# Electrical DC Motor

## Working Princi**ple **

A machine that converts DC electrical power into mechanical power are known as an electrical DC motors. Its operation is based on the principle that when a current carrying conductor is placed in a magnetic field, the conductor experiences a mechanical force. The direction of this force is given by Fleming’s left hand rule and its magnitude is given by:

F = B×I×*l* newtons

Basically, there is no constructional difference between an electrical DC motors and DC generators. The same machine can be run as a DC generator or DC motor.

# Working Of DC Motor

Consider a multi-polar DC motor as shown below. When the motor terminals are connected to an external DC supply:

- the field magnet are excited developing alternate N and S poles.
- the armature conductors carries current. All conductors under N pole carries current in one direction, while those under S pole carries current in opposite direction.

Let us suppose conductors under N pole carries current into the plane of the paper and those under S pole carries current out of the plane of the paper. Since, each armature conductor is carrying current and is placed in magnetic field, a mechanical force acts on it. Applying Fleming’s left hand rule, that shows that the force on each conductor is tending to rotate the armature in anticlockwise direction. All these forces adds together to produce a driving torque which sets the armature rotating. When the conductor moves from one side of the brush to other, the current in that conductor is reversed and at the same time it comes under the influence of next pole which is of opposite polarity. Consequently, the force on the conductor remains the same.

# Back Or Counter E.M.F

When the armature of a DC motor rotates under the influence of the driving torque, the armature conductor move through the magnetic field and hence e.m.f is induced in them as in a generator. The induced e.m.f acts as in opposite direction to the applied voltage V and is known as back or counter e.m.f E_{b}. The back e.m.f E_{b} is always less than the applied voltage V, although this difference is small when the motor is running under normal conditions.

Consider a motor shown below,when the DC voltage is applied across the terminals, the field magnets are excited and armature conductors are supplied with current. Therefore, driving torque acts on an armature which begins to rotate. As the armature rotates back e.m.f E_{b} is induced which opposes the applied voltage V. The applied voltage has to force current through the armature against the back e.m.f. The electric work done is overcoming and causing the current to flow against is converted in to mechanical energy develop in the armature. It follows, therefore, that energy conversion in a DC motor is only possible due to the production of back e.m.f E_{b}.

Net voltage across armature circuit = V – E_{b}

If R_{a} is the armature circuit resistance, then,

I_{a} = (V – E_{b})/R_{a}

Since, V and R_{a} are usually fixed, the value of E_{b} will determine the current drawn by the motor. If the speed of the motor is high, Then back e.m.f E_{b} (= PΦZN/60A) is large and hence the motor will draw less armature current.

# Significance Of Back E.M.F

The presence of back e.m.f makes a DC motor a self regulating machine i.e. its make the motor to draw as much as armature current as is just sufficient to develop the torque required by the load.

Armature current, I_{a} = (V – E_{b})/R_{a}

- When the motor is running on no load, small torque is required to overcome the friction and windage losses. Therefore, the armature current I
_{a}is small and the back e.m.f. is nearly equal to the applied voltage. - If the motor is suddenly loaded, the first effect is to cause the armature to slow down. Therefore, the speed at which the armature conductors move through the field is reduced and hence the back e.m.f. E
_{b}falls. The decreased back e.m.f allows a larger current to flow through the armature and larger current means increased driving torque. Thus, the driving torque increases as the motor slows down. The motor will stop slowing down when the armature current is just sufficient to produce the increased torque required by the load. - If the load on to the motor is decreased, the driving torque is momentarily in excess of the requirement so that armature is accelerated. As the armature speed increases, the back e.m.f E
_{b}also increases and causes the armature current I_{a}to decrease. The motor will stop accelerating when the armature current is just sufficient to produce the reduced torque required by the load.

*“It follows, therefore, that back e.m.f in an electrical DC motors regulates the flow of armature current i.e. it automatically changes the armature current to meet the load requirement.”*

# Voltage Equation Of An Electrical DC Motors

Let in a DC motor

V = Applied Voltage

Eb = Back E.M.F.

