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POWER ELECTRONICS Devices, Circuits, and Applications FOURTH EDITION
CHAPTER CHAPTER
6
DC–AC Converters
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
Copyright ©2014 by Pearson Education, Inc. All rights reserved.
Learning Outcomes
After completing this chapter, students should be able to do the following: Describe the switching techniques for dc–ac converters known as inverters and list the types of inverters. Explain the operating principal of inverters. List and determine the performance parameters of inverters. List the different types of modulation techniques to obtain a near sinusoidal output waveform and the techniques to eliminate certain harmonics from the output. Design and analyze inverters. Evaluate the performances of inverters by using PSpice simulations. Evaluate the effects of load impedances on the load current.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Symbols and Their Meanings
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 6.1
Input and output relationship of a dc–ac converter.
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Equations 6.1 and 6.1a
Performance Parameters
• The output power is given by
where Vo and Io are the rms load voltage and load current, θ is the angle of the load impedance, and R is the load resistance.
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Equation 6.2
Performance Parameters
• The ac input power of the inverter is
where VS and IS are the average input voltage and input current.
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Equation 6.3
Performance Parameters
• The rms ripple content of the input current
is
where Ii and Is are the rms and average values of the dc supply current.
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Equations 6.4 and 6.5
Performance Parameters
• The ripple factor of the input current is
• The harmonic factor is defined as
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Equations 6.6 and 6.7
Performance Parameters
• The total harmonic distortion is defined as
• DF is a measure of effectiveness in
reducing unwanted harmonics without having to specify the values of a secondorder load filter and is defined as
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Equation 6.8
Performance Parameters
• The DF of an individual (or nth) harmonic
component is defined as
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Figure 6.2
Single-phase half-bridge inverter.
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Equations 6.10 and 6.11
Single-phase Half-bridge Inverter
• The instantaneous output voltage νo is
• The rms value of fundamental component
is
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Equation 6.12
Single-phase Half-bridge Inverter
• For an RL load, the instantaneous load
current i0 can be found by dividing the instantaneous output voltage by the load impedance Z = R + jnωL.
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Equation 6.13
Single-phase Half-bridge Inverter
• If I01 is the rms fundamental load current,
the fundamental output power (or n = 1) is
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Figure 6.3
Single-phase full-bridge inverter.
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Equations 6.15 and 6.16
Single-phase Full-bridge Inverter
• The rms output voltage can be found from
• Equation (6.10) can be extended to
express the instantaneous output voltage in a Fourier series as
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Equations 6.17 and 6.18
Single-phase Full-bridge Inverter
• Eq. (6.16) gives the rms value of
fundamental component as
• Using Eq. (6.12), the instantaneous load
current i0 for an RL load becomes
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Equation 6.19
Single-phase Full-bridge Inverter
• Because the dc supply voltage remains
constant νs(t) = Vs, we get
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Table 6.1
Switch States for a Single-Phase Full-Bridge VoltageSource Inverter
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Figure 6.4
Waveforms for Example 6.3.
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Figure 6.5
Three-phase inverter formed by three single-phase inverters.
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Figure 6.6
Three-phase bridge inverter.
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Figure 6.7
Delta- and Υ-connected load.
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Table 6.2
Switch States for Three-Phase Voltage-Source Inverter
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Figure 6.8
Equivalent circuits for Υ-connected resistive load.
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Equations 6.20a, 6.20b, and 6.20c
• The instantaneous line-to-line voltage νab
(for a Υ-connected load) is
• Both νbc and νca can be found from Eq.
(6.20a) by phase shifting νab by 120° and 240°, respectively,
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Equations 6.23 and 6.24
• The rms fundamental line voltage.
• The rms value of line-to-neutral voltages
can be found from the line voltage,
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Figure 6.9
Three-phase inverter with RL load.
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Equations 6.25a, 6.25b, and 6.25c
• The instantaneous phase voltages (for a
Υ-connected load) are
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Equation 6.26
• Using Eq. (6.25a), the line current ia for
an RL load is given by
where θn = tan−1(nωL/R).
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Figure 6.10
Gating signals for 120° conduction.
