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CHAPTER 2
SINGLE PHASE PULSE WIDTH MODULATED INVERTERS
2.1 Introduction
The dc-ac converter, also known as the inverter, converts dc power to ac power at desired output voltage and frequency. The dc power input to the inverter is obtained from an existing power supply network or from a rotating alternator through a rectifier or a battery, fuel cell, photovoltaic array or magneto hydrodynamic generator. The filter capacitor across the input terminals of the inverter provides a constant dc link voltage. The inverter therefore is an adjustable-frequency voltage source. The configuration of ac to dc converter and dc to ac inverter is called a dc- link converter. Inverters can be broadly classified into two types, voltage source and current source inverters. A voltage–fed inverter (VFI) or more generally a voltage–source inverter (VSI) is one in which the dc source has small or negligible impedance. The voltage at the input terminals is constant. A current–source inverter (CSI) is fed with adjustable current from the dc source of high impedance that is from a constant dc source. A voltage source inverter employing thyristors as switches, some type of forced commutation is required, while the VSIs made up of using GTOs, power transistors, power MOSFETs or IGBTs, self commutation with base or gate drive signals for their controlled turn-on and turn-off.
A standard single-phase voltage or current source inverter can be in the half- bridge or full-bridge configuration. The single-phase units can be joined to have three-phase or multiphase topologies. Some industrial applications of inverters are for adjustable-speed ac drives, induction heating, standby aircraft power supplies, UPS (uninterruptible power supplies) for computers, HVDC transmission lines, etc. In this chapter single-phase inverters and their operating principles are analyzed in detail. The concept of Pulse Width Modulation (PWM) for inverters is described with analyses extended to different kinds of PWM strategies. Finally the simulation results for a single-phase inverter using the PWM strategies described are presented. 2.2 Voltage Control in Single - Phase Inverters
The schematic of inverter system is as shown in Figure 2.1, in which the battery or rectifier provides the dc supply to the inverter. The inverter is used to control the fundamental voltage magnitude and the frequency of the ac output voltage. AC loads may require constant or adjustable voltage at their input terminals, when such loads are fed by inverters, it is essential that the output voltage of the inverters is so controlled as to fulfill the requirement of the loads. For example if the inverter supplies power to a magnetic circuit, such as a induction motor, the voltage to frequency ratio at the inverter output terminals must be kept constant. This avoids saturation in the magnetic circuit of the device fed by the inverter.
adjusting the on and the off periods of the inverter components. The advantages of the PWM control scheme are [10]: a) The output voltage control can be obtained without addition of any external components. b) PWM minimizes the lower order harmonics, while the higher order harmonics can be eliminated using a filter. The disadvantage possessed by this scheme is that the switching devices used in the inverter are expensive as they must possess low turn on and turn off times, nevertheless PWM operated are very popular in all industrial equipments. PWM techniques are characterized by constant amplitude pulses with different duty cycles for each period. The width of these pulses are modulated to obtain inverter output voltage control and to reduce its harmonic content. There are different PWM techniques which essentially differ in the harmonic content of their respective output voltages, thus the choice of a particular PWM technique depends on the permissible harmonic content in the inverter output voltage.
2.2.2 Sinusoidal-Pulse Width Modulation (SPWM)
The sinusoidal PWM (SPWM) method also known as the triangulation, sub harmonic, or suboscillation method, is very popular in industrial applications and is extensively reviewed in the literature [1-2]. The SPWM is explained with reference to Figure 2.2, which is the half-bridge circuit topology for a single-phase inverter.
S (^11)
S (^12)
V d
V d
V d
C
C
V o
Figure 2.2: Schematic diagram for Half-Bridge PWM inverter. For realizing SPWM, a high-frequency triangular carrier wave is
compared with a sinusoidal reference of the desired frequency. The intersection of
and waves determines the switching instants and commutation of the modulated
pulse. The PWM scheme is illustrated in Figure 2.3 a, in which v is the peak value of
triangular carrier wave and v that of the reference, or modulating signal. The figure
shows the triangle and modulation signal with some arbitrary frequency and magnitude. In the inverter of Figure 2.2 the switches and are controlled based
on the comparison of control signal and the triangular wave which are mixed in a comparator. When sinusoidal wave has magnitude higher than the triangular wave the comparator output is high, otherwise it is low.
