AC DRIVES

15.1 INTRODUCTION
AC drives is a term used to refer to equipment designed to control the
speed of an a.c. motor. They receive a.c. power and convert it to an
adjustable frequency, adjustable voltage output for controlling motor
operation. Inverters and other types of frequency changers are typical
examples of modern a.c. drives which are also called adjustable frequency
drives.
A typical inverter receives 400 V a.c., three-phase, 50 Hz input power
and in turn provides the proper voltage and frequency for a given speed to
the motor. The three common inverter types are the variable voltage
inverter (VVI), current source inverter (CSI), and pulse width modulation
(PWM). Another type of a.c. drive is a cycloconverter. These are
commonly used for very large motors used in steel industry and mils. The
cycloconverters is an arrangement of poly-phase rectifiers in which the
firing delay is cyclically varied to synthesise an a.c. output, instead of the
set delay for producing a controllable d.c. for the d.c. motor previously
mentioned.
A feature of a.c. drives is the ability to increase or decrease the voltage
and frequency to a motor gradually. This accelerates the motor smoothly
with less stress on the motor and connected load. Smoothing is a feature
that can be added to the acceleration/ deceleration operation. This feature
smoothes the transition between starting and steady-state operation.
There are several types of a.c. motors used in industrial applications
that need real drives to suit a given task. In all types of drives, motors and
load have stored energy which can be either regenerated or dissipated as
the load speed falls. One third of the world's electricity consumption is
used for running induction motors driving pumps, fans, compressors,
elevators and machinery of various types. In general, the speed of a.c.
motors depends on the frequency of the supply voltage and the number of
magnetic poles per phase in the stator. Early speed controllers depended
 on switching in different numbers of poles and control was only available
manually and in crude steps. Modern electronic drives make continuously
variable frequency supplies possible permitting closed-loop speed control.
This chapter is intended to provide a basic understanding of a.c. drive
terms, types and theory of operations.

Benefits of AC drive

 Large energy savings at lower speed.

 Increased life of rotating components due to lower operating speed.
 Reduced noise and vibration level.
 Reduction of thermal and mechanical stresses.
 Lower kVA.
 High power factor.

15.2 TYPES OF AC MOTORS

The types of a.c. motor available in industry are classified according to

the supply as single-phase and poly-phase motors. These two types may
also be classified according to the principle of operation into induction
type and synchronous type as follows:

1- single-phase motors
(a) Induction type-squirrel cage
(i) Split-phase
(ii) Capacitor start
(iii) Permanent split capacitor
(iv) Capacitor start / capacitor run
(v) Split-phase start / capacitor run
(vi) Shaded pole
(b) Induction type-wound rotor
(i) Repulsion
(ii) Repulsion start
(iii) Repulsion induction
(c) Single-phase synchronous
(i) Hysteresis
(ii) Reluctance
(iii) Permanent magnet
(d) Single-phase universal motor (AC and DC)

2- Poly-phase motors
(a) Induction type
(i) Wound rotor
(ii) Squirrel – cage
(b) Synchronous
 These motors are summarised in the following diagram (Fig.15.1) ,

Split –phase
Capacitor start
Squirrel Permanent capacitor
-cage Capacitor start capacitor run
Shaded -pole
Single-phase Induction

Wound Repulsion
Universal rotor Repulsion start
Repulsion induction
AC and DC
Synchronous
Hysteresis
AC Reluctance
Motors Permanent magnet

Induction
Wound rotor
Poly-phase

Synchronous Squirrel -cage

Fig.15.1 Types of a.c. motors.

15.3 THREE-PHASE INDUCTION MOTOR :

REVISION OF EQUATIONS
The three-phase induction motors ( also called asynchronous motors)
are the most widely used electric motors in industry. The popularity of
this type of motors in most industrial applications is because of their
simple, robust construction because they can build without slip-rings or
commutator, rugged, relatively cheap, require little maintenance and have
self-starting torque. An induction motor of a medium size may have an
efficiency as high as 90 percent and power factor of nearly 0.9. The
physical size of such a motor for a given output rating is small as
compared with d.c. and a.c. synchronous motors of same rating. There is
another distinguishing feature of induction motor is that it is a singly
excited machine, i.e. only the stator winding is connected to the a.c.
supply, no electrical connection from the supply to the rotor is needed.
Finally It can be manufactured with characteristics to suit most industrial
requirements.
 Beside the above numerous advantages of the induction motor, it has
two main inherent disadvantages:
1. Its starting torque is inferior to d.c. shunt motor.
2. It is essentially a constant speed motor and its speed cannot be
changed easily.
The speed of an induction motor is determined by the supply frequ-
ency and number of poles, with a few percent regulation from no-load to
full-load. However, the speed is frequency dependent and consequently
these motors are not easily adapted to speed control. A wide range of
speed control is only possible by using expensive power electronic circuit
with advanced digital control. We usually prefer d.c. motors when large
speed variations are required due to its inexpensive methods of control.

15.3.1 Basic Principles of Three-Phase Induction Motor with Sinusoidal

Supply Voltages
Like any electric motor, a three-phase induction motor has a stator
and a rotor. The stator carries a three-phase winding (called stator
winding) while the rotor carries a short-circuited winding (called rotor
winding). Only the stator winding is fed from three-phase supply. The
rotor winding derives its voltage and power from the externally energized
stator winding through electromagnetic induction and hence such a
machine is often called the induction machine. The induction motor may
be considered to be a transformer with a rotating secondary in the sense
that the power is transferred from the stator (primary) to the rotor
(secondary) winding only by mutual induction. Hence, it can, therefore,
be described as a “transformer type” a.c. machine in which electrical
energy is converted into mechanical energy.
When the stator windings are connected to a set of balanced three-
phase voltages and the rotor circuit is closed, the resulting three-phase
current establish a rotating mmf wave that results in a flux wave of
constant amplitude rotating at constant speed known as the synchronous
speed. The value of the synchronous speed is fixed by two parameters:

(a)The supply frequency, (Hertz),

(b)The number of poles p for which the primary is wound.

The synchronous speed of the rotating magnetic field is given by

The number of poles p must be an even integer since for every north
pole there is a corresponding south pole. The following Table-15.1
shows motors speeds for motors with different numbers of poles
working with different a.c. supply frequencies.
 Table 15.1. Synchronous speed of induction motor for different
number of poles.

