2

Pulse Width-Modulated DC/AC Inverters

DC/AC inverters are quickly developed with knowledge of the power switching circuits applied in industrial applications in comparison with other power switching circuits. In the past century, plenty of topologies of DC/AC inverters have been created. DC/AC inverters are mainly used in AC motor adjustable speed drives (ASDs), as shown in Figure 2.1. Power DC/AC inverters have been widely used in other industrial applications since the late 1980s. Semiconductor manufacture development allowed high-power devices such as IGBTs and MOSFETs to operate at higher switching frequencies (e.g., from tens of kHz up to a few MHz). Conversely, some devices such as thyristors (SCRs), GTOs, triacs, and BTs, with lower switching frequency and higher power rate, the IGBT and MOSFET may have both high power rate and high switching frequency [1,5].

Square waveform DC/AC inverters were used well before the 1980s and the thyristor, GTO, and triac could be used in low-frequency switching operations. The power BT and IGBT were produced for high frequency operation. The corresponding equipment implementing the pulsewidth-modulation (PWM) technique has a large range of output voltage and frequency and low THD.

Nowadays, two DC/AC inversion techniques are popular in this area: PWM and MLM. Most DC/AC inverters are still PWM DC/AC inverters in different prototypes. We will introduce PWM inverters in this chapter and MLM inverters in Chapter 8.

2.1    Introduction

DC/AC inverters are used for converting a DC power source into AC power applications. They are generally used in the following applications:

1.  Variable voltage/variable frequency AC supplies in adjustable speed drive (ASD), devices such as induction motor drives, synchronous machine drives, and so on

2.  Constant regulated voltage AC power supplies, such as uninterruptible power supplies (UPSs)

FIGURE 2.1
A standard adjustable speed drive (ASD) scheme.

3.  Static variability (reactive power) compensations

4.  Passive/active series and parallel filters

5.  Flexible AC transmission systems (FACTSs)

6.  Voltage compensations

Adjustable speed induction motor drive systems are widely applied in industrial applications. These systems require DC/AC power supply with variable frequency usually from 0 Hz to 400 Hz in fractional horsepower (HP) to hundreds of HP. A large number of DC/AC inverters are in the world market. The typical block circuit of an ASD is shown in Figure 2.1. From this block diagram, we can see that the power DC/AC inverter produces variable frequency and voltage to implement ASD.

The PWM technique is different from pulse amplitude modulation (PAM) and pulse phase modulation (PPM). In this technique, all pulses have adjustable width with constant amplitude and phase. The corresponding circuit is called the pulse width modulator. Typical input and output waveforms of a pulse width modulator are shown in Figure 2.2. The output pulse train has the pulses of the same amplitude and different widths, which corresponds to the input signal at the sampling instants.

FIGURE 2.2
Typical input and output waveforms of a pulse width modulator.

2.2    Parameters Used in PWM Operation

Some parameters specially used in PWM operation are introduced in this section.

2.2.1    Modulation Ratios

The modulation ratio is usually obtained from a uniform amplitude triangle (carrier) signal with amplitude V_tri-m. The maximum amplitude of the input signal is assumed to be V_in-m. We define the amplitude modulation ratio m_a for a single-phase inverter as follows:

$m_a=\frac{V_{in-m}}{V_{tri-m}}$

(2.1)

We also define the frequency modulation ratio m_f as follows:

$m_f=\frac{f_{tri-m}}{f_{in-m}}$

(2.2)

FIGURE 2.3
Single-leg switch-mode inverter.

A single-leg switch-mode inverter is shown in Figure 2.3. The DC-link voltage is V_d. Two large capacitors are used to establish the neutral point N. The AC output voltage from point a to N is V_AO, and its fundamental component is (V_AO)₁. We denote ${({\widehat{V}}_{AO})}_1$ to show the maximum amplitude of (V_AO)₁. The waveforms of the input (control) signal and triangle signal, and the spectrum of the PWM pulse train are shown in Figure 2.4.

If the maximum amplitude ${({\widehat{V}}_{AO})}_1$ of the input signal is smaller than or equal to half of the DC-link voltage V_d/2 and the modulation ratio m_a is smaller than or equal to the unity. In this case, the fundamental component (V_AO)₁ of the output AC voltage is proportional to the input voltage. The voltage control by varying m_a for a single-phase PWM is split in three areas, which are shown in Figure 2.5.

2.2.1.1    Linear Range (m_a ≤ 1.0)

The condition ${({\widehat{V}}_{AO})}_1=m_a\frac{V_d}{2}$ determines the linear region. It is a sinusoidal PWM where the amplitude of the fundamental frequency voltage varies linearly with the amplitude modulation ratio m_a. The PWM pushes the harmonics into a high-frequency range around the switching frequency and its multiples. However, the maximum available amplitude of the fundamental frequency component may not be as high as desired.

