Chapter 36: Complex Waveforms
36.1 Introduction
A waveform that is not sinusoidal is said to be a complex waveform [3953]. A periodic function satisfies f(t) = f(t + T) for all values of t, where T is the interval between successive repetitions (the period) [3954][3958]. A complex periodic waveform can be resolved into the sum of sinusoidal waveforms, each having a different frequency, amplitude, and phase [3960][3962].
The initial sine wave component has a frequency equal to the frequency of the complex wave and is called the fundamental frequency [3964][3968]. The other sine wave components, having frequencies which are integer multiples of the fundamental frequency, are called harmonics [3969][3978][3980]. If the fundamental frequency is f, the second harmonic is 2f, the third is 3f, and so on [3972][3973].
36.2 The General Equation of a Complex Waveform
The instantaneous value of a complex voltage wave acting in a linear circuit may be represented by the general equation [3985]:
Here, V1m sin(ωt + φ1) represents the fundamental component, where V1m is the maximum peak value, the frequency is ω/(2π), and φ1 is the phase angle with respect to time t=0 [3989][3990]. Similarly, the nth harmonic component is Vnm sin(nωt + φn) [3991].
36.6 R.M.S Value, Mean Value, and Form Factor
If the instantaneous value of a complex current is given by [4553][4555]:
The R.M.S value of the first harmonic of current is I1 = I1m / √2 [4559][4560]. When multiplying harmonics together, all products of different frequencies average to zero over a complete cycle [4563][4567]. It follows that the mean value of i2 is [4593]:
Taking the square root gives the overall RMS current (I) [4596]:
The RMS value for voltage follows the exact same pattern [4596]:
The RMS value of a complex wave is not affected by the relative phase angles of the harmonic components [4599]. If a DC component I0 exists, the total RMS current is [4607]:
Form Factor: The form factor of a complex wave (whose negative half cycle is similar in shape to its positive half cycle) is defined as the RMS value divided by the mean value [4625][4627].
36.7 Power Associated With Complex Waves
The power in any circuit is given by the product of voltage and current, or I2R [4696][4700]. To get the average power supplied, we multiply the complex component of current by the complex component of voltage [4704]. The product of these complex components over an average cycle is zero when they are of a different frequency. Therefore, only products of voltages and currents of the same frequency matter [4707][4709][4711].
Where V and I are RMS values for each respective harmonic, and φ is the phase difference between them [4729][4730]. If there is a DC component to the voltage and current, add V0I0 to the total power P [4739].
Overall Power Factor: When dealing with harmonics, the total power factor is defined as [3654]:
36.8 Harmonics in Single-Phase Circuits
When a complex waveform containing harmonics is applied to a single-phase circuit containing resistors, capacitors, and/or inductors, the resulting current is also a complex harmonic wave [3721][3722].
(a) Pure Resistance
If the circuit has an impedance of pure resistance, there is no phase change with respect to the current [3727]. The impedance of pure resistance is independent of the frequency. The general expression is [3728]:
(b) Pure Inductance
The impedance of a pure inductive reactance XL = 2πfL increases linearly with frequency [3744]. Also, in every harmonic, the current will lag the voltage by 90° (π/2 rads) [3745]:
(c) Pure Capacitance
Capacitive reactance is given by XC = 1/(2πfC) = 1/(ωC). It varies inversely with frequency [3748]. The capacitive current leads the voltage by 90° (π/2 rads) [3749]:
If a DC component is contained in a pure capacitance circuit, the DC current will not flow through the pure capacitor (as it blocks DC) [3769][3829].
Resonance Due to Harmonics
In industrial circuits at power frequencies, the values of L and C may cause resonance at the fundamental frequency [3874]. However, if the waveform is not a pure sine wave, it is quite possible to achieve resonance at one of the harmonics [3879][3880]. This can cause dangerous voltage drops across L and C [3884].
When resonance occurs at harmonic frequencies, the effect is called selective or harmonic resonance [3885][3892]. For resonance at the nth harmonic [3893]:
Solving for n gives the harmonic at which resonance occurs [3933]:
General Conclusion on Odd and Even Harmonics
Odd Harmonics: Whenever odd harmonics are added to a fundamental, the positive and negative half cycles of the resulting complex wave are identical in shape (except inverted) [4098][4101][4103].
Even Harmonics: Whenever even harmonics are added to a fundamental component, the positive and negative half cycles are dissimilar (not identical in shape) [4106][4113][4123].
36.4 Fourier Series of Periodic and Non-Periodic Functions
A periodic function is defined as f(x) = f(x + T) for all values of x, where T is the period [4212][4214][4215]. A great advantage of the Fourier series is that it can be applied to functions with finite discontinuities (sudden jumps) [4241][4244][4248].
A trigonometric series defined in terms of a convergent Fourier series takes the form [4253][4255][4256]:
Where a0, an, and bn are the Fourier coefficients, determined by [4263][4265]:
an = (1 / π) ∫-ππ f(x) cos(nx) dx [4262]
bn = (1 / π) ∫-ππ f(x) sin(nx) dx [4262]
36.5 Even and Odd Fourier Series
Even Functions: These have symmetry about the y-axis, meaning f(-x) = f(x) [4464][4465][4472]. The Fourier series of an even periodic function will only contain cosine terms, and may contain a constant term a0 [4474]. Hence, bn = 0.
Odd Functions: A function y = f(x) is said to be odd if f(-x) = -f(x) for all values of x [4480]. Graphs of odd functions are always symmetrical about the origin [4481]. The Fourier series for an odd periodic function contains only sine terms (and does not contain a constant term) [4486][4487][4488]. Hence, a0 = 0 and an = 0.
Expansion over any period L
If f(x) is a periodic function with a period of L, then the Fourier series is given by [4494][4495][4496]:
The limits of integration for finding a0, an, and bn would span the interval L (e.g., from -L/2 to L/2, or from 0 to L) [4502][4506].