Rusko A., Thompson M. Power Quality in Electrical Systems

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−

(a)

(t)

(b)

Harmonic amplitude

of line current

Harmonic number (N)

(c)

7 9 11 13 15

1.0

Figure 5.2 Full-wave rectifier with current source load. (a) The circuit. (b)

Waveforms, showing line-voltage and line current. Line current is a square

wave. (c) The spectrum of the line current, showing the characteristic 1/N

rolloff in spectral component amplitude with harmonic number.

BUS

−

Figure 5.3 A full-wave rectifier with current source load, I

, and capac-

itive filter, C

BUS

Harmonic Current Sources 65

current source load and capacitive filter (Figure 5.4). In this case, we have

a 120-V AC source, a DC load of 5 A, and a capacitive output filter. With

a 1000 F bus capacitor, the load voltage ripple is about 25-V peak-peak

(Figure 5.5a). Note that the line current waveform has significant harmonic

distortion (Figure 5.5b and Figure 5.5c).

Next, we can reduce the load voltage ripple by increasing the filter

capacitor from 1000 µF to 10,000 µF. As shown in Figure 5.6a, note the

peak-peak load voltage ripple is reduced, but at the expense of higher

peak line current (Figure 5.6b) and higher levels of harmonic distortion

in the line current (Figure 5.6c).

In Table 5.1, we see the spectra in both cases, tabulated. The THD for

the rectifier with a 1000 F capacitive filter is approximately 140 per-

cent, and the peak-peak output ripple is 25 V. By increasing the filter

66 Chapter Five

vin+

BUS

1000uF

vin−

vload+

vload−

Figure 5.4 A full-wave rectifier with current source load and capacitive filter

showing PSPICE circuit with bus capacitor 1000 µF and a load that draws a

DC current of 5 A.

−40 V

0 ms

10 ms

V(vload+)

− V(vload−)

20 ms 30 ms 40 ms 50 ms

Time

(a)

60 ms 70 ms 80 ms 90 ms 100 ms

80 V

40 V

0 V

120 V

160 V

200 V

Figure 5.5 A full-wave rectifier with current source load and capacitive filter. (a) The

waveform of the load voltage, which has a ripple due to the load current and the finite

value of the filter capacitor. (b) The line current, which is very “spikey.” (c) The spectrum

of the line current.

capacitor to 10,000 F, we reduce the output ripple to approximately

4 V, but the line current is much more spikey and the THD of the line

current is approximately 265 percent. This result illustrates one trade-off

with this type of rectifier.

Harmonic Current Sources 67

−40 A

100 ms

110 ms 120 ms 130 ms 140 ms 150 ms

(b)

160 ms 170 ms 180 ms 190 ms 200 ms

−20 A

0 A

20 A

40 A

I(vs)

0 A

0 Hz 0.1 KHz 0.2 KHz 0.3 KHz 0.4 KHz 0.5 KHz

Frequency

(c)

0.6 KHz 0.7 KHz 0.8 KHz 0.9 KHz 1.0 KHz

4 A

2 A

6 A

8 A

10 A

I(vs)

Figure 5.5 (Continued)

−40 V

0 ms

10 ms

V(vload+)

− V(vload−)

20 ms 30 ms 40 ms 50 ms

Time

(a)

60 ms 70 ms 80 ms 90 ms 100 ms

80 V

40 V

0 V

120 V

160 V

200 V

Capacitor voltage

−150 A

I(Vs)

130 ms120 ms110 ms100 ms 140 ms 150 ms

Time

(b)

160 ms 170 ms 180 ms 190 ms 200 ms

0 A

−50 A

−100 A

50 A

100 A

150 A

0 A

0 Hz

0.1 KHz 0.2 KHz 0.3 KHz 0.4 KHz 0.5 KHz

Frequency

(c)

0.6 KHz 0.7 KHz 0.8 KHz 0.9 KHz 1.0 KHz

4 A

2 A

6 A

8 A

10 A

I(Vs)

Figure 5.6 A full-wave rectifier with current source load and increased capacitive filter to

10,000 µF. (a) The waveform of the load voltage, which has a reduced ripple compared to

the 1000 µF case. (b) The line current, which is very “spikey.” (c) The spectrum of the line

current.

Three-Phase Rectiﬁers

The six-pulse rectiﬁer

A typical application using a three-phase, six-pulse rectifier is an

adjustable speed drive (Figure 5.7). Three-phase power (labeled phases

a, b, and c) is full-wave rectified by the six-pulse rectifier. The rectified

voltage is filtered by the high-voltage bus capacitor, C

bus

, generating a

DC voltage, which is used by the subsequent inverter. The three-phase

inverter generates the three-phase currents necessary to drive the motor.

A six-pulse rectifier is shown in Figure 5.8a. Assuming that the load

approximates a current source (with very large load inductance), the line

current drawn from the rectifier shows a THD of 31 percent, and an

absence of a third-harmonic and all triplen harmonics (Figure 5.8b). We

can show that the harmonic amplitudes of the phase currents of the ideal

six-pulse rectifier with current source load, I

, are [5.1]:

The spectrum of the AC line currents is shown in Figure 5.8c.

sina

b sina

Harmonic Current Sources 69

TABLE 5.1 Results of the PSPICE Simulation Showing the Fourier Spectrum of Line

Current for Example 5.1

Harmonic current with Harmonic current with

Harmonic number 1000 µF filter capacitor 10,000 µF filter capacitor

1 10.0 10.0

3 9.4 10.0

5 6.9 9.8

7 4.8 9.5

9 3.0 9.0

11 2.2 8.25

13 2.1 7.75

15 2.0 7.0

17 1.65 6.5

19 1.35 6.3

21 1.25 5.0

23 1.2 4.5

25 1.1 3.8

27 1.0 3.3

Diode

rectifier

DC link bus

capacitor PWM inverter

Motor

Figure 5.7 Adjustable speed drive for a three-phase induction motor.