2.2.1.2    Over Modulation (1.0 < m_a ≤ 3.24)

The condition $\frac{V_d}{2}<{({\widehat{V}}_{AO})}_1\le \frac{4}{\pi }\frac{V_d}{2}$ determines the overmodulation region. When the amplitude of the fundamental frequency component in the output voltage increases beyond 1.0, it reaches overmodulation. In the overmodulation range, the amplitude of the fundamental frequency voltage no longer varies linearly with m_a.

FIGURE 2.4
Pulse width modulation.

Overmodulation causes the output voltage to contain many more harmonics in the sidebands as compared with the linear range. The harmonics with dominant amplitudes in the linear range may not be dominant during overmodulation.

2.2.1.3    Square Wave (Sufficiently Large m_a > 3.24)

The condition ${({\widehat{V}}_{AO})}_1>\frac{4}{\pi }\frac{V_d}{2}$ determines the square wave region. The inverter voltage waveform degenerates from a pulse width modulated waveform into a square wave. Each switch of the inverter leg in Figure 2.3 is on for one half-cycle (180°) of the desired output frequency.

FIGURE 2.5
Voltage control by varying m_a.

2.2.1.4    Small m_f (m_f ≤ 21)

Usually the triangle waveform frequency is much larger than the input signal frequency to obtain small THD. For the situation with a small mf< 21, two points have to be mentioned:

•  Synchronous PWM. For small values of m_f, the triangle waveform signal and the input signal should be synchronized to each other (synchronous PWM). This synchronous PWM requires that m_f be an integer. Synchronous PWM is used because asynchronous PWM (where m_f is not an integer) results in subharmonics (of the fundamental frequency) that are very undesirable in most applications. This implies that the triangle waveform frequency varies with the desired inverter frequency (e.g., if the inverter output frequency and hence the input signal frequency are 65.42 Hz and m_f = 15, the triangle wave frequency should be exactly 15 × 65.42 = 981.3 Hz).

•  m_f ≤ 21 and should be an odd integer. As discussed previously, m_f should be an odd integer except in single-phase inverters with PWM unipolar voltage switching.

2.2.1.5    Large m_f (m_f > 21)

The amplitudes of subharmonics due to asynchronous PWM are small at large values of m_f. Therefore, at large values of m_f, asynchronous PWM can be used where the frequency of the triangle waveform is kept constant, whereas the input signal frequency varies, resulting in nonintegral values of m_f (so long as they are large). However, if the inverter is supplying a load such as an AC motor, the subharmonics at zero or close to zero frequency, even though small in amplitude, will result in large current, which is highly undesirable. Therefore, asynchronous PWM should be avoided.

It is very important to determine the harmonic components of the output voltage. Referring to Figure 2.4c, we have the fast Fourier transform (FFT) spectrum and the harmonics. Choosing the frequency modulation ratio m_f as an odd integer and amplitude modulation ratio m_a < 1, we have the generalized harmonics of the output voltage shown in Table 2.1.

The rms voltages of the output voltage harmonics are calculated by the following formula:

${(V_O)}_h=\frac{V_d}{\sqrt{2}}\frac{{({\widehat{V}}_{AO})}_h}{V_d/2}$

(2.3)

where (V_O)_h is the hth harmonic rms voltage of the output voltage, V_d is the DC link voltage, and ${({\widehat{V}}_{AO})}_h/(V_d/2)$/(V_d/2) or ${({\widehat{V}}_{AO})}_h/(V_d/2)$/(V_d/2) is tabulated as a function of m_a.

TABLE 2.1
Generalized Harmonics of V_O (or V_AO) for Large m_f

Note: (VAO)h/(Vd/2) or (VAO)h/(Vd/2) is tabulated as a function of m_a.

If the input (control) signal is a sinusoidal wave, we usually call this inversion sinusoidal pulse width modulation (SPWM). The typical waveforms of an SPWM are also shown in Figures 2.4a and 2.4b.

Example 2.1

A single-phase half-bridge DC/AC inverter is shown in Figure 2.3 to implement an SPWM with V_d = 200 V, m_a = 0.8, and m_f = 27. The fundamental frequency is 50 Hz. Determine the rms value of the fundamental frequency and some of the harmonics in output voltage using Table 2.1.

