Chapter 36: Complex Waveforms
Date: 12-8-19
36.1 Introduction
A waveform that is not sinusoidal is said to be a complex waveform. For a complex waveform, f(t+T) = f(t) for all values of t, where T is the interval between two successive repetitions. T is called the period of the function.
A complex wave can be resolved into the sum of a number of sinusoidal waveforms, and each of the sine waves can have a different frequency, amplitude, and phase. The initial sine wave component has a frequency equal to the frequency of the complex wave, and this frequency is called the fundamental frequency. The other sine wave components are having frequencies which are integer multiples of the fundamental frequency; these are known as harmonics.
If the fundamental frequency of a wave is f, then the second harmonic is 2f, the third harmonic is 3f, and so on. For instance, if the fundamental is 50 Hz, the third harmonic is 150 Hz, the fourth is 200 Hz, etc.
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:
Here, V1m sin(ω t + θ1) represents the fundamental component, where V1m is the peak value, frequency f = ω / 2π, and θ1 is the phase angle with respect to time t=0.
Similarly, V2m sin(2ω t + θ2) represents the second harmonic, and Vnm sin(nω t + θn) represents the nth harmonic component, where Vnm is the peak value, frequency = nω / 2π, and θn is the phase angle.
In the same way, the instantaneous value of a complex current i may be represented by the general equation:
The phase angle between the fundamental voltage and current is θ1 – φ1, and between the second harmonic voltage and current is θ2 – φ2.
36.4 Fourier Series of Periodic and Non-Periodic Functions
Fourier series is mainly used for analysing periodic functions into their constituent components. It is used for alternating currents and voltages, acoustic waves, displacement, velocity, and acceleration of mechanisms, typical examples in engineering and science.
Periodic Functions
A function f(x) is said to be periodic if f(x + T) = f(x) for all values of x, where T is the period. For example, y = sin(x) has a period of 2π.
A great advantage of the Fourier series is it can be applied to functions which are discontinuous as well as functions which are continuous.
The Fourier series of a periodic function f(x) defined in the interval -π to π can be written as:
Where a_0, a_n, and b_n are the Fourier coefficients determined by:
- a0 = (1 / 2π) ∫-ππ f(x) dx
- an = (1 / π) ∫-ππ f(x) cos(nx) dx
- bn = (1 / π) ∫-ππ f(x) sin(nx) dx
36.5 Even and Odd Fourier Series
Even Functions
These have symmetry about the y-axis, so f(-x) = f(x). cos(x) is an example. The Fourier series of an even periodic function f(x) having period 2π will only contain cosine terms and may contain a constant term:
Odd Functions
A function y = f(x) is said to be odd if f(-x) = -f(x) for all values of x. Graphs of odd functions are symmetrical about the origin; y = sin(x) is an odd function. The Fourier series for an odd periodic function f(x) having a period of 2π will contain only sine terms, and will not contain a constant term:
36.6 RMS Value, Mean Value, and Form Factor of a Complex Wave
The RMS value of a complex wave is given by taking the square root of the sum of the squares of the RMS values of its individual harmonic components, including any DC component:
Similarly, for voltage:
From the above equations, it can be seen that the RMS value of a complex wave is not affected by the relative phase angles of the harmonic components.
Form Factor
The form factor of a complex wave whose negative half cycle is similar in shape to its positive half cycle is defined by:
36.7 Power Associated With Complex Waves
In order to get the power supplied, we multiply the complex component of current by the complex component of voltage. The product of 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 contribute to the average power.
Where V_n and I_n are the RMS values of the nth harmonic, and \phi_n is the phase angle between the voltage and current for that harmonic.
Power Factor
When dealing with harmonics, the total power factor is defined as overall power factor:
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 wave. The following scenarios can apply:
- Pure Resistance: There is no phase change. The resistance is independent of frequency. The percentage of harmonic in the current wave is the same as that of the voltage wave.
- Pure Inductance: The inductive reactance XL = 2π fL increases linearly with frequency. The current will lag the voltage by 90° for every harmonic. The nth harmonic current is scaled down by a factor of n.
- Pure Capacitance: The capacitive reactance XC = 1 / (2π fC) decreases with frequency. The capacitive current leads the voltage by 90°. The nth harmonic current is scaled up by a factor of n.
36.10 Resonance Due to Harmonics
If the applied voltage waveform is not a pure sine wave, it is quite possible to achieve resonance at one of the harmonics. The magnitude of the resonant harmonic current can be large, causing dangerous voltage drops across the components.
When resonance occurs at one of the harmonic frequencies, the effect is called selective or harmonic resonance. The condition for resonance at the nth harmonic is:
Chapter 36: Complex Waveforms
Date: 12-8-19
36.1 Introduction
A waveform that is not sinusoidal is said to be a complex waveform. For a complex waveform, $f(t+T) = f(t)$ for all values of $t$, where $T$ is the interval between two successive repetitions. $T$ is called the period of the function.
