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866 M. F. Rahman et al.

FIGURE 33.13 Steady-state equivalent circuit of an induction motor.

From this representation, the following power relationships

in terms of motor parameters and the rotor slip can be found.

Power in the rotor circuit, P

= 3I

3sR

(

sω

)

(33.13)

Output power, P

= P

−3I

= (1 −s)P

= ω

T =

(1 −s)ω

T (33.14)

where

slip, s =

−ω

−pω

(33.15)

P = number of pole pairs

2πN

rad/s; N is the rotor speed in rev/min,

= rotor speed in electrical rad/s,

and

= 2πf

rad/s (electrical), f

being the supply frequency.

The developed torque, T =

Nm (33.16)

The slip frequency, sf

, is the frequency of the rotor current

and the airgap voltage E

is given by

= ω

(33.17)

where λ

is the stator ﬂux linkage due to the airgap ﬂux. If the

stator impedance is negligible compared to E

, which is true

when f

is near the rated frequency f

≈ E

= 2πf

(33.18)

and

T =

(

sω

)



2πf



(

sω

)

(33.19)

33.3.3 Characteristics and Methods of Control

The above analysis suggests several speed-control methods.

The following are the widely used methods:

(1) Stator voltage control,

(2) Slip power control,

(3) Variable-voltage, variable-frequency (V–f ) control,

(4) Variable-current, variable-frequency (I–f ) control.

These methods are sometimes called scalar controls to dis-

tinguish them from vector controls, which are described in

Section 33.3.4. The torque–speed characteristics of the motor

differ signiﬁcantly under different types of control, as will be

evident in the following sections.

33.3.3.1 Stator Voltage Control

In this method of control, back-to-back thyristors are used to

supply the motor with variable ac voltage, as indicated in the

converter circuit diagram of Fig. 33.14a.

The analysis of Section 33.3.1 implies that the developed

torque varies inversely as the square of the input root mean

square (RMS) voltage to the motor, as indicated in Fig. 33.14b.

This makes such a drive suitable for fan- and impeller-type

loads for which the torque demand rises faster with speed. For

other type of loads, the suitable speed range is very limited.

Motors with high rotor resistance may offer an extended speed

range. It should be noted that this type of drive with back-to-

back thyristors with the ﬁring-angle control suffers from poor

power and harmonic distortion factors when operated at low

speed.

If unbalanced operation is acceptable, the thyristors in one

or two supply lines to the motor may be bypassed. This offers

the possibility of dynamic braking or plugging, desirable in

some applications.

33.3.3.2 Slip Power Control

Variable-speed, three-phase, wound-rotor (or slip-ring)

induction motor drives with slip power control may take sev-

eral forms. In a passive scheme, the rotor power is rectiﬁed

and dissipated in a liquid resistor or in a multi-tapped resistor

that may be adjustable and forced cooled. In a more popular

scheme, which is widely used in medium- to large-capacity

pumping installations, the rectiﬁed rotor power is returned to

the ac mains by a thyristor converter operating in a naturally

commutated inversion mode. This static Scherbius scheme is

indicated in Fig. 33.15. In this scheme, the rotor terminals are

connected to a three-phase diode bridge which rectiﬁes the

rotor voltage. This rotor output is then inverted into mains

frequency ac by a fully controlled thyristor converter operating

from the same mains as the motor stator.

The converter in the rotor circuit handles only the rotor slip

power, so that the cost of the power converter circuit can be

much less than that of an equivalent inverter drive, albeit at

33 Motor Drives 867

Torque, Nm

V =1 pu

V =0.75 pu

V =0.5 pu

Load T-w

Motor T-w

(a) (b)

FCC

Speed,

rev/min

FIGURE 33.14 (a) Stator voltage controller and (b) motor and load torque–speed characteristics under voltage control.

MAINS

ref

DC Reactor

FCC

−

FIGURE 33.15 The static Scherbius drive scheme of slip power control.

the cost of the more expensive motor. The dc link current,

smoothed by a reactor, may be regulated by controlling the

ﬁring angle of the converter in order to maintain the developed

torque at the level required by the load. The current controller

(CC) and speed controller (SC) are also indicated in Fig. 33.15.

The current controller output determines the converter ﬁring

angle α from the ﬁring control circuit (FCC).

