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# Speed and Torque Control

Module by: NGUYEN Phuc. E-mail the author

Summary: - The objective of this chapter is to discuss various techniques for the control of electric machines. Basic techniques for speed and torque control will be illustrated with typical configurations of drive electronics that are used to implement the control algorithms. - The discussion of this chapter is limited to steady-state operation. The steady-state picture presented here is quite adequate for a wide variety of electricmachine applications. - In the discussion of torque control for synchronous and induction machines, the techniques of field-oriented or vector control are introduced and the analogy is made with torque control in dc motors. This material is somewhat more sophisticated mathematically than the speed-control discussion and requires application of the dq0 transformations.

## CONTROL OF DC MOTORS

### Speed Control

• The three most common speed-control methods for dc motors are adjustment of the flux, usually by means of field-current control, adjustment of the resistance associated with the armature circuit, and adjustment of the armature terminal voltage.

Field-Current Control: In part because it involves control at a relatively low power level, field-current control is frequently used to control the speed of a dc motor with separately excited or shunt field windings. The equivalent circuit for a separately excited dc machine is in Fig. 10.1. The shunt field current can be adjusted by means of a variable resistance in series with the shunt field. Alternatively, the field current can be supplied by power-electronic circuits which can be used to rapidly change the field current in response to a wide variety of control signals.

• Figure 10.2a shows a switching scheme for pulse-width modulation of the field voltage. It consists of a rectifier which rectifies the ac input voltage, a dc-link capacitor which filters the rectified voltage, producing a dc voltage VdcVdc size 12{V rSub { size 8{ ital "dc"} } } {}, and a pulse-width modulator.
• In this system, because only a unidirectional field current is required, the pulsewidth modulator consists of a single switch and a free-wheeling diode. Assuming both the switch and diode to be ideal, the average voltage across the field winding will be equal to

Vf=DVdcVf=DVdc size 12{V rSub { size 8{f} } = ital "DV" rSub { size 8{ ital "dc"} } } {}(10.1)

where D is the duty cycle of the switching waveform; i.e., D is the fraction of time that the switch S is on. Figure 10.2b shows the resultant field current.

• In the steady-state the average voltage across the inductor must equal zero, the average field current IfIf size 12{I rSub { size 8{f} } } {} will thus be equal to

If=VfRf=DVdcRfIf=VfRf=DVdcRf size 12{I rSub { size 8{f} } = { {V rSub { size 8{f} } } over {R rSub { size 8{f} } } } =D left [ { {V rSub { size 8{ ital "dc"} } } over {R rSub { size 8{f} } } } right ]} {}(10.2)

Figure 10.1 Equivalent circuit for a separately excited dc motor.

Figure 10.2 (a) Pulse-width modulation system for a dc-machine field winding.

(b) Field-current waveform.

• Thus, the field current can be controlled simply by controlling the duty cycle of the pulse-width modulator. If the field-winding time constant Lf/RfLf/Rf size 12{L rSub { size 8{f} } /R rSub { size 8{f} } } {}is long compared to the switching time, the ripple current ΔifΔif size 12{Δi rSub { size 8{f} } } {} will be small compared to the average current IfIf size 12{I rSub { size 8{f} } } {}.
• To examine the effect of field-current control, let us begin with the case of a dc motor driving a load of constant torque TloadTload size 12{T rSub { size 8{ ital "load"} } } {}. The generated voltage:

Ea=KfIfωmEa=KfIfωm size 12{E rSub { size 8{a} } =K rSub { size 8{f} } I rSub { size 8{f} } ω rSub { size 8{m} } } {}( 10.3)

where IfIf size 12{I rSub { size 8{f} } } {} is the average field current, ωmωm size 12{ω rSub { size 8{m} } } {} is the angular velocity in rad/sec, and Kf=KaPdNfKf=KaPdNf size 12{K rSub { size 8{f} } =K rSub { size 8{a} } P rSub { size 8{d} } N rSub { size 8{f} } } {} is a geometric constant which depends upon the dimensions of the motor, the properties of the magnetic material used to construct the motor, as well as the number of turns in the field winding.

• The electromagnetic torque is given as

Tmech=EaIaωm=KfIfIaTmech=EaIaωm=KfIfIa size 12{T rSub { size 8{ ital "mech"} } = { {E rSub { size 8{a} } I rSub { size 8{a} } } over {ω rSub { size 8{m} } } } =K rSub { size 8{f} } I rSub { size 8{f} } I rSub { size 8{a} } } {}(10.4)

and the armature current is given by

Ia=VaEaRaIa=VaEaRa size 12{I rSub { size 8{a} } = { {V rSub { size 8{a} } - E rSub { size 8{a} } } over {R rSub { size 8{a} } } } } {}(10.5)

• Setting the motor torque equal to TloadTload size 12{T rSub { size 8{ ital "load"} } } {}, solve for ωmωm size 12{ω rSub { size 8{m} } } {}

ωm=VaIaRaKfIf=VaTloadRaKfIfKfIfωm=VaIaRaKfIf=VaTloadRaKfIfKfIf size 12{ω rSub { size 8{m} } = { {V rSub { size 8{a} } - I rSub { size 8{a} } R rSub { size 8{a} } } over {K rSub { size 8{f} } I rSub { size 8{f} } } } = { {V rSub { size 8{a} } - { {T rSub { size 8{ ital "load"} } R rSub { size 8{a} } } over {K rSub { size 8{f} } I rSub { size 8{f} } } } } over {K rSub { size 8{f} } I rSub { size 8{f} } } } } {}(10.6)

• From Eq. 10.6, recognizing that the armature resistance voltage drop IaRaIaRa size 12{I rSub { size 8{a} } R rSub { size 8{a} } } {} is generally quite small in comparison to the armature voltage VaVa size 12{V rSub { size 8{a} } } {}, for a given load torque, the motor speed will increase with decreasing field current and decrease as the field current is increased.
• The lowest speed obtainable is that corresponding to maximum field current (the field current is limited by heating considerations); the highest speed is limited mechanically by the mechanical integrity of the rotor and electrically by the effects of armature reaction under weak-field conditions giving rise to poor commutation.
• Armature current is typically limited by motor cooling capability. In many dc motors, cooling is aided by a shaft-driven fan whose cooling capacity is a function of motor speed. Under constant-terminal-voltage operation with varying field current, the EaIaEaIa size 12{E rSub { size 8{a} } I rSub { size 8{a} } } {} product, and hence the allowable motor output power, remain substantially constant as the speed is varied. A dc motor controlled in this fashion is referred to as a constant-power drive.
• Torque, however, varies directly with field flux and therefore has its highest allowable value at the highest field current and hence lowest speed.
• Field-current control is thus best suited to drives requiring increased torque at low speeds.

Armature-Circuit Resistance Control: Armature-circuit resistance control provides a means of obtaining reduced speed by the insertion of external series resistance in the armature circuit.

• It can be used with series, shunt, and compound motors; for the last two types, the series resistor must be connected between the shunt field and the armature, not between the line and the motor.
• Depending upon the value of the series armature resistance, the speed may vary significantly with load, since the speed depends on the voltage drop in this resistance and hence on the armature current demanded by the load.
• The disadvantage of poor speed regulation may not be important in a series motor, which is used only where varying-speed service is required or can be tolerated.
• A significant disadvantage of this method of speed control is that the power loss in the external resistor is large, especially when the speed is greatly reduced.
• Armature-resistance control results in a constant-torque drive because both the field-flux and, to a first approximation, the allowable armature current remain constant as speed changes.
• A variation of this control scheme is given by the shunted-armature method, which may be applied to a series motor, as in Fig. 10.3a, or a shunt motor, as in Fig. 10.3b.

Figure 10.3 Shunted-armature method of speed control applied

to (a) a series motor and (b) a shunt motor.

Armature-Terminal Voltage Control Armature-terminal voltage control can be accomplished with the use of power-electronic systems.

• In Fig. 10.4a, a phase-controlled rectifier in combination with a dclink filter capacitor can be used to produce a variable dc-link voltage which can be applied directly to the armature terminals of the dc motor.
• In Fig. 10.4b, a constant dc-link voltage is produced by a diode rectifier in combination with a dc-link filter capacitor. The armature terminal voltage is then varied by a pulse-width modulation scheme in which switch S is alternately opened and closed.
• When switch S is closed, the armature voltage is equal to the dc-link voltage VdcVdc size 12{V rSub { size 8{ ital "dc"} } } {}, and when the switch is opened, current transfers to the freewheeling diode, essentially setting the armature voltage to zero. Thus the average armature voltage under this condition is equal to

Va=DVdcVa=DVdc size 12{V rSub { size 8{a} } = ital "DV" rSub { size 8{ ital "dc"} } } {}(10.7)

Figure 10.4 Three typical configurations for armature-voltage control.

(a) Variable dc-link voltage (produced by a phase-controlled rectifier)

applied directly to the dc-motor armature terminals.

(b) Constant dc-link voltage with single-polarity pulse-width modulation.

(c)Constant dc-link voltage with a full H-bridge.

where

Va=Va= size 12{V rSub { size 8{a} } ={}} {} average armature voltage (V)

Vdc=Vdc= size 12{V rSub { size 8{ ital "dc"} } ={}} {} dc-link voltage (V)

D = PWM duty cycle (fraction of time that switch S is closed)

• Figure 10.4c shows an H-bridge configuration. Note that if switch S3 is held closed while switch S4 remains open, this configuration reduces to that of Fig. 10.4b. However, the H-bridge configuration is more flexible because it can produce both positive- and negative-polarity armature voltage.
• For example, with switches S 1 and S3 closed, the armature voltage is equal to VdcVdc size 12{V rSub { size 8{ ital "dc"} } } {} while with switches S2 and S4 closed, the armature voltage is equal to VdcVdc size 12{ - V rSub { size 8{ ital "dc"} } } {}.
• Using such an H-bridge configuration in combination with an appropriate choice of control signals to the switches allows this PWM system to achieve any desired armature voltage in the range VdcVaVdcVdcVaVdc size 12{ - V rSub { size 8{ ital "dc"} } <= V rSub { size 8{a} } <= V rSub { size 8{ ital "dc"} } } {}.
• Advantages of armature-voltage control: because the voltage drop across the armature resistance is relatively small, a change in the armature terminal voltage of a shunt motor is accompanied in the steady state by a substantially equal change in the speed voltage. Thus, motor speed can be controlled directly by means of the armature terminal voltage.
• Frequently the control of motor voltage is combined with field-current control in order to achieve the widest possible speed range.
• With such dual control, base speed can be defined as the normal-armature-voltage, full-field speed of the motor.
• Speeds above base speed are obtained by reducing the field current; speeds below base speed are obtained by armature-voltage control. The range above base speed is that of a constant-power drive.
• The range below base speed is that of a constant-torque drive because, as in armature resistance control, the flux and the allowable armature current remain approximately

constant.

• The overall output limitations are shown in Fig. 10.6a for approximate allowable torque and in Fig. 10.6b for approximate allowable power.
• The constant-torque characteristic is well suited to many applications in the machinetool industry, where many loads consist largely of overcoming the friction of moving parts and hence have essentially constant torque requirements.

