# Motional EMF of DC Motor

When the armature is stationary, no motional e.m.f. is induced in it. But when the rotor turns, the armature conductors cut the radial magnetic flux and an e.m.f. is induced in them. As far as each individual coil on the armature is concerned, an alternating e.m.f. will be induced in it when the rotor rotates.

For the coil **ab** in Figure 1, for example, side **a **will be moving upward through the flux if the rotation is clockwise, and an e.m.f. directed out-of-the-plane of the paper will be generated.

At the same time the ‘return’ side of the coil b will be moving downwards, so the same magnitude of e.m.f. will be generated, but directed into the paper. The resultant e.m.f. in the coil will therefore be twice that in the coil-side, and this e.m.f. will remain constant for almost half a revolution, during which time the coil sides are cutting a constant flux density.

For the comparatively short time when the coil is not cutting any flux, the e.m.f. will be zero and then the coil will begin to cut through the flux again, but now each side is under the other pole, so the e.m.f. is in the opposite direction.

The resultant e.m.f. waveform in each coil is therefore a rectangular alternating wave, with magnitude and frequency proportional to the speed of rotation. The coils on the rotor are connected in series, so if we were to look at the e.m.f. across any given pair of diametrically opposite commutator segments, we would see a large alternating e.m.f. (We would have to station ourselves on the rotor to do this, or else make sliding contacts using slip-rings.)

The fact that the induced voltage in the rotor is alternating may come as a surprise, since we are talking about a d.c. motor rather than an a.c. motor. But any worries we may have should be dispelled when we ask what we will see by way of induced e.m.f. when we ‘look in’ at the brushes. We will see that the brushes and commutator effect a remarkable transformation, bringing us back into the reassuring world of d.c.

The first point to note is that the brushes are stationary. This means that although a particular segment under each brush is continually being replaced by its neighbour, the circuit lying between the two brushes always consists of the same number of coils, with the same orientation with respect to the poles. As a result the e.m.f. at the brushes is direct (i.e. constant), rather than alternating.

The magnitude of the e.m.f. depends on the position of the brushes around the commutator, but they are invariably placed at the point where they continually ‘see’ the peak value of the alternating e.m.f. induced in the armature.

In effect, the commutator and brushes can be regarded as mechanical rectifier, which converts the alternating e.m.f. in the rotating reference frame to a direct e.m.f. in the stationary reference frame. It is a remarkably clever and effective device, its only real drawback being that it is a mechanical system, and therefore subject to wear and tear.

We saw earlier that to obtain smooth torque it was necessary to have a large number of coils and commutator segments, and we find that much the same considerations apply to the smoothness of the generated e.m.f.

If there are only a few armature coils, the e.m.f. will have a noticeable ripple superimposed on the mean d.c. level. The higher we make the number of coils, the smaller the ripple, and the better the d.c. we produce.

The small ripple we inevitably get with a finite number of segments is seldom any problem with motors used in drives, but can sometimes give rise to difficulties when a d.c. machine is used to provide a speed feedback signal in a closed-loop system.

When a conductor of length **l **moves at velocity **v** through a flux density B, the motional e.m.f. induced is given by

e = Blv ………(equation 1)

In the complete machine we have many series-connected conductors; the linear velocity (v) of the primitive machine is replaced by the tangential velocity of the rotor conductors, which is clearly proportional to the speed of rotation (n); and the average flux density cut by each conductor (B) is directly related to the total flux (F).

If we roll together the other influential factors (number of conductors, radius, active length of rotor) into a single constant (K_{E}), it follows that the magnitude of the resultant e.m.f. (E) which is generated at the brushes is given by

E = K_{E}ɸn …….(equation 2)

This equation reminds us of the key role of the flux, in that until we switch on the field no voltage will be generated, no matter how fast the rotor turns. Once the field is energised, the generated voltage is directly proportional to the speed of rotation, so if we reverse the direction of rotation, we will also reverse the polarity of the generated e.m.f.

