3. Pulse width modulation

To introduce the concept of PWM, let us consider a basic configuration of one-phase two-level inverter leg shown in Figure 1. It consists of two switches, S<sup>1</sup> and S2, and two diodes, D<sup>1</sup> and D2. Switches S<sup>1</sup> and S<sup>2</sup> are operating alternately at high frequency to generate a quasiperiodic output voltage vaðtÞ, whose low-frequency components are intended to deliver a prescribed AC supply. When the switch S<sup>1</sup> (S2) is ON, a positive voltage, þVdc, (respectively, negative voltage, −Vdc) is supplied to a load at the connection point a.

The essential concept of a two-level pulse-width-modulated converter system is that a lowfrequency target waveform is compared against a high-frequency carrier waveform, and the comparison result is used to control the state of a switched phase leg. In case of the inverter in Figure 1, the phase leg is switched to the upper DC rail when the target waveform is greater than the carrier waveform, otherwise to the lower DC rail. As a result, a sequence of pulses switching between the upper and the lower DC rails is generated, which contains the target waveform as the fundamental component but also a series of unwanted harmonics arising due to the switching process.

Figure 1. Half-bridge one phase two-level inverter leg.

The most well-known analytical method of determining the harmonic components of a PWM switched phase leg was first developed by Bowes and Bullough [1], who adopted an analysis approach originally developed for communication systems by Bennet [6] and Black [7] to modulated converter systems.

The analysis is based on the existence of two time variables xðtÞ ¼ ωct and yðtÞ ¼ ω0t, where ω<sup>0</sup> and ω<sup>c</sup> are the angular frequencies of the fundamental (target, sinusoid) low-frequency modulated waveform and the carrier high-frequency modulating waveform, ω<sup>0</sup> ≪ ωc.Variables xðtÞ and yðtÞ are considered to be independently periodic. If the ratio ωc=ω<sup>0</sup> is integer, the generated pulse width trail will be periodic [4].

The problem of finding a PWM for the modulated periodic waveform fðtÞ can be solved by exploring a unit cell which identifies contours within which fðtÞ remains constant for cyclic variations of xðtÞ and yðtÞ and is equal to the phase leg output voltage. Thus, a three-dimensional (3D) unit cell is a plot of two time variables function with z assuming values of fðx, yÞ where x and y vary from −π to π. Contours of fðx, yÞ within the unit cell depend on a particular PWM strategy which will be discussed below.

### 3.1. Carrier-based PWM schemes

### 3.1.1. Carrier waveforms and unit cells

Since the target waveform is usually a sinusoid, PWM schemes can be categorized based on the carrier waveform: saw-tooth leading edge (Figure 2a), saw-tooth trailing edge (Figure 2b), and double edge (Figure 2c).

Let the modulated waveform of a phase be given vid <sup>a</sup> ¼ M cos y, where M is the modulation index, 0 < M < 1. For the one-phase two-level inverter leg shown in Figure 1, unit cells with contour plots for each carrier waveform modulation are presented in Figure 3. The output of the modulated waveform assumes either þVdc or −Vdc, and the regions of the constant output are bounded by reference waveforms Ωðy޼�πM cos y. For saw-tooth modulations, one of

Figure 2. Carrier waveform: (a) saw-tooth leading edge; (b) saw-tooth trailing edge; (c) double edge.

Figure 3. Contour plots for a sine modulated reference waveform and different carrier modulating waveform: (a) sawtooth leading edge; (b) saw-tooth trailing edge; (c) double edge.

switching time instances (within a period of the carrier waveform) is independent of the reference waveform resulting in only one side of the contour plot to be sinusoid. The doubleedge PWM both sides of the switched output are modulated providing better harmonic performance unlike saw-tooth modulations [4]. Hereinafter, only double-edge modulation is considered.

To determine the harmonics content and the output waveform of a particular PWM, the double Fourier series coefficients Cmn (or, equivalently, Amn and Bmn) are to be found using Eq. (2). To solve the problem, the periodic function fðx, yÞ is to be integrated over the unit cell of the PWM scheme.

### 3.1.2. PWM sampling schemes

The most well-known analytical method of determining the harmonic components of a PWM switched phase leg was first developed by Bowes and Bullough [1], who adopted an analysis approach originally developed for communication systems by Bennet [6] and Black [7] to

The analysis is based on the existence of two time variables xðtÞ ¼ ωct and yðtÞ ¼ ω0t, where ω<sup>0</sup> and ω<sup>c</sup> are the angular frequencies of the fundamental (target, sinusoid) low-frequency modulated waveform and the carrier high-frequency modulating waveform, ω<sup>0</sup> ≪ ωc.Variables xðtÞ and yðtÞ are considered to be independently periodic. If the ratio ωc=ω<sup>0</sup> is integer, the

The problem of finding a PWM for the modulated periodic waveform fðtÞ can be solved by exploring a unit cell which identifies contours within which fðtÞ remains constant for cyclic variations of xðtÞ and yðtÞ and is equal to the phase leg output voltage. Thus, a three-dimensional (3D) unit cell is a plot of two time variables function with z assuming values of fðx, yÞ where x and y vary from −π to π. Contours of fðx, yÞ within the unit cell depend on a particular

Since the target waveform is usually a sinusoid, PWM schemes can be categorized based on the carrier waveform: saw-tooth leading edge (Figure 2a), saw-tooth trailing edge (Figure 2b),

index, 0 < M < 1. For the one-phase two-level inverter leg shown in Figure 1, unit cells with contour plots for each carrier waveform modulation are presented in Figure 3. The output of the modulated waveform assumes either þVdc or −Vdc, and the regions of the constant output are bounded by reference waveforms Ωðy޼�πM cos y. For saw-tooth modulations, one of

<sup>a</sup> ¼ M cos y, where M is the modulation

modulated converter systems.

generated pulse width trail will be periodic [4].