Rotor Speed (rpm)

Number of
2 4 6 8 10 12
poles (p)
Frequency
3000 1500 1000 750 600 500
f = 50 Hz
Frequency
3600 1800 1200 900 720 600
f = 60 Hz

An induction motor runs at a shaft speed n that is less than the synchr-
onous speed at which the stator rotating field is rotate. The speed
difference is called the slip speed. The ratio of slip speed to
synchronous speed is the most important variable in induction motor
operation and is called the per-unit slip s, and is given by:

where s is the slip in per unit, ns is the synchronous speed in rpm, and n is
the rotor speed. Since the rotor current is proportional to the relative
motion between the rotating field and the rotor speed, the rotor current
and hence the torque are both directly proportional to the slip. For
particular cases, the slip of the motor will have the following special
values:

 When the motor is running at synchronous speed , i.e. and

.
 At standstill and .
 If the motor is rotating at synchronous speed in the reverse
direction, then and .

Squirrel cage motors are built with the slip ranging from about
3 – 20%. Motors with a slip of 5% or higher are used for hard-to-start
applications. A motor with a slip of 5% or less is called a normal slip
motor. A normal slip motor is often referred to as a constant speed motor
because the speed changes very little with variations in load. At full load
the per-unit slip usually 5% for a small motor because .
15.3.2 Development of Circuit Model (Equivalent Circuit)
Standstill Operation
At standstill, the motor can be considered as a static transformer with
primary (stator) winding and secondary (rotor) winding. If the stator is fed
from a three-phase supply with voltage V1 is the phase voltage, the air gap
field produced rotates at synchronous speed ns . This field induces emfs E1
and E2 in both the stator and rotor winding respectively. The magnitudes
of these emfs are given by, assuming unity winding factor (kw =1) ,

where is the effective transformation (turns) ratio between the stator

and rotor winding which is usually greater than unity and is typically in
the range 1.1 – 1.3. In slip ring wound rotor motor, is easily defined
exactly. However, in cage rotor motor because there is no distinct winding
on the cage, the rotor quantities can also be referred to the stator side by
taking as unity. The induced voltage E1 (back emf) will be differs
from V1 by the voltage drop in the stator leakage impedance Z1= R1+ jX1
as shown in Fig. 15.2.

Fig. 15.2 Induction motor per-phase equivalent circuit at standstill.

The stator current I1 can be resolved into two components: a load
component I2 and an exciting (magnetizing) component Iφ. The load
component I2 produces the rotor mmf. The exciting component Iφ is the
additional stator current required to create the resultant air-gap flux.
The exciting current Iφ (also called no-load current) is large compared
with the transformer because of the air gap ( 20% - 30% of the full load
current for small motors, and 30% - 50% of the full load current for large
motors ). Iφ can be resolved into a core-loss component Ic in-phase with
E1 and a magnetizing component Im lagging E1 by 90°.
The rotor circuit at standstill consists of E2 and the rotor leakage
impedance Z‟2 = R‟2 + j X‟2 as shown in Fig.15.2,where in this figure:

V1 = stator line-to-neutral terminal voltage

E1 = back emf (line-to-neutral) generated by the resultant air-gap flux
stator current Il
R1 = stator effective resistance
X1 = stator leakage reactance at standstill = 2πf1L1
Rc = core loss resistance
Xm = magnetizing branch reactance
X‟2 = rotor leakage reactance at standstill = 2πf1L2
R‟2 = rotor effective resistance at standstill.

Equivalent circuit at running operation

If the rotor conductors rotate at speed and cut the constant rotating
stator flux (which rotates at speed ), then at a speed the induced
emf and current in the rotor are of frequency , where

Since the flux in the air gap is constant, the secondary emf at slip s
is proportional to the time rate of flux cutting. Hence,

Hence, the exact equivalent circuit at running operation is shown in

Fig.15.3.

Fig.15.3 The exact equivalent circuit at running operation.

It is clear that, in the equivalent circuit of the rotor of the induction motor
shown in Fig.15.3, both the rotor leakage reactance sX‟2 and the back
emf sE2 , depend on the rotor frequency. But the rotor effective resistance
R‟2 doesn't depend on the frequency. Dividing all elements of the equiv-
alent circuit by s, we can obtain the circuit shown in Fig.15.4.

where
R‟2 = rotor effective resistance
X‟2 = rotor leakage reactance
I‟2 = rotor current
E2 = back emf (line-to-neutral) generated by the resultant air-gap flux.

Fig.15.4 Modified exact equivalent circuit at running operation.

To study the performance of induction motor, it is recommended to

refer the rotor circuit to the stator circuit similar to that of the transformer,
therefore, the overall exact equivalent circuit of the induction motor
viewed from the stator is shown in Fig.15.5.

Fig.15.5 Exact equivalent circuit per-phase of a three-phase induction

referred to the stator.
15.3.3 The Approximate Equivalent Circuit
In Fig.65.5, the resistance (R2 / s) can be divided into two resistances
R2 and R2 (1-s) / s, where R2 represents the referred rotor resistance and
R2 (1-s) / s represents the mechanical load connected to the motor shaft as
shown in Fig.15.6. Now, if the magnetizing branch of Rc and Xm is
moved towards the terminal voltage, one can obtain the approximate
equivalent circuit as depicted also in Fig.65.6. The equivalent rotor circuit
per-phase referred to the stator side is also depicted in Fig.15.7.

Fig.15.6 Approximate equivalent circuit per-phase of induction motor

referred to stator.

Fig.15.7 The equivalent rotor circuit per-phase referred to the stator.

15.3.4 Power and Torque in Induction Motor

An induction motor can be basically described as a rotating transf-
ormer. Its input is a three-phase system of voltages and currents. For an
ordinary transformer, the output is electric power from the secondary
windings. The secondary windings in an induction motor (the rotor) are
shorted out, so no electrical output exists from normal induction motors.
Instead, the output is mechanical. The relationship between the input
electric power and the output mechanical power of this motor is shown in
Fig.15.8.
The input power to an induction motor Pin is in the form of three-phase
electric voltages and currents. The first losses encountered in the machine
are I2R losses in the stator windings (the stator copper loss PSCL). Then,
 Fig.15.8 Power flow diagram of a three-phase induction motor.