Ra = Armature Resistance

Ia = Armature current

Since, back e.m.f. acts in opposition to the applied voltage V, the net voltage across the armature circuit is V-E_{b}. The armature current I_{a} is given by:

Ia = (V – E_{b})/ R_{a}

V = E_{b} +I_{a}R_{a}

This is known as voltage equation of the DC motor.

# Power Equation For An Electrical DC Motors

The power equation for an electrical DC motors can be calculated as below:

Power is given by:

P = V×I

Here, V= E_{b} +I_{a}R_{a} and Ia = (V – E_{b})/ R_{a}

So, by rearranging, we get

P = E_{b}I_{a}+I^{2}_{a}R_{a}

This is known as power equation of an electrical DC motor.

Where,

VI_{a} = electric power supplied to armature. (armature input)

E_{b}I_{a} = power developed by armature. (armature output)

I^{2}_{a}R_{a} = electric power wasted in armature. (armature Cu loss)

Thus out of armature input, a small portion is wasted as I^{2}_{a}R_{a} and the remaining portion E_{b}I_{a} is converted in to mechanical power within the armature.

# Condition For Maximum Mechanical Power

The mechanical power developed by the motor is P_{m} = E_{b}I_{a}

Since V and Ra are fixed, power developed by the motor depends up armature current. For maximum power, dP_{m}/dI_{a} should be zero.

dP_{m}/dI_{a} = V – 2I_{a}R_{a} = 0

IaRa = V/2

Now, I_{a}R_{a} = V/2

V = E_{b} + I_{a}R_{a} = E_{b} + V/2

E_{b} = V/2

Hence mechanical power developed by the motor is maximum when back e.m.f is equal to half the applied voltage.

Limitations. In practice, we never aim at achieving maximum power due to the following reasons.

- The armature current under this condition is very large-much excess of rated current of the machine.
- Half of the input power is wasted in the armature circuit. In fact, if we take into account other losses, the efficiency will be well below 50%.

# Types Of Electrical DC Motors

Like generators, there are three types of an electrical DC motors characterized by the connection of field winding in relation to the armature.

**Shunt Wound Motor** in which the field winding is connected in parallel with the armature.

**Series Wound Motor** in which the field winding is connected in series with the armature.

**Compound Wound Motor** which has two field winding, one connected in parallel with the armature and the other in series with it. There are two types of compound motor connections. When the shunt field winding is directly connected across the armature terminals, it is called **short shunt connection**. When the shunt field winding is so connected that it shunts the series combination of armature and series fields, it is called **long shunt connection**.

# Armature Torque Of An Electrical DC Motors

Torque is the turning moment of a force about an axis and is measured by the product of force (F) and radius (r) at right angle to which the force acts. i.e.

T = F×r

In an electrical DC motors, each conductor are acted upon by a circumferential force F at a distance r (the radius of the armature). Therefore, each conductor exerts a torque, tending to rotate the armature. The sum of the torque due to all armature conductors is known as gross or armature torque (T).

Let in a DC motor,

r = average radius of armature in meter

*l* = effective length of each conductor in meter

Z = total number of armature conductor

A = number of parallel paths

i = current in each conductor = I_{a}/A

B = average flux density in Wb/m^{2}

Φ = flux per pole in Weber

P = number of poles

Force on each conductor, F = B×i×*l *newtons

Torque due to one conductor = F×r newton-meter

Total armature torque, T = Z×F×r newton-meter = Z×B×i×l×r

Now i= Ia/A, B=Φ/*a* where *a *is the cross sectional area of flux path per pole at radius r. Clearly, a = 2×π×r×*l/*P*.*

T_{a} = Z×(Φ/*a*) × (I_{a}/A)×*l*×r

= Z×( (Φ/(2×π×r×*l/*P))×(Ia/A)×*l*×r ) = (ZΦI_{a}P/2πA) N-m

T_{a} = 0.159 ZΦI_{a}(P/A) N-m

Since Z, P and A are fixed for a given machine,

T_{a} is directly proportional to ΦI_{a}

Hence torque in a DC motor is directly proportional to flux per pole and armature current.

