11.2.1 DC Motors

DC motors were widely used in the past for all types of drive applications, and they continue to be used for controlling speed and position. There are two designs of DC machines: stators consisting of either permanent magnets or a field winding. The power-processing units can also be classified into two categories, either switch-mode power converters that operate at a high switching frequency, as discussed in the next chapter, or line-commutated thyristor converters, which are discussed later, in Chapter 14. In this chapter, we will describe permanent-magnet DC motors, which are usually supplied by switch-mode power electronic converters.

Figure 11.2 shows a cut-away view of a DC motor. It shows a permanent-magnet stator, a rotor that carries a winding, a commutator, and the brushes. In DC machines, the stator establishes a uniform flux in the air gap in the radial direction (the subscript “f” is for field). If permanent magnets like those shown in Figure 11.2 are used, the air gap flux density established by the stator remains constant. A field winding whose current can be varied can be used to achieve an additional degree of control over the air gap flux density.

As shown in Figure 11.2, the rotor slots contain a winding, called the armature winding, which handles electrical power for conversion to (or from) mechanical power at the rotor shaft. In addition, there is a commutator affixed to the rotor. On its outer surface, the commutator contains copper segments, which are electrically insulated from each other by means of mica or plastic. The coils of the armature winding are connected to these commutator segments so that a stationary DC source can supply voltage and current to the rotating commutator by means of stationary carbon brushes that rest on top of the commutator. The wear due to the mechanical contact between the commutator and the brushes requires periodic maintenance, which is the main drawback of DC machines.

In a DC machine, the magnitude and the direction of the electromagnetic torque depend on the armature current . Therefore, in a permanent-magnet DC machine, the electromagnetic torque produced by the machine is linearly related to a machine torque constant , which is unique to a given machine and is specified in its data sheets,

(11.1)

Similarly, the induced emf depends only on the rotational speed , and can be related to it by a voltage constant , which is unique to a given machine and specified in its data sheets,

(11.2)

Equating the mechanical power () to the electrical power (), the torque constant and the voltage constant are exactly the same numerically in MKS units,

(11.3)

The direction of the armature current determines the direction of currents through the conductors. Therefore, the direction of the electromagnetic torque produced by the machine also depends on the direction of . This explains how a DC machine, while rotating in a forward or reverse direction, can be made to go from motoring mode (where the speed and torque are in the same direction) to its generator mode (where the speed and torque are in the opposite direction) by reversing the direction of

Applying a reverse-polarity DC voltage to the armature terminals makes the armature current flow in the opposite direction. Therefore, the electromagnetic torque is reversed, after some time reversing the direction of rotation and the polarity of induced emfs in conductors, which depends on the direction of rotation.

The above discussion shows that a DC machine can easily be made to operate as a motor or as a generator in the forward or in the reverse direction of rotation.

It is convenient to discuss a DC machine in terms of its equivalent circuit, presented in Figure 11.3, which shows the conversion between electrical and mechanical power. The armature current produces the electromagnetic torque necessary to rotate the mechanical load at the desired speed. Across the armature terminals, the rotation at the speed of induces a voltage, called the back-emf , represented by a dependent voltage source.

The applied voltage from the PPU overcomes the back-emf and causes the current to flow. Recognizing that there is a voltage drop across both the armature winding resistance (which includes the voltage drop across the carbon brushes) and the armature winding inductance , we can write the equation of the electrical side as

(11.4)

On the mechanical side, the electromagnetic torque produced by the motor overcomes the mechanical-load torque to produce acceleration:

(11.5)

where is the total effective value of the combined inertia of the DC machine and the mechanical load.