From the equivalent circuit of Fig. 33.13 and ignoring the

stator impedance, the RMS voltage per phase in the rotor

circuit is given by

sω

(33.20)

where ω

and ω

are the angular frequencies of the voltages in

the stator and rotor circuits, respectively, and n is the ratio of

the equivalent stator to rotor turns. The dc-link voltage at the

rectiﬁer terminals of the rotor, v

is given by

√

Assuming that the transformer interposed between the

inverter output is and the ac supply has the same turns ratio

n as the effective stator-to-rotor turns of the motor,

=−

√

cos α (33.21)

868 M. F. Rahman et al.

The negative sign arises because the thyristor converter

develops negative dc voltage in the inverter mode of oper-

ation. The dc-link inductor is mainly to ensure continuous

current through the converter so that the expression (33.21)

holds for all conditions of operation. Combining the preceding

three equations gives

sω

=−ω

cos α so that, s =−n cos α

and the rotor speed

(

1 −s

)

(

1 +n cos α

)

rad/s (33.22)

Thus, the motor speed can be controlled by adjusting the

ﬁring angle α. By varying α between 180

◦

and 90

◦

, the speed of

the motor can be varied from zero to full speed, respectively.

For a motor with low rotor resistance and with the assump-

tions taken earlier, it can be shown that the developed torque

of the motor is given by

T = 3P

≈ 3Pλ

Nm (33.23)

where i

is the dc-link current. Thus, the inner torque control

loop of a variable-speed drive using the Scherbius scheme nor-

mally employs a dc-link current loop as the innermost torque

loop. Figure 33.16 shows the transient responses of the dc-

link, rotor and stator currents of such a drive when the motor

is accelerated between two speeds.

The drive is normally started with a short-time-rated liquid

resistor, and the thyristor speed controller is started when the

drive reaches a certain speed.

By replacing the diode rectiﬁer of Fig. 33.16 with another

thyristor bridge, power can be made to ﬂow to and from the

Step response of dc link current

control loop

Rotor current during step change

Stator current during step change

−50

−44

025

0.25 0.5 0.75 1.0

50 75 100

t/ms

FIGURE 33.16 Transient responses of a slip power controlled drive under acceleration.

rotor circuit, allowing the motor to operate at a rate higher

than synchronous speed. For very large drives, a cycloconverter

may also be used in the rotor circuit with direct conversion of

frequency between the ac supply and the rotor and driving the

motor above and below synchronous speed.

33.3.3.3 Variable-voltage, Variable-frequency (V–f )

Control

33.3.3.3.1 SPWM Inverter Drive When an induction motor

is driven from an ideal ac voltage source, its normal operating

speed is less than 5% below the synchronous speed, which

is determined by the ac source frequency and the number of

motor poles. With a sinusoidally modulated (SPWM) inverter,

indicated in Fig. 33.17, the supply frequency to the motor can

be easily adjusted for variable speed. Equation (33.18) implies

that, if rated airgap ﬂux is to be maintained at its rated value

at all speeds, the supply voltage V

to the motor should be

varied in proportion to the frequency f

. The block diagram of

Fig. 33.18a shows how the frequency f

and the output voltage

of the SPWM inverter are proportionately adjusted with

the speed reference. The speed reference signal is normally

passed through a ﬁlter that only allows a gradual change in

the frequency f

. This type of control is widely referred to as

the V–f inverter drive. Control of the stator input voltage V

as a function of the frequency f

is readily arranged within

the inverter by modulating the switches T

–T

. At low speed,

however, where the input voltage V

is low, most of the input

voltage may drop across the stator impedance, leading to a

reduction in airgap ﬂux and loss of torque.

Compensation for the stator resistance drop, as indicated in

Fig. 33.18b, is often employed. However, if the motor becomes

lightly loaded at low speed, the airgap ﬂux may exceed the rated

value, causing the motor to overheat.

33 Motor Drives 869

Mains

−V

FIGURE 33.17 V–f drive with SPWM inverter.

1+T

Speed

reference

Compensated

(a) (b)

Ideal

FIGURE 33.18 (a) Input reference ﬁlter and voltage and frequency reference generation for the V–f inverter drive and (b) voltage compensation at

low speed.

