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POWER ELECTRONICS Devices, Circuits, and Applications FOURTH EDITION
CHAPTER CHAPTER
14
Dc Drives
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 basic characteristics of dc motors and their control characteristics. List the types of dc drives and their operating modes. List the control requirements of four-quadrant drives. Describe the parameters of the transfer function of converter-fed dc motors. Determine the performance parameters of single-phase and three-phase converter drives. Determine the performance parameters of dc–dc converter drives. Determine the closed-loop and open-loop transfer functions of dc motors. Determine the speed and torque characteristics of converter-fed drives. Design and analyze a feedback control of a motor drive. Determine the optimized parameters of the current- and speed feedback controller.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Symbols and Their Meanings
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Symbols and Their Meanings (continued)
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Figure 14.1
Controller rectifier- and dc–dc converter-fed drives.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 14.2
Equivalent circuit of separately excited dc motors.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equations 14.3, 14.4, and 14.5
Separately Excited Dc Motor
• Under steady-state conditions, the time
derivatives in these equations are zero and the steady-state average quantities are
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equations 14.6 and 14.7
Separately Excited Dc Motor
• The developed power is
• From Eq. (14.3), the speed of a separately
excited motor can be found from
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Figure 14.3
Magnetization characteristic.
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Equation 14.2
Magnetization Characteristic
• Under steady-state conditions, the time
derivatives in these equations are zero and the steady-state average quantities are
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 14.4
Characteristics of separately excited motors.
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Equations 14.4 and 14.5
Magnetization Characteristic
• Under steady-state conditions, the time
derivatives in these equations are zero and the steady-state average quantities are
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Equation 14.6
Separately Excited Dc Motor
• The developed power is
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 14.5
Equivalent circuit of dc series motors.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equations 14.8, 14.9, 14.10, and 14.11
Series-Excited Dc Motor
• The steady-state average quantities are
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Equation 14.12
Series-Excited Dc Motor
• The speed of a series motor can be
determined from Eq. (14.10):
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Figure 14.6
Characteristics of dc series motors.
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Figure 14.7
Schematic for a gearbox between the motor and the load.
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Equations 14.13 and 14.14
Gear Ratio
• Assuming zero losses in the gearbox, the
power handled by the gear is the same on both sides. That is,
• The speed on each side is inversely
proportional to its tooth number. That is,
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Equations 14.15, 14.16, and 14.17
Gear Ratio
• Substituting Eq. (14.14) into Eq. (14.13)
gives
• Similar to a transformer, the load inertia J1
and the load bearing constant B1 can be reflected to the motor side by
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Figure 14.8
Operating modes.
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Figure 14.8 (continued)
Operating modes.
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Figure 14.8 (continued)
Operating modes.
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Figure 14.8 (continued)
Operating modes.
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Figure 14.9
Conditions for four quadrants.
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Figure 14.10
Basic circuit arrangement of a single-phase dc drive.
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Figure 14.11
Field and armature reversals using contactors.
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Figure 14.12
Single-phase semiconverter drive.
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Equations 14.18 and 14.19
Gear Ratio
• With a single-phase semiconverter in the
armature circuit, the average armature voltage is can given by [12]
• With a semiconverter in the field circuit,
Eq. (10.52) gives the average field voltage as
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Figure 14.13
Single-phase full-converter drive.
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Equations 14.20 and 14.21
Single-Phase Full-Converter Drives
• With a single-phase full-wave converter in
the armature circuit, Eq. (10.1) gives the average armature voltage as
• With a single-phase full-converter in the
field circuit, Eq. (10.5) gives the field voltage as
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Figure 14.14
Single-phase dual-converter drive.