Figure 10.6 (a) Torque and (b) power limitations of combined armature-voltage

and field-current methods of speed control.

Figure 10.7 Block diagram for a speed-control system

for a separately excited or shunt-connected dc motor.

• Figure 10.7 shows a block diagram of a feedback-control system that can be used to regulate the speed of a separately excited or shunt-connected dc motor. The inputs to the dc-motor block include the armature voltage and the field current as well as the load torque TloadTload size 12{T rSub { size 8{ ital "load"} } } {}. The resultant motor speed ωmωm size 12{ω rSub { size 8{m} } } {} is fed back to a controller block which represents both the control logic and power electronics and which controls the armature voltage and field current applied to the dc motor, based upon a reference speed signal ωrefωref size 12{ω rSub { size 8{ ital "ref"} } } {}.
• Depending upon the design of the controller, with such a scheme it is possible to control the steady-state motor speed to a high degree of accuracy independent of the variations in the load torque.

### Torque Control

• The electromagnetic torque in the case of a separately excited or shunt motor

Tmech=KfIfIaTmech=KfIfIa size 12{T rSub { size 8{ ital "mech"} } =K rSub { size 8{f} } I rSub { size 8{f} } I rSub { size 8{a} } } {} (10.8)

and Tmech=KmIaTmech=KmIa size 12{T rSub { size 8{ ital "mech"} } =K rSub { size 8{m} } I rSub { size 8{a} } } {}(10.9)

in the case of a permanent-magnet motor.

• Torque can be controlled directly by controlling the armature current. Fig. 10.8 shows three possible configurations.
• In Fig. 10.8a, a phase-controlled rectifier, in combination with a dc-link filter inductor, can be used to create a variable dc-link current which can be applied directly to the armature terminals of the dc motor.
• In Fig. 10.8b, a constant dc-link current is produced by a diode rectifier. The armature terminal voltage is then varied by a pulse-width modulation scheme in which switch S is alternately opened and closed. When switch S is opened, the current IdcIdc size 12{I rSub { size 8{ ital "dc"} } } {} flows into the dc-motor armature while when switch S is closed, the armature is

Figure 10.8 Three typical configurations for armature-current control.

(a) Variable dc-link current (produced by a phase-controlled rectifier)

applied directly to the dc-motor armature terminals.

(b) Constant dc-link current with single-polarity pulse-width modulation.

(c) Constant dc-link current with a full H-bridge.

Figure 10.9 Block diagram of a dc-motor

speed-control system using direct-control of motor torque.

shorted and IaIa size 12{I rSub { size 8{a} } } {} decays. Thus, the duty cycle of switch S will control the average current into the armature.

• Fig 10.8c shows an H-bridge configuration .Appropriate control of the four switches S 1 through S4 allows this PWM system to achieve any desired armature average current in the range IdcIaIdcIdcIaIdc size 12{ - I rSub { size 8{ ital "dc"} } <= I rSub { size 8{a} } <= I rSub { size 8{ ital "dc"} } } {}.
• Note that in each of the PWM configurations of Fig. 10.8b and c, rapid changes in instantaneous current through the dc machine armature can give rise to large voltage spikes, which can damage the machine insulation as well as give rise to flashover and voltage breakdown of the commutator. In order to eliminate these effects, a practical system must include some sort of filter across the armature terminals (such as a large capacitor) to limit the voltage rise and to provide a low-impedance path for the high-frequency components of the drive current.
• Figure 10.9 shows a typical configuration in which the torque control is surrounded by a speed-feedback loop. Instead of controlling the armature voltage, in this case the output of the speed controller is a torque reference signal TrefTref size 12{T rSub { size 8{ ital "ref"} } } {} which in turn serves as the input to the torque controller.

## CONTROL OF SYNCHRONOUS MOTORS

### Speed Control

• Synchronous motors are essentially constant-speed machines, with their speed being determined by the frequency of the armature currents. The synchronous angular velocity ωs=2polesωeωs=2polesωe size 12{ω rSub { size 8{s} } = left [ { {2} over { ital "poles"} } right ]ω rSub { size 8{e} } } {}(10.10)

where

ωs=ωs= size 12{ω rSub { size 8{s} } ={}} {} synchronous spatial angular velocity of the air-gap mmf wave [rad/sec]

ωe=2πfe=ωe=2πfe= size 12{ω rSub { size 8{e} } =2πf rSub { size 8{e} } ={}} {} angular frequency of the applied electrical excitation [rad/sec]

fe=fe= size 12{f rSub { size 8{e} } ={}} {} applied electrical frequency [Hz]

• The simplest means of synchronous motor control is speed control via control of the frequency of the applied armature voltage, driving the motor by a polyphase voltage-source inverter shown in Fig. 10.10. This inverter can either be used to supply stepped ac voltage waveforms of amplitude VdcVdc size 12{V rSub { size 8{ ital "dc"} } } {} or the switches can be controlled to produce pulse-widthmodulated ac voltage waveforms of variable amplitude. The dc-link voltage VdcVdc size 12{V rSub { size 8{ ital "dc"} } } {} can itself be varied, for example, through the use of a phase-controlled rectifier.

Figure 10.10 Three-phase voltage-source inverter.

• The frequency of the inverter output waveforms can of course be varied by controlling the switching frequency of the inverter switches. For ac-machine applications, coupled with this frequency control must be control of the amplitude of the applied voltage.
• The air-gap component of the armature voltage in an ac machine is proportional to the peak flux density in the machine and the electrical frequency. If we neglect the voltage drop across the armature resistance and leakage reactance,

Va=fefratedBpeakBratedVratedVa=fefratedBpeakBratedVrated size 12{V rSub { size 8{a} } = left [ { {f rSub { size 8{e} } } over {f rSub { size 8{ ital "rated"} } } } right ] left [ { {B rSub { size 8{ ital "peak"} } } over {B rSub { size 8{ ital "rated"} } } } right ]V rSub { size 8{ ital "rated"} } } {}(10.11)

where VaVa size 12{V rSub { size 8{a} } } {} is the amplitude of the armature voltage, fefe size 12{f rSub { size 8{e} } } {} is the operating frequency, and BpeakBpeak size 12{B rSub { size 8{ ital "peak"} } } {}is the peak air-gap flux density. VratedVrated size 12{V rSub { size 8{ ital "rated"} } } {}, fratedfrated size 12{f rSub { size 8{ ital "rated"} } } {}, and BratedBrated size 12{B rSub { size 8{ ital "rated"} } } {} are the corresponding rated-operating-point values.

• Consider a situation in which the frequency of the armature voltage is varied while its amplitude is maintained at its rated value ( Va=VratedVa=Vrated size 12{V rSub { size 8{a} } =V rSub { size 8{ ital "rated"} } } {}). Under these conditions,

Bpeak=fratedfeBratedBpeak=fratedfeBrated size 12{B rSub { size 8{ ital "peak"} } = left [ { {f rSub { size 8{ ital "rated"} } } over {f rSub { size 8{e} } } } right ]B rSub { size 8{ ital "rated"} } } {}(10.12)

For a given armature voltage, the machine flux density is inversely proportional to frequency and thus as the frequency is reduced, the flux density will increase. A significant drop in frequency will increase the flux density to the point of potential machine damage due both to increased core loss and to the increased machine currents required to support the higher flux density.

• As a result, for frequencies less than or equal to rated frequency, it is typical to operate a machine at constant flux density. From Eq. 10.11, with Bpeak=BratedBpeak=Brated size 12{B rSub { size 8{ ital "peak"} } =B rSub { size 8{ ital "rated"} } } {}

Va=fefratedVratedVa=fefratedVrated size 12{V rSub { size 8{a} } = left [ { {f rSub { size 8{e} } } over {f rSub { size 8{ ital "rated"} } } } right ]V rSub { size 8{ ital "rated"} } } {}(10.13)

Vafe=VratedfratedVafe=Vratedfrated size 12{ { {V rSub { size 8{a} } } over {f rSub { size 8{e} } } } = { {V rSub { size 8{ ital "rated"} } } over {f rSub { size 8{ ital "rated"} } } } } {}(10.14)

• From Eq. 10.14, constant-flux operation can be achieved by maintaining a constant ratio of armature voltage to frequency. This is referred to as constant volts- per-hertz (constant V/Hz) operation. It is typically maintained from rated frequency down to the low frequency at which the armature resistance voltage drop becomes a significant component of the applied voltage.
• If the machine is operated at frequencies in excess of rated frequency with the voltage at its rated value, the air-gap flux density will drop below its rated value. In order to maintain the flux density at its rated value, it would be necessary to increase the terminal voltage for frequencies in excess of rated frequency. In order to avoid insulation damage, it is common to maintain the machine terminal voltage at its rated value for frequencies in excess of rated frequency.
• Figure 10.11 shows a plot of maximum power and maximum torque versus speed for a synchronous motor under variable-frequency operation. The operating regime below rated frequency and speed is referred to as the constant-torque regime and that above rated speed is referred to as the constant-power regime.

Figure 10.11 Variable-speed operating regimes for a synchronous motor.

• Although during steady-state operation the speed of a synchronous motor is determined by the frequency of the drive, speed control by means of frequency control is of limited use in practice. This is due in most part to the fact that it is difficult for the rotor of a synchronous machine to track arbitrary changes in the frequency of the applied armature voltage.
• Problems with changing speed result from the fact that, in order to develop torque, the rotor of a synchronous motor must remain in synchronism with the stator flux. Control of synchronous motors can be greatly enhanced by control algorithms in which the stator flux and its relationship to the rotor flux are controlled directly.

### Torque Control

Direct torque control in an ac machine, which can be implemented in a number of different ways, is commonly referred to as field-oriented control or vector control. To facilitate our discussion of field-oriented control, it is helpful to return to the discussion of Section 5.6.1. Under this viewpoint, which is formalized in Appendix C, stator quantities (flux, current, voltage, etc.) are resolved into components which rotate in synchronism with the rotor. Direct-axis quantities represent those components which are aligned with the field-winding axis, and quadrature-axis components are aligned perpendicular to the field-winding axis.

Section C.2 of Appendix C derives the basic machine relations in dq0 variables for a synchronous machine consisting of a field winding and a three-phase stator winding. The transformed flux-current relationships are found to be

λ=Ldid+Lafifλ=Ldid+Lafif size 12{λ=L rSub { size 8{d} } i rSub { size 8{d} } +L rSub { size 8{ ital "af"} } i rSub { size 8{f} } } {}(10.15)

λq=Lqiqλq=Lqiq size 12{λ rSub { size 8{q} } =L rSub { size 8{q} } i rSub { size 8{q} } } {}(10.16)

λf=32Lafid+Lffifλf=32Lafid+Lffif size 12{λ rSub { size 8{f} } = { {3} over {2} } L rSub { size 8{ ital "af"} } i rSub { size 8{d} } +L rSub { size 8{ ital "ff"} } i rSub { size 8{f} } } {}(10.17)

where the subscripts d, q, and f refer to armature direct-, quadrature-axis, and fieldwinding quantities respectively. Note that throughout this chapter we will assume balanced operating conditions, in which case zero-sequence quantities will be zero and hence will be ignored.