- For a
**shunt motor**, flux Φ is practically constant.

T_{a} is directly proportional to I_{a}

- For a
**series motor**, flux Φ is directly proportional to armature current I_{a}provided magnetic saturation does not take place.

T_{a} is directly proportional to I^{2}_{a}

Alternative expression for T_{a},

E_{b} = PΦZN/60A

PΦZ/A = 60E_{b}/N

We get,

Ta = 0.159 × ( 60×E_{b}/N) × I_{a}

Ta = 9.55 × E_{b}I_{a}/N N-m

# Shaft Torque (T_{sh})

The torque which is available at the motor shaft for doing useful work is known as shaft torque. The total torque (T_{a}) developed in armature of a motor is not available at the shaft as a part of it is lost in overcoming the iron and frictional losses in the motor. Therefore, shaft Torque (T_{sh}) is somewhat less than the total armature torque (T_{a}). The differences T_{a} – T_{sh} is known as lost torque.

T_{sh} = 9.55 × Output/N N-m

Lost torque = 9.55 × Iron and friction losses/N N-m

# Brake Horse Power (B.H.P)

The horse power developed by the shaft torque is known as brake horse power (B.H.P). If the motor is running at N r.p.m and the shaft torque is T_{sh} newton meter, then

= F×2πr = 2π × T_{sh} J

W.D./minute = 2π×N×T_{sh} J

W.D./second = 2π×N×T_{sh}/60 watts = 2π×N×T_{sh}/60×746 H.p

Useful Output power = 2π×N×T_{sh}/60×746 H.p

# Speed Of An Electrical DC Motors

E_{b} = V – I_{a}R_{a}

E_{b} = P×Φ×Z×N/60×A

P×Φ×Z×N/60×A = V – I_{a}R_{a}

N = (V – I_{a}R_{a})/Φ × 60×A/P×Z

N = K (V – I_{a}R_{a})/Φ where K= 60×A/P×Z

But V – I_{a}R_{a} = E_{b}

N = K×(E_{b}/Φ)

OR

N is directly proportional to (E_{b}/Φ)

Therefore, in a DC motor, speed is directly proportional to back E.M.F. E_{b} and inversely proportional to flux per pole Φ.

# Speed Relations In An Electrical DC Motors

If a DC motor has initial values of speed, flux per pole and back e.m.f as N_{1}, Φ_{1} and E_{b1} respectively and the corresponding final values are N_{2}, Φ_{2} and E_{b2}. Then,

N_{1} is directly proportional to E_{b1}/Φ_{1}

and N_{2} is directly proportional to E_{b2}/Φ_{2}

N_{1}/N_{2} = E_{b1}/E_{b2} × Φ_{2}/Φ_{1}

- For a shunt motor, flux practically remains constant so that Φ1 = Φ2

N_{1}/N_{2} = E_{b1}/E_{b2}

- For a series motor, Φ is proportional to Ia prior to saturation.

N_{1}/N_{2} = E_{b1}/E_{b2 }× Ia2/Ia1

Where, I_{a1} = initial armature current

I_{a2} = final armature current

# Speed Regulation

The speed regulation of a motor is the change in speed from full load to no load and is expressed as a percentage of the speed at full load. i.e.

= ( N_{0} – N )/N × 100

Where, N_{0} = no load speed

N = full load speed

# Losses In An Electrical DC Motors

The losses occurring in a DC motor are the same as in a DC generator. These are:

- Copper Losses
- Mechanical Losses
- Iron Losses

As in a generator, these losses causes

- An increases of temperature
- Reduction in efficiency of the DC motor.