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Equation 14.22
Single-Phase Full-Converter Drives
• If converter 1 operates with a delay angle
of αa1, Eq. (10.11) gives the armature voltage as
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Equations 14.23 and 14.24
Single-Phase Full-Converter Drives
• If converter 2 operates with a delay angle
of αa2, Eq. (10.12) gives the armature voltage as
• where αa2 = π − αa1. With a full converter
in the field circuit, Eq. (10.1) gives the field voltage as
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Equations 14.25 and 14.26
Three-Phase Semiconverter Drives
• With a three-phase semiconverter in the
armature circuit, Eq. (10.69) gives the armature voltage as
• With a three-phase semiconverter in the
field circuit, Eq. (10.69) gives the field voltage as
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Equations 14.27 and 14.28
Three-Phase Full-Converter Drives
• With a three-phase full-wave converter in
the armature circuit, Eq. (10.15) gives the armature voltage as
• With a three-phase full converter in the
field circuit, Eq. (10.15) gives the field voltage as
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Equations 14.29 and 14.30
Three-Phase Dual-Converter Drives
• If converter 1 operates with a delay angle
of αa1, Eq. (10.15) gives the average armature voltage as
• If converter 2 operates with a delay angle
of αa2, Eq. (10.15) gives the average armature voltage as
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Equation 14.31
Three-Phase Dual-Converter Drives
• With a three-phase full converter in the
field circuit, Eq. (10.15) gives the average field voltage as
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Figure 14.15
Converter-fed dc drive in power control.
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Equations 14.32 and 14.33
Dc–Dc Converter Drives
• The average armature voltage is
• where k is the duty cycle of the dc–dc
converter. The power supplied to the motor is
where Ia is the average armature current of the motor and it is ripple free. Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equations 14.34 and 14.35
Dc–Dc Converter Drives
• The average value of the input current is
• The equivalent input resistance of the dc–
dc converter drive seen by the source is
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Equation 14.36
Dc–Dc Converter Drives
• For a finite armature circuit inductance,
Eq. (5.29) can be applied to find the maximum peak-to-peak ripple current as
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Figure 14.16
Regenerative braking of dc separately excited motors.
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Equations 14.37 and 14.38
Regenerative Brake Control
• The average voltage across the dc–dc
converter is
• If Ia is the average armature current, the
regenerated power can be found from
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Equations 14.40 and 14.42
Regenerative Brake Control
• Therefore, the equivalent load resistance
of the motor acting as a generator is
• The minimum braking speed of the motor
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Equation 14.43
Regenerative Brake Control
• The maximum braking speed of a series
motor can be found from Eq. (14.41):
and ω ≤ ωmax.
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Figure 14.17
Rheostatic braking of dc separately excited motors.
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Equations 14.44 and 14.45
Rheostatic Braking
• The average current of the braking
resistor,
• and the average voltage across the
braking resistor,
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Equations 14.46 and 14.47
Rheostatic Braking
• The equivalent load resistance of the
generator,
• The power dissipated in the resistor Rb is
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Figure 14.18
Combined regenerative and rheostatic braking.
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Figure 14.19
Two-quadrant transistorized dc–dc converter drive.
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Figure 14.20
Four-quadrant transistorized dc–dc converter drive.
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Figure 14.21
Multiphase dc–dc converters.
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Equation 14.48
Multiphase Dc–dc Converters
• For u dc–dc converters in multiphase
operation, it can be proved that Eq. (5.29) is satisfied when k = 1/2u and the maximum peak-to-peak load ripple current becomes
where Lm and Rm are the total armature inductance and resistance, respectively.
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Equation 14.49
Multiphase Dc–dc Converters
• For 4ufLm
Rm, the maximum peak-to-peak load ripple current can be approximated to
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Equation 14.50
Multiphase Dc–dc Converters
• If an LC-input filter is used, Eq. (5.125)
can be applied to find the rms nth harmonic component of dc–dc convertergenerated harmonics in the supply
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Equation 14.51
Multiphase Dc–dc Converters
• If (nuf/f0)
1, the nth harmonic current in the supply becomes
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Figure 14.22
Block diagram of a closed-loop converter-fed dc motor drive.