The twelve-pulse rectiﬁer

The twelve-pulse rectifier (Figure 5.9a) is comprised of two six-pulse rec-

tifiers fed from separate transformers. One six-pulse is fed from a Y/Y

transformer, and the other is fed from a ∆/Y transformer. The two recti-

fier voltages are phase shifted 30 from one another. The resultant phase

current waveform (Figure 5.9b) more closely mimics an ideal sine wave

70 Chapter Five

(a)

−

(t)

−I

90° 180° 270°

(b)

360°

Spectrum for ideal three-phase bridge

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Harmonic number (N)

(c)

Amplitude

Figure 5.8 A six-pulse rectifier. (a) The circuit. (b) The line current for L/R  period

of line frequency. (c) The spectrum of the line current, assuming I

 1.

than in the six-pulse case. This is because the 12-pulse topology elimi-

nates the 5th, 7th, 17th, and 19th harmonics, leaving the 11th, 13th,

23rd, and 25th harmonics.

High-Frequency Fluorescent Ballasts

High-frequency fluorescent ballasts (Figure 5.10) are a rich source of high-

frequency voltage and current harmonics. The line waveform is rectified

by a power-factor corrected boost converter. The rectified DC voltage

powers a high-frequency inverter that generates the high voltage needed

Harmonic Current Sources 71

(a)

Y/Y

Six-pulse

rectifier

∆/Y

Six-pulse

rectifier

LOAD

(b)

(t)

Figure 5.9 A twelve-pulse rectifier. (a) The circuit. (b) The line current for

phase-a for L/R  period of line frequency.

Inverter Lamp

BUS

Power

factor

correction

boost

Figure 5.10 High-frequency switching fluorescent ballast with a power

factor correction front end.

by the lamp. The power factor correction circuit switches at a frequency

much higher than line frequency. Usually, the line side has passive filter-

ing to help reduce the high-frequency harmonics drawn from the AC line.

Transformers

One source of harmonics using transformers with iron cores is the non-

linear B/H curve associated with these devices. Figure 5.11 shows a

representative B/H curve for electrical steel that may be used in a trans-

former core. Not shown here, purposely, is the hysteresis loop, since for

simplicity we’ll consider a single-valued B/H curve. Note that the core

material saturates at a magnetic flux density B ~ 2 Tesla or so.

The slope of the B/H curve at any point is the magnetic permeability

of the core material, or

From this equation, note that as the core material saturates, the mag-

netic permeability decreases. For economic reasons, power transformers

are often operated at a relatively high magnetic flux density, above the

B/H curve “knee.”

Shown in Figure 5.12a is a magnetic circuit without an airgap. The

magnetic circuit is comprised of N turns, the core material has magnetic

permeability m

, and the cross-sectional area of the core is A with a char-

acteristic length, l

, which is the mean path length of the flux lines inside

72 Chapter Five

B/H curve for electrical steel

0.5

1.5

2.5

1 10 100 1000 10000 100000

H, A-turns/m

B, Tesla

Figure 5.11 A representative nonlinear B/H curve.

the core. From magnetic circuit analysis, we can find the inductance of this

structure as:

We’ll now assume that we excite this circuit with sufficient volt-seconds

so the core material begins saturation. Equivalently, saturation means

the permeability of the core decreases with a corresponding decrease

in the inductance at high current levels. This nonlinear property of

the magnetic core requires that the current has harmonic distortion,

as shown in Figure 5.12b.

Other Systems that Draw Harmonic Currents

High-frequency switch-mode power supplies are covered extensively in

Chapter 7. Other sources of harmonic currents are adjustable speed

drives (ASDs), motors, and arc furnaces. Figure 5.13 illustrates the typ-

ical line current and spectrum for an adjustable speed drive.

L 5

Harmonic Current Sources 73

Mean path length l

Area A

Permeability µ

v(t)

(a)

i(t)

v(t)

i(t)

(b)

Figure 5.12 An inductor that draws nonlinear current. (a) The circuit. The induc-

tor has N turns; a mean path length, l

; core permeability, m

; and a core cross-

sectional area, A

. (b) The waveforms of inductor voltage, v(t), and the inductor

current, i(t).

Summary

In this chapter, we’ve covered equipment that generates harmonics, such

as rectifiers, transformers, and switching power supplies. Harmonics

can cause many detrimental effects, including resonances with power

factor correction capacitors, the heating of neutral conductors, the false

tripping of relaying equipment, and the heating of capacitors. In the

next chapter, we’ll discuss power harmonic filters.

References

[5.1] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, second edi-

tion, Springer, 2001.

[5.2] IEEE, “IEEE Recommended Practice for Monitoring Electric Power Quality,” IEEE

Std. 1159-1995.

74 Chapter Five

−50

−30

−10

40 50 60

Time (ms)

70 80

Mpr : 313.517

Mr : 171.304

Avg : 33.5352

Acr : 315.907

RVS : 54.3862

Of : 4.87557

FF : 1.52507

0 600

ASD input current

Frequency (Hz)

Current (A)

1200 1800

Freq : 50

Fwrd : 19.4713

THD : 41.0054

RMSA : 315.907

RMS : 31.0448

ASUM : 37.3558

TIF : 427.213

IT : 8990.51

Figure 5.13 The input line current and spectrum for an adjustable speed drive [5.2].