From the equivalent circuit of Fig. 33.13 and neglecting the

rotor leakage inductance, the developed torque T and the rotor

current I

are given by

sω

(33.24)

and

T = 3P

(33.25)

where sω

is the slip frequency, which is also the frequency of

the voltages and currents in the rotor. Equation (33.24) implies

that by limiting the slip s, the rotor current can be limited,

which in turn limits the developed torque Eq. (33.25). Con-

sequently, a slip-limited drive is also a torque-limited drive.

Note that this is true only in steady state. A speed-control sys-

tem with such a slip limiter is shown in Fig. 33.19. In this

scheme, the motor speed is sensed and added to a limited

speed error (or limited slip speed) to obtain the frequency (or

speed reference for the V–f drive).

Many applications of the V–f controller, however are open-

loop schemes, in which any demanded variation in V

is passed

through a ramp limiter (or ﬁlter) so that sudden changes in

the slip speed ω

are precluded, thereby allowing the motor to

follow the change in the supply frequency without exceeding

the rotor current and torque limits.

870 M. F. Rahman et al.

−

ref

∗

Speed Controller

FIGURE 33.19 Closed-loop speed controller with an inner slip loop.

From the foregoing analyses, it is obvious that the V–f

inverter drive essentially operates in all four quadrants, with

rotor speed dropping slightly with the load, and develop-

ing full torque at the same slip speed at all speeds. This

assumes that the stator input voltage is properly compensated,

so that the motor is operated with constant (or rated) air-

gap ﬂux at all speed. The motor can be operated above the

base speed by keeping the input voltage V

constant, while

increasing the stator frequency above base frequency in order

to run the motor at speeds higher than the base speed. The

airgap ﬂux and hence the maximum developed torque now

fall with speed, leading to constant power type characteristic.

Figure 33.20 depicts the T–ω characteristics of such a voltage-

and frequency-controlled drive for various operating frequen-

cies. In this ﬁgure, the T–ω characteristic for base speed has

been drawn in full, indicating the maximum developed torque

ω, rad/sec

T, Nm

max

rated

Sequence: a-b-c

Sequence: a-c-b

Base speed

with rated V

and base f

FIGURE 33.20 Typical T–ω characteristics of V–f drive with the input

frequency f

and voltage V

below and above base speed.

max

and the rated torque. Below base speed, the V

–f

ratio is

maintained to keep the airgap ﬂux constant. Above base speed,

is kept constant, while f

increases with speed, thus weaken-

ing the airgap ﬂux. Forward driving in quadrant 1 takes place

with the inverter output voltage sequence of a–b–c, whereas

reverse driving in quadrant 3 takes place with the sequence

a–c–b. Regenerative braking while forward driving takes place

by adjusting the input frequency f

in such a way that the

motor operates in quadrant 2 (quadrant 4 for reverse braking)

with the desired braking characteristic.

Note that the characteristics in Fig. 33.20 are based on the

steady-state equivalent circuit model of the motor. Such a

drive suffers from the poor torque response during transient

operation because of time-dependent interactions between the

stator and rotor ﬂuxes. Figure 33.21 indicates the machine air-

gap ﬂux during acceleration with V-f control obtained from

a dynamic model. Clearly, the airgap ﬂux does not remain

constant during the dynamic operation.

33.3.3.3.2 Cycloconverter Drive For large-capacity induc-

tion motor drives, the variable-frequency supply at variable

voltage is effectively obtained from a cycloconverter in which

a back-to-back thyristor converter pairs are used, one for each

phase of the motor, as indicated in Fig. 33.22. Each thyristor

block in this ﬁgure represents a fully controlled thyristor ac–dc

converter. The maximum output frequency of such a con-

verter can be as high as about 40% of the supply frequency. In

view of the large number of thyristor switches required, cyclo-

converter drives are suitable for large capacity but low-speed

applications.

33.3.3.4 Variable Current–Variable Frequency (I–f)

Control

In this scheme, medium- to large-capacity induction motors

are driven from a variable but stiff current supply that may

be obtained from a thyristor converter and a dc-link induc-

tor as indicated in Fig. 33.23. The frequency of the current

supply to the motor is adjusted by a thyristor converter with

33 Motor Drives 871

Speed, r/minTorque, NmAirgap flux, TeslaStator current, A

1500

100

0.5

−20

−40

1000

500

0 0.2 0.314 0.4 0.6

0 0.2 0.4 0.6

T, Sec.