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Figure 14.23
Converter-fed separately excited dc motor drive.
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Equation 14.52
Transfer Function of Separately Excited Motors
• Assuming a linear power converter of gain
K2, the armature voltage of the motor is
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Equations 14.53, 14.54, 14.55, and 14.56
Transfer Function of Separately Excited Motors
• Assuming that the motor field current If and
the back emf constant Kv remain constant during any transient disturbances, the system equations are
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Figure 14.24
Open-loop block diagram of separately excited dc motor drive.
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Equations 14.66 and 14.67
Open-loop Block Diagram of Separately Excited Dc Motor Drive
• From Eq. (14.63), the motor speed is
where πm = J/B is known as the mechanical time constant of the motor.
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Equation 14.68
Open-loop Block Diagram of Separately Excited Dc Motor Drive
• From Figure 14.24, we can obtain the
speed response due to reference voltage as
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Figure 14.25
Open-loop block diagram for torque disturbance input.
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Equation 14.69
Open-loop Block Diagram for Torque Disturbance Input
• The block diagram for a step change in
load torque disturbance is shown in Figure 14.25.
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Equations 14.70 and 14.71
Open-loop Block Diagram for Torque Disturbance Input
• Using the final value theorem, the steady-
state relationship of a change in speed Δω, due to a step change in control voltage ΔVr, and a step change in load torque ΔTL, can be found from Eqs. (14.68) and (14.69), respectively, by substituting s = 0.
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Equation 14.72
Open-loop Block Diagram for Torque Disturbance Input
• Eqs (14.68) and (14.69) give the speed
response as
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Figure 14.26
Dc–dc converter-fed dc series motor drive.
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Equations 14.73, 14.74, and 14.75
Transfer Function of Series Excited Motors
• Assuming that the back emf constant Kv
does not change with the armature current and remains constant, the system equations are
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Equations 14.76 and 14.77
Transfer Function of Series Excited Motors
• Assuming that the back emf constant Kv
does not change with the armature current and remains constant, the system equations are
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Figure 14.27
Open-loop block diagram of dc–dc converter-fed dc series drive.
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Equations 14.78, 14.79, 14.80, 14.81, and 14.82
Block Diagram of Dc–dc Converter-fed Dc Series Drive
• Transforming these equations in the
Laplace domain gives us
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Figure 14.28
Block diagram for reference voltage and load torque distributions.
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Equation 14.83
Converter Control Models
• The gain of a dc–dc converter can be
expressed as
where Vc is the control signal voltage (say 0 to 10 V) and Vcm is the maximum value of the control signal voltage (10 V).
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Equation 14.86
Converter Control Models
• Kc is the gain of the single-phase
converter as
where Vs = Vm/√2 is the rms value of the single-phase ac supply voltage.
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Equation 14.88
Converter Control Models
• Kr is the gain of the three-phase converter
as
where Vs = Vm/√2 is the rms value of the ac supply voltage per phase.
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Equations 14.89 and 14.90
Converter Control Models
• Therefore, a converter can be modeled
with a transfer function Gc(s) of a certain gain and phase delay as described by
• which can be approximated as a first-
order lag function given by
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Figure 14.29
Block diagram for closed-loop control of separately excited dc motor.
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Equations 14.92
Closed-Loop Transfer Function
• The closed-loop step response due to a
change in reference voltage can be found from Figure 14.24 with TL = 0.
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Equations 14.93
Closed-Loop Transfer Function
• The response due to a change in the load
torque TL can also be obtained from Figure 14.29 by setting Vr to zero.
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Equations 14.94 and 14.95
Closed-Loop Transfer Function
• The steady-state change in speed Δω, due
to a step change in control voltage Vr and a step change in load torque TL, can be found from Eqs. (14.92) and (14.93), respectively, by substituting s = 0.