The corresponding transformed voltage equations are

vd=Raid+ddtωmeλqvd=Raid+ddtωmeλq size 12{v rSub { size 8{d} } =R rSub { size 8{a} } i rSub { size 8{d} } + { {dλ rSub { size 8{d} } } over { ital "dt"} } - ω rSub { size 8{ ital "me"} } λ rSub { size 8{q} } } {}(10.18)

vq=Raiq+qdt+ωmeλdvq=Raiq+qdt+ωmeλd size 12{v rSub { size 8{q} } =R rSub { size 8{a} } i rSub { size 8{q} } + { {dλ rSub { size 8{q} } } over { ital "dt"} } +ω rSub { size 8{ ital "me"} } λ rSub { size 8{d} } } {}(10.19)

vf=Rfif+fdtvf=Rfif+fdt size 12{v rSub { size 8{f} } =R rSub { size 8{f} } i rSub { size 8{f} } + { {dλ rSub { size 8{f} } } over { ital "dt"} } } {}(10.20)

where ωme=(poles/2)ωmωme=(poles/2)ωm size 12{ω rSub { size 8{ ital "me"} } = $$ital "poles"/2$$ ω rSub { size 8{m} } } {} is the electrical angular velocity of the rotor.

Finally, the electromagnetic torque acting on the rotor of a synchronous motor is shown to be Tmech=32poles2(λdiqλqid)Tmech=32poles2(λdiqλqid) size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] $$λ rSub { size 8{d} } i rSub { size 8{q} } - λ rSub { size 8{q} } i rSub { size 8{d} }$$ } {}(10.21)

Under steady-state, balanced-three-phase operating conditions, ωme=ωeωme=ωe size 12{ω rSub { size 8{ ital "me"} } =ω rSub { size 8{e} } } {} where ωeωe size 12{ω rSub { size 8{e} } } {} is the electrical frequency of the armature voltage and current in rad/sec. Because the armature-produced mmf and flux waves rotate in synchronism with the rotor and hence with respect to the dq reference frame, under these conditions an observer in the dq reference frame will see constant fluxes, and hence one can set d/dt = 0

Letting the subscripts F, D, and Q indicate the corresponding constant values of field-, direct- and quadrature-axis quantities respectively, the flux-current relationships of Eqs. 10.15 through 10.17 then become

λD=LdiD+LafiFλD=LdiD+LafiF size 12{λ rSub { size 8{D} } =L rSub { size 8{d} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} } } {}(10.22)

λQ=LqiQλQ=LqiQ size 12{λ rSub { size 8{Q} } =L rSub { size 8{q} } i rSub { size 8{Q} } } {}(10.23)

λF=32LafiD+LffiFλF=32LafiD+LffiF size 12{λ rSub { size 8{F} } = { {3} over {2} } L rSub { size 8{ ital "af"} } i rSub { size 8{D} } +L rSub { size 8{ ital "ff"} } i rSub { size 8{F} } } {}(10.24)

Armature resistance is typically quite small, and, if we neglect it, the steady-state voltage equations (Eqs. 10.18 through 10.20) then become

vD=ωeλQvD=ωeλQ size 12{v rSub { size 8{D} } = - ω rSub { size 8{e} } λ rSub { size 8{Q} } } {}(10.25)

vQ=ωeλDvQ=ωeλD size 12{v rSub { size 8{Q} } =ω rSub { size 8{e} } λ rSub { size 8{D} } } {}(10.26)

vF=RfiFvF=RfiF size 12{v rSub { size 8{F} } =R rSub { size 8{f} } i rSub { size 8{F} } } {}(10.27)

Finally, we can write Eq. 10.21 as

Tmech=32poles2(λDiQλQiD)Tmech=32poles2(λDiQλQiD) size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] $$λ rSub { size 8{D} } i rSub { size 8{Q} } - λ rSub { size 8{Q} } i rSub { size 8{D} }$$ } {}(10.28)

From this point on, we will focus our attention on machines in which the effects of saliency can be neglected. In this case, the direct- and quadrature-axis synchronous inductances are equal and we can write

Ld=Lq=LsLd=Lq=Ls size 12{L rSub { size 8{d} } =L rSub { size 8{q} } =L rSub { size 8{s} } } {}(10.29)

where LsLs size 12{L rSub { size 8{s} } } {} is the synchronous inductance. Substitution into Eqs. 10.22 and 10.23 and then into Eq. 10.28 gives

T mech = 3 2 poles 2 [ ( L s i D + L af i F ) i Q L s i Q i D ] T mech = 3 2 poles 2 [ ( L s i D + L af i F ) i Q L s i Q i D ] size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] $$$L rSub { size 8{s} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} }$$ i rSub { size 8{Q} } - L rSub { size 8{s} } i rSub { size 8{Q} } i rSub { size 8{D} }$ } {}

=32poles2LafiFiQ=32poles2LafiFiQ size 12{ {}= { {3} over {2} } left [ { { ital "poles"} over {2} } right ]L rSub { size 8{ ital "af"} } i rSub { size 8{F} } i rSub { size 8{Q} } } {}(10.30)

Equation 10.30 shows that torque is produced by the interaction of the field flux (proportional to the field current) and the quadrature-axis component of the armature current, in other words the component of armature current that is orthogonal to the field flux. By analogy, we see that the direct-axis component of armature current, which is aligned with the field flux, produces no torque.

This result is fully consistent with the generalized torque expressions which are derived in Chapter 4. Consider for example the equation which expresses the torque in terms of the product of the stator and rotor mmfs ( FsFs size 12{F rSub { size 8{s} } } {} and FrFr size 12{F rSub { size 8{r} } } {} respectively) and the sine of the angle between them.

T=poles2μ0πDl2gFsFrsinδsrT=poles2μ0πDl2gFsFrsinδsr size 12{T= - left [ { { ital "poles"} over {2} } right ] left [ { {μ rSub { size 8{0} } π ital "Dl"} over {2g} } right ]F rSub { size 8{s} } F rSub { size 8{r} } "sin"δ rSub { size 8{ ital "sr"} } } {}(10.31)

where δrδr size 12{δ rSub { size 8{r} } } {} is the electrical space angle between the stator and rotor mmfs. This shows clearly that no torque will be produced by the direct-axis component of the armature mmf which, by definition, is that component of the stator mmf which is aligned with that of the field winding on the rotor.

Equation 10.31 shows the torque in a nonsalient synchronous motor is proportional to the product of the field current and the quadrature-axis component of the armature current. This is directly analogous to torque production in a dc machine for which the equations can be combined to show that the torque is proportional to the product of the field current and the armature current.

The analogy between a nonsalient synchronous machine and dc machine can be further reinforced. Consider the equation, which expresses the rms value of the line-toneutral generated voltage of a synchronous generator as

Eaf=ωeLafiF2Eaf=ωeLafiF2 size 12{E rSub { size 8{ ital "af"} } = { {ω rSub { size 8{e} } L rSub { size 8{ ital "af"} } i rSub { size 8{F} } } over { sqrt {2} } } } {}(10.32)

Substitution into Eq. 10.30 gives

Tmech=32poles2EafiQωeTmech=32poles2EafiQωe size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over { sqrt {2} } } right ] { {E rSub { size 8{ ital "af"} } i rSub { size 8{Q} } } over {ω rSub { size 8{e} } } } } {}(10.33)

This is directly analogous to Eq. Tmech=EaIa/ωmTmech=EaIa/ωm size 12{T rSub { size 8{ ital "mech"} } =E rSub { size 8{a} } I rSub { size 8{a} } /ω rSub { size 8{m} } } {} for a dc machine in which the torque is proportional to the product of the generated voltage and the armature current.

The brushes and commutator of a dc machine force the commutated armature current and armature flux along the quadrature axis such that IdId size 12{I rSub { size 8{d} } } {} = 0 and it is the interaction of this quadrature-axis current with the direct-axis field flux that produces the torque. A field-oriented controller which senses the position of the rotor and controls the quadrature-axis component of armature current produces the same effect in a synchronous machine.

Although the direct-axis component of armature current does not play a role in torque production, it does play a role in determining the resultant stator flux and hence the machine terminal voltage, as can be readily shown. Specifically, from the transformation equations of Appendix C,

va=vDcos(ωet)vQsin(ωet)va=vDcos(ωet)vQsin(ωet) size 12{v rSub { size 8{a} } =v rSub { size 8{D} } "cos" $$ω rSub { size 8{e} } t$$ - v rSub { size 8{Q} } "sin" $$ω rSub { size 8{e} } t$$ } {}(10.34)

And thus the rms amplitude of the armature voltage is equal to

V a = v D 2 + v Q 2 2 = ω e λ D 2 + λ Q 2 2 V a = v D 2 + v Q 2 2 = ω e λ D 2 + λ Q 2 2 size 12{V rSub { size 8{a} } = sqrt { { {v rSub { size 8{D} } rSup { size 8{2} } +v rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } =ω rSub { size 8{e} } sqrt { { {λ rSub { size 8{D} } rSup { size 8{2} } +λ rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } } {}

=ωe(LsiD+LafiF)2+(LsiQ)22=ωe(LsiD+LafiF)2+(LsiQ)22 size 12{ {}=ω rSub { size 8{e} } sqrt { { { $$L rSub { size 8{s} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} }$$ rSup { size 8{2} } + $$L rSub { size 8{s} } i rSub { size 8{Q} }$$ rSup { size 8{2} } } over {2} } } } {}(10.35)

Dividing VaVa size 12{V rSub { size 8{a} } } {} by the electrical frequency ωeωe size 12{ω rSub { size 8{e} } } {}, we get an expression for rms armature flux linkages

(λa)rms=λD2+λQ22=(LsiD+LafiF)2+(LsiQ)22(λa)rms=λD2+λQ22=(LsiD+LafiF)2+(LsiQ)22 size 12{ $$λ rSub { size 8{a} }$$ rSub { size 8{ ital "rms"} } = sqrt { { {λ rSub { size 8{D} } rSup { size 8{2} } +λ rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } = sqrt { { { $$L rSub { size 8{s} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} }$$ rSup { size 8{2} } + $$L rSub { size 8{s} } i rSub { size 8{Q} }$$ rSup { size 8{2} } } over {2} } } } {}(10.36)

Similarly, the transformation equations of Appendix C can be used to show that the rms amplitude of the armature current is equal to

Ia=iD2+iQ22Ia=iD2+iQ22 size 12{I rSub { size 8{a} } = sqrt { { {i rSub { size 8{D} } rSup { size 8{2} } +i rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } } {}(10.37)

From Eq.10.33 we see that torque is controlled by the product iFiQiFiQ size 12{i rSub { size 8{F} } i rSub { size 8{Q} } } {} of the field current and thee quadrature-axis component of the armature current. Thus, simply specifying a desired torque is not sufficient to uniquely determine either iFiF size 12{i rSub { size 8{F} } } {} or iQiQ size 12{i rSub { size 8{Q} } } {}.