# Efficiency of An Electrical DC Motors

Like a DC generator, the efficiency of a DC motor is the ratio of output power to the input power. i.e.

Efficiency η = Output/Input × 100 = Output/(Input+losses)× 100

As for a generator, the efficiency of a DC motor will be maximum when:

Variable Losses = Constant Losses

# Power Stages In An Electrical DC Motors

The power stages in a DC motor are represented as:

A-B = Copper losses

B-C = Iron and friction losses

Overall efficiency = η_{c} = C/A

Electrical efficiency = η_{e} = B/A

Mechanical efficiency = η_{m} = C/B

# Electrical DC Motors Characteristics

The preference of a DC motor can be judged from its characteristic curves known as motor characteristics. Following are the three important characteristics of a DC motor.

**Torque and armature current characteristic (T _{a}/I_{a})**

It is the curve between armature torque (T_{a}) and armature current I_{a} of a DC motor. It is also known as electrical characteristic of the motor.

**Speed and armature current characteristics (N/I _{a}) **

It is the curve between speed N and armature current Ia of a DC motor. It is very important characteristics as it is often the deciding factor in the selection of the motor for a particular application.

**Speed and torque characteristics (N/T _{a})**

It is the curve between speed N and armature torque T_{a} of a DC motor. It is known as mechanical characteristics.

# Characteristics Of DC Shunt Motors

The field current I_{sh} in a DC shunt motor is constant since the field winding is directly connected to the supply voltage V which is assumed to be constant. Hence, the flux in a shunt motor is approximately constant.

**T _{a}/I_{a }**

**Characteristics**

We know that in a DC motor.

T_{a} is directly proportional to Φ I_{a}

Since the motor is operating from a constant supply voltage, flux Φ is constant.

T_{a} is directly proportional to I_{a}

**N/I _{a}**

**Characteristics**

The speed N of the DC motor is given by:

N is directly proportional to I_{a}

The flux Φ and back e.m.f. in a shunt motor are almost constant under normal conditions. Therefore, speed of a shunt motor will remain constant as the armature current varies. When load is increased, E_{b} (V-I_{a}R_{a}) and Φ decrease due to the armature resistance drop and armature reaction respectively. However, E_{b} decreases slightly more than Φ so that speed of the motor decreases slightly with load.

**N/T _{a}**

**Characteristics**

The speed falls somewhat as the load torque increases.

Following two **conclusions** are drawn from the above characteristics

- There is a slight change in the speed of a shunt motor from no load to full load. Hence, it is essentially a constant speed motor.
- The starting torque is not high because T
_{a}is directly proportional to I_{a}

# Characteristics Of DC Series Motor

As we know that, the current passing through the field winding is the same as that in the armature. If the mechanical load on the motor increases, the armature current also increases. Hence, the flux in a series motor increases with the increase in armature current.

**T _{a}/I_{a }**

**Characteristics**

We know that:

T_{a} is directly proportional to Φ I_{a}

**Up to**** magnetic saturation**, Φ is directly proportional to I

_{a}so that T

_{a}is directly proportional to I

^{2}

_{a}

**After**** magnetic saturation**, Φ is constant so that T

_{a}is directly proportional to I

_{a}

Thus up to magnetic saturation, the armature torque is directly proportional to the square of armature current. If I_{a} is doubled, T_{a} is almost quadrupled.

**N/I _{a}**

**Characteristics**

The speed N of a series motor is given by:

N is directly proportional to E_{b}/Φ where Eb = V-I_{a}(R_{a}+R_{se})

When the armature current increases, the back e.m.f. decreases due to I_{a}(R_{a}+R_{se}) drop while the flux Φ increases. However, I_{a}(R_{a}+R_{se}) drop is quite small under normal conditions and may be neglected.

N is directly proportional to 1/Φ

is directly proportional to 1/I_{a} (Up to magnetic saturation)

**N/T _{a}**

**Characteristics**

Series motor develops high torque at low speed. It is because an increase in torque requires an increase in armature current, which is also the field current. The result is that flux is strengthened and hence the speed drops (N is directly proportional to 1/I_{a}).

