7.1 INTRODUCTION

An electric machine is a link between electrical and mechanical systems. The conversion of electrical to mechanical systems is based on transferring voltage and current electrical quantities to mechanical quantities as torque and angular speed and vice versa. Generators and motors can be used interchangeably. Feeding a generator from the mains will convert it to a motor, while driving a motor with a prime mover will convert it to a generator. Generators or motors can be AC or DC machines.

DC and AC machines are subject to electromechanical energy conversion, which satisfies Lenz's, Faraday's or Lorentz's fundamental laws. We present herein a simple illustration of two principles on which such machines are developed:

• conductor moving in a uniform magnetic field
• current-carrying conductor in a uniform magnetic field

7.2 CONDUCTOR MOVING IN A UNIFORM MAGNETIC FIELD

The movement of a conductor in a magnetic field gives induced voltage in a conductor in the magnetic field, as shown in Figure 7.1 [1],

(7.1)

where B is the flux density in Tesla or Weber/m², l is the conductor length (m), and V is the velocity of its movement in the magnetic field (ms⁻¹).

7.3 CURRENT-CARRYING CONDUCTOR IN A UNIFORM MAGNETIC FIELD

When a current-carrying conductor is in a magnetic field of density B Tesla, as shown in Figure 7.2 [1], the conductor experiences a force F given by

(7.2)

A tuning force is impacted on the conductor, causing it to rotate in the magnetic field; this is also due to Lorentz's law.

Of all the rotating machines defined so far, synchronous, induction, and DC machines all use these fundamental principles to operate. To generalize these two principles, machines are designed based on two major components, namely a rotating (moving) part called the rotor and a stationary part called the stator.

The windings are made of copper, which allows current to flow through or allows for induced voltage based on the two principles. In general, the armature winding used to denote rotor and field winding is also used to denote stator winding for same class of machines.

Each of the machines by principle follows the concept—either electric-power conversion to mechanical torque, referred to as motor action, or mechanical-power conversion to electrical power, known as generator action.

7.4 DC-MACHINE CONSTRUCTION AND NAMEPLATE PARAMETERS

DC machines are built with salient magnetic poles on the stator and have commutating poles between the main poles. The coils are situated in the interpolar region with contact through carbon brushes. The windings are based on loop configuration and allow for pairs of poles in the wave winding. The brush holder holds the machines against the commutator surface by springs to maintain a constant pressure and smooth running speed.

A general schematic of a DC machine is shown in Figure 7.3. It is important to note that compensating winding is in the main poles, whereas the interpole is inserted in the commutating poles, which allows for reversal of current. So, we can claim here that a DC machine can be modeled in three configurations based on the location of the shunt or series winding of the interconnected windings of rotor and stator. Electric motors are described as shunt, series, or compound wound-type configuration depending on exciting current.

7.5 DC MACHINE PERTINENT NAMEPLATE PARAMETERS

In specifying machines for practical applications, it is important that the parameters are clearly spelled out to ensure easy equipment selection and accurate execution of tasks. Consequently, manufacturers usually include a nameplate on the body of the machine that spells out the machine's important parameters. Important parameters of DC machines that may be found on the nameplate include:

• armature and shunt field current (amps)
• armature resistance R_a (ohm)
• horsepower rating (HP)
• interpole winding and brushes resistance (ohm)
• machine type by winding (series, shunt, or compound wound)
• nominal power rating P (kW)
• number of poles P
• rotating losses
• speed N (rpm)
• terminal voltage V (volts)

7.6 DEVELOPMENT AND CONFIGURATION OF EQUIVALENT CIRCUITS OF DC MACHINES

From the principles of a rotating conductor, as detailed in Section 7.3, DC-machine armature rotates in a magnetic field produced by the stator magnetic poles. By applying Lenz's law, the voltage induced in the armature winding is given by:

(7.3)

where l is the length of conductor in the slot of the armature, φ is the flux per pole ω_m is the mechanical speed, r is the distance of conductor from , and A is the area per pole .

Consequently, we can recast Equation 7.3 as

(7.4)

This leads to computation of terminal voltage as a function of the number of turns connected in series with the armature, given as

(7.5)

where a is the number of parallel paths and N is the total number of turns in the winding.

(7.6)

(7.7)

Therefore, we observe that E_a is a function of φ and ω_m, given represented by a constant k for a specific machine.