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Figure 6.11
Equivalent circuits for Υ-connected resistive load.
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Equations 6.28a, 6.28b, and 6.28c
Equivalent Circuits
• The line-to-neutral voltages that are
shown in Figure 6.10 can be expressed in Fourier series as
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Equations 6.29a, 6.29b, and 6.29c
Equivalent Circuits
• The instantaneous line-to-line voltages
(for a Υ-connected load) are
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Figure 6.12
Multiple-pulse-width modulation.
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Equations 6.31 and 6.34
Multiple-pulse-width Modulation
• If δ is the width of each pulse, the rms
output voltage can be found from
• The coefficient Bn of Eq. (6.32) can be
found by adding the effects of all pulses,
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Figure 6.13
Harmonic profile of multiple-pulse-width modulation.
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Equations 6.35a, 6.35b, and 6.35c
Harmonic profile
• The mth time tm and angle αm of intersection
can be determined from
• Because all widths are the same, we get the
pulse width d (or pulse angle δ) as where Ts = T/2p. Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 6.14
Sinusoidal pulse-width modulation.
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Equation 6.36
Sinusoidal pulse-width Modulation
• If δm is the width of mth pulse, Eq. (6.31)
can be extended to find the rms output voltage by summing the average areas under each pulse as
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Equation 6.37
Sinusoidal pulse-width Modulation
• Equation (6.34) can also be applied to
determine the Fourier coefficient of output voltage as
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Figure 6.15
Harmonic profile of sinusoidal pulse-width modulation.
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Equations 6.38a, 6.38b, and 6.38c
Sinusoidal pulse-width Modulation
• The mth time tm and angle δm of
intersection can be determined from
where tx can be solved from
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Figure 6.16
Peak fundamental output voltage versus modulation index M.
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Figure 6.17
Modified sinusoidal pulse-width modulation.
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Equations 6.42a, 6.42b, and 6.42c
Modified Sinusoidal Pulse-width Modulation
• The mth time tm and angle δm of
intersection can be determined from
where tx can be solved from
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Figure 6.18
Harmonic profile of modified sinusoidal pulse-width modulation.
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Figure 6.19
Phase-displacement control.
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Equations 6.45 and 6.46
Phase-displacement Control
• The instantaneous output voltage can be
simplified to
• The rms value of the fundamental output
voltage is
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Figure 6.20
Sinusoidal pulse-width modulation for three-phase inverter.
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Equations 6.50 and 6.51
Sinusoidal PWM for Three-phase Inverter
• The harmonics in the ac output voltage
appear at normalized frequencies fh centered around mf and its multiples, specifically, at
• For nearly sinusoidal ac load current, the
harmonics in the dc-link current are at frequencies given by
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Equation 6.52
Sinusoidal PWM for Three-phase Inverter
• One can write the peak amplitude as
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Figure 6.21
Square-ware operation.
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Equations 6.54 and 6.55
Overmodulation
• The fundamental ac line voltage is given by
• The ac line output voltage contains the
harmonics fn and their amplitudes are inversely proportional to their harmonic order n.
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Figure 6.22
Output waveform for 60° PWM.
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Figure 6.23
Output waveform for third-harmonic PWM.
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Equations 6.57, 6.60a, and 6.60b
Third-harmonic PWM
• This is shown in Figure 6.24. A rotating
space vector(s) u(t) in complex notation is then given by
• The coordinate transformation from the a–
b–c-axis to the x–y axis can be written as
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Equation 6.64
Third-harmonic PWM
• Then, using Eq. (6.57), we get the space
vector representation as
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Figure 6.24
Three-phase coordinate vectors and space vector u(t).
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Figure 6.25
The on and off states of the inverter switches. [Ref. 13]
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Equations 6.67 and 6.70
On and Off State
• We can derive all six vectors as
• The normalized peak value of the nth line
voltage vector can be found from
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Equation 6.72
On and Off State
• If the output voltages are purely
sinusoidal, then the performance vector U becomes
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Figure 6.26
The space vector representation.