Following **conclusions** are drawn from the above characteristics of a series motor:

- It has a high starting torque because initially T
_{a}is directly proportional to I^{2}_{a} - It is a variable speed motor i.e. automatically adjusts the speed as the load changes. Thus, if the load decreases, its speed is automatically raised.
- At no load, the armature current is very small and so is the flux. Hence, the speed rises to an excessive high value. This is dangerous for the machine which may be destroyed due to centrifugal forces set up in the rotating parts. Therefore, a series motor should never be started on no load. However, to start a series motor, mechanical load is first put and then the motor is started.

# Compound Motors

A compound motor has both series and shunt field. The shunt field is always stronger than the series field. Compound motors are of two types.

- Cumulative compound motors in which series field aids the shunt field.
- Differential compound motors in which series field opposes the shunt field.

Differential compound motors are rarely used due to their poor characteristics at heavy load.

## Cumulative Compound Motor Characteristics

**T _{a}/I_{a }**

**Characteristics**

As the load increases, the series field increases but shunt field strength remains constant. Consequently, total flux is increases and hence the armature torque (T_{a} is directly proportional to I_{a}). It may be noted the torque of a cumulative compound motor is greater than that of a shunt motor for a given armature current due to series field.

**N/I _{a}**

**Characteristics**

As we know that, as the load increases, the flux per pole also increases. Consequently, the speed (N is directly proportional to 1/Φ) of the motor as the load increases. It may be noted that as the load is added, the increased amount of the flux causes the speed to decrease more than does the speed of the shunt motor. Thus, the speed regulation of a cumulative compound is poorer than that of a shunt motor.

**N/T _{a}**

**Characteristics**

In cumulative compound motor for a given armature current, the torque of a cumulative compound motor is more than that of a shunt motor but less than that of a series motor.

**Conclusion**: A cumulative compound motor has characteristics intermediate between series and shunt motors.

- Due to the presence of shunt field, the motor is prevented from running away at no load.
- Due to the presence of series field, the starting torque is increased.

# Comparison Of Three Types Of Motors

- The speed regulation of a shunt motor is better than that of a series motor. However, speed regulation of a cumulative compound motor lies between shunt and series motors.
- For a given current, the starting torque of a series motor is more than that of a shunt motor. However, the starting torque of a cumulative compound motor lies between series and shunt motors.
- Both shunt and cumulative compound motors have definite no load speed. However, a series motor has dangerously high speed at no load.

# Applications Of Electrical DC Motors

## Shunt Motors

The characteristics of a shunt motor reveals that it is an approximately constant speed motor. It is therefore, used

- Where the speed is required to remain almost constant from no load to full load.
- Where the load has to driven at a number of speed and any of which is required to remain nearly constant.

## Series Motors

It is a variable speed motor i.e. speed is low at high torque. However, at light or no load, the motor tends to attain dangerously high speed, the motor has a high starting torque. It is therefore, used

- Where large starting torque is required.
- Where the load is subjected to heavy fluctuations and the speed is automatically required to reduce at high torques.

## Compound Motors

Differential compound motors are rarely used because of their poor torque characteristics. However, cumulative compound motors are used where a fairly constant speed is required with irregular loads or suddenly applied heavy loads.

# Speed Control Of Electrical DC Motors

The speed of an electrical DC motors can be given as:

N is directly proportional to E_{b}/Φ

N = k (V-I_{a}R)/Φ r.p.m

where R = R_{a} for shunt motor

= R_{a}+R_{se}

From equation, it can be seen that there are two methods of controlling the speed of a DC motors, namely:

- By varying the flux per pole (Φ). This is known as
.**flux control method** - By varying the resistance in an armature circuit. This is known as
.**armature control method**

## Speed Control Of DC Shunt Motors

The speed of a DC shunt motor can be controlled by flux control method or armature control method. The former method is frequently used because it is simple and inexpensive.