In fact, under the field-oriented-control viewpoint presented here, there are actually three independent variables, iF,iQiF,iQ size 12{i rSub { size 8{F} } ,i rSub { size 8{Q} } } {} and iDiD size 12{i rSub { size 8{D} } } {}, and , in general, three constraints will be required to uniquely determine them. In addition to specifying the desired torque, a typical controller will implement additional contraints flux-linkages and current using the basic relationships found in Eqs.10.36 and 10.37.

Figure 10.12a shows a typical field-oriented torque-control system in block-diagram form. The torque-controller block has two inputs, TrefTref size 12{T rSub { size 8{ ital "ref"} } } {}, the reference value or set point for torque and (iF)ref(iF)ref size 12{ $$i rSub { size 8{F} }$$ rSub { size 8{ ital "ref"} } } {}, the reference value or set point for the field current, which is also supplied to the power supply which supplies the current iFiF size 12{i rSub { size 8{F} } } {} to the motor field winding. (iF)ref(iF)ref size 12{ $$i rSub { size 8{F} }$$ rSub { size 8{ ital "ref"} } } {} is determined by an auxiliary controller which also determines

Figure 10.12 (a) Block diagram of a field-oriented torque-control system for

a synchronous motor. (b) Block diagram of a synchronous-motor speed-control

loop built around a field-oriented torque control system.

the reference value (iD)ref(iD)ref size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } } {} of the direct-axis current, based upon desired values for the armature current and voltage. The torque controller calculates the desired quadrature axis current from Eq. 10.30 based upon TrefTref size 12{T rSub { size 8{ ital "ref"} } } {} and (iF)ref(iF)ref size 12{ $$i rSub { size 8{F} }$$ rSub { size 8{ ital "ref"} } } {}

(iQ)ref=232polesTrefLaf(iF)ref(iQ)ref=232polesTrefLaf(iF)ref size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } = { {2} over {3} } left [ { {2} over { ital "poles"} } right ] { {T rSub { size 8{ ital "ref"} } } over {L rSub { size 8{ ital "af"} } $$i rSub { size 8{F} }$$ rSub { size 8{ ital "ref"} } } } } {}(10.38)

Note that a position sensor is required to determine the angular position of the rotor in order to implement the dq0 to abc transformation.

In a typical application, the ultimate control objective is not to control motor torque but to control speed or position. Figure 10.12a shows how the torque-control system of Fig. 10.12b can be used as a component of a speed-control loop, with speed feedback forming an outer control loop around the inner torque-control loop.

As we have discussed, a practical field-oriented control must determine values for all three currents iFiF size 12{i rSub { size 8{F} } } {}, iDiD size 12{i rSub { size 8{D} } } {}, and iQiQ size 12{i rSub { size 8{Q} } } {}. In Example 10.8 two of these values were chosen relatively arbitrarily ( iFiF size 12{i rSub { size 8{F} } } {} = 2.84 and iDiD size 12{i rSub { size 8{D} } } {} = 0) and the result was a control that achieved the desired torque but with a terminal voltage 30 percent in excess of the motorrated voltage. In a practical system, additional constraints are required to achieve an acceptable control algorithm.

One such algorithm would be to require that the motor operate at rated flux and at unity terminal power factor. Such an algorithm canbe derived with reference to the

Figure 10.13 Phasor diagram for unity-powerfactor

field-oriented-control algorithm

phasor diagram of Fig 10.13 and can be implemented using the following steps:

Step 1. Calculate the line-to-neutral armature voltage corresponding to rated flux as

Va=(Va)ratedωm(ωm)ratedVa=(Va)ratedωm(ωm)rated size 12{V rSub { size 8{a} } = $$V rSub { size 8{a} }$$ rSub { size 8{ ital "rated"} } left [ { {ω rSub { size 8{m} } } over { $$ω rSub { size 8{m} }$$ rSub { size 8{ ital "rated"} } } } right ]} {}(10.39)

where ***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.*** is the rated line-to-neutral armature voltage at rated motor speed, ωmωm size 12{ω rSub { size 8{m} } } {} is the desired motor speed, and (ωm)rated(ωm)rated size 12{ $$ω rSub { size 8{m} }$$ rSub { size 8{ ital "rated"} } } {} is its rated speed.

Step 2. Calculate the rms armature current from the desired torque TrefTref size 12{T rSub { size 8{ ital "ref"} } } {} as

Ia=Pref3Va=Trefωm3VaIa=Pref3Va=Trefωm3Va size 12{I rSub { size 8{a} } = { {P rSub { size 8{ ital "ref"} } } over {3V rSub { size 8{a} } } } = { {T rSub { size 8{ ital "ref"} } ω rSub { size 8{m} } } over {3V rSub { size 8{a} } } } } {}(10.40)

where PrefPref size 12{P rSub { size 8{ ital "ref"} } } {} is the mechanical power corresponding to the desired torque.

Step 3. Calculate the angle δδ size 12{δ} {} based upon the phasor diagram of Fig 11.14

δ=tan1ωeLsIaVaδ=tan1ωeLsIaVa size 12{δ= - "tan" rSup { size 8{ - 1} } left [ { {ω rSub { size 8{e} } L rSub { size 8{s} } I rSub { size 8{a} } } over {V rSub { size 8{a} } } } right ]} {}(10.41)

where ωe=ωme=(poles/2)ωmωe=ωme=(poles/2)ωm size 12{ω rSub { size 8{e} } =ω rSub { size 8{ ital "me"} } = $$ital "poles"/2$$ ω rSub { size 8{m} } } {} is the electrical frequency corresponding to the desired motor speed.

Step 4. Calculate (iQ)ref(iQ)ref size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } } {} and (iD)ref(iD)ref size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } } {}

(iQ)ref=2Iacosδ(iQ)ref=2Iacosδ size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } = sqrt {2} I rSub { size 8{a} } "cos"δ} {}(10.42)

(iD)ref=2Iasinδ(iD)ref=2Iasinδ size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } = sqrt {2} I rSub { size 8{a} } "sin"δ} {}(10.43)

Step 5. Calculate (iF)ref(iF)ref size 12{ $$i rSub { size 8{F} }$$ rSub { size 8{ ital "ref"} } } {}

(iF)ref=232polesTrefLaf(iQ)ref(iF)ref=232polesTrefLaf(iQ)ref size 12{ $$i rSub { size 8{F} }$$ rSub { size 8{ ital "ref"} } = { {2} over {3} } left [ { {2} over { ital "poles"} } right ] { {T rSub { size 8{ ital "ref"} } } over {L rSub { size 8{ ital "af"} } $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } } } } {}(10.44)

The discussion of this section has focused upon synchronous machines with field windings and the corresponding capability to control the field excitation. The basic concept, of course, also applies to synchronous machines with permanent magnets on the rotor. However, in the case of permanent-magnet synchronous machines, the effective field excitation is fixed and, as a result, there is one less degree of freedom for the field-oriented control algorithm.

For a permanent-magnet synchronous machine, since the effective equivalent field current is fixed by the permanent magnet, the quadrature-axis current is determined directly by the desired torque. Consider a three-phase permanent-magnet motor whose rated rms, line-to-neutral open-circuit voltage is (Eaf)rated(Eaf)rated size 12{ $$E rSub { size 8{ ital "af"} }$$ rSub { size 8{ ital "rated"} } } {} at electrical frequency (ωe)rated(ωe)rated size 12{ $$ω rSub { size 8{e} }$$ rSub { size 8{ ital "rated"} } } {}. From Eq. 10.32 we see that the equivalent LafIfLafIf size 12{L rSub { size 8{ ital "af"} } I rSub { size 8{f} } } {} product for this motor, which we will refer to by the symbol ΛPMΛPM size 12{Λ rSub { size 8{ ital "PM"} } } {}, is

ΛPM=2(Eaf)rated(ωe)ratedΛPM=2(Eaf)rated(ωe)rated size 12{Λ rSub { size 8{ ital "PM"} } = { { sqrt {2} $$E rSub { size 8{ ital "af"} }$$ rSub { size 8{ ital "rated"} } } over { $$ω rSub { size 8{e} }$$ rSub { size 8{ ital "rated"} } } } } {} (10.45)

Thus, the direct-axis flux-current relationship for this motor, corresponding to Eq. 10.22, becomes

λD=LdiD+ΛPMλD=LdiD+ΛPM size 12{λ rSub { size 8{D} } =L rSub { size 8{d} } i rSub { size 8{D} } +Λ rSub { size 8{ ital "PM"} } } {}(10.46)

and the torque expression becomes

Tmech=32poles2ΛPMiQTmech=32poles2ΛPMiQ size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ]Λ rSub { size 8{ ital "PM"} } i rSub { size 8{Q} } } {}(10.47)

From Eq. 10.47 we see that, for a permanent-magnet sychronous machine under field-oriented control, the quadrature-axis current is uniquely determined by the desired torque and Eq. 10.38 becomes

(iQ)ref=232polesTrefΛPM(iQ)ref=232polesTrefΛPM size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } = { {2} over {3} } left [ { {2} over { ital "poles"} } right ] { {T rSub { size 8{ ital "ref"} } } over {Λ rSub { size 8{ ital "PM"} } } } } {}(10.48)

Once (iQ)ref(iQ)ref size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } } {} has been specified, the only remaining control choice remains to determine the desired value for the direct-axis current, (iD)ref(iD)ref size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } } {}.One possibility is simply to set (iD)ref=0(iD)ref=0 size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } =0} {}.

Figure 10.14 Block diagram of a field-oriented torque-control system

for a permanent-magnet synchronous motor.

This will clearly result in the lowest possible armature current for a given torque. However, as we have seen in Example 10.8, this is likely to result in terminal voltages in excess of the rated voltage of the machine. As a result, it is common to supply direct-axis current so as to reduce the direct-axis flux linkage of Eq. 10.22, which will in turn result in reduced terminal voltage. This technique is commonly referred to as flux weakening and comes at the expense of increased armature current. In practice, the chosen operating point is determined by a trade-off between reducing the armature voltage and an increase in armature current. Figure 10.14 shows the block diagram for a field-oriented-control system for use with a permanent-magnet motor.

## CONTROL OF INDUCTION MOTORS

### Speed Control

Induction motors supplied from a constant-frequency source admirably fulfill the requirements of substantially constant-speed drives. Many motor applications, however, require several speeds or even a continuously adjustable range of speeds. From the earliest days of ac power systems, engineers have been interested in the development of adjustable-speed ac motors.

The synchronous speed of an induction motor can be changed by (a) changing the number of poles or (b) varying the line frequency. The operating slip can be changed by (c) varying the line voltage, (d) varying the rotor resistance, or (e) applying voltages of the appropriate frequency to the rotor circuits. The salient features of speed-control methods based on these five possibilities are discussed in the following five sections.

Figure 10.15 Principles of the pole-changing winding.

Pole-Changing Motors In pole-changing motors, the stator winding is designed so that, by simple changes in coil connections, the number of poles can be changed in the ratio 2 to 1. Either of two synchronous speeds can then be selected. The rotor is almost always of the squirrel-cage type, which reacts by producing a rotor field having the same number of poles as the inducing stator field. With two independent sets of stator windings, each arranged for pole changing, as many as four synchronous speeds can be obtained in a squirrel-cage motor, for example, 600, 900, 1200, and 1800 r/min for 60-Hz operation.