Note that the electric and the mechanical systems are coupled. The torque in the mechanical system (Equation 11.1) depends on the electrical current The back-emf in the electrical system (Equation 11.2) depends on the mechanical speed . The electrical power absorbed from the electrical source by the motor is converted into mechanical power and vice versa. In a DC steady state, with a voltage applied to the armature terminals and a load torque being supplied to the load, the inductance in the equivalent circuit of Figure 11.3 does not play a role. Hence, in Figure 11.3,

(11.6)

(11.7)

11.2.1.1 Requirements Imposed by DC Machines on the PPU

Based on Equations 11.6 and 11.7, the steady-state torque-speed characteristics for various values of are plotted in Figure 11.4a. Neglecting the voltage drop across the armature resistance in Equation 11.7, the terminal voltage varies linearly with the speed , as plotted in Figure 11.4b. At low values of speed, the voltage drop across the armature resistance can be a significant component of the terminal voltage, and hence the relationship in Figure 11.4a in this region is shown dotted.

In summary, in DC-motor drives, the voltage rating of the power-processing unit is dictated by the maximum speed, and the current rating is dictated by the maximum load torque being supplied.

11.2.2 Permanent-Magnet AC Machines

Sinusoidal-waveform, permanent-magnet AC (PMAC) drives are used in applications where high efficiency and high power density are required in controlling the speed and position. In trade literature, they are also called “brushless DC” drives. We will examine these machines for speed and position control applications, usually in small (<10 kW) power ratings, where these drives have three-phase AC stator windings, and the rotor has DC excitation in the form of permanent magnets. In such drives, the stator windings of the machine are supplied by controlled currents, which require a closed-loop operation, as described later in this section.

We will consider 2-pole machines, like the one shown schematically in Figure 11.5a. This analysis can be generalized to p-pole machines where . The stator contains three-phase, wye-connected windings, each of which produces a sinusoidally distributed flux-density distribution in the air gap when supplied by a current.

The permanent-magnet pole pieces mounted on the rotor surface are shaped to ideally produce a sinusoidally distributed flux density in the air gap. The rotor flux-density distribution (represented by a vector ) peaks at an axis that is at an angle with respect to the a-axis (the phase axis of winding a), as shown in Figure 11.5b. As the rotor turns, the entire rotor-produced flux density distribution in the air gap rotates with it and “cuts” the stator-winding conductors and produces emf in phase windings that are sinusoidal functions of time. In the AC steady state, these voltages can be represented by phasors.

Considering phase-a as the reference in Figure 11.5b, the induced voltage in it due to the rotor flux cutting it can be expressed as

(11.8)

It should be noted that by Faraday’s law (, where is the flux-linkage of the coil), the induced voltage in phase-a peaks when the rotor-flux peak in Figure 11.5a is pointing downward, before it reaches the phase-a magnetic axis. Since the permanent magnets on the rotor produce a constant amplitude flux, the RMS magnitude of the induced voltage in each stator phase is linearly related to the rotational speed by a per-phase voltage-constant ,

(11.9)

An important characteristic of the machines under consideration is that they are supplied through a current-regulated power-processing unit, which controls the currents , , and supplied to the stator at any instant of time. To optimize such that the maximum torque-per-ampere is produced, each phase current in AC steady-state is controlled in phase with the induced voltage. Therefore, with the voltage expressed in Equation 11.8, the current in AC steady state is

(11.10)

In AC steady state, accounting for all three phases, the input electric power, supplied by the current in opposition to the induced back-emf, equals the mechanical output power. Using Equations 11.8 through 11.10,

(11.11)

The torque contribution of each phase is , so from Equation 11.11,

(11.12)

where the constant that relates the RMS current to the per-phase torque is the per-phase torque constant and similar to that in DC machines, in MKS units,

(11.13)

At this point, we should note that PMAC drives constitute a class, which we will call self-synchronous motor drives, where the term “self” is added to distinguish these machines from the conventional synchronous machines. The reason for this is as follows: in PMAC drives, the stator phase currents are synchronized to the mechanical position of the rotor such that, for example, the current into phase-a, in order to be in phase with the induced emf, peaks when the rotor-flux peak is pointing downward, before it reaches the phase-a magnetic axis. This explains the necessity for the rotor-position sensor, as shown in the block diagram of such drives in Figure 11.6.

