapter 5 an example of the harmonics generated by thyristor-controlled reactors. Cycloconverters, frequency converters, and inverters are among the less common solid-state power converters found in power systems in large sizes, and these all contribute harmonics in varying degrees.

Unlike the harmonics of static power converters (such as rectifiers). which can be calculated from periodic waveforms, the harmonics generat- ed by electric arc furnaces are unpredictable because of the cycle-by-cycle variation of the arc, particularly when boring into new, scrap. The arc current is nonperiodic, and analysis reveals a continuous spectrum of har- monic frequencies of both integer and noninteger orders. Even so, har- monic measurements have shown that integer-order harmonic frequen- cies. particularly the and predominate over the noninteger ones, and that the amplitude of the harmonics decreases with order. Fourier analysis of typical magnetic tape recordings taken of arc furnace currents yielded the harmonic percentages in Table 1 of Chapter 9. It will be noted from that table, which was for selected maximum arc activity periods of five consecutive 5-cycle periods, that low order harmonics pre- vail, and that even harmonics are present. Later, when the arc is steady, and the surface of the molten steel is flat, the harmonic magnitudes de- crease considerably and the even harmonics virtually disappear. Figure 4 is a trace of the spectral response measured by means of a FFT- computing spectrum analyzer in the primary of a furnace transformer while the arc is boring into new scrap (active arcing). Distinct lines for integer orders, as well as a high threshold of noninteger orders. are seen. The frequency range is 0-500 Hz and the magnitudes, displayed on a log- arithmic scale, are peaks averaged over several seconds. In most cases the harmonic magnitudes just reported would not by themselves be trou- blesome to power systems were it not for the possibility of their amplification by resonance of the power capacitors almost always associat- ed with the inherently low power-factor arc furnaces.

FIGURE 4. Spectrum of arc furnace line current during active arcing.