The basic principles of the pole-changing winding are shown in Fig. 10.15, in which aa and a'a'a'a' size 12{ { {a}} sup { ' } { {a}} sup { ' }} {} are two coils comprising part of the phase-a stator winding. An actual winding would, of course, consist of several coils in each group. The windings for the other stator phases (not shown in the figure) would be similarly arranged. In Fig. 10.15a the coils are connected to produce a four-pole field; in Fig. 10.15b the current in the a'a'a'a' size 12{ { {a}} sup { ' } { {a}} sup { ' }} {} coil has been reversed by means of a controller, the result being a two-pole field.

Figure 10.16 shows the four possible arrangements of these two coils: they can be connected in series or in parallel, and with their currents either in the same direction (four-pole operation) or in the opposite direction (two-pole operation). Additionally, the machine phases can be connected either in Y or ΔΔ size 12{Δ} {}, resulting in eight possible combinations.

Note that for a given phase voltage, the different connections will result in differing levels of air-gap flux density. For example, a change from a ΔΔ size 12{Δ} {} to a Y connection will reduce the coil voltage (and hence the air-gap flux density) for a given coil arrangement by 33 size 12{ sqrt {3} } {}. Similarly, changing from a connection with two coils in series to two in parallel will double the voltage across each coil and therefore double the magnitude of the air-gap flux density. These changes in flux density can, of course, be compensated for by changes in the applied winding voltage. In any case, they must be considered, along with corresponding changes in motor torque, when the configuration to be used in a specific application is considered.

Armature-Frequency Control: The synchronous speed of an induction motor can be controlled by varying the frequency of the applied armature voltage. This method

Figure 10.16 Four possible arrangements of phase-a stator coils

in a pole-changing induction motor: (a) series-connected, four-pole;

(b) series-connected, two-pole; (c)parallel-connected, four-pole;

(d) parallel-connected, two-pole.

of speed control is identical to that discussed in Section 10.2.1 for synchronous machines. In fact, the same inverter configurations used for synchronous machines, such as the three-phase voltage-source inverter of Fig. 10.10, can be used to drive induction motors. As is the case with any ac motor, to maintain approximately constant flux density, the armature voltage should also be varied directly with the frequency (constant-volts-per-hertz).

The torque-speed curve of an induction motor for a given frequency can be calculated by using the methods of Chapter 6 within the accuracy of the motor parameters at that frequency. Consider the torque expression which is repeated here.

Tmech=1ωsnphV1,eq2(R2/s)(R1,eq+(R2/s))2+(X1,eq+X2)2Tmech=1ωsnphV1,eq2(R2/s)(R1,eq+(R2/s))2+(X1,eq+X2)2 size 12{T rSub { size 8{ ital "mech"} } = { {1} over {ω rSub { size 8{s} } } } left [ { {n rSub { size 8{ ital "ph"} } V rSub { size 8{1, ital "eq"} } rSup { size 8{2} } $$R rSub { size 8{2} } /s$$ } over { $$R rSub { size 8{1, ital "eq"} } + \( R rSub { size 8{2} } /s$$ \) rSup { size 8{2} } + $$X rSub { size 8{1, ital "eq"} } +X rSub { size 8{2} }$$ rSup { size 8{2} } } } right ]} {}(10.49)

where ωs=(2/poles)ωeωs=(2/poles)ωe size 12{ω rSub { size 8{s} } = $$2/ ital "poles"$$ ω rSub { size 8{e} } } {} and ωeωe size 12{ω rSub { size 8{e} } } {} is the electrical excitation frequency of the motor in rad/sec,

Vˆ1,eq=Vˆ1jXmR1+j(X1+Xm)Vˆ1,eq=Vˆ1jXmR1+j(X1+Xm) size 12{ { hat {V}} rSub { size 8{1, ital "eq"} } = { hat {V}} rSub { size 8{1} } left [ { { ital "jX" rSub { size 8{m} } } over {R rSub { size 8{1} } +j $$X rSub { size 8{1} } +X rSub { size 8{m} }$$ } } right ]} {}(10.50)

and

R1,eq+jX1,eq=jXm(R1+jX1)R1+j(X1+Xm)R1,eq+jX1,eq=jXm(R1+jX1)R1+j(X1+Xm) size 12{R rSub { size 8{1, ital "eq"} } + ital "jX" rSub { size 8{1, ital "eq"} } = { { ital "jX" rSub { size 8{m} } $$R rSub { size 8{1} } + ital "jX" rSub { size 8{1} }$$ } over {R rSub { size 8{1} } +j $$X rSub { size 8{1} } +X rSub { size 8{m} }$$ } } } {}(10.51)

To investigate the effect of changing frequency, we will assume that R1R1 size 12{R rSub { size 8{1} } } {} is negligible. In this case,

Vˆ1,eq=Vˆ1XmX1+XmVˆ1,eq=Vˆ1XmX1+Xm size 12{ { hat {V}} rSub { size 8{1, ital "eq"} } = { hat {V}} rSub { size 8{1} } left [ { {X rSub { size 8{m} } } over {X rSub { size 8{1} } +X rSub { size 8{m} } } } right ]} {}(10.52)

R1,eq=0R1,eq=0 size 12{R rSub { size 8{1, ital "eq"} } =0} {}(10.53)

And

X1,eq=XmX1X1+XmX1,eq=XmX1X1+Xm size 12{X rSub { size 8{1, ital "eq"} } = { {X rSub { size 8{m} } X rSub { size 8{1} } } over {X rSub { size 8{1} } +X rSub { size 8{m} } } } } {}(10.54)

Let the subscript 0 represent rated-frequency values of each of the induction motor parameters. As the electrical-excitation frequency is varied, we can then write

(X1,eq+X2)=ωeωe0(X1,eq+X2)0(X1,eq+X2)=ωeωe0(X1,eq+X2)0 size 12{ $$X rSub { size 8{1, ital "eq"} } +X rSub { size 8{2} }$$ = left [ { {ω rSub { size 8{e} } } over {ω rSub { size 8{e0} } } } right ] $$X rSub { size 8{1, ital "eq"} } +X rSub { size 8{2} }$$ rSub { size 8{0} } } {}(10.55)

Under constant-volts-per-hertz control, we can also write the equivalent source voltage as

Vˆ1=ωeωe0(Vˆ1)0Vˆ1=ωeωe0(Vˆ1)0 size 12{ { hat {V}} rSub { size 8{1} } = left [ { {ω rSub { size 8{e} } } over {ω rSub { size 8{e0} } } } right ] $${ hat {V}} rSub { size 8{1} }$$ rSub { size 8{0} } } {}(10.56)

And hence, since Vˆ1,eqVˆ1,eq size 12{ { hat {V}} rSub { size 8{1, ital "eq"} } } {}is equal to Vˆ1Vˆ1 size 12{ { hat {V}} rSub { size 8{1} } } {} multiplied by a reactance ratio which stays constant with changing frequency,

Vˆ1,eq=ωeωe0(Vˆ1,eq)0Vˆ1,eq=ωeωe0(Vˆ1,eq)0 size 12{ { hat {V}} rSub { size 8{1, ital "eq"} } = left [ { {ω rSub { size 8{e} } } over {ω rSub { size 8{e0} } } } right ] $${ hat {V}} rSub { size 8{1, ital "eq"} }$$ rSub { size 8{0} } } {}(10.57)

Figure 10.17 A family of typical induction-motor speed-torque curves for a four-pole motor for various values of the electrical supply frequency. (a) R1R1 size 12{R rSub { size 8{1} } } {}sufficiently small so that its effects are negligible.(b) R1R1 size 12{R rSub { size 8{1} } } {}not negligible.

Finally, we can write the motor slip as

s=ωsωmωs=poles2Δωmωes=ωsωmωs=poles2Δωmωe size 12{s= { {ω rSub { size 8{s} } - ω rSub { size 8{m} } } over {ω rSub { size 8{s} } } } = { { ital "poles"} over {2} } left [ { {Δω rSub { size 8{m} } } over {ω rSub { size 8{e} } } } right ]} {}(10.58)

where Δωm=ωsωmΔωm=ωsωm size 12{Δω rSub { size 8{m} } =ω rSub { size 8{s} } - ω rSub { size 8{m} } } {} is the difference between the synchronous and mechanical angular velocities of the motor.

Substitution of Eqs. 10.55 through 10.58 into Eq. 10.49 gives

Tmech=nph[(V1,eq)0]2(R2/Δω)e0poles(R2/Δω)2+[(X1,eq+X2)0]2Tmech=nph[(V1,eq)0]2(R2/Δω)e0poles(R2/Δω)2+[(X1,eq+X2)0]2 size 12{T rSub { size 8{ ital "mech"} } = { {n rSub { size 8{ ital "ph"} } $$$V rSub { size 8{1, ital "eq"} }$$ rSub { size 8{0} }$ rSup { size 8{2} } $$R rSub { size 8{2} } /Δω$$ } over { left [ { {2ω rSub { size 8{e0} } } over { ital "poles"} } $$R rSub { size 8{2} } /Δω$$ right ] rSup { size 8{2} } + $$$X rSub { size 8{1, ital "eq"} } +X rSub { size 8{2} }$$ rSub { size 8{0} }$ rSup { size 8{2} } } } } {}(10.59)

Equation 10.59 shows the general trend in which we see that the frequency dependence of the torque-speed characteristic of an induction motor appears only in the term R2/ΔωR2/Δω size 12{R rSub { size 8{2} } /Δω} {}. Thus, under the assumption that R1R1 size 12{R rSub { size 8{1} } } {} is negligible, as the electrical supply frequency to an induction motor is changed, the shape of the speed-torque curve as a function of ΔωΔω size 12{Δω} {} (the difference between the synchronous speed and the motor speed) will remain unchanged. As a result, the torque-speed characteristic will simply shift along the speed axis as ωe(fe)ωe(fe) size 12{ω rSub { size 8{e} } $$f rSub { size 8{e} }$$ } {}is varied.

A set of such curves is shown in Fig. 10.17a. Note that as the electrical frequency (and hence the synchronous speed) is decreased, a given value of ΔωΔω size 12{Δω} {} corresponds to a larger slip.

Figure 10.17(continued)

Thus, for example, if the peak torque of a four-pole motor driven at 60 Hz occurs at 1638 r/min, corresponding to a slip of 9 percent, when driven at 30 Hz, the peak torque will occur at 738 r/min, corresponding to a slip of 18 percent.

In practice, the effects of R1R1 size 12{R rSub { size 8{1} } } {} may not be fully negligible, especially for large values of slip. If this is the case, the shape of the speed-torque curves will vary somewhat with the applied electrical frequency. Figure 10.17b shows a typical family of curves for this case.

Line-Voltage Control The internal torque developed by an induction motor is proportional to the square of the voltage applied to its primary terminals, as shown by the two torque-speed characteristics in Fig. 10.18. If the load has the torque-speed characteristic shown by the dashed line, the speed will be reduced from n1n1 size 12{n rSub { size 8{1} } } {} to n2n2 size 12{n rSub { size 8{2} } } {}. This method of speed control is commonly used with small squirrel-cage motors driving fans, where cost is an issue and the inefficiency of high-slip operation can be tolerated. It is characterized by a rather limited range of speed control.