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Equations 6.74 and 6.75
Space Vector
• The vectors of three-phase line
modulating signals can be represented by the complex vector U* = υr = [υ]αβ = [υrαυrβ]T as given by
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Figure 6.27
Determination of state times.
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Equations 6.76a, 6.76b, and 6.77
Determination of State Times
• We can equate the volt time of the
reference vector to the SVs as
• Equation (6.67) gives the space vectors in
sector 1 as
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Figure 6.28
Pattern of SVM.
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Equations 6.79a, 6.79b, and 6.79c
Pattern of SVM
• Solving for T1, T2, and Tz in sector 1 (0 ≤
θ ≤ π/3), we get
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Equation 6.83
Pattern of SVM
• After substituting in Eq. (6.82)
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Table 6.3
Relationship between the Dwell Times and the Space Vector Angle θ for Sector 1
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Figure 6.29
Three-phase waveforms for space vector modulation (M = 0.8, fsn = 18).
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Equations 6.87a, 6.87b, and 6.87c
Waveforms for Space Vector Modulation
• The instantaneous phase voltages can be
found by time averaging of the SVs during one switching period for sector 1 as given by
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Figure 6.30
Overmodulation. [Ref. 20, R. Valentine]
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Equations 6.88a, 6.88b, and 6.88c
Overmodulation
• The portions of the circle outside the
hexagon are limited by the boundaries of the hexagon and the corresponding time states Tn and Tn+1 can be found from [20]
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Table 6.4
Switching Segments for all SVM Sectors
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Figure 6.31
Block diagram for digital implementation of the SVM algorithm.
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Equations 6.89a and 6.89b
Digital Implementation of the SVM Algorithm
• Find magnitude Vr and the angle θ of the
reference vector.
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Table 6.5
Summary of Modulation Techniques
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Figure 6.32
Output voltage with two bipolar notches per half-wave.
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Equation 6.94
Bipolar Notches
• Equation (6.92) can be extended to m
notches per quarter-wave:
where α1 < α2 < . . . < αk < π/2.
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Figure 6.33
Unipolar output voltage with two notches per half-cycle.
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Equation 6.96
Unipolar Output Voltage
• Equation (6.95) can be extended to m
notches per quarter-wave as
where α1 < α2 < . . . < αk < π/2.
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Figure 6.34
Output voltage for modified sinusoidal pulse-width modulation.
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Equation 6.97
60-Degree Modulation
• The coefficient Bn is given by
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Figure 6.35
Elimination of harmonics by transformer connection.
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Figure 6.36
Single-phase current source.
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Equations 6.99 and 6.100
Single-phase Current Source
• From Eq. (6.20a), the instantaneous
current for phase a of a Υ-connected load can be expressed as
• From Eq. (6.25a), the instantaneous
phase current for a delta-connected load is given by
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Table 6.6
Switch States for a Full-Bridge Single-Phase CurrentSource Inverter (CSI)
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Figure 6.37
Three-phase current source transistor inverter.
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Figure 6.38
Variable dc-link inverter.
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Equation 6.103
Variable Dc-link Inverter
• The output voltage is sinusoidal as given
by
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Figure 6.39
Principle of boost inverter.
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Figure 6.40
Boost inverter consisting of two boost converters. [Ref. 22, R. CaCeres]
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Figure 6.41
Equivalent circuit of converter A.
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Equations 6.104 and 6.105
Equivalent Circuit of Converter
• The average output voltage of converter
A, which operates under the boost mode, can be found from
• The average output voltage of converter
B, which operates under the buck mode, can be found from
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Figure 6.42
Equivalent circuits during modes of operation.
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Equation 6.110
Equivalent Circuits
• The inductor current IL that depends on
the load resistance R and the duty cycle k can be found from
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Figure 6.43
Gain characteristics of the boost inverter.
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Equations 6.106 and 6.109
Gain characteristics
• The dc gain of the boost inverter is given
as
• The ac voltage gain is given as
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Figure 6.44
Buck–boost inverter. [Ref. 23, R. CaCeres]
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Figure 6.45
Output filters.
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Figure 6.46
Single-phase inverter for PSpice simulation.