v c v r v c vr c
12
r
S 11 S
v (^) r > vc S 11 is on , Vout = V 2^ d (2.1) and vr < vc S 12 is on , Vout = − V 2^ d (2.2)
The comparator output is processesed in a trigger pulse generator in such a manner that the output voltage wave of the inverter has a pulse width in agreement
with the comparator output pulse width. The magnitude ratio of vv cr^ is called the
modulation index ( ) and it controls the harmonic content of the output voltage
waveform. The magnitude of fundamental component of output voltage is proportional to. The amplitude of the triangular wave is generally kept
constant. The frequency–modulation ratio is defined as
m i
mi vc m f
m f t f m = f (2.3)
To satisfy the Kirchoff’s Voltage law (KVL) constraint, the switches on the same leg are not turned on at the same time, which gives the condition S 11 + S 12 = 1 (2.4)
for each leg of the inverter. This enables the output voltage to fluctuate between
2
V d (^) and 2 − V^ d as shown in Figure 2.4 for a dc voltage of 200 V.
Figure 2.4: Output voltage of the Half-Bridge inverter.
2.3 Single-Phase Inverters
A single-phase inverter in the full bridge topology is as shown in Figure 2.5, which consists of four switching devices, two of them on each leg. The full- bridge inverter can produce an output power twice that of the half-bridge inverter with the same input voltage. Three different PWM switching schemes are discussed in this section, which improve the characteristics of the inverter. The objective is to add a zero sequence voltage to the modulation signals in such a way to ensure the clamping of the devices to either the positive or negative dc rail; in the process of which the voltage gain is improved, leading to an increased load fundamental voltage, reduction in total current distortion and increased load power factor. In Figure 2.5, the top devices are assigned to be S 11 and S 21 while the bottom devices as S 12 and S22, the voltage equations for this converter are as given in the following equations.
Equations 2.8 and 2.9 give the general expression for the modulation signals for single-phase dc-ac converters. The various types of modulation schemes presented in the literature can be obtained from these equations using appropriate definition for , V and V. Making use of this concept different modulation schemes have been
proposed some of which are explained in detail in the following sections.
V an (^) bn no
2.3.1 SPWM With Bipolar Switching
In this scheme the diagonally opposite transistors S11, S 22 and S 21 and S 12 are turned on or turned off at the same time. The output of leg A is equal and opposite to the output of leg B. The output voltage is determined by comparing the control signal, and the triangular signal, V as shown in Figure 2.6(a) to get the switching pulses
for the devices , and the switching pattern is as follows.
V r (^) c
V r > V (^) c , S 11 is on => Vao =^ Vd 2 and S 22 is on => Vbo =−^ Vd 2 ; (2.10)
V r < V (^) c , S 12 is on => Vao =−^ Vd 2 and S 21 is on => Vbo =^ Vd 2 ; (2.11)
hence Vbo ( t )= − Vao ( t ) (2.12)
(a)
(b)
(c)
Figure 2.6:Bipolar PWM (a) Sine-triangle comparison (b) Switching pulses for S 11 /S 22 (c) Switching pulses for S 12 /S 21
(a)
(b)
(c)
Figure 2.7: Bipolar PWM scheme (a) Modulation signal for leg ‘a’ (b) output line-line voltage (c) load current
For the bipolar PWM switching scheme there is only one modulation signal and the switches are turned ‘on’ or turned ‘off’ according to the pattern given in Equations 2.10 and 2.11. The input dc voltage was 200 V and the modulation index (mi) was taken to be 0.8. The switching frequency for the carrier, which is the triangle, is 10 kHz.
2.3.2 SPWM With Unipolar Switching
In this scheme, the devices in one leg are turned on or off based on the comparison of the modulation signal V with a high frequency triangular wave. The
devices in the other leg are turned on or off by the comparison of the modulation signal with the same high frequency triangular wave. Figure 2.8 and 2.9 show
the unipolar scheme for a single –phase full bridge inverter, with the modulation signals for both legs and the associated comparison to yield switching pulses for both the legs.
r
− V r
In Figure 2.8 the simulation results show the sine triangle comparison, the switching pulses for S 11 and S 21 are shown. The switching for the other two devices is obtained as S 12 = 1 – S 11 and S 22 = 1- S 21. Figure 2.9 shows the phase voltages , line- to-line voltages obtained from a unipolar PWM scheme , also shown is the load current. The simulation was carried out for an RL load of R = 10Ω and L = 0.125H. The dc voltage is 200 V and the switching frequency is 10kHz. The modulation signal has a magnitude of 0.8, i.e mi = 0.8.