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Figure 14.30
Converter-fed dc motor with current-control loop.
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Equations 14.99 and 14.100
Closed-loop Control of Separately Excited Dc Motor
• The blocks in Figure 14.30b can be
simplified
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Figure 14.31
Motor drive with current- and speed-control loops.
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Equations 14.101 and 14.102
Converter-fed Dc Motor with Current-control Loop
• Using proportional-integral type of control,
the transfer functions of the current controller Gc(s) and the speed controller Gs(s) can be represented as
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Equation 14.103
Converter-fed Dc Motor with Current-control Loop
• The transfer function of the speed
feedback filter can be expressed as
where Kω and τω are the gain and time constants of the speed feedback loop.
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Figure 14.32
Closed-loop speed control with inner current loop and field weakening.
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Figure 14.33
Outer speed-control loop.
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Equations 14.116 and 14.117
Outer Speed-control Loop
• The loop gain function is expressed as
• Assuming 1 + sτm ≈ sm and combining the
delay time τs of the speed controller with the delay time τω of the speed feedback filter to an equivalent delay time te = τs + τω, Eq. (14.116) can be approximated to
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Equation 14.120
Outer Speed-control Loop
• The closed-loop transfer function of the
speed in response to speed reference signal is given by
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Figure 14.34
Adding a pole-zero compensating network.
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Figure 14.35
Dc–dc converter-fed speed and position control system with an IGBT bridge.
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Equation 14.127
Speed and Position Control System
• For forward driving, the transistors T1 and
T4 and diode D2 are used as a buck converter that supplies a variable voltage, νa, to the armature given by
where VDC is the dc supply voltage to the converter and δ is the duty cycle of the transistor T1. Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 14.36
Typical profile of a dc–dc converter-fed four-quadrant drive.
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Figure 14.37
Phase-locked-loop control system.
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Figure 14.38
Schematic diagram of computer-controlled four-quadrant dc drive.
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CHAPTER CHAPTER
14
Dc Drives
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Learning Outcomes
After completing this chapter, students should be able to do the following: Describe the basic characteristics of dc motors and their control characteristics. List the types of dc drives and their operating modes. List the control requirements of four-quadrant drives. Describe the parameters of the transfer function of converter-fed dc motors. Determine the performance parameters of single-phase and three-phase converter drives. Determine the performance parameters of dc–dc converter drives. Determine the closed-loop and open-loop transfer functions of dc motors. Determine the speed and torque characteristics of converter-fed drives. Design and analyze a feedback control of a motor drive. Determine the optimized parameters of the current- and speed feedback controller.
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Symbols and Their Meanings
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Symbols and Their Meanings (continued)
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Figure 14.1
Controller rectifier- and dc–dc converter-fed drives.
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Figure 14.2
Equivalent circuit of separately excited dc motors.
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Equations 14.3, 14.4, and 14.5
Separately Excited Dc Motor
• Under steady-state conditions, the time
derivatives in these equations are zero and the steady-state average quantities are
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Equations 14.6 and 14.7
Separately Excited Dc Motor
• The developed power is
• From Eq. (14.3), the speed of a separately
excited motor can be found from
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Figure 14.3
Magnetization characteristic.
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Equation 14.2
Magnetization Characteristic
• Under steady-state conditions, the time
derivatives in these equations are zero and the steady-state average quantities are
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Figure 14.4
Characteristics of separately excited motors.
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Equations 14.4 and 14.5
Magnetization Characteristic
• Under steady-state conditions, the time
derivatives in these equations are zero and the steady-state average quantities are
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Equation 14.6
Separately Excited Dc Motor
• The developed power is
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Figure 14.5
Equivalent circuit of dc series motors.
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Equations 14.8, 14.9, 14.10, and 14.11
Series-Excited Dc Motor
• The steady-state average quantities are
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Equation 14.12
Series-Excited Dc Motor
• The speed of a series motor can be
determined from Eq. (14.10):
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Figure 14.6
Characteristics of dc series motors.
