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

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Harmonics and Interharmonics 45

Harmonic amplitude

123456

Harmonic number

789

3π

5π

7π

9π

(b)

Figure 4.1 (Continued)

−1

v(t)

(a)

Harmonic amplitude

123456

Harmonic number

789

9π

25π

(b)

Figure 4.2 Periodic waveforms. (a) A triangle wave. (b) The spec-

trum of a triangle wave.

We see that the harmonics of the triangle wave (Figure 4.2b) fall off

at a faster rate than those of a square wave. This makes sense since the

triangle wave more closely resembles a pure sine wave than a square

wave, and therefore has fewer harmonics than the square wave.

Next, we’ll build up a square wave from its constitutive harmonics.

Shown in Figure 4.3 are the first three harmonics of a square wave (top

three traces) and the resultant wave when the three harmonics are

added (bottom trace).

Another waveform often encountered in power systems is the trape-

zoidal waveform (Figure 4.4). This waveform models a switching wave-

form with a finite risetime and falltime. The Fourier series for this

waveform is given by [4.3]:

The spectrum for this switching waveform (Figure 4.4) has frequency

components at multiples of the switching frequency f

, where f

is the

inverse of the switching period, or f

 1/T. The amplitude of the har-

monics falls off at a rate of –20 dB/decade in the frequency range

between f

and f

, while above f

the harmonic amplitudes fall off at a

std 5 a



N51,2,3c

sin pN a

pN a

≤±

sin pN a

pNpN a

≤cos a

2pNt

46 Chapter Four

−1

0 0.005 0.01 0.015 0.02 0.025 0.03

−1

0 0.005 0.01 0.015 0.02 0.025 0.03

−1

0 0.005 0.01 0.015 0.02 0.025 0.03

−1

0 0.005 0.01 0.015 0.02 0.025 0.03

Figure 4.3 The first three harmonics that make up a square wave. Shown are the first har-

monic at 60 Hz (top trace), third and fifth harmonics, and the total waveform (bottom trace)

that is the sum of the three harmonics.

This equation assumes the risetime and falltime of the trapezoid are the same.

rate of 40 dB/decade. It can be shown that the two corner frequencies f

and f

are found by [4.4], [4.5]:

Root-mean square

Root-mean square is a measure of the heating value of a periodic waveform

when this periodic waveform drives a resistive load. Mathematically, the

Harmonics and Interharmonics 47

(a)

10%

(t)

90%

−20 dB

Decade

(b)

10f

11f

−40 dB

Decade

Figure 4.4 Trapezoidal waveform. (a) Current waveform that may

be produced by a DC/DC converter or other electronic system.

(b) Spectrum.

root-mean square or rms of a periodic voltage waveform is expressed as:

where we see inside the radical that we first square the waveform, and

then take the mean value (or average) of the waveform over one period.

For a sine wave of peak value V

, the rms value is

For a square wave (with no DC value) as shown in Figure 4.1a, the

rms value is the peak value of the square wave. The rms value of a

waveform can be interpreted by considering power dissipation. Looking

at Figure 4.5, we see a 120-V DC battery driving a 10- load, and a

120-V AC source (with rms value 120 V) driving a 10- load. The

power dissipation in both loads is the same at 1440 W.

In the following, we’ll discuss a few commonly encountered wave-

forms in power systems and power electronics, and their corresponding

rms (root-mean square) values [4.6]. Remember that the rms value of a

periodic waveform is the square root of the average value of the square

of the waveform over a period. For a periodic current i(t), the corre-

sponding rms value is

rms

stddt

rms

[vstd]

48 Chapter Four

120 V DC

P = 1440 W P = 1440 W

120 V AC

−

10 Ω 10 Ω

Figure 4.5 Illustration of the meaning of rms. The power dissipation in both

cases is the same.

i(t)

Figure 4.6 DC current.

DC current

A DC current (Figure 4.6) has an rms value equal to the steady-state

current, or:

Pure sine wave

A pure sine wave (Figure 4.7) has an rms value equal to the peak value

divided by the square root of two. In the case of a sine wave with peaks

at 170 V, the rms value is

Square wave

A 50 percent square wave (Figure 4.8) can be generated by full-bridge

and half-bridge power converters. The rms value of this waveform is

rms

5 I

rms

170

5 120 V

rms

5 I

Harmonics and Interharmonics 49

−170

+170

v(t)

Figure 4.7 A pure sine wave.