some amount of power is lost as hysteresis and eddy currents in the stator
(Pcore). The power remaining at this point is transferred to the rotor of the
machine across the air gap between the stator and rotor. This power is
called the air gap power PAG of the machine. After the power is
transferred to the rotor, some of it is lost as I2R losses (the rotor copper
loss PRCL), and the rest is converted from electrical to mechanical form
(Pconv = Pm). Finally, friction and windage losses PF&W and stray losses
Pmisc are subtracted. The remaining power is the output of the motor
which is mechanical Pout =ωTL . However, one may simplify the power
flow diagram to the form shown in Fig.15.8. This can be validated by
considering opposite variations of mechanical loss and rotor iron loss with
speed.
By examining the per-phase equivalent circuit, the power and torque
equations governing the operation of the motor can be derived. The input
current to a phase of the motor is:

√
Thus, the stator copper losses, the core losses, and the rotor copper losses
can be found.
The stator copper losses in the three phases are: PSCL = 3 I12 R1
The core losses: Pcore = 3 E12/ Rc
The air-gap power: Pg = Pin – PSCL - Pcore

Also, the only element in the equivalent circuit where the air-gap power
can be consumed is in the resistor R2 / s. Thus, the air-gap power:
The total actual resistive losses in the rotor circuit are given by:

PRCL = 3 (I„2) 2 R‟2

Since power is unchanged when referred across an ideal transformer, the

rotor copper losses can also be expressed as:

PRCL = 3 I22 R2 = s Pg

where

√( √
)

After stator copper losses, core losses and rotor copper losses are
subtracted from the input power to the motor, the remaining power is
converted from electrical to mechanical form. The power converted,
which is called developed mechanical power is given as:

( )

( )

The rotor copper losses can be given to be equal to the air-gap power
times the slip : PRCL = s Pg . Hence, the lower the slip of the motor, the
lower the rotor losses. Also, if the rotor is not turning, the slip is s =1 and
the air-gap power is entirely consumed in the rotor. This is logical, since
if the rotor is not turning, the output power Pout ( = ωm TL ) must be zero.
Since Pconv = Pg – PRCL , this also gives another relationship between the
air-gap power and the power converted from electrical and mechanical
form:
 Pm = Pg – PRCL
= Pg – s Pg
Pm = (1- s) Pg (15.13)

Note that the proportion of the above quantities is fixed by “s”.

Finally, if the friction and windage losses and the stray losses are known,
the output power:
Po = Pm – PF&W – Pmisc (15.14)
The induced torque in a machine was defined as the torque generated
by the internal electric to mechanical power conversion. This torque
differs from the torque actually available at the terminals of the motor by
an amount equal to the friction and windage torques in the machine.
Hence, the developed torque is:

or it can be expressed as

The output or load torque TL per-phase can be found from Eq.(15.15) as

Substitute for from Eq.(15.11),

sE
s

From Eq.(15.18), it is revealed that:

(i) At synchronous speed, i.e. when slip s is zero, the torque is zero
so the torque-speed curve or the external characteristic of the
induction motor start from the origin as shown in Fig.15.9.
 (ii) At very low slip, i.e. the motor speed near the synchronous speed,
the term sX2 is very small and can be neglected. Therefore, the
torque is approximately proportional to the slip and the relation
between the torque and speed is approximately straight line.
(iii) When the motor is loaded, the speed will drop and the slip
increases and for further increase in the load, torque will reach its
maximum value Tm which also called the breakdown torque.

Fig.15.9 Torque-speed curve of an induction motor.

(iv) With further drop in speed due to increase in the load ,slip will
increase and if the load increases beyond maximum torque that
the motor cold tolerate , the motor start slows down and finally it
becomes at stopping position. The value of the torque that results
in motor stopping is called the pull-up torque as shown in
Fig.15.9.

From the approximate circuit of Fig.15.4, the current I2 can be expressed

in terms of the primary voltage V1 as

√( )

Substituting Eq.(15.19) into Eq.(15.18) gives

 In Eq.(15.20) slip s is the only variable, hence maximum torque can
be found by differentiating this equation with respect to s and equating to
zero to give the result the slip sm at which the peak torque occurs as

where X = motor total reactance = X1 +X2 .

Substituting Eq.(15.21) into Eq.(15.20) gives an expression maximum
torque Tm (sometimes called the peak torque or breakdown torque)

15.4 SPEED CONTROL OF INDUCTION MOTOR

The three-phase induction motor runs at a speed slightly less than

synchronous speed and is load dependents. Therefore, it is an inherently a
constant speed motor and its output mechanical power depends on the slip
s ( ). So it is difficult to control its speed. The
speed control of induction motor is done at the cost of decrease in
efficiency and low electrical power factor. Before discussing the methods
to control the speed of three-phase induction motor one should know the
basic formulas of speed and torque of three-phase induction motor as the
methods of speed control depends upon these formulae.
The synchronous speed was given by Eq.(15.1) as , where
f = frequency and p is the number of pole. The speed of induction motor is
given by,

Hence, the speed of the induction motor can be changed either from the
stator or from the rotor sides. Therefore, from Eq.(15.1), the speed control
of three-phase induction motor from stator side are classified as:

1. Changing the number of stator poles (p).

2. Stator voltage control (controlling the supply voltage (V1)).
3. Supply frequency changing:
 (i) Variable-voltage, variable-frequency (V/ f ) control.
(ii) Variable-current, variable-frequency (I / f ) control.

The speed controls of three-phase induction motor from rotor side are
further classified as:

1. Adding external resistance on rotor side.

2. Rotor injected voltage / slip energy recovery.
3. Cascade control method.

These methods are sometimes called scalar controls to distinguish them

from vector controls. The torque-speed characteristics of the motor
differ significantly under different types of control.

15.4.1 Speed Control from Stator Side

(1) Changing the number of stator poles (p)

The stator poles can be changed by three methods

(i) Method of varying the number of consequent poles

(ii) Multiple stator winding method
(iii) Pole amplitude modulation method (PAM)

Method of varying the number of consequent poles: In this method,

the number of poles can be changed in the ratio of 2:1 by changing the
connection of the coils. Fig.15.10 shows the stator connections for two-
speed operation of the thee-phase induction motor. The windings can be
connected in series or in parallel.

Fig.15.10 Stator windins connections for two-speed operation of

induction motor: (a) Series connection, (b) Parallel
connection.
 Multiple-stator winding method: In this method of speed control
of three-phase induction motor, the stator is provided by two separate
winding. These two stator windings are electrically isolated from each
other and are wound for two different pole numbers. Using switching
arrangement, at a time, supply is given to one winding only and hence
speed control is possible.
The disadvantage of this method is that, it enable speed changes in
terms of 2:1 ratio steps, hence to obtained variations in speed, multiple-
stator windings has to be applied. Multiple-stator windings have extra
sets of windings that may be switched in or out to obtain the required
number of poles. Unfortunately this would an expensive alternative.