(a)
(b)
(c)
Figure 2.8: Unipolar PWM voltage switching scheme (a) Sine triangle comparison (b) switching pulses for S 11 (c) switching pulses for S 21.
(a)
(b)
(c)
(d)
Figure 2.9: Unipolar PWM voltage switching scheme (a) phase voltage ‘a’ (b) phase voltage ‘b’ (c) line to line voltage Vab (d) load current
Table 2.1. Switching state of the unipolar PWM and the corresponding voltage levels. S 11 S 12 S 21 S (^) 22 VAn VBn Vo = VAn − VBn ON - - ON (^) Vd (^0) Vd
- ON ON - (^0) Vd - Vd ON - ON - (^) Vd Vd 0
- ON - ON 0 0 0
The fundamental component of the output voltage is given as Vo = miV d ( mi ≤ 1. 0 ) (2.21)
Vd < Vo < π^4 V d ( mi > 1. 0 ). (2.22)
2.3.3 SPWM With Modified Bipolar Switching Scheme (MBPWM)[14]
In the inverter employing the bipolar switching scheme, switches are operated in such a way that during the positive half of the modulation signal one of the top devices in one of the switching leg is kept on and the two other switching devices in the other leg are PWM operated, and during the negative half of the modulation signal one of the bottom switching device is kept on continuously while the other two switching devices in the other leg are PWM operated. The output voltage is determined by comparing the control signal Vrand the triangular wave.
The switching pattern along with the sine-triangle comparison is as shown in Figure 2.10. The switching pattern for positive values of modulating signal V (^) m is as given
V (^) r > V (^) c , S 21 is on (2.23)
and V (^) r < V (^) c , S 22 is on.
(a)
(b)
(c)
(d)
(e)
Figure 2.10: Modified bipolar PWM (a) Sine-triangle comparison (b), (c), (d), and (e) switching pulses for devices S 11 , S12, S 21 and S 22.
Table 2.2. Switching state of the modified bipolar PWM and the correspondingvoltage. S 11 S 12 S (^) 21 S 22 VAn VBn Vo = VAn − VBn ON - - ON (^) Vd (^0) Vd
- ON ON - (^0) Vd - Vd ON - ON - (^) Vd Vd 0
- ON - ON 0 0 0
From Table 2.2 it can be observed that when the two top or the two bottom devices are turned on the output voltage is zero. In the modified bipolar switching scheme the output voltage level changes between either 0 to - V or from 0 to + V. Since the sign of the modulation signal
decides the switching pattern the analysis of this switching scheme is complex. The relationship between input and output voltage is given as [14],
d d
Vo = mV d
(2.25)
where m = 0. 5 ( mi + π^4 ) ( mi < 1. 0 ). (2.26)
Thus from the above equation it can be observed that the fundamental component of the voltage as obtained from the MBPWM is the maximum when compared to the other switching schemes even in the linear modulation region; that is when the modulation index is less than unity.
2.3.4 Generalized Carrier-based PWM
In the inverter shown in Figure 2.5, the output voltage and the input current are given as
- 5 Vd ( S 11 − S 12 )= Vao = Van + V no (2.27)
- 5 Vd ( S 21 − S 22 )= Vbo = Vbn + V no (2.28) I (^) d = Ia ( S 11 − S 21 ) (2.29) Vab = V (^) an − V bn. (2.30)
The voltages V and V are the output voltages from phases ‘a’ and ‘b’ to a arbitrary
point while V is the neutral voltage between the point ‘n’ and the mid-point of the
DC source. The generalized carrier-based PWM scheme is obtained by defining the quantity using the concept of q-d Space Vector representation. A special q-d
reference frame transformation to transform the two phase voltages to orthogonal q-d voltage components is defined as
an no
bn
V no
Vq = 0. 5 ( Van + Vbn ) (2.31) Vd = (^0). 5 ( Van − Vbn ) (2.32)
where and are the q-axis and the d-axis voltages in an orthogonal coordinate
system. The q-d voltages for each of the possible switching instant are shown in Table 2.3.
V q Vd