Figure 10.18 Speed control by means of line voltage.

Rotor-Resistance Control The possibility of speed control of a wound-rotor motor by changing its rotor-circuit resistance has already been pointed out in Section 6.7.1.

The torque-speed characteristics for three different values of rotor resistance are shown in Fig. 10.19. If the load has the torque-speed characteristic shown by the dashed line, the speeds corresponding to each of the values of rotor resistance are n1,n2 and n3n1,n2 and n3 size 12{n rSub { size 8{1} } ,n rSub { size 8{2} } " and "n rSub { size 8{3} } } {}. This method of speed control has characteristics similar to those of dc shunt-motor speed control by means of resistance in series with the armature.

The principal disadvantages of both line-voltage and rotor-resistance control are low efficiency at reduced speeds and poor speed regulation with respect to change in load. In addition, the cost and maintenance requirements of wound-rotor induction motors are sufficiently high that squirrel-cage motors combined with solid-state drives have become the preferred option in most applications.

### Torque Control

In Section 10.2.2 we developed the concept of field-oriented-control for synchronous machines. Under this viewpoint, the armature flux and current are resolved into two components which rotate synchronously with the rotor and with the air-gap flux wave.

Figure 10.19 Speed control by means of rotor resistance.

The components of armature current and flux which are aligned with the field-winding are referred to as direct-axis components while those which are perpendicular to this axis are referred to as quadrature-axis components.

It turns out that the same viewpoint which we applied to synchronous machines can be applied to induction machines. As is discussed in Section 6.1, in the steadystate the mmf and flux waves produced by both the rotor and stator windings of an induction motor rotate at synchronous speed and in synchronism with each other.

Thus, the torque-producing mechanism in an induction machine is equivalent to that of a synchronous machine. The difference between the two is that, in an induction machine, the rotor currents are not directly supplied but rather are induced as the induction-motor rotor slips with respect to the rotating flux wave produced by the stator currents.

To examine the application of field-oriented control to induction machines, we begin with the dq0 transformation of Section C.3 of Appendix C. This transformation transforms both the stator and rotor quantities into a synchronously rotating reference frame. Under balanced three-phase, steady-state conditions, zero-sequence quantities will be zero and the remaining direct- and quadrature-axis quantites will be constant.

Hence the flux-linkage current relationships of Eqs. C.52 through C.58 become

λD=LSiD+LmiDRλD=LSiD+LmiDR size 12{λ rSub { size 8{D} } =L rSub { size 8{S} } i rSub { size 8{D} } +L rSub { size 8{m} } i rSub { size 8{ ital "DR"} } } {}(10.60)

λQ=LSiQ+LmiQRλQ=LSiQ+LmiQR size 12{λ rSub { size 8{Q} } =L rSub { size 8{S} } i rSub { size 8{Q} } +L rSub { size 8{m} } i rSub { size 8{ ital "QR"} } } {}(10.61)

λDR=LmiD+LRiDRλDR=LmiD+LRiDR size 12{λ rSub { size 8{ ital "DR"} } =L rSub { size 8{m} } i rSub { size 8{D} } +L rSub { size 8{R} } i rSub { size 8{ ital "DR"} } } {}(10.62)

λQR=LmiQ+LRiQRλQR=LmiQ+LRiQR size 12{λ rSub { size 8{ ital "QR"} } =L rSub { size 8{m} } i rSub { size 8{Q} } +L rSub { size 8{R} } i rSub { size 8{ ital "QR"} } } {}(10.63)

In these equations, the subscripts D, Q, DR, and QR represent the constant values of the direct- and quadrature-axis components of the stator and rotor quantities respectively. It is a straight-forward matter to show that the inductance parameters can be determined from the equivalent-circuit parameters as

Lm=Xm0ωe0Lm=Xm0ωe0 size 12{L rSub { size 8{m} } = { {X rSub { size 8{m0} } } over {ω rSub { size 8{e0} } } } } {}(10.64)

LS=Lm+X10ωe0LS=Lm+X10ωe0 size 12{L rSub { size 8{S} } =L rSub { size 8{m} } + { {X rSub { size 8{"10"} } } over {ω rSub { size 8{e0} } } } } {}(10.65)

LR=Lm+X20ωe0LR=Lm+X20ωe0 size 12{L rSub { size 8{R} } =L rSub { size 8{m} } + { {X rSub { size 8{"20"} } } over {ω rSub { size 8{e0} } } } } {}(10.66)

where the subscript 0 indicates the rated-frequency value.

The transformed voltage equations Eqs. C.63 through C.68 become

vD=RaiDωeλQvD=RaiDωeλQ size 12{v rSub { size 8{D} } =R rSub { size 8{a} } i rSub { size 8{D} } - ω rSub { size 8{e} } λ rSub { size 8{Q} } } {}(10.67)

vQ=RaiQ+ωeλDvQ=RaiQ+ωeλD size 12{v rSub { size 8{Q} } =R rSub { size 8{a} } i rSub { size 8{Q} } +ω rSub { size 8{e} } λ rSub { size 8{D} } } {}(10.68)

0=RaRiDR(ωeωme)λQR0=RaRiDR(ωeωme)λQR size 12{0=R rSub { size 8{ ital "aR"} } i rSub { size 8{ ital "DR"} } - $$ω rSub { size 8{e} } - ω rSub { size 8{ ital "me"} }$$ λ rSub { size 8{ ital "QR"} } } {}(10.69)

0=RaRiQR+(ωeωme)λDR0=RaRiQR+(ωeωme)λDR size 12{0=R rSub { size 8{ ital "aR"} } i rSub { size 8{ ital "QR"} } + $$ω rSub { size 8{e} } - ω rSub { size 8{ ital "me"} }$$ λ rSub { size 8{ ital "DR"} } } {}(10.70)

where one can show that the resistances are related to those of the equivalent circuit as

and

Ra=R1Ra=R1 size 12{R rSub { size 8{a} } =R rSub { size 8{1} } } {}(10.71)

RaR=R2RaR=R2 size 12{R rSub { size 8{ ital "aR"} } =R rSub { size 8{2} } } {}(10.72)

For the purposes of developing a field-oriented-control scheme, we will begin with the torque expression of Eq. C.70

Tmech=32poles2LmLR(λDRiqλQRid)Tmech=32poles2LmLR(λDRiqλQRid) size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] left [ { {L rSub { size 8{m} } } over {L rSub { size 8{R} } } } right ] $$λ rSub { size 8{ ital "DR"} } i rSub { size 8{q} } - λ rSub { size 8{ ital "QR"} } i rSub { size 8{d} }$$ } {}(11.73)

For the derivation of the dq0 transformation in Section C.3, the angular velocity of the reference frame was chosen to the synchronous speed as determined by the stator electrical frequency ωeωe size 12{ω rSub { size 8{e} } } {}. It was not necessary for the purposes of the derivation to specify the absolute angular location of the reference frame. It is convenient at this point to choose the direct axis of the reference frame aligned with the rotor flux.

If this is done

λQR=0λQR=0 size 12{λ rSub { size 8{ ital "QR"} } =0} {}(10.74)

and the torque expression of Eq. 11.75 becomes

Tmech=32poles2LmLRλDRiQTmech=32poles2LmLRλDRiQ size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] left [ { {L rSub { size 8{m} } } over {L rSub { size 8{R} } } } right ]λ rSub { size 8{ ital "DR"} } i rSub { size 8{Q} } } {}(10.75)

From Eq. 10.69 we see that

iDR=0iDR=0 size 12{i rSub { size 8{ ital "DR"} } =0} {}(10.76)

and thus

λDR=LmiDλDR=LmiD size 12{λ rSub { size 8{ ital "DR"} } =L rSub { size 8{m} } i rSub { size 8{D} } } {}(10.77)

and

λD=LSiDλD=LSiD size 12{λ rSub { size 8{D} } =L rSub { size 8{S} } i rSub { size 8{D} } } {}(10.78)

From Eqs. 11.77 and 11.78 we see that by choosing set λQR=0λQR=0 size 12{λ rSub { size 8{ ital "QR"} } =0} {} and thus aligning the synchronously rotating reference frame with the axis of the rotor flux, the directaxis rotor flux (which is, indeed, the total rotor flux) as well as the direct-axis flux are determined by the direct-axis component of the armature current. Notice the direct analogy with a dc motor. In a dc motor, the field- and direct-axis armature fluxes are determined by the field current and in this field-oriented control scheme, the rotor and direct-axis armature fluxes are determined by the direct-axis armature current.

In other words, in this field-oriented control scheme, the direct-axis component of armature current serves the same function as the field current in a dc machine.

The torque equation, Eq. 10.75, completes the analogy with the dc motor. We see that once the rotor direct-axis flux λDRλDR size 12{λ rSub { size 8{ ital "DR"} } } {} is set by the direct-axis armature current, the torque is then determined by the quadrature-axis armature current just as the torque is determined by the armature current in a dc motor.

In a practical implementation of the technique which we have derived, the directand quadrature-axis currents iD and iQiD and iQ size 12{i rSub { size 8{D} } " and "i rSub { size 8{Q} } } {} must be transformed into the three motor phase currents ia(t),ib(t),and ic(t)ia(t),ib(t),and ic(t) size 12{i rSub { size 8{a} } $$t$$ ,i rSub { size 8{b} } $$t$$ ,"and "i rSub { size 8{c} } $$t$$ } {}. This can be done using the inverse dq0 transformation of Eq. C.48 which requires knowledge of θSθS size 12{θ rSub { size 8{S} } } {}, the electrical angle between the axis of phase a, and the direct-axis of the synchronously rotating reference frame.