(a)

(b)

(c)

(d)

FIGURE 33.21 Transient response of torque, speed, current, and airgap ﬂux during acceleration from standstill using a V–f inverter drive.

auxiliary diodes and capacitors. The diodes in each inverter

leg and the capacitors across them are needed for turning off

the thyristors when current is to be commutated from one to

the next in sequence. The motor current waveforms are nor-

mally six-step, or quasi-square, as indicated in Fig. 33.24. The

switching states of the inverter thyristors are also indicated in

this ﬁgure. The motor voltage waveforms are determined by

the load. These waveforms are more nearly sinusoidal than the

current waveforms.

The thyristor converter supplying the quasi-square current

waveforms to the motor has ﬁring-angle control, in order to

regulate the dc-link current to the inverter. The dynamics

of the dc-link current control is such that this current may

be considered to be constant during the time, the inverter

switches commutate the dc-link current from one switch to the

next. Such a current-source drive offers four-quadrant oper-

ation, with independent control of the dc-link current and

output frequency. One drawback is that the motor voltage

waveforms have voltage spikes due to commutation.

From the analysis of Section 33.3.2, if the higher order har-

monics of the current waveforms in Fig. 33.24 are neglected,

and it is assumed that the motor voltage and current wave-

forms are taken to be sinusoidal, the magnetizing current I

in Fig. 33.13 can be kept constant (for constant-airgap ﬂux

872 M. F. Rahman et al.

TRANSFORMER BANK

FCC

MOTOR

AC SUPPLY

FIGURE 33.22 V–f drive with cycloconverter drive.

FCC

Inverter

Rectifier

Mains

FIGURE 33.23 DC-link current-source thyristor inverter drive.

33 Motor Drives 873

−I

FIGURE 33.24 Motor-current waveforms and the thyristor switching states for a current-source drive.

Stator current (A)

−5 −4 −3 −2 −10

Rotor frequency (Hz)

12345

FIGURE 33.25 Stator current vs rotor (slip) frequency for constant-

airgap ﬂux operation.

operation) if the RMS value of the stator supply current I

is deﬁned according to Eq. (33.26). This relationship is also

shown in graphical form in Fig. 33.25.

= I





2πf



+2πf

(

)



1/2

= constant (33.26)

The control scheme for variable-speed operation with a

current-source drive is indicated in the block diagram of

Fig. 33.26. The speed reference deﬁnes the stator current ref-

erence according to Eq. (33.26) and the frequency reference

is obtained by adding the rotor frequency to the actual speed

of the motor. The inverter drive may consist of the thyris-

tor current source and the inverter of Fig. 33.23 or a diode

rectiﬁer supplied SPWM transistor inverter of Fig. 33.17 with

independent current regulators, one for each phase.

The dynamic performance of such current-controlled

induction motor drives is not very satisfactory, just as for

−

INVERTER

ref

FIGURE 33.26 Variable-current–variable-frequency inverter drive

scheme.

the voltage-source inverters. Furthermore, the current-source

inverter drive cannot normally be operated open-loop, like

the V–f inverter drive. For high dynamic performance, vector-

controlled drives are becoming popular.

33.3.4 Vector Controls

The foregoing scalar control methods are only suitable for

adjustable speed applications in which the load speed or posi-

tion is not controlled like in a servo system. Instantaneous

frequency control with a view to control motor speed or posi-

tion cannot be deﬁned, and therefore, instantaneous torque

control cannot be addressed by scalar control methods. Vec-

tor control technique allows a squirrel-cage induction motor

to be driven with high dynamic performance, comparable to

that of a dc motor. In vector control, machine dynamic model

rather than steady-state model (on which scalar controllers are

based on), is used to design the controller. For this, the con-

troller needs to know the rotor speed (in the indirect method)

or the airgap ﬂux vector (in the direct method) accurately,

using sensors. The latter method is not practical because of

the requirement of attaching airgap ﬂux sensors. The indi-

rect method, which is being widely accepted in recent years,

requires the controller to be matched with the motor being

driven. This is because the controller also needs to know

874 M. F. Rahman et al.

some rotor parameter(s), which may vary according to the

conditions of operation, continuously.