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Figure 6.47
PSpice plots for Example 6.8.
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CHAPTER CHAPTER
6
DC–AC Converters
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Learning Outcomes
After completing this chapter, students should be able to do the following: Describe the switching techniques for dc–ac converters known as inverters and list the types of inverters. Explain the operating principal of inverters. List and determine the performance parameters of inverters. List the different types of modulation techniques to obtain a near sinusoidal output waveform and the techniques to eliminate certain harmonics from the output. Design and analyze inverters. Evaluate the performances of inverters by using PSpice simulations. Evaluate the effects of load impedances on the load current.
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Symbols and Their Meanings
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Figure 6.1
Input and output relationship of a dc–ac converter.
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Equations 6.1 and 6.1a
Performance Parameters
• The output power is given by
where Vo and Io are the rms load voltage and load current, θ is the angle of the load impedance, and R is the load resistance.
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Equation 6.2
Performance Parameters
• The ac input power of the inverter is
where VS and IS are the average input voltage and input current.
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Equation 6.3
Performance Parameters
• The rms ripple content of the input current
is
where Ii and Is are the rms and average values of the dc supply current.
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Equations 6.4 and 6.5
Performance Parameters
• The ripple factor of the input current is
• The harmonic factor is defined as
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Equations 6.6 and 6.7
Performance Parameters
• The total harmonic distortion is defined as
• DF is a measure of effectiveness in
reducing unwanted harmonics without having to specify the values of a secondorder load filter and is defined as
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Equation 6.8
Performance Parameters
• The DF of an individual (or nth) harmonic
component is defined as
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Figure 6.2
Single-phase half-bridge inverter.
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Equations 6.10 and 6.11
Single-phase Half-bridge Inverter
• The instantaneous output voltage νo is
• The rms value of fundamental component
is
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Equation 6.12
Single-phase Half-bridge Inverter
• For an RL load, the instantaneous load
current i0 can be found by dividing the instantaneous output voltage by the load impedance Z = R + jnωL.
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Equation 6.13
Single-phase Half-bridge Inverter
• If I01 is the rms fundamental load current,
the fundamental output power (or n = 1) is
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Figure 6.3
Single-phase full-bridge inverter.
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Equations 6.15 and 6.16
Single-phase Full-bridge Inverter
• The rms output voltage can be found from
• Equation (6.10) can be extended to
express the instantaneous output voltage in a Fourier series as
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Equations 6.17 and 6.18
Single-phase Full-bridge Inverter
• Eq. (6.16) gives the rms value of
fundamental component as
• Using Eq. (6.12), the instantaneous load
current i0 for an RL load becomes
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Equation 6.19
Single-phase Full-bridge Inverter
• Because the dc supply voltage remains
constant νs(t) = Vs, we get
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Table 6.1
Switch States for a Single-Phase Full-Bridge VoltageSource Inverter
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Figure 6.4
Waveforms for Example 6.3.
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Figure 6.5
Three-phase inverter formed by three single-phase inverters.
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Figure 6.6
Three-phase bridge inverter.
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Figure 6.7
Delta- and Υ-connected load.
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Table 6.2
Switch States for Three-Phase Voltage-Source Inverter
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Figure 6.8
Equivalent circuits for Υ-connected resistive load.
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Equations 6.20a, 6.20b, and 6.20c
• The instantaneous line-to-line voltage νab
(for a Υ-connected load) is
• Both νbc and νca can be found from Eq.
(6.20a) by phase shifting νab by 120° and 240°, respectively,
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Equations 6.23 and 6.24
• The rms fundamental line voltage.
• The rms value of line-to-neutral voltages
can be found from the line voltage,
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Figure 6.9
Three-phase inverter with RL load.
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Equations 6.25a, 6.25b, and 6.25c
• The instantaneous phase voltages (for a
Υ-connected load) are
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Equation 6.26
• Using Eq. (6.25a), the line current ia for
an RL load is given by
where θn = tan−1(nωL/R).
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Figure 6.10
Gating signals for 120° conduction.
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Figure 6.11
Equivalent circuits for Υ-connected resistive load.