Figure 4.8 A 50 percent duty-cycle

square wave.

−I

i(t)

DC waveform ⫹ ripple

A DC waveform with a finite peak-peak ripple (Figure 4.9) is generated

by a variety of switching circuits, including motor drives and DC/DC

converters. This waveform has a DC value I

and a peak-peak ripple of

∆i

. The rms value of this waveform is

Triangular ripple

A triangular ripple waveform (Figure 4.10) is typical of the capacitor cur-

rent in a variety of switching circuits. In this case, the rms value of the

current is

rms

i

2!3

rms

5 I

1 1 a

i

50 Chapter Four

i(t)

(a)

∆i

(b)

i(t)

Figure 4.9 DC + ripple. (a) Waveform. (b) DC/DC converter circuit that pro-

duces this waveform. In this case, i(t) is the inductor current.

Pulsating waveform

The rms value of a pulsating waveform (Figure 4.11) with duty cycle

D is

rms

5 I

Harmonics and Interharmonics 51

i(t)

(a)

∆i

−

i(t)

(b)

Figure 4.10 Triangular ripple. (a) A waveform. (b) The DC/DC converter cir-

cuit that produces this waveform. In this case, i(t) is the capacitor current.

i(t)

DT T

Figure 4.11 Pulsating waveform

with duty cycle D.

Pulsating waveform with ripple

The rms value of a pulsating waveform (Figure 4.12) with duty cycle D

and peak-peak ripple ∆i

Triangular waveform

The rms value of a triangular pulsating waveform (Figure 4-13) with

peak value I

and duty cycle D is

Piecewise Calculation

Let’s assume we have a periodic waveform i(t) that can be broken up

into different frequency components I

, I

, . . . etc. The rms value of

is I

RMS.

The rms value of the total waveform made up of the sum of

rms

5 I

rms

5 I 2D

1 1 a

i

52 Chapter Four

i(t)

DT T

∆i

Figure 4.12 Pulsating waveform

with duty cycle D and peak-peak

ripple ∆i

i(t)

DT T

Figure 4.13 A triangular waveform.

individual currents is the sum of the squares of the rms values of the

individual components, or:

Total Harmonic Distortion

Total harmonic distortion or THD is a measure of how much harmonic

content there is in a waveform. The total harmonic distortion of a wave-

form is

where V

RMS

is the rms value of the total waveform, and V

1,RMS

is the rms

value of the first harmonic. The THD of a sine wave is 0 percent, and

the THD of a square wave is 48 percent.

Crest Factor

Crest factor is another term sometimes used in power systems analysis,

and represents the ratio of the peak value to the rms value of a waveform.

For a sine wave (Figure 4.14a), the peak value is 1.0 and the rms value is

0.707. Thus, the crest factor is 1.414. For a square wave (Figure 4.14b),

the peak and rms values are both 1.0—hence, the crest factor is 1.0.

Example 4.1: A truncated square wave. A square wave with peak value +1

has the Fourier series:

A truncated Fourier series approaches the ideal waveform. We’ll next

find the rms value and total harmonic distortion for a square wave

waveform with harmonics present up to the seventh. The total waveform

for this example is

The rms value of the first harmonic is

1,rms

π!2

5 0.9

vstd 5 a

bsinsvtd 1 a

bsins3vtd 1 a

bsins5vtd 1 a

bsins7vtd

vstd 5



n51

bsin s2pntd

THD 5

rms

2 V

1,rms

rms

5 2I

1,rms

1 I

2,rms

1 I

3,rms

1 ???

Harmonics and Interharmonics 53

The rms value of this total waveform is

The total harmonic distortion is

This waveform is shown in Figure 4.15.

THD 5

rms

2 V

1,rms

s0.974d

2 s0.9d

s0.9d

5 0.414 5 41.4%

5 a

p!2

1 1 a

1 a

5 0.974

rms

p!2

1 a

3p!2

1 a

5p!2

1 a

7p!2

54 Chapter Four

−1.0

+1.0

(a)

−1.0

+1.0

(b)

Figure 4.14 Waveforms illustrating the crest factor. (a) A

sine wave with peak value 1.0. (b) A square wave with

peak value 1.0.