Pole amplitude modulation method (PAM): In this method, the original

sinusoidal mmf wave is modulated by another sinusoidal mmf wave
having different number of poles. To explain the method, let :

f1(θ) be the original mmf wave of induction motor whose speed is to

be controlled.
f2(θ) be the modulation mmf wave.
P1 be the number of poles of induction motor whose speed is to be
controlled.
P2 be the number of poles of modulation wave so that,

After modulation, i.e. multiplying by , the resultant mmf

wave is
( ) ( )

Apply formula for: 2sin A sin B

So we get, resultant mmf wave

Therefore, the resultant mmf wave will have two different numbers of
poles,i.e.

Hence, by changing the number of poles we can easily change the speed
of three-phase induction motor.
(2) Controlling supply voltage (Variation of stator voltage)
It is seen from Eq.(15.17) that at any fixed speed, if we neglect the
mechanical losses, the developed torque TL (=Td) is proportional to the
square of the applied stator voltage V12. As the stator voltage is reduced
the rotor speed decreases and the maximum torque available from the
motor also decreases, Eq.(15.19). If the stator voltage is varied to control
the speed then the speed range of this method is limited with a constant-
torque load. This can be proved as follows:
The torque produced by running three-phase induction motor was
given by Eq.(15.17) as
sE
s
In low slip region (sX2)2 is very small as compared to (R2)2, hence it can
be neglected. Therefore the torque becomes,

sE

Since rotor resistance, R2 is constant so the equation of torque further

reduces to

sE
We know that rotor induced emf E2 V1, the supply voltage. So,
s
From the equation above, it is clear that if the supply voltage is
decreased voltage by one half the torque reduces to one quarter.Therefore,
the low speed performance of the motor with this method is poor because
motor current at a given slip is also proportional to the applied voltage
whereas the torque varies as the square of the voltage.This means that the
torque per ampere becomes lower at reduced speed as large currents are
required to develop a sufficient torque. However, in fan or pump drives,
the load torque varies approximately as the square of the speed. Hence the
torque required for low speed operation and starting is small and may
obtained without excessive overheating from a voltage controlled induc-
tion motor.

Methods of reducing stator voltage V1

1- Rheostatic control : The stator voltage can be reduced by conecting

external variable resistance or impedance between stator terminals and the
a.c. supply as shown in Fig.15.11. Saturable reactors have been used in
the past to perform this function.
Fig.15.11 Speed control of three-phase induction motor by adding
rheostat in the stator circuit.

A high resistance modifies the torque-speed characteristics and a wide

range of speed control is obtained. The ohmic losses of this method of
speed control are excessive and particularly at low speeds. Since the
torque produced in an induction motor is proportional to the square of
the supply voltage (T s V12), then if we decrease V1, the supply voltage,
torque will also decrease. However, this method is not efficient from
energy saving point of view and it is rarely used nowadays because small
change in speed requires large reduction in voltage, and hence
the current drawn by motor increases, which cause overheating of induc-
tion motor.
.
2- Electronic control: Nowadays, reduction of stator voltage is perfor-
med by using thyristors (or triacs) that offers several advantages. With
thyristors different techniques can be used to control the rms voltage
applied to the motor. Thyristor can be used as:
 AC regulators
 Transformer adjustable tap changers
 Controllers for multi-winding transformer secondary
Reduction of stator voltage of induction motor using three-phase thyristor
a.c. regulator is shown schematically in Fig.15.12. When the motor is
supplied by balance three-phase voltages of constant frequency, the
torque speed characteristics of the motor have the shape shown in
Fig.15.13. Now if the supply voltage is reduced by one half, the
maximum torque reduces to one quarter of its original value since the
maximum torque is also proportional to | | as described previously.
The operating point of an induction motor can be located on the torque
speed characteristics diagram and it is defined by the point of intersection
between the motor characteristics and the load characteristic as shown in
Fig.15.13. For small reduction in supply voltage the speed variation will
be very small, so that, for example, point 2 in Fig.15.13 is not shown.
Fig.15.12 Reduction of stator voltage of induction motor using three-
phase thyristor a.c. regulator.

Fig .15.13 Torque-speed characteristic of three-phase induction motor for

voltage control with fan load.
The waveform of the motor currents for the connection of Fig.15.12 are
very similar to corresponding waveforms for passive series R-L load
discussed in Chapter Five .The performance analysis of the motor with its
thyristor controller would be very complex due to the interaction between
the motor and its controller. The accurate analysis would require solution
of several nonlinear differential equations for the voltage, speed and
electromagnetic torque. The general solution is only possible using
computer simulation techniques such as Matlab and other computer
programmes. However, for steady-state solution using the approximate
equivalent circuit of Fig.15.6 one can find the performance of the three-
phase induction motor when speed controlled by voltage variation
technique as illustrated in the following example.

Example 15.1

A 7.5 kW, three-phase, 400 V, 50 Hz, 4-pole, 1400 rpm, star-connected

induction motor has the following parameters referred to the stator side:

R1 Ω , R2 = 6 Ω , X1 = 5.75 Ω , X2 = 4.25 Ω , Xm = Very high

The speed of the motor is controlled by voltage variation method using

pair of inverse parallel connected thyristors in each line with symmetrical
phase angle triggering mode. The delay angles of the thyristors are set to
give a line to line voltage of 250 V across the motor windings. Calculate
the motor speed, current and torque when driving a fan load its
characteristic is given by:

TL = 60 (1-s) 2

Solusion

Using Eq.(15.20), the torque of the three-phase induction motor for the
three phases is

Synchronous speed in rpm = 120 f / p = 120 50 /4 =1500.

At steady-state, T = TL , hence
From which ;

Which gives ; s = 0.2005

The torque produced by the motor is

The speed of the motor at 250V is

The line current is calculated from Eq.(15.8) ,since Xm is very high , thus
I1 = I2 ,

√( )

√( )

Approximate method of solution

It is seen from the above example that the equation of the slip s obtain
is of high order that is mathematically difficult to solve. However, an
approximate method of solution for steady-state operation can be used
over a range of average speeds to determine the corresponding range of
thyristor firing angles. This approximate method uses the motor funda-
mental equivalent circuit together with the curves giving the relation
between the per unit current and the firing angles for both particular speed
and load angles. These curves are shown in Fig.15.14(a) and can be
approximated by straight line as depicted in Fig. 15.14(b).
For star-connected motors with large phase angle ϕ, the approximated
straight line relationship between the current and firing angle α can be
represented mathematically as
 (a) (b)

Fig. 15.14 Current and the firing angles relationship for three-phase star
connected R-L load : (a) Rms line current versus α,
(b) Straight line approximation of current (p.u.) for three-wire
star-connected induction motor.