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Figure 14.7
Schematic for a gearbox between the motor and the load.
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Equations 14.13 and 14.14
Gear Ratio
• Assuming zero losses in the gearbox, the
power handled by the gear is the same on both sides. That is,
• The speed on each side is inversely
proportional to its tooth number. That is,
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Equations 14.15, 14.16, and 14.17
Gear Ratio
• Substituting Eq. (14.14) into Eq. (14.13)
gives
• Similar to a transformer, the load inertia J1
and the load bearing constant B1 can be reflected to the motor side by
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Figure 14.8
Operating modes.
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Figure 14.8 (continued)
Operating modes.
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Figure 14.8 (continued)
Operating modes.
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Figure 14.8 (continued)
Operating modes.
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Figure 14.9
Conditions for four quadrants.
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Figure 14.10
Basic circuit arrangement of a single-phase dc drive.
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Figure 14.11
Field and armature reversals using contactors.
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Figure 14.12
Single-phase semiconverter drive.
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Equations 14.18 and 14.19
Gear Ratio
• With a single-phase semiconverter in the
armature circuit, the average armature voltage is can given by [12]
• With a semiconverter in the field circuit,
Eq. (10.52) gives the average field voltage as
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Figure 14.13
Single-phase full-converter drive.
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Equations 14.20 and 14.21
Single-Phase Full-Converter Drives
• With a single-phase full-wave converter in
the armature circuit, Eq. (10.1) gives the average armature voltage as
• With a single-phase full-converter in the
field circuit, Eq. (10.5) gives the field voltage as
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Figure 14.14
Single-phase dual-converter drive.
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Equation 14.22
Single-Phase Full-Converter Drives
• If converter 1 operates with a delay angle
of αa1, Eq. (10.11) gives the armature voltage as
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Equations 14.23 and 14.24
Single-Phase Full-Converter Drives
• If converter 2 operates with a delay angle
of αa2, Eq. (10.12) gives the armature voltage as
• where αa2 = π − αa1. With a full converter
in the field circuit, Eq. (10.1) gives the field voltage as
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Equations 14.25 and 14.26
Three-Phase Semiconverter Drives
• With a three-phase semiconverter in the
armature circuit, Eq. (10.69) gives the armature voltage as
• With a three-phase semiconverter in the
field circuit, Eq. (10.69) gives the field voltage as
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Equations 14.27 and 14.28
Three-Phase Full-Converter Drives
• With a three-phase full-wave converter in
the armature circuit, Eq. (10.15) gives the armature voltage as
• With a three-phase full converter in the
field circuit, Eq. (10.15) gives the field voltage as
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Equations 14.29 and 14.30
Three-Phase Dual-Converter Drives
• If converter 1 operates with a delay angle
of αa1, Eq. (10.15) gives the average armature voltage as
• If converter 2 operates with a delay angle
of αa2, Eq. (10.15) gives the average armature voltage as
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Equation 14.31
Three-Phase Dual-Converter Drives
• With a three-phase full converter in the
field circuit, Eq. (10.15) gives the average field voltage as
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Figure 14.15
Converter-fed dc drive in power control.
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Equations 14.32 and 14.33
Dc–Dc Converter Drives
• The average armature voltage is
• where k is the duty cycle of the dc–dc
converter. The power supplied to the motor is
where Ia is the average armature current of the motor and it is ripple free. Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equations 14.34 and 14.35
Dc–Dc Converter Drives
• The average value of the input current is
• The equivalent input resistance of the dc–
dc converter drive seen by the source is
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Equation 14.36
Dc–Dc Converter Drives
• For a finite armature circuit inductance,
Eq. (5.29) can be applied to find the maximum peak-to-peak ripple current as
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Figure 14.16
Regenerative braking of dc separately excited motors.