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Equations 6.28a, 6.28b, and 6.28c
Equivalent Circuits
• The line-to-neutral voltages that are
shown in Figure 6.10 can be expressed in Fourier series as
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Equations 6.29a, 6.29b, and 6.29c
Equivalent Circuits
• The instantaneous line-to-line voltages
(for a Υ-connected load) are
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Figure 6.12
Multiple-pulse-width modulation.
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Equations 6.31 and 6.34
Multiple-pulse-width Modulation
• If δ is the width of each pulse, the rms
output voltage can be found from
• The coefficient Bn of Eq. (6.32) can be
found by adding the effects of all pulses,
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Figure 6.13
Harmonic profile of multiple-pulse-width modulation.
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Equations 6.35a, 6.35b, and 6.35c
Harmonic profile
• The mth time tm and angle αm of intersection
can be determined from
• Because all widths are the same, we get the
pulse width d (or pulse angle δ) as where Ts = T/2p. Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 6.14
Sinusoidal pulse-width modulation.
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Equation 6.36
Sinusoidal pulse-width Modulation
• If δm is the width of mth pulse, Eq. (6.31)
can be extended to find the rms output voltage by summing the average areas under each pulse as
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Equation 6.37
Sinusoidal pulse-width Modulation
• Equation (6.34) can also be applied to
determine the Fourier coefficient of output voltage as
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Figure 6.15
Harmonic profile of sinusoidal pulse-width modulation.
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Equations 6.38a, 6.38b, and 6.38c
Sinusoidal pulse-width Modulation
• The mth time tm and angle δm of
intersection can be determined from
where tx can be solved from
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Figure 6.16
Peak fundamental output voltage versus modulation index M.
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Figure 6.17
Modified sinusoidal pulse-width modulation.
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Equations 6.42a, 6.42b, and 6.42c
Modified Sinusoidal Pulse-width Modulation
• The mth time tm and angle δm of
intersection can be determined from
where tx can be solved from
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Figure 6.18
Harmonic profile of modified sinusoidal pulse-width modulation.
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Figure 6.19
Phase-displacement control.
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Equations 6.45 and 6.46
Phase-displacement Control
• The instantaneous output voltage can be
simplified to
• The rms value of the fundamental output
voltage is
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Figure 6.20
Sinusoidal pulse-width modulation for three-phase inverter.
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Equations 6.50 and 6.51
Sinusoidal PWM for Three-phase Inverter
• The harmonics in the ac output voltage
appear at normalized frequencies fh centered around mf and its multiples, specifically, at
• For nearly sinusoidal ac load current, the
harmonics in the dc-link current are at frequencies given by
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Equation 6.52
Sinusoidal PWM for Three-phase Inverter
• One can write the peak amplitude as
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Figure 6.21
Square-ware operation.
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Equations 6.54 and 6.55
Overmodulation
• The fundamental ac line voltage is given by
• The ac line output voltage contains the
harmonics fn and their amplitudes are inversely proportional to their harmonic order n.
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Figure 6.22
Output waveform for 60° PWM.
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Figure 6.23
Output waveform for third-harmonic PWM.
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Equations 6.57, 6.60a, and 6.60b
Third-harmonic PWM
• This is shown in Figure 6.24. A rotating
space vector(s) u(t) in complex notation is then given by
• The coordinate transformation from the a–
b–c-axis to the x–y axis can be written as
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Equation 6.64
Third-harmonic PWM
• Then, using Eq. (6.57), we get the space
vector representation as
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Figure 6.24
Three-phase coordinate vectors and space vector u(t).
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Figure 6.25
The on and off states of the inverter switches. [Ref. 13]
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Equations 6.67 and 6.70
On and Off State
• We can derive all six vectors as
• The normalized peak value of the nth line
voltage vector can be found from
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Equation 6.72
On and Off State
• If the output voltages are purely
sinusoidal, then the performance vector U becomes
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Figure 6.26
The space vector representation.
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Equations 6.74 and 6.75
Space Vector
• The vectors of three-phase line
modulating signals can be represented by the complex vector U* = υr = [υ]αβ = [υrαυrβ]T as given by
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Figure 6.27
Determination of state times.