For branch-delta connected motor, the approximate relation is found to

be roughly as

Example 15.2

A variable speed drive is used to drive a water pump which has a torque-
speed curves described by the equation SI units, where
is the speed of the pump motor. The drive employs a three-phase,
240V, six-pole, 50 Hz, star-connected induction motor controlled by pairs
of inverse-parallel connected thyristors in each supply line. The per-phase
equivalent circuit parameters of the motor, referred to primary turns are
The required
speed range is 975 - 600 rpm. Use performance curves of current versus
firing-angle to calculate, approximately, the necessary ranges of thyristor
firing-angles.
Solution

The synchronous speed of the motor ns

The slip is given by Eq.(15.2) as

Hence

From Eq.(15.13) , the output power for the three phases of the motor is

( )

From the equivalent circuit of Fig.15.6, neglecting the magnetising

branch,

Ω
Ω

√ √
 √

From Fig,15.14 (a) ,

√ √

From Fig,15.14 (a),

Therefore, the range of the delay angles is:

It is obvious that, with this method of speed control, the variation of speed
is not great (if the voltage reduced to ) . It generates harmonics and
electromagnetic interferences. However, the method for obtaining speed
change is simple and energy saving is possible.

Example 15.3

A fuel pump has load characteristics represented in the speed range by a

line given by

where motor speed.

A three-phase, 240 V, four-pole, 50 Hz, star-connected, squirrel-cage
induction motor is to be used for the speed control of the fuel pump. Per-
phase equivalent circuit parameters of the motor, referred to primary
turns, are Ω Ω Ω very
large. The motor terminal voltages are to be controlled by pairs of
inverse-parallel connected thyristors in the supply lines. If steady-state
speed control is required in the range 750 - 1450 rpm, calculate the
necessary range of thyristor firing-angles.

Solution

The synchronous speed of the motor ns

The slip is given by Eq.(15.2) as

Hence

From Eq.(15.10), the output power for the three phases of the motor is

( )

From the equivalent circuit of Fig.15.6, neglecting the magnetising

branch,

Ω
 Ω

√ √

√ √

From Fig,15.14 (a),

√ √

From Fig.15.14 (a),

 3- Supply frequency changing: This method of speed control of
induction motor is considered as the most efficient one. The motor is
supplied from variable voltage variable frequency source. This is
because, in the three-phase induction motor , emf is induced by induction
similar to that of transformer which is given by

Where K is the winding constant, N is the number of turns per phase and
f is frequency. Now since (4.44 K N) is a constant value for any induction
motor, therefore the above equation can be written as,

⁄ ⁄

It is clear from the above equation that, if we change the frequency. the
synchronous speed will change (ns= 120 f / p). So if the frequency is
decreased the flux will increase and this change in flux causes saturation
of rotor and stator cores that causes increase in no load current of the
motor. Hence, it is important to maintain flux, φ constant and this is only
possible if the value of the voltage V is changed to keep the ratio of (V / f )
as constant. Hence, this technique is known as constant (V / f ) method.
For controlling the speed of three-phase induction motor by (V/ f ) method
it is necessary to supply variable voltage and frequency which is easily
obtained by using converter using solid-state devices / power electronics
which has the ability of providing such requirement.
So far, we have calculated torque-speed relationships at single supply
frequencies, now we need to find how the torque changes with changing
frequency. Consider the circuit diagram shown in Fig.15.15, which shows
the induction machine equivalent circuit in terms of inductances, rather
than reactances at any effective frequency fe :

Fig.15.15 Equivalent circuit per-phase for the induction motor in terms of

inductace.
Now, from analysis with a constant frequency supply we know that the
torque is given by:

sE E
s

where
angular frequency of the supply
= number of poles
= number of pole pairs = 2 .

At small values of slip it is reasonable to say that

and, from Eq.(15.31(b)) , we can re-write this in terms of

electrical supply frequency:

At this point, it is useful to introduce the concepts of slip frequency and

slip speed.

Slip Speed and Slip Frequency

Define simply, slip speed and slip frequency are:
 Slip speed = slip multiplied by synchronous speed
 Slip frequency = slip multiplied by supply frequency
Slip speed may be defined in either rpm (sns) or mechanical radians per
second (sωs).
Slip frequency is usually given in electrical radians per second as

Note that slip frequency has its own symbol, while slip speed is actually
written as the product of slip and synchronous speed.
Now multiplying top and bottom of the torque Eq. (15.32) by ωe yields

E
( )

Torque as a function of slip speed

From the above, we can now re-write the torque as:

E
( )

It can be seen that, if the ratio E2 /ωe is constant the torque will be
proportional to slip frequency. Considering another approach to define
E 2 from the equivalent circuit:

| |

where λ is the magnetizing flux in the machine. Substituting we get:

This is an important result: At small slips torque is proportional to flux

squared times slip speed

It is clear from Eq.(15.37) that the maximum torque is independent on

frequency at a given flux . The torque-speed characteristics of induction
motor operates with variable voltage variable frequency source is depicted
in Fig.15.16. One great advantage of this method is that the value of the
maximum torque for any frequency f1, f2, ……, fn remains constant. This
feature is very desirable in many industrial application for speed control.
 T

Tm
f1
f4 f2
f3 TL

Speed
0 ωs (rated)
ωs
Fig.15.16 Torque-frequency relationship.

Example 15.1

A 400 V, 50 Hz, 4-pole motor has rated speed of 1450 rpm and rated
torque of 10 Nm. If a torque of 10 Nm is needed at a mechanical speed of
1250 rpm, find the synchronous speed, supply frequency and line-to-line
supply voltage.