Solution:

From Equation (2.3) we have the general rms values

${(V_O)}_h=\frac{V_d}{\sqrt{2}}\frac{({\widehat{V}}_{AO})h}{V_d/2}=\frac{200}{\sqrt{2}}\frac{({\widehat{V}}_{AO})h}{V_d/2}=141.42\frac{({\widehat{V}}_{AO})h}{V_d/2}V$

(2.4)

Checking the data from Table 2.1, we can get rms values as follows:

(V_O)₁ = 141.42 × 0.8 = 113.14 V at 50 Hz

(V_O)₂₃ = 141.42 × 0.81 8 = 115.68V at 1150 Hz

(V_O)₂₅ = 141.42 × 0.22 = 31.1 1 V at 1250 Hz

(V_O)₂₇ = 141.42 × 0.81 8 = 115.68V at 1350 Hz

(V_O)₅₁ = 141.42×0.139 = 19.66 V at 2550 Hz

(V_O)₅₃ = 141.42 × 0.314 = 44.41 V at 2650 Hz

(V_O)₅₅ = 141.42 × 0.314 = 44.41 V at 2750 Hz

(V_O)₅₇ = 141.42 × 0.139 = 19.66 V at 2850 Hz

etc.

2.2.2    Harmonic Parameters

Refer to Figure 2.4c, in which various harmonic parameters such as HF_n, THD, and WTHD, which are used in PWM operation, were introduced in Chapter 1.

2.3    Typical PWM Inverters

DC/AC inverters have three typical supply methods:

•  Voltage source inverter (VSI)

•  Current source inverter (CSI)

•  Impedance source inverter (z-source inverter or ZSI)

Generally, the main power circuits of various PWM inverters can be the same. The difference between them is the type of power supply source or network (voltage source, current source, or impedance source).

2.3.1    Voltage Source Inverter (VSI)

A voltage source inverter is supplied by a DC voltage source. In an ASD, the DC source is usually an AC/DC rectifier. There is a large capacitor used to keep the DC-link voltage stable. Usually, a VSI has buck operation function. Its output voltage peak value is lower than the DC link voltage.

It is necessary to avoid a short circuit across the DC voltage source during the operation. If a VSI operates in bipolar mode, that is, the upper switch and lower switch in a leg work to provide a PWM output waveform, the control circuit and interface have to be designed to leave small gaps between switching signals to the upper switch and lower switch in the same leg. For example, if the output voltage frequency is in the 0–400 Hz range, and the PWM carrying frequency is in the 2–20 kHz range, the gaps are usually set as 20–100 ns. This requirement is not very convenient for the control circuit and interface design. Therefore, unipolar operation is implemented in most industrial applications.

2.3.2    Current Source Inverter (CSI)

A current source inverter is supplied by a DC current source. In an ASD, the DC current source is usually an AC/DC rectifier with a large inductor to keep the supplying current stable. Usually, a CSI has a boost operation function. Its output voltage peak value is higher than the DC link voltage.

Since the source is a DC current source, it is necessary to avoid open circuit of the inverter during operation. The control circuit and interface have to be designed for small overlaps between switching signals to the upper switches and lower switches at least in one leg. For example, if the output voltage frequency is in the range 0–400 Hz, and the PWM carrying frequency is in the 2–20 kHz range, the overlaps are usually set as 20–100 ns. This requirement is easy to implement for the control circuit and interface design.

2.3.3    Impedance Source Inverter (z-Source Inverter—ZSI)

An impedance source inverter (ZSI) is supplied by a voltage source or current source via an x-shaped impedance network formed by two capacitors and two inductors, which is called a z-network. In an ASD, the DC impedance source is usually an AC/DC rectifier. A z-network is located between the rectifier and the inverter. Since there are two inductors and two capacitors to be set in front of the chopping legs, no restriction to avoid the legs opened or short-circuited. A ZSI has both buck and boost operation function. Its output voltage peak value can be higher or lower than the DC link voltage.

2.3.4    Circuits of DC/AC Inverters

The generally used DC/AC inverters are as follows:

1.  Single-phase half-bridge voltage source inverter (VSI)

2.  Single-phase full-bridge VSI

3.  Three-phase full-bridge VSI

4.  Three-phase full-bridge current source inverter (CSI)

5.  Multistage PWM inverter

6.  Soft switching inverter

7.  Impedance source inverter (ZSI)

References

1.  Mohan, N., Undeland, T. M., and Robbins, W. P. 2003. Power Electronics: Converters, Applications and Design (3rd edition). New York: John Wiley & Sons.

2.  Holtz, J. 1992. Pulsewidth modulation—A survey. IEEE Trans. Ind. Electron., pp. 410–420.

3.  Peng, F. Z. 2003. Z-source inverter. IEEE Trans. Ind. Appl., pp. 504–510.

4.  Trzynadlowski, A. M. 1998. Introduction to Modern Power Electronics. New York: John Wiley & Sons.

5.  Anderson, J. and Peng, F. Z. 2008. Four quasi-Z-Source inverters. Proc. IEEE PESC’2008, pp. 2743–2749.