Since it is not possible to measure the axis of the rotor flux directly, it is necessary to calculate θSθS size 12{θ rSub { size 8{S} } } {}, where θS=ωet+θ0θS=ωet+θ0 size 12{θ rSub { size 8{S} } =ω rSub { size 8{e} } t+θ rSub { size 8{0} } } {} as given by Eq. C.46. Solving Eq. 10.70 for ωeωe size 12{ω rSub { size 8{e} } } {} gives ωe=ωmeRaRiQRλDRωe=ωmeRaRiQRλDR size 12{ω rSub { size 8{e} } =ω rSub { size 8{ ital "me"} } - R rSub { size 8{ ital "aR"} } left [ { {i rSub { size 8{ ital "QR"} } } over {λ rSub { size 8{ ital "DR"} } } } right ]} {}(10.79)

From Eq. 10.63 with λQRλQR size 12{λ rSub { size 8{ ital "QR"} } } {} = 0 we see that

iQR=LmLRiQiQR=LmLRiQ size 12{i rSub { size 8{ ital "QR"} } = - left [ { {L rSub { size 8{m} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{Q} } } {} (10.80)

Eq. 10.80 in combination with Eq. 10.77 then gives

ωe=ωme+RaRLRiQiD=ωme+1τRiQiDωe=ωme+RaRLRiQiD=ωme+1τRiQiD size 12{ω rSub { size 8{e} } =ω rSub { size 8{ ital "me"} } + { {R rSub { size 8{ ital "aR"} } } over {L rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ]=ω rSub { size 8{ ital "me"} } + { {1} over {τ rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ]} {}(10.81)

where τR=LR/RaRτR=LR/RaR size 12{τ rSub { size 8{R} } =L rSub { size 8{R} } /R rSub { size 8{ ital "aR"} } } {} is the rotor time constant. We can now integrate Eq. 10.81 to find

θˆS=ωme+1τRiQiDt+θ0θˆS=ωme+1τRiQiDt+θ0 size 12{ { hat {θ}} rSub { size 8{S} } = left [ω rSub { size 8{ ital "me"} } + { {1} over {τ rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ] right ]t+θ rSub { size 8{0} } } {}(11.82)

where θˆSθˆS size 12{ { hat {θ}} rSub { size 8{S} } } {} indicates the calculated value of θSθS size 12{θ rSub { size 8{S} } } {} (often referred to as the estimated value of θSθS size 12{θ rSub { size 8{S} } } {}). In the more general dynamic sense

θˆS=otωme+1τRiQiDdt'+θ0θˆS=otωme+1τRiQiDdt'+θ0 size 12{ { hat {θ}} rSub { size 8{S} } = Int rSub { size 8{o} } rSup { size 8{t} } { left [ω rSub { size 8{ ital "me"} } + { {1} over {τ rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ] right ]} d { {t}} sup { ' }+θ rSub { size 8{0} } } {}(10.83)

Note that both Eqs. 10.82 and 10.83 require knowledge of θ0θ0 size 12{θ rSub { size 8{0} } } {}, the value of θSθS size 12{θ rSub { size 8{S} } } {} at t = 0. Although we will not prove it here, it turns out that in a practical implementation, the effects of an error in this initial angle decay to zero with time, and hence it can be set to zero without any loss of generality.

Figure 10.20 a shows a block diagram of a field-oriented torque-control system for an induction machine. The block labeled "Estimator" represents the calculation of Eq. 10.83 which calculates the estimate of θSθS size 12{θ rSub { size 8{S} } } {} required by the transformation from dq0 to abc variables.

Note that a speed sensor is required to provide the rotor speed measurement required by the estimator. Also notice that the estimator requires knowledge of the rotor time constant τR=LR/RaRτR=LR/RaR size 12{τ rSub { size 8{R} } =L rSub { size 8{R} } /R rSub { size 8{ ital "aR"} } } {}. In general, this will not be known exactly, both due to uncertainty in the machine parameters as well as due to the fact that the rotor

Figure 10.20 (a) Block diagram of a field-oriented torque-control system

for an induction motor. (b) Block diagram of an induction-motor speed-control

loop built around a field-oriented torque control system.

resistance RaRRaR size 12{R rSub { size 8{ ital "aR"} } } {} will undoubtedly change with temperature as the motor is operated. It can be shown that errors in τRτR size 12{τ rSub { size 8{R} } } {} result in an offset in the estimate of θSθS size 12{θ rSub { size 8{S} } } {}, which in turn will result in an error in the estimate for the position of the rotor flux with the result that the applied armature currents will not be exactly aligned with the direct- and quadrature-axes. The torque controller will still work basically as expected, although there will be corresponding errors in the torque and rotor flux.

As with the synchronous motor, the rms armature flux-linkages can be found from Eq. 10.36 as

(λa)rms=λD2+λQ22(λa)rms=λD2+λQ22 size 12{ $$λ rSub { size 8{a} }$$ rSub { size 8{ ital "rms"} } = sqrt { { {λ rSub { size 8{D} } rSup { size 8{2} } +λ rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } } {}(10.84)

Combining Eqs. 10.61 and 10.80 gives

λQ=LSiQ+LmiQR=LSLm2LRiQλQ=LSiQ+LmiQR=LSLm2LRiQ size 12{λ rSub { size 8{Q} } =L rSub { size 8{S} } i rSub { size 8{Q} } +L rSub { size 8{m} } i rSub { size 8{ ital "QR"} } = left [L rSub { size 8{S} } - { {L rSub { size 8{m} } rSup { size 8{2} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{Q} } } {}(10.85)

Substituting Eqs. 10.78 and 10.85 into Eq. 11.84 gives

(λa)rms=LS2iD2+LsLm2LR2iQ22(λa)rms=LS2iD2+LsLm2LR2iQ22 size 12{ $$λ rSub { size 8{a} }$$ rSub { size 8{ ital "rms"} } = sqrt { { {L rSub { size 8{S} } rSup { size 8{2} } i rSub { size 8{D} } rSup { size 8{2} } + left [L rSub { size 8{s} } - { {L rSub { size 8{m} } rSup { size 8{2} } } over {L rSub { size 8{R} } } } right ] rSup { size 8{2} } i rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } } {}(10.86)

Finally, as discussed in the footnote to Eq. 10.35, the rms line-to-neutral armature voltage can be found as

V a = v D 2 + v Q 2 2 = ( R a i D ω e λ Q ) 2 + ( R a i Q + ω e λ D ) 2 2 V a = v D 2 + v Q 2 2 = ( R a i D ω e λ Q ) 2 + ( R a i Q + ω e λ D ) 2 2 size 12{V rSub { size 8{a} } = sqrt { { {v rSub { size 8{D} } rSup { size 8{2} } +v rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } = sqrt { { { $$R rSub { size 8{a} } i rSub { size 8{D} } - ω rSub { size 8{e} } λ rSub { size 8{Q} }$$ rSup { size 8{2} } + $$R rSub { size 8{a} } i rSub { size 8{Q} } +ω rSub { size 8{e} } λ rSub { size 8{D} }$$ rSup { size 8{2} } } over {2} } } } {}

=RaiDωeLSLm2LRiQ2+(RaiQ+ωeLSiD)22=RaiDωeLSLm2LRiQ2+(RaiQ+ωeLSiD)22 size 12{ {}= sqrt { { { left [R rSub { size 8{a} } i rSub { size 8{D} } - ω rSub { size 8{e} } left [L rSub { size 8{S} } - { {L rSub { size 8{m} } rSup { size 8{2} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{Q} } right ] rSup { size 8{2} } + $$R rSub { size 8{a} } i rSub { size 8{Q} } +ω rSub { size 8{e} } L rSub { size 8{S} } i rSub { size 8{D} }$$ rSup { size 8{2} } } over {2} } } } {}(10.87)

These equations show that the armature flux linkages and terminal voltage are determined by both the direct- and quadrature-axis components of the armature current.

Thus, the block marked "Auxiliary Controller" in Fig. 10.20 a, which calculates the reference values for the direct- and quadrature-axis currents, must calculate the reference currents (iD)ref(iD)ref size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } } {} and (iQ)ref(iQ)ref size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } } {} which achieve the desired torque subject to constraints on armature flux linkages (to avoid saturation in the motor), armature current, (Ia)rms=(iD2+iQ2)/2(Ia)rms=(iD2+iQ2)/2 size 12{ $$I rSub { size 8{a} }$$ rSub { size 8{ ital "rms"} } = sqrt { $$i rSub { size 8{D} } rSup { size 8{2} } +i rSub { size 8{Q} } rSup { size 8{2} }$$ /2} } {} (to avoid excessive armature heating) and armature voltage (to avoid potential insulation damage).

Note that, as we discussed with regard to synchronous machines in Section 10.2.2, the torque-control system of Fig. 10.20 a is typically imbedded within a larger control loop. One such example is the speed-control loop of Fig. 10.20 b.

The ability to independently control the rotor flux and the torque has important control implications. Consider, for example, the response of the direct-axis rotor flux to a change in direct-axis current. Equation C.66, with λqR=0λqR=0 size 12{λ rSub { size 8{ ital "qR"} } =0} {}, becomes

0=RaRidR+dRdt0=RaRidR+dRdt size 12{0=R rSub { size 8{ ital "aR"} } i rSub { size 8{ ital "dR"} } + { {dλ rSub { size 8{ ital "dR"} } } over { ital "dt"} } } {}(10.88)

Substituting for idRidR size 12{i rSub { size 8{ ital "dR"} } } {} in terms of λdRλdR size 12{λ rSub { size 8{ ital "dR"} } } {}

idR=λdRLmidLRidR=λdRLmidLR size 12{i rSub { size 8{ ital "dR"} } = { {λ rSub { size 8{ ital "dR"} } - L rSub { size 8{m} } i rSub { size 8{d} } } over {L rSub { size 8{R} } } } } {}(10.89)

gives a differential equation for the rotor flux linkages λDRλDR size 12{λ rSub { size 8{ ital "DR"} } } {}

dRdt+RaRLRλdR=LmLRiddRdt+RaRLRλdR=LmLRid size 12{ { {dλ rSub { size 8{ ital "dR"} } } over { ital "dt"} } + left [ { {R rSub { size 8{ ital "aR"} } } over {L rSub { size 8{R} } } } right ]λ rSub { size 8{ ital "dR"} } = left [ { {L rSub { size 8{m} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{d} } } {}(10.90)

From Eq. 10.90 we see that the response of the rotor flux to a step change in direct axis current id is relatively slow; λdRλdR size 12{λ rSub { size 8{ ital "dR"} } } {} will change exponentially with the rotor time constant of τR=LR/RaRτR=LR/RaR size 12{τ rSub { size 8{R} } =L rSub { size 8{R} } /R rSub { size 8{ ital "aR"} } } {}. Since the torque is proportional to the product λdRiqλdRiq size 12{λ rSub { size 8{ ital "dR"} } i rSub { size 8{q} } } {} we see that fast torque response will be obtained from changes in iqiq size 12{i rSub { size 8{q} } } {}. Thus, for example, to implement a step change in torque, a practical control algorithm might start with a step change in (iQ)ref(iQ)ref size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } } {} to achieve the desired torque change, followed by an adjustment in (iD)ref(iD)ref size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } } {} (and hence λdRλdR size 12{λ rSub { size 8{ ital "dR"} } } {}) to readjust the armature current or terminal voltage as desired. This adjustment in (iD)ref(iD)ref size 12{ $$i rSub { size 8{D} }$$ rSub { size 8{ ital "ref"} } } {} would be coupled with a compensating adjustment in (iQ)ref(iQ)ref size 12{ $$i rSub { size 8{Q} }$$ rSub { size 8{ ital "ref"} } } {} to maintain the torque at its desired value.

## CONTROL OF VARIABLE-RELUCTANCE

### MOTORS

Unlike dc and ac (synchronous or induction) machines, VRMs cannot be simply "plugged in" to a dc or ac source and then be expected to run. As is dicussed in Chapter 8, the phases must be excited with (typically unipolar) currents, and the timing of these currents must be carefully correlated with the position of the rotor poles in order to produce a useful, time-averaged torque. The result is that although the VRM itself is perhaps the simplest of rotating machines, a practical VRM drive system is relatively complex.