Solution

At rated torque, the slip speed will be the rated value. For a 4-pole 50 Hz
machine, synchronous speed is 1500 rpm, therefore, rated slip speed =
1500-1450 = 50 rpm. When operating at 1250 rpm, 10 Nm, slip speed will
still be 50 rpm and the synchronous speed is given by

With the synchronous speed, the supply frequency can be found

ns =120fe / p
fe =ns p/120 =1300×4/120 = 43.33 Hz
Finally, if V/f is constant, the supply voltage must be
Methods of obtaining supply frequency changing

Thyristor circuits can be used to produce variable frequency to control

the speed of induction motors.In general the currently available methods
of obtaining a variable frequency power output from the constant public
supply can be divided into two main techniques:-

1. Indirect frequency conversion using d.c. link (Inverters).

2. Direct frequency conversion (Cycloconverters).

These two frequency changing techniques when applied to speed control

of a.c motors are called: variable frequency drives (VFD).These types of
drive perform two main functions:

 Controls the speed of an a.c. motor by varying the frequency

supplied to the motor.
 Regulates the output voltage in proportion to the output frequency
to provide constant ratio of voltage to frequency (V/Hz), required
by the characteristics of the a.c. motor to produce adequate torque
as discussed before.

(A) Induction motor control using d.c link inverter drive:

Inverter drives are of two types:
 Voltage source inverter drives (VSI)
 Current source inverter drives (CSI)
The voltage source inverter has two stages of power conversion,
a rectifier and an inverter. A block diagram of voltage source inverter
drives is shown in Fig.15.17. The rectifier converts the fixed a.c. voltage a
to either fixed or adjustable d.c. voltage. The inverter produces a contr-
ollable a.c. output voltage at the desired frequency. The term “Inverter” is
also used to refer to the entire drive.

Fig.15.17 Block diagram of voltage source inverter drives.

 There are three basic types of inverters commonly employed in
adjustable a.c. drives:
(1) The variable voltage inverter (VVI), or square-wave six-step
voltage source inverter (VSI), receives d.c. power from a fixed or
adjustable voltage source and adjusts the frequency and voltage.
A controlled rectifier transforms supply a.c. to variable voltage d.c. as
shown in Fig.15.18. The converter can be an SCR (silicon-controlled
rectifier) bridge or a diode bridge rectifier with a d.c. chopper to adjusts
d.c. bus voltage to motor requirements. The typical output voltage and
current waveforms of VVI inverter are shown in Fig.15.19.
The output frequency in the VV I inverter is controlled by switching
transistors or thyristors in six steps as shown in Fig.15.20(a), whereas the
VVI inverters control voltage in a separate section from the frequency
generation output.

Fig.15.18 VVI – Variable Voltage Inverter.

(a)

(b)

Fig. 15.19 VVI-Variable Voltage Inverter : (a) Phase voltage waveform,

(b) Motor line current waveform.
The VVI inverter produces nearly sine current waveform as depicted in
Fig.15.19(b). It is considered as the simplest adjustable frequency drive
and most economical; however, it has the poorest output voltage
waveform. It requires the most filtering to the inverter. The ranges
available are typically up to 370 kW or 500 hp.
(2) The current source inverter (CSI) receives d.c. power from an
adjustable current source and adjusts the frequency and current. AC
current transformers are used to adjust the controlled rectifier. Input
converter is similar to the VVI drive. A current regulator presets d.c. bus
current. The inverter delivers six step current frequency pulses, which the
voltage waveform follows. Switches in the inverter can be transistors,
SCR thyristors or gate turnoff thyristors (GTOs). The schematic diagram
of typical current source inverter drive is shown in Fig.15.20. The output
voltage and current waveforms of the CSI inverter are shown in
Fig.15.21.
Features of CSI inverter drives
Because it is difficult to control the motor by current only, the CSI
requires a large filter inductor and complex regulator. The capacitor in the
inverter must match to motor size, and the voltage exhibits commutation
spikes when the thyristors fire. The CSI drives are short circuit proof
because of a constant circuit with the motor. Also they are not suitable for
parallel motor operation, however, power is returned to the supply easily
during braking.
The CSI drive‟s main advantage is in its ability to control current and,
therefore, control torque. This applies in variable torque applications.
CSI-type drives have a higher kW range than VVI and PWM (typically up
to 3750 kW).

Fig.15.20 Schematic diagram of typical current source inverter drive.

 Fig.15.21 CSI drive – motor voltage and current waveforms.
(3) The pulse width modulated (PWM) inverter, this is the most comm-
only used type of inverters in practice. It receives d.c. power from a fixed
voltage source and adjusts the frequency and voltage within the switches
of the inverter itself. Block diagram for a typical PWM drive is shown in
Fig.15.22.

Fig.15.22 Block diagram for a typical PWM drive.

Features of PWM inverter drives:

With PWM inverter drive, motors run smoothly at high and low speed
(no cogging); however, they are current limited. PWM drives can run
multiple parallel motors with acceleration rate matched to total motor
load. At low speeds, PWM drives may require a voltage boost to generate
required torque. However, PWM is the most costly of the three main a.c.
VSD (Variable Speed Drives) types.
The PWM drive‟s main advantage is it requires less filtering to
produce nearly sinusoidal waveforms for both the voltage and current
(PWM types cause the least harmonic noise). The range of PWM
inverters is typically up to 2250 kW. The output voltage and current
waveforms of the PWM inverter are shown in Fig.15.23. Of the three
most common inverter systems, the pulse width modulated inverter
produces output current waveforms that have the least amount of
distortion.

Fig.15.23 PWM inverter : Voltage and current waveforms with motor

load.
(B) Induction motor control using direct AC to AC converter
AC to AC direct frequency changers used in AC drive systems are of
two types :
(i) Cycloconverter drives
(ii) Load-commutated inverters (LCIs) drives
These are used only for large motor speed control applications (nearly
1000kW and above). Both can be used with induction or synchronous
motors. The cycloconverters provide variable frequency variable voltage
supply using large number of power switching devices. They are mainly
used in low frequency applications such as steel rolling mill end tables,
cement mill furnaces, mine hoists and ship propulsion drives. These
drives are also called gearless drives since low speed operation is obtained
without a reduction gear thus reducing the cost compared to the conven-
tional drives.
Cycloconverters are capable of producing output voltage and current
waveforms at frequencies below the mains frequency. This fact make it
possible to manufacture large induction or synchronous motors with high-
number of poles (e.g. 18) hence, a very low-speed direct (gearless) drive
becomes practicable. An 18-pole motor, for example, will have a
synchronous speed of only 33.3 rev/min at 5 Hz, making it suitable for
mine winders, kilns, crushers, etc. These drives are called gearless drives
since low speed operation is obtained without a reduction gear.
 The main advantage of the cycloconverter is that naturally commutated
devices such as thyristors can be used instead of self-commutating
devices, which means that the cost of each device is lower and higher
powers can be achieved. The cycloconverters can have different combin-
ation of input and output phase numbers, but in practice the three-phase to
three-phase version is used for drive of rating 1 MW and above.
A full-wave cycloconverter drive configuration with two three-phase
thyristor bridges per motor phase is shown in Fig.15.24. The output of
a poly-phase controlled rectifier is approximately Vd = Vdo cos α, where
Vdo is the output of the rectifier with zero firing angle, and α is the delay
angle. When α ˃ 90˚ the mean output is negative but the output current
cannot reverse so that the converter is then returning power from the load
back to the supply. A reverse connected converter, bridge – B is used for
the reverse half cycle of load current. On an induction motor the power
factor presented to the frequency changer is variable so that the change-
over of converter circuits cannot be predicted, a current transformer is
used to sense the current zero and inhibit the unwanted firing pulses. The
waveforms are generated in a converter which produces frequency propo-
rtional to control voltage and also imparts the required amplitude /
frequency characteristic on the outputs.