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Equations 6.76a, 6.76b, and 6.77
Determination of State Times
• We can equate the volt time of the
reference vector to the SVs as
• Equation (6.67) gives the space vectors in
sector 1 as
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Figure 6.28
Pattern of SVM.
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Equations 6.79a, 6.79b, and 6.79c
Pattern of SVM
• Solving for T1, T2, and Tz in sector 1 (0 ≤
θ ≤ π/3), we get
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Equation 6.83
Pattern of SVM
• After substituting in Eq. (6.82)
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Table 6.3
Relationship between the Dwell Times and the Space Vector Angle θ for Sector 1
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Figure 6.29
Three-phase waveforms for space vector modulation (M = 0.8, fsn = 18).
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Equations 6.87a, 6.87b, and 6.87c
Waveforms for Space Vector Modulation
• The instantaneous phase voltages can be
found by time averaging of the SVs during one switching period for sector 1 as given by
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Figure 6.30
Overmodulation. [Ref. 20, R. Valentine]
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Equations 6.88a, 6.88b, and 6.88c
Overmodulation
• The portions of the circle outside the
hexagon are limited by the boundaries of the hexagon and the corresponding time states Tn and Tn+1 can be found from [20]
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Table 6.4
Switching Segments for all SVM Sectors
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Figure 6.31
Block diagram for digital implementation of the SVM algorithm.
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Equations 6.89a and 6.89b
Digital Implementation of the SVM Algorithm
• Find magnitude Vr and the angle θ of the
reference vector.
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Table 6.5
Summary of Modulation Techniques
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Figure 6.32
Output voltage with two bipolar notches per half-wave.
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Equation 6.94
Bipolar Notches
• Equation (6.92) can be extended to m
notches per quarter-wave:
where α1 < α2 < . . . < αk < π/2.
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Figure 6.33
Unipolar output voltage with two notches per half-cycle.
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Equation 6.96
Unipolar Output Voltage
• Equation (6.95) can be extended to m
notches per quarter-wave as
where α1 < α2 < . . . < αk < π/2.
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Figure 6.34
Output voltage for modified sinusoidal pulse-width modulation.
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Equation 6.97
60-Degree Modulation
• The coefficient Bn is given by
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Figure 6.35
Elimination of harmonics by transformer connection.
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Figure 6.36
Single-phase current source.
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Equations 6.99 and 6.100
Single-phase Current Source
• From Eq. (6.20a), the instantaneous
current for phase a of a Υ-connected load can be expressed as
• From Eq. (6.25a), the instantaneous
phase current for a delta-connected load is given by
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Table 6.6
Switch States for a Full-Bridge Single-Phase CurrentSource Inverter (CSI)
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Figure 6.37
Three-phase current source transistor inverter.
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Figure 6.38
Variable dc-link inverter.
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Equation 6.103
Variable Dc-link Inverter
• The output voltage is sinusoidal as given
by
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Figure 6.39
Principle of boost inverter.
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Figure 6.40
Boost inverter consisting of two boost converters. [Ref. 22, R. CaCeres]
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Figure 6.41
Equivalent circuit of converter A.
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Equations 6.104 and 6.105
Equivalent Circuit of Converter
• The average output voltage of converter
A, which operates under the boost mode, can be found from
• The average output voltage of converter
B, which operates under the buck mode, can be found from
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Figure 6.42
Equivalent circuits during modes of operation.
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Equation 6.110
Equivalent Circuits
• The inductor current IL that depends on
the load resistance R and the duty cycle k can be found from
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Figure 6.43
Gain characteristics of the boost inverter.
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Equations 6.106 and 6.109
Gain characteristics
• The dc gain of the boost inverter is given
as
• The ac voltage gain is given as
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Figure 6.44
Buck–boost inverter. [Ref. 23, R. CaCeres]
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Figure 6.45
Output filters.
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Figure 6.46
Single-phase inverter for PSpice simulation.
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Figure 6.47
PSpice plots for Example 6.8.
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