So we may now write

(7.8)

where k varies with each machine configuration depending on N, P, π, and a.

Assume

then

(7.9)

E_a is now fully defined as voltage in generator operation and E_a is given as back emf in motor operation.

Typically, we develop the equivalent circuit (Figure 7.4) as follows under steady-state operations. As a generator (DC machine) we write

(7.10)

and

(7.11)

or simply

(7.12)

Combining these two equations

(7.13)

where k_a depends on number of conductor and pole pairs, i.e.,

(7.14)

Based on the second fundamental equation of machine operation (Equation 7.2), torque is developed based on resulting force on the conductor by Lorentz's equation. Torque is produced by armature winding with current in a magnetic-field winding of the stator.

The force armature conductor is calculated as

(7.15)

(7.16)

(7.17)

which gives the average torque

(7.18)

where k_a is a design constant that depends on .

Therefore, in the case of motor operation:

• The terminal voltage generated is greater than emf and the electromagnetic torque produces rotation against a load.
• Alternatively, when the terminal voltage V_t is less than the generated voltage in the case of a generator, the electromagnetic torque opposes the torque applied to the shaft by the prime mover.

Simply, in the case of a motor

where Tω_m is the developed power within the mechanical system. The converse is also true for a motor or generator operation. So, we can write

(7.19)

where K_a, W_m, φ, and I_a are already defined.

Example 1

A shunt generator delivers 450A at 230 V and the resistance of the shunt field and armature are 50 ohm and 0.03 ohm, respectively. Calculate the generated emf.

Solution:

Current through shunt field winding

Load current

Armature current

Armature voltage drop

Example 2

A four-pole DC machine has an armature of radius 12.5 cm, effective length of 25 cm. The poles cover 75 percent of the armature periphery. The armature windings consist of thirty-three coils each having seven turns. The coils are accommodated in thirty-three slots. The average flux density under each pole is 0.75T. If the armature is wound, determine:

1. armature constant k_a
2. induced-armature voltage when the armature rotates at 1,000 rpm
3. current and electromagnetic torque developed when the armature is 400A
4. power developed by the armature

If the armature is wave wound, repeat (a)–(d) above.

Solution:

pole area

or

wave wound

7.7 CLASSIFICATION OF DC MACHINES

The field current and armature current can be interconnected in many ways to provide various performance characteristics for DC-machine configuration.

1. Field poles in DC machines are configured by four methods (as shown in Figure 7.5):
   1. series-field winding (connected in series)
   2. shunt-field winding (connected in parallel)
   3. short-shunt compound winding
   4. long-shunt compound winding

The compound-machine configuration combines series and shunt winding and these two windings have different performance. The rheostat helps control the field current to achieve the desired performance.

Given these configurations, we now develop analysis for DC generators using the circuit equivalent.

Consider the machine to be separately excited (Figure 7.6), then the relevant equations are:

and

where

These equations define the termed or external V_t of the separately excited DC generator.

The relationship between V_t and I_t for different operating conditions is shown in Figure 7.7 [2].

From the terminal characteristics of a separately excited DC generator, there is a voltage drop due to armature reaction (ΔV_AR) from the machine internal characteristics to external characteristics. The operating point is the intersection between the generator's external characteristics and the load characteristics. The operating point gives the operating values of load current and terminal voltage.

Example 1

A shunt generator delivers 450A at 230 V and the resistance of the shunt field and armature are 50 ohm and 0.03 ohm, respectively. Calculate the generated emf.

Solution:

Current through shunt field winding = I_sh = 230/50 = 4.6A

Load current

Armature current

Armature voltage drop

Example 2

A long-shunt compound generator delivers a load current of 50A at 500 V and has armature, series-field, and shunt-field resistances of 0.05 ohm, 0.03 ohm, and 250 ohm, respectively. Calculate the generated voltage and the armature current. Allow 1 V per brush for contact drop.

Solution:

Current through armature and series winding = 50 + 2 = 52A

Voltage drop on series field winding = 52 × 0.03 = 1.56V

Armature voltage drop = I_aR_a = 52 × 0.05 = 2.6V

Drop at brushes = 2 × 1 = 2V

Now,

Example 3

A short-shunt compound generator delivers a load current of 30A at 220 V and has armature, series-field, and shunt-field resistance of 0.05 ohm, 0.3 ohm, and 200 ohm, respectively. Calculate the induced emf and armature current. Allow 1.0 V per brush for contact drop.