We should also remember that the e.m.f. depends only on the flux and the speed, and is the same regardless of whether the rotation is provided by some external source (i.e. when the machine is being driven as a generator) or when the rotation is produced by the machine itself (i.e. when it is acting as a motor).

It has already been mentioned that the flux is usually constant at its full value, in which case equations (1) and (2) can be written in the form

T = k_{t}*I* ……….(equation 3)

E = k_{e}ω ……..(equation 4)

Where k_{t} is the motor torque constant, k_{e} is the e.m.f. constant, and ω is the angular speed in rad/s.

In the SI system (which succeeded the MKS (meter, kilogram, second) system, the units for k_{t} are the units of torque (newton-metre) divided by the unit of current (ampere), i.e. Nm/A; and the units of k_{e} units are volts/rad/s. (Note, however, that k_{e} is more often given in volts/1000 rev/min.)

It is not at all clear that the units for the torque constant (Nm/A) and the e.m.f. constant (V/rad/s), which on the face of it measure very different physical phenomena, are in fact the same, i.e. 1 Nm/A = 1 V/rad/s.

Some readers will be content simply to accept it, others may be puzzled, a few may even find it obvious. Those who are surprised and puzzled may feel more comfortable by progressively replacing one set of units by their equivalent, to lead us in the direction of the other, e.g.

This still leaves us to ponder what happened to the ‘radians’ in k_{e}, but at least the underlying unity is demonstrated, and after all a radian is a dimensionless quantity.

Delving deeper, we note that 1 volt x 1 second = 1 weber, the unit of magnetic flux. This is hardly surprising because the production of torque and the generation of motional e.m.f. are both brought about by the catalytic action of the magnetic flux.

Returning to more pragmatic issues, we have now discovered the extremely convenient fact that in SI units, the torque and e.m.f. constants are equal, i.e. k_{t} = k_{e }= k.

The torque and e.m.f. equations can thus be further simplified as

T = k*I* ……..(equation 5)

E = kω …..(equation 6)

We will make use of these two delightfully simple equations time and again in the subsequent discussion. Together with the armature voltage equation, they allow us to predict all aspects of behaviour of a d.c. motor. There can be few such versatile machines for which the fundamentals can be expressed so simply.

Though attention has been focused on the motional e.m.f. in the conductors, we must not overlook the fact that motional e.m.f.s are also induced in the body of the rotor.

If we consider a rotor tooth, for example, it should be clear that it will have an alternating e.m.f. induced in it as the rotor turns, in just the same way as the e.m.f. induced in the adjacent conductor.

In the machine shown in Figure 2, for example, when the e.m.f. in a tooth under a N pole is positive, the e.m.f in the diametrically opposite tooth (under a S pole) will be negative.

Given that the rotor steel conducts electricity, these e.m.f.s will tend to set up circulating currents in the body of the rotor, so to prevent this happening, the rotor is made not from a solid piece but from thin steel laminations (typically less than 1 mm thick), which have an insulated coating to prevent the flow of unwanted currents. If the rotor was not laminated the induced current would not only produce large quantities of waste heat, but also exert a substantial braking torque.

## Equivalent Circuit of DC Motor

The equivalent circuit of d.c. motor is shown in Figure 3. The voltage **V** is the voltage applied to the armature terminals (i.e. across the brushes), and **E** is the internally developed motional e.m.f. The resistance and inductance of the complete armature are represented by **R **and **L **in Figure 3.

The sign convention adopted is the usual one when the machine is operating as a motor. Under motoring conditions, the motional e.m.f. **E** always opposes the applied voltage **V**, and for this reason it is referred to as ‘back e.m.f.’ For current to be forced into the motor, **V** must be greater than **E**, the armature circuit voltage equation being given by

V = E + IR + L (dI/dt) ……(equation 7)

The last term in equation (7) represents the inductive volt-drop due to the armature self-inductance. This voltage is proportional to the rate of change of current, so under steady-state conditions (when the current is constant), the term will be zero and can be ignored.

We will see later that the armature inductance has an unwelcome effect under transient conditions, but is also very beneficial in smoothing the current waveform when the motor is supplied by a controlled rectifier.