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Equations 14.37 and 14.38
Regenerative Brake Control
• The average voltage across the dc–dc
converter is
• If Ia is the average armature current, the
regenerated power can be found from
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Equations 14.40 and 14.42
Regenerative Brake Control
• Therefore, the equivalent load resistance
of the motor acting as a generator is
• The minimum braking speed of the motor
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Equation 14.43
Regenerative Brake Control
• The maximum braking speed of a series
motor can be found from Eq. (14.41):
and ω ≤ ωmax.
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Figure 14.17
Rheostatic braking of dc separately excited motors.
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Equations 14.44 and 14.45
Rheostatic Braking
• The average current of the braking
resistor,
• and the average voltage across the
braking resistor,
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Equations 14.46 and 14.47
Rheostatic Braking
• The equivalent load resistance of the
generator,
• The power dissipated in the resistor Rb is
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Figure 14.18
Combined regenerative and rheostatic braking.
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Figure 14.19
Two-quadrant transistorized dc–dc converter drive.
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Figure 14.20
Four-quadrant transistorized dc–dc converter drive.
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Figure 14.21
Multiphase dc–dc converters.
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Equation 14.48
Multiphase Dc–dc Converters
• For u dc–dc converters in multiphase
operation, it can be proved that Eq. (5.29) is satisfied when k = 1/2u and the maximum peak-to-peak load ripple current becomes
where Lm and Rm are the total armature inductance and resistance, respectively.
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Equation 14.49
Multiphase Dc–dc Converters
• For 4ufLm
Rm, the maximum peak-to-peak load ripple current can be approximated to
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Equation 14.50
Multiphase Dc–dc Converters
• If an LC-input filter is used, Eq. (5.125)
can be applied to find the rms nth harmonic component of dc–dc convertergenerated harmonics in the supply
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Equation 14.51
Multiphase Dc–dc Converters
• If (nuf/f0)
1, the nth harmonic current in the supply becomes
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Figure 14.22
Block diagram of a closed-loop converter-fed dc motor drive.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 14.23
Converter-fed separately excited dc motor drive.
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Equation 14.52
Transfer Function of Separately Excited Motors
• Assuming a linear power converter of gain
K2, the armature voltage of the motor is
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Equations 14.53, 14.54, 14.55, and 14.56
Transfer Function of Separately Excited Motors
• Assuming that the motor field current If and
the back emf constant Kv remain constant during any transient disturbances, the system equations are
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Figure 14.24
Open-loop block diagram of separately excited dc motor drive.
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Equations 14.66 and 14.67
Open-loop Block Diagram of Separately Excited Dc Motor Drive
• From Eq. (14.63), the motor speed is
where πm = J/B is known as the mechanical time constant of the motor.
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Equation 14.68
Open-loop Block Diagram of Separately Excited Dc Motor Drive
• From Figure 14.24, we can obtain the
speed response due to reference voltage as
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Figure 14.25
Open-loop block diagram for torque disturbance input.
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Equation 14.69
Open-loop Block Diagram for Torque Disturbance Input
• The block diagram for a step change in
load torque disturbance is shown in Figure 14.25.
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Equations 14.70 and 14.71
Open-loop Block Diagram for Torque Disturbance Input
• Using the final value theorem, the steady-
state relationship of a change in speed Δω, due to a step change in control voltage ΔVr, and a step change in load torque ΔTL, can be found from Eqs. (14.68) and (14.69), respectively, by substituting s = 0.
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Equation 14.72
Open-loop Block Diagram for Torque Disturbance Input
• Eqs (14.68) and (14.69) give the speed
response as
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Figure 14.26
Dc–dc converter-fed dc series motor drive.
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Equations 14.73, 14.74, and 14.75
Transfer Function of Series Excited Motors
• Assuming that the back emf constant Kv
does not change with the armature current and remains constant, the system equations are
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Equations 14.76 and 14.77
Transfer Function of Series Excited Motors
• Assuming that the back emf constant Kv
does not change with the armature current and remains constant, the system equations are
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Figure 14.27
Open-loop block diagram of dc–dc converter-fed dc series drive.
