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ENGNG 2024 Electrical Engineering

E Levi, 2002 2

force (not shown in Fig. 1) and the equilibrium is established when the conductor travels at a

constant speed. Under this condition the electromagnetic force and the mechanical force are

mutually equal, but act in the opposite directions.

Consider next Fig. 2, where the same conductor is placed in the same flux density.

However, the conductor is now not connected to the electric source; instead, the electric

circuit is closed by using, say, an external resistance. The conductor is now dragged throughthe flux density using mechanical force at certain speed and this is the origin of the movementin this case. The sequence of events now reverses. An electromotive force, given with (2), is

at first induced in the conductor. Since the circuit is closed, a current starts flowing.Interaction of the current and the flux density causes creation of the electromagnetic force.

This force again acts in the opposite direction to the mechanical force and the equilibrium is

established when the two forces are equal but act in the opposite direction. Note that in this

case the source of motion is the supplied mechanical energy. The mechanical energy is now

converted into electrical energy and the process is called generation.

B

I

Fe

v

Fig. 1 Illustration of motoring.

B

I

Fm

v

Fe

Fig. 2 Illustration of generation.

It is important to note here that the process of electromechanical energy conversion is

reversible. This means that either electric energy can be converted into mechanical energy, ormechanical energy can be converted into electric energy, by means of the same physical

assembly. Note as well that both the expression for electromagnetic force acting on a

conductor and the expression for induced electromotive force due to relative movement of

conductor with respect to flux density, which are vectorial, reduce to very simple expressions

due to the relative position of flux density vector, conductor and speed of motion. This is

exactly the situation that is encountered in electric machines. Therefore equations (1) and (2),which contain scalar and vectorial multiplications, reduce to a very simple form of Fe = IlB

and e = vBl.Nothing changes in principle when rotational movement is under consideration instead

of the linear movement. Table I gives the analogy between the linear and the rotational

movement. Creation of torque in the case of rotational movement is illustrated in Fig. 3.

Table I Analogy between linear and rotational movement.

Speed Source of motion Road travelled Power

Linear motion Linear, v [m/s] Force, [N] Linear, s [m] F v

Rotational motion Angular, [rad/s] Torque, [Nm] Angle, [rad] T

Suppose once more that there is a certain flux density, in which a structure is placed.

This structure can rotate and is of radius r. Assume that there are two conductors placed on

the structure, 180 degrees apart, as shown in Fig. 3, and let these two conductors carry current

in designated (mutually opposite) directions. An electromagnetic force, Fe = IlB, is created oneach of the two conductors. However, one of these two forces acts to the left, while the other

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one acts to the right (due to opposite directions of the current flow in the two conductors).

Now, a torque is created on each of the two conductors, that equals the product of the force

and the radius. However, since forces act in opposite directions at opposite sides of the

structure, the torques will both act in anticlockwise direction, initiating the rotation of the

structure in anticlockwise direction. The total electromagnetic torque will in general be the

sum of all the individual torques acting on individual conductors.

Current in

Current out

B

Fe

Fe

Fig. 3 Torque creation in the rotating structure.

Every electric machine consists of ferromagnetic iron cores and windings mounted on

the iron cores, these elements being of essential importance for electromechanical conversion.An electric machine consists of a stationary element, called stator, and a rotating element

(such as the one in Fig. 3) called rotor. The winding is placed in slots of the stationary stator

and/or in slots of rotational rotor. The winding consists of an appropriate number of turns. A

turn is composed of two conductors which are placed in such a way that the induced

electromotive forces in them sum up. The current therefore flows in the opposite direction, as

illustrated in Fig. 3.

As already noted and explained, the operation of electric machines relies on Faraday's

law of electromagnetic induction and on Bio-Savar's law of electromagnetic force (torque).One important point to note is that the induced emf will be described with (2) only if the

current in the system is pure constant DC current. A more general expression for the inducedemf says that, if the total flux through the electric circuit is changed, an electromotive force is

induced,

( )

d

dLi

dt

diLe

dtdddLidtdiLdtdLidtdiLe

Lidtde

+=

==

==

(3)

The first term in this expression will exist only in circuits with AC currents and it is called

transformer emf. The second term is what corresponds to (2) and it is the induced emf due to

the movement of a conductor in certain flux density. It is called rotational emf. Note that,

according to (3), a rotational emf will be induced only if the inductance of an electromagnetic

structure is a function of the rotor position . This may sound awkward but will be clarifiedlater on. In deriving (3) the use was made of the correlation between the angle travelled by therotor and its speed of rotation,

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= dt (4)

that reduces for a constant speed of rotation to = t. Chain differentiation rule was appliedas well. The total flux of the winding is called flux linkage and is denoted with in (3). Itdepends on the flux seen by each conductor and on the number of turns N. Flux linkage is

= N.Electromotive force in an electric machine is induced either due to rotation of awinding in the flux density, or due to rotation of the flux density with respect to a stationary

winding. Change of flux linkage can be caused either by mechanical motion or by change of

current in time. This is reflected in (3) and will be elaborated in detail later on.

Let us further clarify the two operating regimes of electric machines, generating and

motoring. Generating is discussed first. Due to the action of the prime mover (which delivers

mechanical energy to the machines shaft) rotational part of the machine is forced to rotate(Fig. 4). Consequently, the speed of rotation is constant (n = const.) and Te = TPM. Voltage at

machine terminals and induced emf differ because of the voltage drop in the winding; for

generating induced emf is greater than terminal voltage (in the sense of rms values in AC

machines, i.e. in the sense of average values in DC machines). Note that in generationdirection of the speed of rotation coincides with the direction of the mechanical (prime

movers) torque, while the electromagnetic torque of the machine opposes motion.

During motoring (Fig. 4) created electromagnetic torque, which is a consequence of

electric energy delivered to the machine, acts as the source of motion, i.e. it causes the rotor

rotation. In this case the direction of speed and the electromagnetic torque coincide, while the

mechanical torque (that is now load torque) acts against the direction of rotation. Once morethe speed of rotation is constant (n = const.) and Te = TL. During motoring induced emf has

the opposite polarity since it balances the applied voltage. It is therefore usually called

counter-electromotive force. The terminal voltage is greater than the counter-emf in motoring.

n n

TPM

TLTe

Te

Fig. 4 Torque and speed directions in generation (left) and motoring (right).

In what follows a generalised electromechanical converter is discussed at first. The

analysis is valid for any type of electric machine; the only constraint is that there is only one

degree of freedom for mechanical motion (i.e. rotor can rotate along one axis only).

2. GENERAL MODEL OF AN ELECTRIC MACHINE

2.1 Losses and efficiency

Efficiency of an electric machine is defined in the same way as for any other device,as ratio of the output to input power

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