33.3.4.1 Basic Principles

The methods of vector control are based on the dynamic equiv-

alent circuit of the induction motor. There are at least three

ﬂuxes (rotor, airgap, and stator) and three currents or mmfs

(stator, rotor, and magnetizing) in an induction motor. For

high dynamic response, interactions among current, ﬂuxes,

and speed, must be taken into account in determining appro-

priate control strategies. These interactions are understood

only via the dynamic model of the motor.

All ﬂuxes rotate at synchronous speed. The three-phase

currents create mmfs (stator and rotor) that also rotate at

synchronous speed. Vector control aligns axes of an mmf and

a ﬂux orthogonally at all times. It is easier to align the stator

current mmf orthogonally to the rotor ﬂux.

Any three-phase sinusoidal set of quantities in the stator can

be transformed to an orthogonal reference frame by





αs

βs











cos θ cos(θ −

2π

) cos(θ −

4π

)

sin θ sin(θ −

2π

) sin(θ −

4π

)















(33.27)

where θ is the angle of the orthogonal set α–β–0 with respect

to any arbitrary reference. If the α–β–0 axes are stationary and

the α-axis is aligned with the stator a-axis, then θ = 0atall

times, Thus





αs

βs











1 −

−

√















(33.28)

If the orthogonal set of reference rotates at the synchronous

speed ω

, its angular position at any instant is given by

θ =



t +θ

(33.29)

The orthogonal set is then referred to as d–q–0 axes. The

three-phase rotor variables, transformed to the synchronously

rotating frame, are















cos

(

−ω

)

t cos



(

−ω

)

t −

2π



cos



(

−ω

)

t −

4π



sin

(

−ω

)

t sin



(

−ω

)

t −

2π



sin



(

−ω

)

t −

4π

















(33.30)

It should be noted that the difference ω

−ω

is the relative

speed between the synchronously rotating reference frame and

the frame attached to the rotor. This difference is also the slip

frequency, ω

, which is the frequency of the rotor variables.

By applying these transformations, voltage equations of the

motor in the synchronously rotating frame reduce to



















+pL

−ω

+pL

−ω

(ω

−ω

+pL

(ω

−ω

−(ω

−ω

) pL

−(ω

−ω

+pL



















(33.31)

where the speed of the reference frame, ω

, is equal to ω

and

= L

, L

= L

Subscripts l and m stand for leakage and magnetizing,

respectively, and p represents the differential operator d/dt.

The equivalent circuits of the motor in this reference frame

are indicated in Figs. 33.27a and b.

The stator ﬂux linkage equations are:

= L

) = L

(33.32)

= L

) = L

(33.33)





+λ



(33.34)

The rotor ﬂux linkages are given by

= L

) = L

(33.35)

= L

) = L

(33.36)





+λ



(33.37)

The airgap ﬂux linkages are given by

= L





(33.38)

= L

(

)

(33.39)





mqs

+λ

mds



(33.40)

The torque developed by the motor is given by

T =



−λ



(33.41)

33 Motor Drives 875

−

(a)

(b)

−

- w

FIGURE 33.27 Motor dynamic equivalent circuits in the synchronously rotating: (a) q- and (b) d-axes.

From Eq. (33.31), the rotor voltage equations are

= 0 = L

(

−ω

)





(

−ω

)

r (33.42)

= 0 = L

(

−ω

)





(

−ω

)

(33.43)

Using Eqs. (33.35) and (33.36),

dλ

+(ω

−ω

)λ

= 0 (33.44)

and

dλ

+(ω

−ω

)λ

= 0 (33.45)

Also from Eqs. (33.35) and (33.36),

−

(33.46)

−

(33.47)

The rotor currents i

and i

can be eliminated from

Eqs. (33.44) and (33.45) by using Eqs. (33.46) and (33.47).

Thus

dλ

−

+(ω

−ω

)λ

= 0 (33.48)

dλ

−

+(ω

−ω

)λ

= 0 (33.49)

The elimination of transients in rotor ﬂux and the coupling

between the two axes occurs when

= 0 and

= λ

(33.50)

The rotor ﬂux should also remain constant so that

dλ

= 0 =

dλ

(33.51)

From Eq. (33.48) to Eq. (33.51),

−ω

= ω

(33.52)

and

= L

(33.53)