DC MACHINES
1.0 Basic principles of operation
DC machines, like other electromechanical energy conversion devices, have two sets of electrical windings; field and amarture windings. The field winding is on stator, and the amarture windings is on the rotor. The general arrangement of a DC machine is shown in Figure 1.
Figure 1: General arrangement of a dc machine
The magnetic field of the field winding is approximately sinusoidal. Thus, AC voltage is induced in the armature winding as the rotor turns under the magnetic fields of the stator. This induced or generated voltage is also approximately sinusoidal. Since the armature windings are distributed over armature periphery, the generated voltages of the armature turns reach their maxima at different times. The commutator and brush combination converts the AC generated voltages to DC. The commutator is located on the same shaft as the armature and rotates together with armature windings. The brushes are stationary and are in located so that commutation occurs when the coil sides are in the neutral zone; that is, the potentials of the conductor loops that leaves the brush and loop that comes in contact with the brush are the same.
The axis of the armature mmf is 900 from the axis of the field winding. Denoting the stator (field) winding and rotor (armature) winding mmfs by Fs and Fr, respectively, these mmfs are shown in Figure 2.
Figure 2: Stator and rotor mmf vector representation
The electromagnetic torque Te is produced by the interaction of the stator and rotor mmfs. Since they are in quadrature with each other, Te may be expressed as follows:
(1)
The rotor mmf Fr is a linear function of the armature current IA, and the stator mmf Fs is similarly a linear function of the field current IF. Hence, Equation (1) may be written as follows:
(2)
Where KT = torque constant
The DC induced voltage EA appearing between the brushes is a function of the field current IF and the speed of rotation ω of the machine. This generated voltage is given by
(3)
where = voltage constant
If the losses of the DC machine are neglected, from the energy conservation principle, the electrical power is equal to the mechanical power:
(4)
where
EAIA = electrical power
ωTm = mechanical power
At steady state, the mechanical torque Tm is equal to the electromagnetic torque, Te
2.0 Generation of unidirectional voltage
A DC generator with two poles in the stator and a single conducting loop on the rotor is shown in Figure 3.
Figure 3: A two pole DC generator
As the rotor is rotated at an angular velocity, ω, the armature flux linkages change and a voltage eaa’ is induced between terminals a and a’. The expression for the voltage induced is given by Faraday’s Law as
(5)
This induced voltage is plotted against time in Figure 4b, where at time t = 0 the conductors a and a’ are as shown in Figure 3. The plot of the flux Φaa’ and the plot of the rectified voltage (across brushes B1 and B2) are given in Figures 4a and 4c, respectively.
It is seen from Figure 4 that although the flux and the coil voltage are both sinusoidal functions of time, the voltage across the brushes is a unidirectional voltage.
Figure 4: a) Flux linkage of coil aa’; b) induced voltage; c) rectified voltage
Suppose that a second windings bb’ is placed on the armature displaced from the aa’ winding by 900. Two new commutator segments are also added as illustrated in Figure 5
Figure 5: A two pole, two coil DC generator
The induced voltages eaa’ and ebb’ across terminals aa’ and bb’, respectively, and the voltage across the brushes, eB1B2, are plotted in Figure 6.
Figure 6: a) Voltage of coil aa’; b) voltage coil bb’ c) voltage across armature terminals
(between brushes)
It may be seen from Figure 6c that the armature voltage is closer to a DC voltage for the generator of Figure 4. Therefore, it may be concluded that by increasing the number of conducting loops on the armature and correspondingly increasing the number of commutator segments, the quality of the armature terminal voltage is greatly improved. In the limit, a pure DC voltage between brushes is obtained as shown in Figure 7.
Figure 7: DC terminal voltage
The generated voltage of a DC machine having p poles and Z conductors on the armature with a parallel path between brushes is given in equation 6.
(6)
where K = pZ /(2πa) = machine constant
By substituting equation (6) into (4), the mechanical torque, which also equal to the electromagnetic torque, is found as follows:
(7)
In the case of a generator, Tm is the input mechanical torque, which is converted to electrical power. For the motor, Te is the developed electromagnetic torque, which is used to drive the mechanical load.
Example 1:
A six pole DC machine has flux per pole of 30mWb. The armature has 536 conductors connected as a lap winding in which the number of parallel paths a is equal to the number of poles p. The DC machine runs at 1500 rpm, and it delivers a rated armature current of 225A to a load connected to its terminals. Calculate the following:
a) Machine constant, K
b) Generated voltage EA
c) Conductor current Ic
d) Electromagnetic torque, Te
e) Power delivered, Pdev, by the armature
Example 2:
Repeat Example 1 if the armature is reconnected as a wave winding such that the rated conductor current remains the same. For a wave winding, there are two parallel paths; that is, a = 2.
3.0 DC motor
DC motors are classified according to the electrical connections of the armature winding and field windings. The operating characteristics of the specific DC machine being considered depend on the particular interconnection of the armature and field windings.
There are generally five major types of DC motors:
1. The separately excited dc motor
2. The shunt dc motor
3. The permanent magnet dc motor
4. The series dc motor
5. The compounded dc motor
3.1 The equivalent circuit of a DC motor
The equivalent circuit of a dc motor is shown in Figure 8.
Figure 8: The equivalent circuit of a dc motor
where the armature is represented by an ideal voltage source EA and a resistor RA. The brush voltage drop is represented by a small battery Vbrush opposing the direction of the current flow in the machine. The field coils, which produce the magnetic flux in the generator, are represented by inductor LF and RF. The separate resistor Radj represents an external variable resistor used to control the amount of current in the filed circuit.