Solution:

Example 4

A 12 kW, 100 V, 1,000 rpm DC shunt generator has armature resistance R_a = 0.1 ohm, shunt-field winding resistance R_f = 80 ohm, and N_f = 1, 200 turns/pole. Rated field current is 1A. Find V_t at full load if E_a is 98 V.

If armature resistance is at full load of 0.06A, determine full load at V_t.

Solution:

Given that

Example 5

Determine the field current required to make the terminal voltage of V_t = 100V at full load.

Solution:

so

7.8 VOLTAGE REGULATION

(7.20)

V_{t, NL} is the no-load terminal voltage, and is the full-load terminal voltage.

Since E_a is equal to V_{t, NL},

(7.21)

7.9 POWER COMPUTATION FOR DC MACHINES

For DC-machine operation, DC current must be provided. Use

(7.22)

at armature reaction

(7.23)

where

(7.24)

Finally, the total resistance

(7.25)

This is used to determine the performance of the machine.

7.9.1 Loss Calculation

(7.26)

(7.27)

(7.28)

where

7.9.2 Torque Computation

Assume a DC machine has V_a, I_a, E_a, K_max, then

This can be written as function of losses

(7.29)

and the corresponding torque

(7.30)

(7.31)

(7.32)

(7.33)

where

so that

(7.34)

n_m is in rad/s.

Using the loss formula, we can determine power flow and efficiency of the machine.

7.10 POWER FLOW AND EFFICIENCY

From the Figure 7.8a and Figure 7.8b, we can compute efficiency

where

(7.35)

(7.36)

Copper loss depends on armature-winding loss (I²_aR_a) or field-winding losses (I²_fR_f) when I_a and I_f are armature- and field-winding currents, respectively, R_a and R_f are armature and field resistances, respectively, brush loss P_b = V_bI_a, and V_b is the voltage drop across brushes, which is typically about 2 V.

7.10.1 Core Loss

The core loss is derived from the eddy-current and hysteresis losses, as discussed in Chapter 2. Core loss depends on the value of the magnetic flux density B and the speed of rotation of the machine. Mechanical losses can be given as a function of friction and windage. Stray losses account for approximately 1 percent of full-load loss. Power losses in DC machines are given or empirically computed.

Example 1

A shunt generator delivers 195A at terminal voltage 250 V. The armature resistance and shunt-field resistance are 0.02Ω and 50Ω, respectively. The iron and frictional losses equal 950 W. Find:

1. emf generated
2. copper loss
3. output of the prime mover
4. commercial, mechanical, and electrical efficiencies

Solution:

7.11 DC MOTORS

Under a steady-state condition, the interaction between voltage and current for DC machines is given as

(7.37)

(7.38)

where we further suggest, as discussed in Section 7.12, that for a motor

(7.39)

from which it can be further shown that

(7.40)

So, as usual

(7.41)

Different configurations can be represented as follows:

1. shunt motors, whose field current can be controlled by use of variable resistance in series with the field winding
2. series motors, for which it is difficult to control the armature and field winding independently, so we treat flux as largely dependent on the armature current and field current
3. compound motors, where the effect of armature current on the flux depends on different issues (short and long winding)

7.12 COMPUTATION OF SPEED OF DC MOTORS

From the fundamental expression

(7.42)

which means that speed, ω_m, depends on the applied voltage V_t, armature current I_a, armature resistance R_a, and flux φ per pole.

The schematic diagram (as seen in Figure 7.9) of each DC-motor configuration with respect to control options is given as follows:

For configuration (a), shunt motor

(7.43)

(7.44)

So, we can write

(7.45)

For configuration (b), series DC motor

(7.46)

(7.47)

So, we can write

(7.48)

For configuration (c), compound motor

(7.49)

(7.50)

But

(7.51)

Then,

(7.52)

(7.53)

The following figures namely Figures 7.10, 7.11, and 7.12 show the torque/armature current characteristics, torque/speed characteristics and speed/power characteristics respectively of the dc motor.

7.12.1 Connection Description

A shunt machine is a constant-speed machine with low-speed regulation. Speed can be controlled by field flux.

For a series motor, recall the equation for T_e given as

(7.54)

If R_a ≈ 0, T_e is a function of flux φ, terminal voltage will be

(7.55)

from which

(7.56)

Also

(7.57)

(7.58)

Thus, the speed/power relationship is as shown in Figure 7.13.