Figure 1. Half-bridge one phase two-level inverter leg.

122 Fourier Transforms - High-tech Application and Current Trends

PWM strategy which will be discussed below.

Let the modulated waveform of a phase be given vid

3.1. Carrier-based PWM schemes 3.1.1. Carrier waveforms and unit cells

and double edge (Figure 2c).

Based on the choice of switching time instances, PWM schemes can be divided into: naturally sampled (NS), symmetrically regularly sampled (SR), and asymmetrically regularly sampled (AR) PWMs.

### 3.1.2.1. Naturally sampled PWM

For NS PWM scheme, switching occurs at time instances corresponding to intersection of the carrier and target waveforms. Switching time instances can also be determined as the intersection between the reference waveform and the solution trajectory y ¼ ðω0=ωcÞx. For example, switching time instances for the NS double-edge modulation of the one-phase two-level inverter leg in Figure 1 are defined from its unit cell in Figure 3c such that fðx, yÞ changes

from −Vdc to Vdc when x ¼ −πM cos y,

from Vdc to −Vdc when x ¼ πM cos y.

### 3.1.2.2. Symmetrically regularly sampled PWM

Switching instances for SR PWM can be determined by the intersection between the sampled sinusoid waveform and the solution trajectory line <sup>y</sup> <sup>¼</sup> <sup>y</sup>′ þ ðω0=ωcÞx. The same switching instances can be determined as the intersection between the continuous sinusoid waveform and a staircase variable y′ which has a constant value within each carrier interval [4]. In general, the value of y′ within each carrier interval can be expressed as

$$y' = \frac{\omega\_0}{\omega\_c} 2p\pi, \quad p = 0, 1, 2, \dots \tag{5}$$

where p represents the pth carrier interval within a fundamental cycle. The staircase variable y′ in terms of continuous variables x and y is given by

$$y' = y - \frac{\omega\_0}{\omega\_c}(\text{x-2p\pi}), \quad p = 0, 1, 2, \dots \tag{6}$$

The double Fourier series coefficients for the case of SR PWM with a triangle carrier can be found analogously to NS PWM with variable y substituted by variable y′ found from Eq. (6).

Considering the previous example with the one-phase two-level inverter leg shown in Figure 1, switching time instances for the SR double-edge modulation are defined such that fðx, yÞ changes

$$\text{from } -V\_{dc} \text{ to } V\_{dc} \text{ when } \mathbf{x} = -\pi M \cos \vec{y'},$$

from Vdc to <sup>−</sup>Vdc when <sup>x</sup> <sup>¼</sup> <sup>π</sup><sup>M</sup> cos <sup>y</sup>′ .

### 3.1.2.3. Asymmetrically regularly sampled PWM

Switching time instances for AR PWM are determined similarly to SR PWM. Unlike SR PWM, switching occurs twice within each carrier interval for AR PWM. The switching time instances can be determined as the intersection between the continuous sinusoid waveform and two staircase variables

$$y\_i^{'} = \frac{\omega\_0}{\omega\_c} \left( 2p\pi + (-1)^i \frac{\pi}{2} \right), \quad i = 1, 2,\tag{7}$$

which can be expressed in terms of continuous variables x and y as

$$y\_i^{'} = y - \frac{\alpha\_0}{\alpha\_c} \left( \text{x-} 2p\pi \text{-} (-1)^i \frac{\pi}{2} \right), \quad i = 1, 2. \tag{8}$$

To write the double Fourier series integral for AR PWM, the switched waveform in each carrier interval must be split into two sections for analysis, and with the results added by superposition, the first section (i ¼ 1) has modulated "rising" edge in the first half carrier interval and a "falling" edge in the center of the carrier interval. The second section (i ¼ 2) has a modulated "rising" edge in the center of the carrier interval and "falling" edge in the second half carrier interval. Mathematically, this behavior can be expressed as a sum of two functions, f <sup>1</sup>ðx, yÞ and f <sup>2</sup>ðx, yÞ, representing "rising" and "falling" edges of the double-edge carrier waveform fðx, yÞ ¼ f <sup>1</sup>ðx, yÞ þ f <sup>2</sup>ðx, yÞ.

In the previous example with the one-phase two-level inverter leg (Figure 1), functions f <sup>1</sup>ðx, yÞ and f <sup>2</sup>ðx, yÞ are defined as follows:

f <sup>1</sup>ðx, yÞ steps from Vdc to −Vdc at x ¼ xðy<sup>1</sup> ′ Þ þ 2pπ and from −Vdc to Vdc at x ¼ 2pπ; f <sup>2</sup>ðx, yÞ steps from Vdc to −Vdc at x ¼ 2pπ and from −Vdc to Vdc at x ¼ xðy<sup>2</sup> ′ Þ þ 2pπ.