VRM drive systems are competitive only because this complexity can be realized easily and inexpensively through power and microelectronic circuitry. These drive systems require a fairly sophisticated level of controllability for even the simplest modes of VRM operation. Once the capability to implement this control is available, fairly sophisticated control features can be added (typically in the form of additional software) at little additional cost, further increasing the competitive position of VRM drives.

In addition to the VRM itself, the basic VRM drive system consists of the following components: a rotor-position sensor, a controller, and an inverter. The function of the rotor-position sensor is to provide an indication of shaft position which can be used to control the timing and waveform of the phase excitation. This is directly analogous to the timing signal used to control the firing of the cylinders in an automobile engine.

The controller is typically implemented in software in microelectronic (microprocessor) circuitry. Its function is to determine the sequence and waveforms of the phase excitation required to achieve the desired motor speed-torque characteristics. In addition to set points of desired speed and/or torque and shaft position (from the shaftposition sensor), sophisticated controllers often employ additional inputs including shaft-speed and phase-current magnitude. Along with the basic control function of determining the desired torque for a given speed, the more sophisticated controllers attempt to provide excitations which are in some sense optimized (for maximum efficiency, stable transient behavior, etc.).

The control circuitry consists typically of low-level electronics which cannot be used to directly supply the currents required to excite the motor phases. Rather its output consists of signals which control an inverter which in turn supplies the phase currents. Control of the VRM is achieved by the application of an appropriate set of currents to the VRM phase windings.

Figures 10.21a to c show three common configurations found in inverter systems for driving VRMs. Note that these are simply H-bridge inverters of the type discussed in Section 10.3. Each inverter is shown in a two-phase configuration. As is clear from the figures, extension of each configuration to drive additional phases can be readily accomplished.

The configuration of Fig. 10.21a is perhaps the simplest. Closing switches S1aS1a size 12{S rSub { size 8{1a} } } {} and S1bS1b size 12{S rSub { size 8{1b} } } {} connects the phase-1 winding across the supply ( v1=V0v1=V0 size 12{v rSub { size 8{1} } =V rSub { size 8{0} } } {}) and causes the winding current to increase. Opening just one of the switches forces a short across

Figure 11.23 Inverter configurations.

(a) Two-phase inverter which uses two switches per phase.

(b) Two-phase inverter which uses a split supply and one switch per phase.

(c) Two-phase inverter with bifilar phase windings and one switch per phase.

the winding and the current will decay, while opening both switches connects the winding across the supply with negative polarity through the diodes ( v1=V0v1=V0 size 12{v rSub { size 8{1} } = - V rSub { size 8{0} } } {}) and the winding current will decay more rapidly. Note that this configuration is capable of regeneration (returning energy to the supply), but not of supplying negative current to the phase winding. However, since the torque in a VRM is proportional to the square of the phase current, there is no need for negative winding current. As discussed in Section 10.3.2, the process of pulse-width modulation, under which a series of switch configurations alternately charge and discharge a phase winding, can be used to control the average winding current. Using this technique, an inverter such as that of Fig. 10.21a can readily be made to supply the range of waveforms required to drive a VRM.

The inverter configuration of Fig. 10.21a is perhaps the simplest of H-bridge configurations which provide regeneration capability. Its main disadvantage is that it requires two switches per phase. In many applications, the cost of the switches (and their associated drive circuitry) dominates the cost of the inverter, and the result is that this configuration is less attractive in terms of cost when compared to other configurations which require one switch per phase.

Figure 10.21b shows one such configuration. This configuration requires a split supply (i.e., two supplies of voltage V0V0 size 12{V rSub { size 8{0} } } {}) but only a single switch and diode per phase.

Closing switch S 1 connects the phase-1 winding to the upper dc source. Opening the switch causes the phase current to transfer to diode D 1, connecting the winding to the bottom dc source. Phase 1 is thus supplied by the upper dc source and regenerates through the bottom source. Note that to maintain symmetry and to balance the energy supplied from each source equally, phase 2 is connected oppositely so that it is supplied from the bottom source and regenerates into the top source.

The major disadvantages of the configuration of Fig. 10.21b are that it requires a split supply and that when the switch is opened, the switch must withstand a voltage of 2 V0V0 size 12{V rSub { size 8{0} } } {}. This can be readily seen by recognizing that when diode D 1 is forward-biased, the switch is connected across the two supplies. Such switches are likely to be more expensive than the switches required by the configuration of Fig. 10.21a. Both of these issues will tend to offset some of the economic advantage which can be gained by the elimination of one switch and one diode as compared with the inverter circuit of Fig. 10.21a.

A third inverter configuration is shown in Fig. 10.21c. This configuration requires only a single dc source and uses only a single switch and diode per phase. This configuration achieves regeneration through the use of bifilar phase windings. In a bifilar winding, each phase is wound with two separate windings which are closely coupled magnetically (this can be achieved by winding the two windings at the same time) and can be thought of as the primary and secondary windings of a transformer.

When switch S 1 is closed, the primary winding of phase 1 is energized, exciting the phase winding. Opening the switch induces a voltage in the secondary winding (note the polarity indicated by the dots in Fig. 11.23c) in such a direction as to forwardbias D 1. The result is that current is transferred from the primary to the secondary winding with a polarity such that the current in the phase decays to zero and energy is returned to the source.

Although this configuration requires only a single dc source, it requires a switch which must withstand a voltage in excess of 2 V0V0 size 12{V rSub { size 8{0} } } {} (the degree of excess being determined by the voltage developed across the primary leakage reactance as current is switched from the primary to the secondary windings) and requires the more complex bifilar winding in the machine. In addition, the switches in this configuration must include snubbing circuitry (typically consisting of a resistor-capacitor combination) to protect them from transient overvoltages. These overvoltages result from the fact that although the two windings of the bifilar winding are wound such that they are as closely coupled as possible, perfect coupling cannot be achieved. As a result, there will be energy stored in the leakage fields of the primary winding which must be dissipated when the switch is opened.

As is discussed in Section 10.3, VRM operation requires control of the current applied to each phase. For example, one control strategy for constant torque production is to apply constant current to each phase during the time that dL/mdL/m size 12{ ital "dL"/dθ rSub { size 8{m} } } {} for that phase is constant. This results in constant torque proportional to the square of the phasecurrent magnitude. The magnitude of the torque can be controlled by changing the magnitude of the phase current.

The control required to drive the phase windings of a VRM is made more complex because the phase-winding inductances change both with rotor position and with current levels due to saturation effects in the magnetic material. As a result, it is not possible in general to implement an open-loop PWM scheme based on a precalculated algorithm. Rather, pulse-width-modulation is typically accomplished through the use of current feedback. The instantaneous phase current can be measured and a switching scheme can be devised such that the switch can be turned off when the current has been found to reach a desired maximum value and turned on when the current decays to a desired minimum value. In this manner the average phase current is controlled to a predetermined function of the rotor position and desired torque.

This section has provided a brief introduction to the topic of drive systems for variable-reluctance machines. In most cases, many additional issues must be considered before a practical drive system can be implemented. For example, accurate rotor-position sensing is required for proper control of the phase excitation, and the control loop must be properly compensated to ensure its stability. In addition, the finite rise and fall times of current buildup in the motor phase windings will ultimately limit the maximum achievable rotor torque and speed.

The performance of a complete VRM drive system is intricately tied to the performance of all its components, including the VRM, its controller, and its inverter.

In this sense, the VRM is quite different from the induction, synchronous, and dc machines discussed earlier in this chapter. As a result, it is useful to design the complete drive system as an integrated package and not to design the individual components (VRM, inverter, controller, etc.) separately. The inverter configurations of Fig. 11.21 are representative of a number of possible inverter configurations which can be used in VRM drive systems. The choice of an inverter for a specific application must be made based on engineering and economic considerations as part of an integrated VRM drive system design.

## SUMMARY

This chapter introduces various techniques for the control of electric machines. The broad topic of electric machine control requires a much more extensive discussion than is possible here so our objectives have been somewhat limited. Most noticeably, the discussion of this chapter focuses almost exclusively on steady-state behavior, and the issues of transient and dynamic behavior are not considered.

Much of the control flexibility that is now commonly associated with electric machinery comes from the capability of the power electronics that is used to drive these machines. This chapter builds therefore on the discussion of power electronics in Chapter 10.

The starting point is a discussion of dc motors for which it is convenient to subdivide the control techniques into two categories: speed and torque control. The algorithm for speed control in a dc motor is relatively straight forward. With the exception of a correction for voltage drop across the armature resistance, the steadystate speed is determined by the condition that the generated voltage must be equal to the applied armature voltage. Since the generated voltage is proportional to the field flux and motor speed, we see that the steady-state motor speed is proportional to the armature voltage and inversely proportional to the field flux.

An alternative viewpoint is that of torque control. Because the commutator/brush system maintains a constant angular relationship between the field and armature flux, the torque in a dc motor is simply proportional to the product of the armature current and the field flux. As a result, dc motor torque can be controlled directly by controlling the armature current as well as the field flux.

Because synchronous motors develop torque only at synchronous speed, the speed of a synchronous motor is simply determined by the electrical frequency of the applied armature excitation. Thus, steady-state speed control is simply a matter of armature frequency control. Torque control is also possible. By transforming the stator quantities into a reference frame rotating synchronously with the rotor (using the dq0 transformation of Appendix C), we found that torque is proportional to the field flux and the component of armature current in space quadrature with the field flux. This is directly analogous to the torque production in a dc motor. Control schemes which adopt this viewpoint are referred to as vector or field-oriented control.

Induction machines operate asynchronously; rotor currents are induced by the relative motion of the rotor with respect to the synchronously rotating stator-produced flux wave. When supplied by a constant-frequency source applied to the armature winding, the motor will operate at a speed somewhat lower than synchronous speed, with the motor speed decreasing as the load torque is increased. As a result, precise speed regulation is not a simple matter, although in most cases the speed will not vary from synchronous speed by an excessive amount.

Analogous to the situation in a synchronous motor, in spite of the fact that the rotor of an induction motor rotates at less than synchronous speed, the interaction between the rotor and stator flux waves is indeed synchronous. As a result, a transformation into a synchronously rotating reference frame results in rotor and stator flux waves which are constant. The torque can then be expressed in terms of the product of the rotor flux linkages and the component of armature current in quadrature with the rotor flux linkages (referred to as the quadrature-axis component of the armature current) in a fashion directly analogous to the field-oriented viewpoint of a synchronous motor. Furthermore, it can be shown that the rotor flux linkages are proportional to the direct-axis component of the armature current, and thus the direct-axis component of armature current behaves much like the field current in a synchronous motor. This field-oriented viewpoint of induction machine control, in combination with the power-electronic and control systems required to implement it, has led to the widespread applicability of induction machines to a wide range of variable-speed applications.

Finally, this chapter ends with a brief discussion of the control of variablereluctance machines. To produce useful torque, these machines typically require relatively complex, nonsinusoidal current waveforms whose shape must be controlled as a function of rotor position. Typically, these waveforms are produced by pulse-width modulation combined with current feedback using an H-bridge inverter of the type discussed in Chapter 10. The details of these waveforms depend heavily upon the geometry and magnetic properties of the VRM and can vary significantly from motor to motor.

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