Fig.15. 24 Cycloconvertor drive circuit for a three-phase induction motor.

 The voltage and current waveforms produced by direct ac-to-ac
conversion systems approximate to pure sine wave due to the large
number of thyristors used to synthesis the output voltage and current
waveforms. Voltage and current waveforms for squirrel cage medium
power motor driven by cycloconverter drive are shown in Fig.15.25. This
type of drive has limitation that waveforms become distorted above 40%
of input frequency (i.e., 20 Hz from 50 Hz supply). However, it has an
advantage that high power factor is obtained when used with synchronous
motors.

Fig.15.25 Output voltage and current waveforms of a typical cycloconverter.

15.4.2 Speed Control from Rotor Side

(1) Speed control by changing rotor-circuit resistance

It has been shown previously that the slip of an induction motor
equals the ratio of rotor copper loss to rotor input. Therefore, changing
total resistance of the rotor circuit can change the slip. This may achieved
by inserting a rheostat in the rotor circuit as shown in Fig.15.26(a). This
method is only possible for wound rotor applications, and not be possible
for squirrel-cage rotor, but with a cost of reduced motor efficiency. The
changing total resistance of the rotor circuit can change the speed can also
be proved as follows:
The equation of torque for three-phase induction motor is given
previously in Eq.(15.17) as
 (a) (b)

Fig.15.26 Three-phase induction motor speed control by changing rotor-

circuit resistance method.
In general, the three-phase induction motor operates in low slip region,
hence the term (sX2)2 becomes very small as compared to , so it can be
neglected. Also if we consider that E2 is constant, then the equation of
torque may be written as,

.
It is clear that from the above equation, the torque is inversely
proportional to the rotor resistance. Hence, if the rotor resistance
R2 increases, torque decreases, but to supply the same load, torque must
remain constant. So, if the slip is increased this will cause further
reduction in rotor speed. Thus by adding additional resistance in rotor
circuit the speed of three-phase induction motor can be decreased. The
main advantage of this method is that with addition of external resistance
starting torque increases. However, this method of speed control of three-
phase induction motor suffers from some disadvantages:

(a) This method can only reduce the speed below the maximum value
correspond to zero external resistance, hence, the speed above the
normal value is not possible. Obviously the method is charact-
erised by low efficiency due to high waste of energy. For example,
to reduce the speed to 50% of its normal value, one has to
dissipate 50% of the power absorbed from the source in the
added resistor. The rheostat, which can dissipate this high energy
with normal temperature rise, is costly.
 (b) Another objection against this method is the departure of the
torque-speed characteristic from its original shape of small slop to
a new characteristic of considerable slop and the speed regulation
is degraded. The slop is dependent on the value of the added
resistance as shown in Fig.15.26(b).
Rotor-circuit resistance variation using choppers

The three-phase resistor shown in Fig.15.26 (a) may be replaced by a

single resistor and d.c. chopper. The slip power from the rotor is
converted to d.c current by rectification. The average resistance across the
rotor slip rings will vary from 0 to R depending on the rate of switching of
the rapidly pulsed thyristor. There is need only for one main thyristor and
an auxiliary thyristor for turn-off. The fact that there is only one resistance
is another advantage and this also provides perfect circuit balancing
between the three phases. Schematic diagram of the method is shown in
Fig.15.27.

Fig.15.27 Speed control by varying rotor resistance using d.c. chopper.

The external resistances Rex = 0 during chopper conduction (γ = 0) , where

γ is the chopper duty cycle and Rex = R during chopper extinction with
variation. Therfore,

Disadvantages: high losses in the commutating circuit at high chopping

frequency. At high motor speeds E2 is low and may be insufficient to
provide commutating voltage. So small range of speed can be achieved.
Example 15.4

A 75 kW, 4-pole, 440 V, 50 Hz, star-connected, three-phase induction

motor has the following parameters per-phase referred to the stator side:
R1 = 0.1 Ω, R2 = 0.083 Ω, X1+ X2 = 1.83 Ω, aeff = Np / Ns = 2.5
If the rotor is star connected, determine the external resistance inserted in
series with the rotor winding per phase such that the motor develops an
output shaft torque of 150 Nm at a speed of 1250 rpm.

Solution
The synchronous speed of the motor is

From Eq.(15.2) , the slip is

The approximate equivalent circuit of the motor referred to the stator side
is shown in Fig.15.28.

Fig. 15.28 Approximate equivalent circuit of the motor.

Let Rext be the additional resistance inserted in each rotor phase at

s = 0.167 such that the new rotor resistance becomes Rx , hence from the
torque equation Eq.(15.17),
 √
* +

This leads to the following quadratic equation

From which we get,

Neglecting the smaller value, hence

, referred to the stator

This resistance referred to the rotor side as

Example 15.5

A 4-pole, 3 hp, 415 V, 50 Hz, star-connected, three-phase induction motor

has the following parameters per-phase referred to the stator side:

R1 = R2 = 0.80 Ω, X1 = X2 = 3.5 Ω , aeff = Np / Ns = 2.5

Friction and windage loss = 170 W

(a) Calculate the slip at full load.

(b) If the rotor is star connected, determine the external resistance
inserted in series with the rotor winding per-phase such that the slip
would increase to four time the value obtained in (a) above with the
full load torque remains constant.