The relationship between speed and torque for each class of DC-machine configuration operating as a motor is as shown in Figure 7.14.

Example 1

A 440 V shunt motor has armature resistance of 0.8 ohm and field resistance of 200 ohm. Determine the back emf when giving an output of 7.46 kW at 85 percent efficiency.

Solution:

Now,

Example 2

The following data is for a DC shunt motor:

At no load, armature resistance = 0.5Ω, speed = 1,000 rpm, armature current = 4A, torque = 10Nm.

1. Find the armature current and the speed of the motor.
2. If the motor is required to develop 10 hp at 1,200 rpm, compute the required value of the external series resistance in the field circuit.

Solution:

Assume field current

with machine on load

We can now obtain

Since

we can recompute

with corresponding speed

For 10 Hp at 1,200 rpm,

So, given

or

Example 3

The input to a 230 V DC shunt motor is 11 kW. Calculate:

1. torque developed
2. efficiency
3. speed at this load

The particulars of the motor are: no-load current is 5A, no-load speed is 1,150 rpm, armature resistance is 0.5 ohm, and shunt-field resistance is 110 ohm.

Solution:

No-load input = 220 × 5 = 1,100 W

with 11 kW input,

7.13 DC-MACHINE SPEED-CONTROL METHODS

Three methods are used mainly for DC machine control:

1. field control
2. armature-resistance control
3. solid-state rectifier control

7.13.1 Field-Control Method

From

(7.59)

V_t is kept constant so ω_m depends on I_f only. φ decreases while speed increases and since I_f is directly proportional to φ, it follows that by varying I_f, the machine's speed can be varied. One way to do this is to insert a variable resistance in the field circuit to control I_f. This is one of the simplest but most efficient methods of DC-machine speed control.

7.13.2 Armature-Resistance Control Method

Terminal voltage and field current are maintained consistent at their rated values. Motor speed is easily controlled by varying armature resistance through an external rheostat. This control method is simple and inexpensive but has the disadvantage of low efficiency.

Again, ω_m changes depend on V_t. It is easy to compute voltage regulation or speed regulation using speed regulation

(7.60)

The method of control here is applicable to shunt-, series-, and compound-motor configurations.

7.13.3 Solid-State Rectifier Control

Silicon-controlled rectifiers (SCRs) are one of the solid-state devices widely used in the control of DC motors. The SCR is used to rectify the voltage and control the armature current of the DC machine by adjusting the firing angle with the application of the appropriate trigger voltage to the gate of the SCR. The firing angle of the SCR determines the amount of DC current supplied to the motor's armature. Thus, by controlling the armature current, the machine is also controlled. This is consistent with the equation

(7.61)

For a DC-series motor it can be shown that

from which we can see that

(7.62)

Thus, increasing the armature current results in a decrease in motor speed, which can be achieved using the SCR.

7.14 WARD LEONARD SYSTEM

For a solid-state control rectifier using a DC machine, the speed-control system is called the Ward Leonard system. It is capable of controlling a DC motor from zero to maximum or maximum to zero.

The configuration of the Ward Leonard system is shown in Figure 7.15 [3]. AC source (induction or synchronous machine) serves as the prime mover for a DC generator. The generator in turn supplies V_t, which supplies a DC motor for operation.

Varying the field current of the DC machine, which varies the speed of the motor, can consequently control the armature voltage in the motor. That is, the motor speed can be reduced or increased by the motor field current with changes in flux. Further, if the current of the generator is reversed, the polarity of the generator-armature voltage will reverse as well. These reversals also change or cause the motor direction of rotation to change. The speed variation occurs as the field current changes back and forth in the opposite direction for the Ward Leonard system.

7.15 CHAPTER SUMMARY

We have presented the concepts and fundamental principles of DC machines in this chapter. The internal structures were briefly discussed. Also, the various configurations and their important characteristics were discussed. We distinguished between DC generators and DC motors and discussed methods of speed control.

DC motors have found wide applications in control systems. As a rotating machine with an inertial mass, it is of interest to us as it applies to the microgrid. The behavior of DC machines as a generator or motor—for instance on a DC microgrid—should be of practical interest. The student is encouraged to learn these principles as an integral part of the knowledge relative to energy conversion in the smart grid and microgrid. Further, this chapter provided principles for studying the performance of DC-machine applications.