The brush drop voltage is often only a very tiny fraction of the generated voltage in the motor. Therefore, in cases where it is not critical, the brush drop voltage may be left out or approximately included in the value of RA. Also, the internal resistance of the filed coils is sometimes lumped together with the variable resistor, and the total is called RF (refer to Figure 9)
Figure 9: A simplified equivalent circuit eliminating the brush voltage drop and
combining Radj with the field resistance.
The internal generated voltage in this motor is
(8)
Therefore, EA is directly proportional to the flux in the motor and speed of the motor.
Refer to Figure 10, the field current in dc motor produces a field magnetomotive force given by F = NF IF. This magnetomotive force produces a flux in the motor in accordance with its magnetization curve.
Figure 10: The magnetization curve of a ferromagnetic material (Φ versus F )
Since the field current is directly proportional to the magnetomotive force and since EA is directly proportional to the flux, it is customary to present the magnetization curve as a plot EA versus field current for a given speed ω0 (refer Figure 11)
Figure 11: The magnetization curve of a dc machine expresses as a plot of EA versus IF,
for a fixed speed ω0
The induced torque developed by the motor is
(9)
3.2 Separately excited and shunt DC motors.
Figure 12 and 13 shows the equivalent circuit of separately excited dc motor and shunt motor.
Figure 12: The equivalent circuit of separately excited dc motor
Figure 13: The equivalent circuit of a shunt dc motor
Refer to Figure 12,
(10)
(11)
(12)
Refer to Figure 13,
(13)
(14)
(15)
3.2.1 Speed-Torque characteristics
Consider the DC shunt motor whose equivalent circuit is shown in Figure 13. From Kirchhoff’s voltage law,
(16)
Substituting the expression for induced voltage given by equation (8) into equation (16) and solving ω yields
(17)
It may be observed that loss of field excitation results in overspeeding for a shunt motor. Thus, care should be taken to prevent the field circuit from getting open.
From equation (9), the armature current may be expressed as follows:
(18)
Substituting equation (18) into equation (17) yields the speed – torque equation of a DC shunt motor.
(19)
This equation is just a straight line with a negative slope. The resulting torque – speed characteristic of a DC shunt motor is shown in Figure 14.
Figure 14: Torque - speed characteristic of a shunt or separately excited DC motor with compensating windings to eliminate armature reaction.
In an actual machine, however, as the load increases, the flux is reduced because of armature reaction. Since the denominator terms decrease, there is less reduction in speed and speed regulation is improved somewhat. Figure 15 show the torque speed characteristic of the shunt motor with armature reaction.
Figure 15: Torque - speed characteristic of the shunt motor with armature reaction.
3.2.2 Speed control of Shunt DC motors
i) Changing the field resistance
Assume that the field resistor increases and observe the response. If the field resistance increases, then the field current decreases (IF = VT/RF ), and as the field current decreases, the flux Φ decreases with it. A decrease in flux causes an instantaneous decrease in the internal generated voltage, EA = KΦ ω, which causes large increase in machine’s armature currents, since
The induced torque in a motor is given by. Since the flux Φ in shunt DC motor decreases while the current IA increases. The increase in current predominates over the decrease in flux, and the induced torque rises:
Since, the motor speed up.
However, as the motor speeds up, the internal generated voltage EA rises, causing IA to fall. As IA falls, the induced torque τind falls too, and finally τind again equals τload at a higher steady state speed than originally.
Summary:
1. Increasing RF causes IF () to decrease.
2. Deceasing IF decreases Φ.
3. Decreasing Φ lowers EA ().
4. Decreasing EA increases IA ().
5. Increasing IA increases τind (), with the change in IA dominant over the change in flux.
6. Increasing τind makes, and the speed ω increases.
7. Increasing to increases EA = KΦ ω again.
8. Increasing EA decreases IA.
9. Decreasing IA decreases τind until at a higher speed ω
The effect of increasing the field resistance on the output characteristic of a shunt motor is shown in Figure 16. Notice that as the flux in the machine decreases, the no load speed of the motor increases, while the slope of the torque speed curve becomes steeper. Naturally, decreasing RF would reverse the whole process, and the speed of the would drop.
Figure 16: The effect of field resistance speed control on a shunt motor’s torque speed
characteristic: over the motor’s normal operating range
ii) Changing the armature voltage
Changing the voltage applied to the armature of the motor without changing the voltage applied to the field. A connection is shown in Figure 17. In effect, the motor must be separately excited to use armature voltage control.
If the voltage VA is increased, then the armature current in the motor must rise [IA = (VA – EA)/RA]. As IA increases, the induced torque τind = KΦIA increases, making, and the speed ω of the motor increases. But as the speed ω increases, the internal generated voltage EA (= KΦω ) increases, causing the armature current to decrease. This decrease in IA decreases the induced torque, causing torque induced to equal torque load at a higher rotational speed ω.
Figure 17: Armature voltage control of a shunt or separately excited DC motor
Summary:
1. An increase in VA increases IA [= (VA – EA)/RA]
2. Increasing IA increases τind (= KΦIA )
3. Increasing τind makes increasing ω.
4. Increasing ω increases EA (=KΦω )
5. Increasing EA decreases IA [ = (VA – EA)/RA]
6. Decreasing IA decreases τind until at a higher ω.
The effect of an increase in VA on the torque speed characteristic of separately excited is shown in Figure 18. Notice that the no load speed of the motor is shifted by this method of speed control, but the slope of the curve remains constant.
Figure 18: The effect of armature voltage speed control on a shunt motor’s torque speed characteristic.