**Flux Control Method**

It is based on the fact that by varying the flux Φ, the motor speed (N is directly proportional to 1/Φ) can be changed and hence the name flux control method. In this method, a variable resistance known as shunt field rheostat is placed in series with shunt field winding. The shunt field rheostat reduces the shunt current I_{sh} and hence the flux Φ. Therefore, we can only raise the speed of the motor above the normal speed. Generally, this method permits to increase the speed in the ratio 3:1. Wider speed ranges tend to produce instability and poor commutation.

**Advantages**

- This is easy and convenient method.
- It is an inexpensive method since very little power is wasted in the shunt field rheostat due to relatively small value of I
_{sh}. - The speed control exercise by this method is independent of load on the machine.

**Disadvantages**

- Only speeds higher than the normal speed can be obtained since the total field circuit resistance cannot be reduced below R
_{sh}(the shunt field winding resistance) - There is a limit to the maximum speed obtainable by this method. It is because if the flux is too much weakened, commutation becomes poorer.

### Armature Control Method

This method is based on the fact that by varying the voltage available across the armature, the back e.m.f and hence the speed of the motor can be changed. This is done by inserting a variable resistance R_{c} (know as controller resistance) in series with the armature.

Due to voltage drop in the controller resistance, the back e.m.f is decreased. Since (N is directly proportional to E_{b}), the speed of the motor is reduced. The highest speed obtainable is that corresponding to R_{c}=0. i.e. normal speed. Hence, this method can only provide speeds below the normal speed.

**Disadvantages**

- A large amount of power is wasted in the controller resistance since it carries full armature current I
_{a}. - The speed varies widely with load since the speed depends upon the voltage drop in the controller resistance and hence on the armature current demanded by the load.
- The output and efficiency of the motor are reduced.
- This method results in poor speed regulation.

## Speed Control Of DC Series Motors

The speed of DC series motors can also be obtained by * flux control method* and

*.*

**armature resistance control method**### Flux Control Method

In this method, the flux produced by the series motor is varied and hence the speed. The variation of flux can be achieved in the following ways.

**Field Diverters**: In this method, a variable resistance called field diverter is connected in parallel with the series field winding. Its effect is to shunt some portion of the line current from the series field winding,thus weakening the field and increasing the speed. The lowest speed obtainable is that corresponding to zero current in the diverter. Obviously, the lowest speed obtainable is the normal speed of the motor. Consequently, this method can only provide speeds above the normal speed. The series field diverter method is often employed in traction work.

**Armature Diverters: **In order to obtain speeds below the normal speed, a variable resistance called armature diverter is connected in parallel with the armature.

The diverter shunts some of the line current, thus reducing the armature current, thus reducing the armature current. Now for a given load, if I a is decreased, the flux Φ must increase. Since, (N is directly proportional to 1/Φ) the motor speed is decreased. By adjusting the armature diverter, any speed lower than the normal speed can be obtained.

**Tapped Field Control:** In this method, the flux is reduced (and hence speed in increased) by decreasing the number of turns of the series field winding , the switch S can short circuit any )art of the field winding, thus decreasing the flux and raising the speed. With full turns of the field winding, the motor runs at the normal speed as the field turns are cut out, speeds higher than normal speed are achieved.

**Armature Resistance Control:** In this method, a variable resistance is directly connected in series with the supply to the complete motor circuit. This reduces the voltage available across the armature and hence the speed falls. By changing the value of variable resistance, any speed below the normal speed can be obtained. This is the most common method employed to control the speed of DC series motors. Although this method has poor speed regulation, this has no significance for series motor because they are used in varying speed applications. The loss of the power in series resistance for many applications of series motors is not too serious since in these applications, the control is unutilized for a large portion of the time for reducing the speed under light load conditions and is only used intermittently when the motor is carrying full load.