Solution
(a) From Eq.(15.9) , the mechanical power is
 [( ) ]

√
[( ) ]

Simplifying the above equation yields

Solving thequadratic equation above ;

The lower value of s can be obtain as :

(b) From the torque equation (15.14), for the torque to have a fixed value,
all other parameters of the equation must be constant. However, if the slip
becomes four times, the quantity must unchanged .i.e.

where , hence 3.2 Ω , therefore,

the extra resistance required is

(2) Injecting slip frequency emf into rotor side

Induction motor drives with full-power control on the stator side are
widely used in industrial applications. Although either a cage-type or
wound-rotor machine can be used in the drive, the former is always
preferred because a wound-rotor machine is heavier, more expensive,
has higher rotor inertia, a higher speed limitation, and maintenance and
reliability problems due to brushes and slip rings. When the speed
control of three-phase induction motor is done by adding resistance in
rotor circuit, some part of power called, the slip power is lost as I2R
losses. Therefore, the efficiency of the motor is reduced by this method of
speed control. This slip power loss can be recovered and supplied back in
order to improve the overall efficiency of motor and this scheme of
recovering the power is called slip power recovery scheme. This is done
by replacing the d.c. chopper and resistor R in Fig.15.27 by a three-phase
bridge converter as shown in Fig.15.29.
The converter operates in inversion mode with firing angles
thereby returning energy to the source. The variation of
the triggering angle α results in variation of speed, hence speed control is
achieved by this technique. Therefore, one feature of wound rotor
machine is that the slip power becomes easily available from the slip
rings, which can be electronically controlled to control speed of the
motor.
The two well known types of converter use the slip energy recovery
technique are:
1. Static Kramer drive: only allows operation at sub-synchronous
speed.
2. Static Scherbius Drive: allows operation above and below
synchronous speed.

Fig.15.29 Slip power recovery (Static Kramer drive).

Static Kramer drive

A static Kramer drive is a method to obtain an injected voltage that

is in phase with the rotor current. The voltage at the slip rings is forced
to be in phase with the rotor currents by the diode rectifier. The
magnitude of the slip ring voltage is set by the d.c. link voltage, which
is in turn set by the inverter connected back to the a.c. supply. The
schematic diagram of the converter is depicted in Fig.15.29. The static
Kramer drive is, therefore, a slip power-controlled drive that permits
only a sub-synchronous range of speed control through a converter
cascade using static power semiconductor devices. It is different from
the old original Kramer drive, where rotating machines were used for
slip energy recovery.
The static Kramer drive has been very popular in large power pump
and fan-type drives, where the range of speed control is limited near, but
below the synchronous speed. The drive system is very efficient and the
converter power rating is low because it has to handle only the slip
power. The additional advantages are that the drive system has d.c.
machine-like characteristics and the control is very simple. These
advantages largely offset the disadvantages of the wound-rotor induction
machine.

Static Scherbius drive

Another technique that employs the principle of slip power returns to
the supply is kown as static Scherbius drive shown in Fig.15.30. In this
system the bridge rectifier in Fig.15.26 is replaced by cycloconverter (or
by three-phase duel converter).

Fig.15.30 Static Scherbius drive.

For limited-range speed control applications, where the slip power is
only a fraction of the total power rating of the machine, Kramer and
Schrebius drives (slip-power recovery drives) have been used in the
following applications:
 Large-capacity pumps and fan drives
 Variable-speed wind energy systems
 Shipboard VSCF (variable-speed/constant-frequency) systems
  Variable-speed hydro pumps/generators
 Utility system flywheel energy storage systems

Simplified analysis of three-phase induction motor

with injected secondary voltage
In slip energy recovery, a voltage is applied to the slip ring terminals
of a wound rotor induction motor, in phase with the rotor current. Such
an external injected voltage must operate at slip frequency for all motor
speeds. Using the equivalent circuit of induction motor, the injected
voltage ViR to the rotor is shown in Fig.15.31. The magnitude of the
secondary current is given by

Fig.15.31 Per-phase equivalent circuit of a three-phase induction motor

with injected secondary voltage.
The injected voltage can be referred to the stator is

In order to simplify the analysis, assume that the magnetising

reactance can be moved to the terminals of the equivalent circuit
resulting in the approximate circuit referred to the stator side as shown
in Fig.15.32. If the injected voltage is in phase with the rotor current,
then the voltages in the equivalent circuit may be written as
Fig.15.32 Approximated equivalent circuit of induction motor with
injected voltage referred to stator side.

By re-arranging Eq.(15.41), the slip may be found as

Power and Torque

The air gap power of the machine may be written as

Breaking this equation into parts, it can be seen that the air gap power is
the sum of resistive losses, power recovered through the slip rings and
the mechanical power.

Using the expression for air gap power, the torque may be written as

Now, substituting the slip expression into the torque expression gives
the result that torque is only a function of rotor current, not slip or
injected voltage:

⁄
[ ]

The expression above means that for a given torque, the rotor current
will always be the same, independent of speed.
No-Load Condition

Consider again the expression for slip given in Eq.(15.42), if the torque
is zero, then the rotor current will also be zero and at zero torque,
therefore the slip is given by

Efficiency

Since some of the power supplied to the motor is recovered from the
rotor circuit, the efficiency cannot be calculated as simply output power
over input power. Instead, in slip energy recovery drive the efficiency is

(3) Cascade control method

In this method of speed control of three-phase induction motor,
two motors are required one of them should be a wound rotor type. These
two motors are connected on common shaft and hence called cascaded
motor as shown in Fig.15.33.

Fig.15.33 Cascade connections of induction motors for speed control.

One motor is the called the main motor and another motor is called the
auxiliary motor. The three-phase supply is given to the stator of the main
motor while the auxiliary motor is derived at a slip frequency from the
slip ring of main motor.
Example 15.6
Two three-phase induction motors are to be speed control by cumulative
cascade arrangement as shown in Fig.15.32. The main motor has four
poles whereas the auxiliary motor has six poles. The supply voltage is
400 V, 50 Hz for the main motor while the frequency in the rotor of the
auxiliary motor is 1.0 Hz. Calculate the slip of each motor and the
combined speed of the whole set.
Solution
Let fRm = Rotor frequency of the main motor
fRa = Rotor frequency of the auxiliary motor
Pm = Number of poles of the main motor
Pa = Number of poles of the auxiliary motor

The synchronous speed of the set is

The slip of the set is

Where n = speed of the set , which can be evaluated as

The synchronous speed of main motor is

The slip of the main motor is

The synchronous speed of auxiliary motor is

But

The slip of the auxiliary motor is