As shown in the block diagram of Figure 11.6, the task of the controller is to enable the power-processing unit to supply the desired currents to the PMAC motors. The reference torque signal is generated from the outer speed and position loops. The rotor position is measured by the position sensor connected to the shaft. Knowing the torque constant allows us to calculate the RMS value of the reference current from Equation 11.12. Knowing the current RMS value and allows the reference currents , , and for the three phases to be calculated, which the PPU delivers at any instant of time.

The electromagnetic torque acts on the mechanical system connected to the rotor, and the resulting speed can be obtained from the equation below:

(11.14)

where is the combined motor-load inertia and is the load torque, which may include friction. The rotor position is

(11.15)

where is the rotor position at time t = 0. Similar to a DC machine, it is convenient to discuss a PMAC machine in terms of its equivalent circuit of Figure 11.7, which shows the conversion between electrical and mechanical power.

In the AC steady state, using phasors, the current is ensured by the feedback control to be in phase with the phase-a induced voltage . The phase currents produce the total electromagnetic torque necessary to rotate the mechanical load at a speed . The induced back-emf , whose RMS magnitude is linearly proportional to the speed of rotation , is represented by a dependent voltage source.

The applied voltage in Figure 11.7a overcomes the back-emf and causes the current to flow. The frequency of the phasors in hertz equals in a 2-pole PMAC machine. There is a voltage drop across both the per-phase stator winding resistance (neglected here) and a per-phase inductance , which is the sum of the leakage inductance caused by the leakage flux of the stator winding and due to the effect of the combined flux produced by the currents flowing in the stator phases. The phasor diagram, neglecting , is shown in Figure 11.7b.

11.2.2.1 Requirements Imposed by PMAC Machines on the PPU

In PMAC machines, the speed is independent of the electromagnetic torque developed by the machine, as shown in Figure 11.8a, and depends on the frequency of voltages and currents applied to the stator phases of the machine.

In the per-phase equivalent circuit of Figure 11.7a and the phasor diagram of Figure 11.7b, the back-emf is proportional to the speed Similarly, since the electrical frequency is linearly related to the voltage drop across is also proportional to . Therefore, neglecting the per-phase stator winding resistance , in the phasor diagram of Figure 11.7b, the voltage phasors are all proportional to , requiring that the per-phase voltage magnitude that the power-processing unit needs to supply is proportional to the speed as plotted in Figure 11.8b.

At very low speeds, this voltage-speed relationship, shown dotted in Figure 11.8b, is approximate, where a higher voltage, called the voltage boost, is needed to overcome the voltage drop across the stator winding resistance ; this voltage drop becomes significant at higher torque loading and hence cannot be neglected.

In summary, in PMAC-motor drives, similar to DC drives, the voltage rating of the power-processing unit is dictated by the maximum speed, and the current rating is dictated by the maximum load torque being supplied.

11.2.3 Induction Machines

Induction motors with squirrel-cage rotors are the workhorses of industry because of their low cost and rugged construction. When operated directly from line voltages (a 50- or 60-Hz utility input at essentially a constant voltage), induction motors operate at a nearly constant speed. However, by means of power electronic converters, it is possible to vary their speed efficiently.

Just as in PMAC motors, the stator of an induction motor consists of three-phase windings distributed in the stator slots. These three windings are displaced by 120° in space with respect to each other, as shown by their axes in Figure 11.9a.

The rotor, consisting of a stack of insulated laminations, has electrically conducting bars of copper or aluminum inserted (molded) through it, close to the periphery in the axial direction. These bars are electrically shorted at each end of the rotor by electrically conducting end-rings, thus producing a conducting cage-like structure, as shown in Figure 11.9b. Such a rotor, called a squirrel-cage rotor, has a low cost and rugged nature.