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Equations 14.78, 14.79, 14.80, 14.81, and 14.82
Block Diagram of Dc–dc Converter-fed Dc Series Drive
• Transforming these equations in the
Laplace domain gives us
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Figure 14.28
Block diagram for reference voltage and load torque distributions.
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Equation 14.83
Converter Control Models
• The gain of a dc–dc converter can be
expressed as
where Vc is the control signal voltage (say 0 to 10 V) and Vcm is the maximum value of the control signal voltage (10 V).
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Equation 14.86
Converter Control Models
• Kc is the gain of the single-phase
converter as
where Vs = Vm/√2 is the rms value of the single-phase ac supply voltage.
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Equation 14.88
Converter Control Models
• Kr is the gain of the three-phase converter
as
where Vs = Vm/√2 is the rms value of the ac supply voltage per phase.
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Equations 14.89 and 14.90
Converter Control Models
• Therefore, a converter can be modeled
with a transfer function Gc(s) of a certain gain and phase delay as described by
• which can be approximated as a first-
order lag function given by
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Figure 14.29
Block diagram for closed-loop control of separately excited dc motor.
Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Equations 14.92
Closed-Loop Transfer Function
• The closed-loop step response due to a
change in reference voltage can be found from Figure 14.24 with TL = 0.
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Equations 14.93
Closed-Loop Transfer Function
• The response due to a change in the load
torque TL can also be obtained from Figure 14.29 by setting Vr to zero.
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Equations 14.94 and 14.95
Closed-Loop Transfer Function
• The steady-state change in speed Δω, due
to a step change in control voltage Vr and a step change in load torque TL, can be found from Eqs. (14.92) and (14.93), respectively, by substituting s = 0.
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Figure 14.30
Converter-fed dc motor with current-control loop.
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Equations 14.99 and 14.100
Closed-loop Control of Separately Excited Dc Motor
• The blocks in Figure 14.30b can be
simplified
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Figure 14.31
Motor drive with current- and speed-control loops.
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Equations 14.101 and 14.102
Converter-fed Dc Motor with Current-control Loop
• Using proportional-integral type of control,
the transfer functions of the current controller Gc(s) and the speed controller Gs(s) can be represented as
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Equation 14.103
Converter-fed Dc Motor with Current-control Loop
• The transfer function of the speed
feedback filter can be expressed as
where Kω and τω are the gain and time constants of the speed feedback loop.
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Figure 14.32
Closed-loop speed control with inner current loop and field weakening.
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Figure 14.33
Outer speed-control loop.
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Equations 14.116 and 14.117
Outer Speed-control Loop
• The loop gain function is expressed as
• Assuming 1 + sτm ≈ sm and combining the
delay time τs of the speed controller with the delay time τω of the speed feedback filter to an equivalent delay time te = τs + τω, Eq. (14.116) can be approximated to
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Equation 14.120
Outer Speed-control Loop
• The closed-loop transfer function of the
speed in response to speed reference signal is given by
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Figure 14.34
Adding a pole-zero compensating network.
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Figure 14.35
Dc–dc converter-fed speed and position control system with an IGBT bridge.
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Equation 14.127
Speed and Position Control System
• For forward driving, the transistors T1 and
T4 and diode D2 are used as a buck converter that supplies a variable voltage, νa, to the armature given by
where VDC is the dc supply voltage to the converter and δ is the duty cycle of the transistor T1. Power Electronics: Devices, Circuits, and Applications, 4e Muhammad H. Rashid
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Figure 14.36
Typical profile of a dc–dc converter-fed four-quadrant drive.
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Figure 14.37
Phase-locked-loop control system.
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Figure 14.38
Schematic diagram of computer-controlled four-quadrant dc drive.
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