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POWER ELECTRONICS Devices, Circuits, and Applications FOURTH EDITION
CHAPTER CHAPTER
11
AC Voltage Controllers
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Learning Outcomes
After completing this chapter, students should be able to do the following: List the types of ac voltage controllers. Describe the operation of ac voltage controllers. Describe the characteristics of ac voltage controllers. List the performance parameters of ac voltage controllers. Describe the operation of matrix converters. Design and analyze ac voltage controllers. Evaluate the performances of controlled rectifiers by using SPICE simulations. Evaluate the effects of load inductance 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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Symbols and Their Meanings (continued)
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Figure 11.1 Input and output relationship of an ac voltage controller. (a) Block diagram, (b) Input supply, (c) Output voltage, and (d) Input current.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 11.2 Single-phase full-wave controller. (a) Circuit, (b) Input supply voltage, (c) Output voltage, (d) Gate pulse for T1, and (e) Gate pulse for T2.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equation 11.1
Single-phase Full-wave Controller
• If υs = √2Vssinωt is the input voltage, and
the delay angles of thyristors T1 and T2 are equal (α2 = π + α1), the rms output voltage can be found from
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equation 11.2
Single-phase Full-wave Controller
• The rms value of load current is Io = Vo/R
= 84.85/10 = 8.485 A and the load power is Po = Io2R = 8.4852 × 10 = 719.95 W. Because the input current is the same as the load current, the input PF is
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equation 11.3
Single-phase Full-wave Controller
• The average thyristor current
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equation 11.4
Single-phase Full-wave Controller
• The rms value of the thyristor current
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 11.3
Single-phase full-wave controller with common cathode.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 11.4 pulse for T1.
Single-phase full-wave controller with one thyristor. (a) Circuit, (b) Input supply voltage, (c) Output current, and (d) Gate
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 11.5 Single-phase full-wave controller with RL load. (a) Circuit, (b) Input supply voltage, (c) Gate pulses for T1 and T2, (d) Current through thyristor T1, (e) Continuous gate pulses for T1 and T2, and (f) Train of gate pulses for T1 and T2.
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Equations 11.8 and 11.9
Single-phase Full-wave Controller with RL Load
• Substitution of A1 from Eq. (11.7) in Eq.
(11.6) yields
• The angle β, when current i1 falls to zero
and thyristor T1 is turned off, can be found from the condition i1(ωt = β) = 0 in Eq. (11.8) and is given by the relation
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Equation 11.11
Single-phase Full-wave Controller with RL Load
• The rms output voltage
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 11.6 Typical waveforms of single-phase ac voltage controller with an RL load. (a) Input supply voltage and output current, (b) Output voltage, and (c) Voltage across thyristor T1.
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Equations 11.12
Typical Waveforms
• The rms thyristor current can be found
from Eq. (11.8) as
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Equations 11.14
Typical Waveforms
• The average value of thyristor current can
also be found from Eq. (11.8) as
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Equations 11.16 and 11.17
Typical Waveforms
• If α = θ, from Eq. (11.9)
• Because the conduction angle δ cannot
exceed π and the load current must pass through zero, the delay angle α may not be less than θ and the control range of delay angle is
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Figure 11.7
Three-phase bidirectional controller.
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Three-phase Bidirectional Controller
• We define the instantaneous input phase
voltages as
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Figure 11.8 Waveforms for three-phase bidirectional controller. (a) Input line voltages, (b) Input phase voltages, (c) Thyristor gate pulses, and (d) Output phase voltage.
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Equation 11.19
Waveforms for Three-phase Bidirectional Controller
• For 0 ≤ α < 60°:
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Equations 11.20
Waveforms for Three-phase Bidirectional Controller
• For 60° ≤ α < 90°:
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Equations 11.20 and 11.21
Waveforms for Three-phase Bidirectional Controller
• For 90° ≤ α ≤ 150°:
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Figure 11.9
Arrangement for three-phase bidirectional tie control.
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Figure 11.10
Delta-connected three-phase controller.
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Delta-connected Three-phase Controller
• Let us assume that the instantaneous line-
to-line voltages are
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Figure 11.11 Waveforms for delta-connected controller. (a) Input line voltages, (b) Thyristor gate pulses, (c) Output phase currents, and (d) Output line currents.
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Equations 11.22 and 11.23
Waveforms for Delta-connected Controller
• For resistive loads, the rms output phase
voltage can be determined from
• The maximum output voltage would be
obtained when α = 0, and the control range of delay angle is
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Equations 11.24 and 11.26
Waveforms for Delta-connected Controller
• The line currents, which can be
determined from the phase currents, are
• The rms line current becomes
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Equation 11.27
Waveforms for Delta-connected Controller
• As a result, the rms value of line current
would not follow the normal relationship of a three-phase system such that
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Figure 11.12
Three-phase three-thyristor controller.
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Figure 11.13
Single-phase transformer connection changer.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Single-phase Transformer Connection Changer
• The gating pulses of thyristors can be
controlled to vary the load voltage. The rms value of load voltage Vo can be varied within three possible ranges:
and
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Figure 11.14 Waveforms for transformer connection changer. (a) Voltage for secondary 1, (b) Voltage for secondary 2, (c) Output voltage for case 1, (d) Output voltage for case 2, and (d) Output voltage for case 3.