A complex wave can be resolved into the sum of a number of sinusoidal waveforms, and each of the sine waves can have a different frequency, amplitude, and phase. The initial sine wave component has a frequency equal to the frequency of the complex wave, and this frequency is called the fundamental frequency. The other sine wave components are having frequencies which are integer multiples of the fundamental frequency; these are known as harmonics.
If the fundamental frequency of a wave is $f$, then the second harmonic is $2f$, the third harmonic is $3f$, and so on. For instance, if the fundamental is 50 Hz, the third harmonic is 150 Hz, the fourth is 200 Hz, etc.
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:
Here, $V_{1m} \sin(\omega t + heta_1)$ represents the fundamental component, where $V_{1m}$ is the peak value, frequency $f = \omega / 2\pi$, and $ heta_1$ is the phase angle with respect to time $t=0$.
Similarly, $V_{2m} \sin(2\omega t + heta_2)$ represents the second harmonic, and $V_{nm} \sin(n\omega t + heta_n)$ represents the nth harmonic component, where $V_{nm}$ is the peak value, frequency $= n\omega / 2\pi$, and $ heta_n$ is the phase angle.
In the same way, the instantaneous value of a complex current $i$ may be represented by the general equation:
The phase angle between the fundamental voltage and current is $ heta_1 – \phi_1$, and between the second harmonic voltage and current is $ heta_2 – \phi_2$.
36.4 Fourier Series of Periodic and Non-Periodic Functions
Fourier series is mainly used for analysing periodic functions into their constituent components. It is used for alternating currents and voltages, acoustic waves, displacement, velocity, and acceleration of mechanisms, typical examples in engineering and science.
Periodic Functions
A function $f(x)$ is said to be periodic if $f(x + T) = f(x)$ for all values of $x$, where $T$ is the period. For example, $y = \sin(x)$ has a period of $2\pi$.
A great advantage of the Fourier series is it can be applied to functions which are discontinuous as well as functions which are continuous.
The Fourier series of a periodic function $f(x)$ defined in the interval $-\pi$ to $\pi$ can be written as:
Where $a_0, a_n,$ and $b_n$ are the Fourier coefficients determined by:
- a0 = (1 / 2π) ∫-ππ f(x) dx
- an = (1 / π) ∫-ππ f(x) cos(nx) dx
- bn = (1 / π) ∫-ππ f(x) sin(nx) dx
36.5 Even and Odd Fourier Series
Even Functions
These have symmetry about the y-axis, so $f(-x) = f(x)$. $\cos(x)$ is an example. The Fourier series of an even periodic function $f(x)$ having period $2\pi$ will only contain cosine terms and may contain a constant term:
Odd Functions
A function $y = f(x)$ is said to be odd if $f(-x) = -f(x)$ for all values of $x$. Graphs of odd functions are symmetrical about the origin; $y = \sin(x)$ is an odd function. The Fourier series for an odd periodic function $f(x)$ having a period of $2\pi$ will contain only sine terms, and will not contain a constant term:
36.6 RMS Value, Mean Value, and Form Factor of a Complex Wave
The RMS value of a complex wave is given by taking the square root of the sum of the squares of the RMS values of its individual harmonic components, including any DC component:
Similarly, for voltage:
From the above equations, it can be seen that the RMS value of a complex wave is not affected by the relative phase angles of the harmonic components.
Form Factor
The form factor of a complex wave whose negative half cycle is similar in shape to its positive half cycle is defined by:
36.7 Power Associated With Complex Waves
In order to get the power supplied, we multiply the complex component of current by the complex component of voltage. The product of 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 contribute to the average power.
Where $V_n$ and $I_n$ are the RMS values of the $n$th harmonic, and $\phi_n$ is the phase angle between the voltage and current for that harmonic.
Power Factor
When dealing with harmonics, the total power factor is defined as overall power factor:
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 wave. The following scenarios can apply:
- Pure Resistance: There is no phase change. The resistance is independent of frequency. The percentage of harmonic in the current wave is the same as that of the voltage wave.
- Pure Inductance: The inductive reactance $X_L = 2\pi fL$ increases linearly with frequency. The current will lag the voltage by 90° for every harmonic. The $n$th harmonic current is scaled down by a factor of $n$.
- Pure Capacitance: The capacitive reactance $X_C = 1 / (2\pi fC)$ decreases with frequency. The capacitive current leads the voltage by 90°. The $n$th harmonic current is scaled up by a factor of $n$.
36.10 Resonance Due to Harmonics
If the applied voltage waveform is not a pure sine wave, it is quite possible to achieve resonance at one of the harmonics. The magnitude of the resonant harmonic current can be large, causing dangerous voltage drops across the components.
When resonance occurs at one of the harmonic frequencies, the effect is called selective or harmonic resonance. The condition for resonance at the $n$th harmonic is: