Fig. 2-55(a) shows a basic sine wave with its most important dimensions and the equa- tion expressing it. A basic cosine wave is illustrated in Fig. 2-55(b). Note that the cosine wave has the same shape as a sine wave but leads the sine wave by 90°. A harmonic is a sine wave whose frequency is some integer multiple of a fundamental sine wave. For example, the third harmonic of a 2-kHz sine wave is a sine wave of 6 kHz. Fig. 2-56 shows the i rst four harmonics of a fundamental sine wave.
What the Fourier theory tells us is that we can take a nonsinusoidal waveform and break it down into individual harmonically related sine wave or cosine wave components.
The classic example of this is a square wave, which is a rectangular signal with equal- duration positive and negative alternations. In the ac square wave in Fig. 2-57, this means that t1 is equal to t2. Another way of saying this is that the square wave has a 50 percent duty cycle D, the ratio of the duration of the positive alteration t1 to the period T expressed as a percentage:
D5 t1
T 3100
Fourier analysis tells us that a square wave is made up of a sine wave at the funda- mental frequency of the square wave plus an ini nite number of odd harmonics. For example, if the fundamental frequency of the square wave is 1 kHz, the square wave can be synthesized by adding the 1-kHz sine wave and harmonic sine waves of 3 kHz, 5 kHz, 7 kHz, 9 kHz, etc.
Fig. 2-58 shows how this is done. The sine waves must be of the correct amplitude and phase relationship to one another. The fundamental sine wave in this case has a value of 20 V peak to peak (a 10-V peak). When the sine wave values are added instantaneously, the result approaches a square wave. In Fig. 2-58(a), the fundamental and third harmonic Fourier analysis
Harmonic
Square wave
Duty cycle D
Figure 2-55 Sine and cosine waves.
0
0° 90° 180° 270° 360°
VP
VP
T
v ⫽ VP sin 2ft v ⫽ VP sin t
0° 90° 180° 270° 360° VP
⫺VP
T
v ⫽ VP cos 2ft v ⫽ VP cos t T ⫽period of one cycle
in seconds f ⫽frequency in Hz
⫽
v ⫽instantaneous value of voltage VP ⫽peak voltage
⫽2f 1 T
are added. Note the shape of the composite wave with the third and i fth harmonics added, as in Fig. 2-58(b). The more higher harmonics that are added, the more the com- posite wave looks like a perfect square wave. Fig. 2-59 shows how the composite wave would look with 20 odd harmonics added to the fundamental. The results very closely approximate a square wave.
Figure 2-56 A sine wave and its harmonics.
Fundamental (f)
Third harmonic (3f)
Fourth harmonic (4f) Second harmonic (2f)
Figure 2-57 A square wave.
(a)
t1 t2
T ⫽ t1 ⫹ t2
(50% duty cyle) Duty cyle ⫽
t
⫹V
0
⫺V
(b)
t1 t2
t
⫹V
0
⫻ 100 t1 T 1 T
t1 ⫽ t2 f ⫽
The implication of this is that a square wave should be analyzed as a collection of harmonically related sine waves rather than a single square wave entity. This is coni rmed by performing a Fourier mathematical analysis on the square wave. The result is the following equation, which expresses voltage as a function of time:
f(t) 54V
π csin 2πa1 Tbt1 1
3 sin 2πa3 Tbt 1 1
5 sin 2πa5 Tbt 1 1
7sin 2πa7
Tbt 1. . .d Figure 2-58
A square wave is made up of a fundamental sine wave and an infi nite number of odd harmonics.
T/4 T/2 3T/4 T t
1 f Fundamental
Fundamental
With 3rd and 5th harmonics
Perfect square wave (infinite harmonics) With 3rd harmonic Composite wave of fundamental
plus 3rd harmonic
Composite wave with 3rd and 5th harmonics added to the fundamental
Period ⫽ T 10 V
v v
0
⫺10 V
10 V
0
⫺10 V
v 10 V V⬁
⫺V⬁ 0
⫺10 V
T/4
T/2
T/2
T
3T/4 T
t
t
⫽
(a) (b)
(c)
Figure 2-59 Square wave made up of 20 odd harmonics added to the fundamental.
⫺10 V 10 V
0 V
where the factor 4V/π is a multiplier for all sine terms and V is the square wave peak voltage. The i rst term is the fundamental sine wave, and the succeeding terms are the third, i fth, seventh, etc., harmonics. Note that the terms also have an amplitude factor. In this case, the amplitude is also a function of the harmonic. For example, the third harmonic has an amplitude that is one-third of the fundamental amplitude, and so on. The expres- sion could be rewritten with f51/T. If the square wave is direct current rather than alternating current, as shown in Fig. 2-57(b), the Fourier expression has a dc component:
f(t)5 V 2 1 4V
π asin 2πft1 1
3 sin 2π3ft1 1
5 sin 2π5ft11
7 sin 2π7ft1. . .b In this equation, V/2 is the dc component, the average value of the square wave. It is also the baseline upon which the fundamental and harmonic sine waves ride.
A general formula for the Fourier equation of a waveform is f(t) 5V
2 1 4V πn a
q
n51
(sin 2πnft)
where n is odd. The dc component, if one is present in the waveform, is V/2.
By using calculus and other mathematical techniques, the waveform is dei ned, ana- lyzed, and expressed as a summation of sine and/or cosine terms, as illustrated by the expression for the square wave above. Fig. 2-60 gives the Fourier expressions for some of the most common nonsinusoidal waveforms.
Example 2-26
An ac square wave has a peak voltage of 3 V and a frequency of 48 kHz. Find (a) the frequency of the i fth harmonic and (b) the rms value of the i fth harmonic. Use the formula in Fig. 2-60(a).
a. 5348 kHz5240 kHz
b. Isolate the expression for the fifth harmonic in the formula, which is
1
5sin 2π(5/T)t. Multiply by the amplitude factor 4V/π. The peak value of the fifth harmonic VP is
VP5 4V π a1
5b54(3) 5π 50.76 rms50.7073peak value
Vrms50.707VP50.707(0.76) 50.537 V
The triangular wave in Fig. 2-60(b) exhibits the fundamental and odd harmonics, but it is made up of cosine waves rather than sine waves. The sawtooth wave in Fig. 2-60(c) contains the fundamental plus all odd and even harmonics. Fig. 2-60(d) and (e) shows half sine pulses like those seen at the output of half and full wave rectii ers. Both have an aver- age dc component, as would be expected. The half wave signal is made up of even harmon- ics only, whereas the full wave signal has both odd and even harmonics. Fig. 2-60(f) shows the Fourier expression for a dc square wave where the average dc component is Vt0/T.