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Equation 11.28
Waveforms for Transformer Connection Changer
• The rms load voltage that can be
determined from Eq. (11.1) load is
and the range of delay angle is 0 ≤ α ≤ π.
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Equation 11.29
Waveforms for Transformer Connection Changer
• The rms load voltage can be found from
and the range of delay angle is 0 ≤ α ≤ π.
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Equation 11.30
Waveforms for Transformer Connection Changer
• The rms load voltage can be found from
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Figure 11.15 Voltage and current waveforms for RL load. (a) Output voltage and current, (b) Output voltage, and (c) Output current and fundamental component.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Voltage and Current Waveforms for RL Load
• The load current would then be
where Z = [R2 + (ωL)2]1/2 and θ = tan−1 (ωL/R).
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Figure 11.16 Single-phase/single-phase cycloconverter. (a) Circuit, (b) Equivalent circuit, (c) Input supply voltage, (d) Output voltage, and (e) Conduction periods for P and N converters.
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Equation 11.33
Single-phase/Single-phase Cycloconverter
• The average output voltage of the positive
converter is equal and opposite to that of the negative converter.
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Equation 11.34
Single-phase/Single-phase Cycloconverter
• Vs = 120 V, fs = 60 Hz, fo = 20 Hz, R =
5Ω, L = 40 mH, αp = 2π/3, ω0 = 2π × 20 = 125.66 rad/s, and XL = ω0L = 5.027Ω. a. For 0 ≤ α ≤ π, Eq. (11.1) gives the rms output voltage
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Equation 11.35
Single-phase/Single-phase Cycloconverter
• Vs = 120 V, fs = 60 Hz, fo = 20 Hz, R =
5Ω, L = 40 mH, αp = 2π/3, ω0 = 2π × 20 = 125.66 rad/s, and XL = ω0L = 5.027Ω. c. The rms input current is Is = I0 = 7.48 A, the VA rating is VA = VsIs = 897.6 VA, and the output power is Po = VoIo cos θ = 53 × 7.48 × cos 45.2° = 279.35 W. Using Eq. (11.1), the input PF,
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Figure 11.17
Cycloconverter with intergroup reactor.
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Figure 11.18 Three-phase/single-phase cycloconverter. (a) Circuit, (b) Line voltages, (c) Output voltage, and (d) Conduction periods for P and N converters.
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Figure 11.19
Three-phase/three-phase cycloconverter.
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Figure 11.20 Generation of thyristor gating signals. (a) Input supply voltage, (b) Reference voltage at output frequency, (c) Conduction periods for P and N converters, (d) Thyristor gate pulses, and (e) Output voltage.
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Equations 11.36 and 11.37
Reduction of Output Harmonics
• The maximum average voltage of a
segment (which occurs for αp = 0) should be equal to the peak value of output voltage; for example, from Eq. (10.1),
• which gives the rms value of output
voltage as
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Equation 11.38
Reduction of Output Harmonics
• The rms input current Is = Io = 10.77 A,
the VA rating is VA = VsIs = 1292.4 VA, and the output power is
The input PF is
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Equation 11.39
Reduction of Output Harmonics
• If we compare Eq. (11.35) with Eq.
(11.38), there is a critical value of delay angle αc, which is given by
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Figure 11.21
Ac voltage controller for PWM control.
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Figure 11.22
Output voltage and load current of ac voltage controller. (a) Output voltage and (b) Output current.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 11.23
(a) Matrix (Φ−Φ) converter circuit with input filter and (b) Switching matrix symbol for converter.
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Equation 11.40
Matrix Converter
• When connected, the voltages νan, νbn, νcn
at the output terminals are related to the input voltages νAN, νBN, νCN as
where SAa through SCc are the switching variables of the corresponding switches. Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equation 11.41
Matrix Converter
• For a balanced linear Υ-connected load at
the output terminals, the input phase currents are related to the output phase currents by
where the matrix of the switching variables in Eq. (11.41) is a transpose of the respective matrix in Eq. (11.40). Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 11.24
Single-phase full converter with RL load.
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Equations 11.45
Design of AC Voltage-Controller Circuits
• for n = 2, 4,...=0
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Equations 11.46
Design of AC Voltage-Controller Circuits
• for n = 2, 4,...=0
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Equation 11.47
Design of AC Voltage-Controller Circuits
• Dividing υo(t) in Eq. (11.42) by load
impedance Z and simplifying the sine and cosine terms give the load current as
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Figure 11.25
Harmonic content as a function of the firing angle for a single-phase voltage controller with RL load.
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Figure 11.26
Equivalent circuit for harmonic current.
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Equivalent Circuit for Harmonic Current
• Figure 11.26 shows the equivalent circuit
for harmonic current. Using the currentdivider rule, the harmonic current through load is given by
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Figure 11.27
Ac thyristor SPICE model.
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Figure 11.28 thyristor T2.
Single-phase ac voltage controller for PSpice simulation. (a) Circuit, (b) Gate pulse for thyristor T1, and (c) Gate pulse for
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Figure 11.29
Plots for Example 11.10.
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Plots for Example 11.10
• From Eq. (10.96), the input PF
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Figure 11.30 Effects of load inductance on load current and voltage. (a) Input voltage, (b) Output voltage and current with load inductance, (c) Output voltage and current without any load inductance.
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