Figure 11.10a shows the stator windings whose voltages are shown in the phasor diagram of Figure 11.10b, where the frequency of the applied line voltages to the motor is in hertz, and is the magnitude in RMS.

(11.16)

By Faraday’s law, the applied stator voltages given in Equation 11.16 establish a rotating flux-density distribution in the air gap by drawing magnetizing currents of RMS value , which are shown in Figure 11.10b:

(11.17)

As these currents vary sinusoidally with time, the combined flux-density distribution in the air gap, produced by these currents, rotates with a constant amplitude at a synchronous speed , where

(11.18)

The rotor in an induction motor turns (due to the electromagnetic torque developed, as will be discussed shortly) at a speed in the same direction as the rotation of the air gap flux-density distribution, such that . Therefore, there is a relative speed between the flux-density distribution rotating at and the rotor conductors at . This relative speed, that is, the speed at which the rotor is “slipping” with respect to the rotating flux-density distribution, is called the slip speed:

(11.19)

By Faraday’s law , voltages are induced in the rotor bars at the slip frequency due to the relative motion between the flux-density distribution and the rotor, where the slip frequency in terms of the frequency of the stator voltages and currents is

(11.20)

Since the rotor bars are shorted at both ends, these induced bar voltages cause slip-frequency currents to flow in the rotor bars. The rotor-bar currents interact with the flux-density distribution established by the stator-applied voltages, and the result is a net electromagnetic torque in the same direction as the rotor’s rotation.

The sequence of events in an induction machine to meet the load torque demand is as follows: At essentially no load, an induction machine operates nearly at the synchronous speed that depends on the frequency of applied stator voltages (Equation 11.18). As the load torque increases, the motor slows down, resulting in a higher value of slip speed. Higher slip speed results in higher voltages induced in the rotor bars and hence higher rotor-bar currents. Higher rotor-bar currents result in a higher electromagnetic torque to satisfy the increased load-torque demand.

Neglecting second-order effects, the air gap flux is totally determined by the applied stator voltages. Hence, the air gap flux produced by the rotor-bar currents is nullified by the additional currents drawn by the stator windings, which are in addition to the magnetizing currents in Equation 11.17.

In this balanced three-phase sinusoidal steady-state analysis, we will neglect second-order effects such as the stator winding resistance and leakage inductance and the rotor circuit leakage inductance. As shown in Figure 11.11a for phase-a in this per-phase circuit, and, similarly for the other two phases, are applied at a frequency , which results in magnetizing currents to establish the air gap flux-density distribution in the air gap.

These magnetizing currents are represented as flowing through a magnetizing inductance . The voltage magnitude and the frequency are such that the magnitude of the flux-density distribution in the air gap is at its rated value, and this distribution rotates counterclockwise at the desired (Equation 11.18) to induce a back-emf in the stationary stator phase windings such that

(11.21)

where is the machine per-phase voltage-constant.

Next, we will consider the effect of the rotor-bar currents. The rotor-bar currents result in additional stator currents (in addition to the magnetizing current), which can be represented on a per-phase basis by a current in-phase with the applied voltage (since the rotor-circuit leakage inductance is ignored), as shown in Figure 11.11a,

(11.22)

Note that the torque on the stator is equal in magnitude and opposite in direction to the torque on the rotor. The flux-density distribution (although produced differently than in PMAC machines) interacts with these stator currents, just like in PMAC machines, and hence the per-phase torque constant equals the per-phase voltage constant,

(11.23)

The per-phase torque can be expressed as

(11.24)

This torque at a mechanical speed results in per-phase electromagnetic power that gets converted to mechanical power, where using Equations 11.23 and 11.24,

(11.25)

In Equation 11.25, can be considered the back-emf , just like in the DC and the PMAC machines, as shown in Figure 11.11a,

(11.26)

In the rotor circuit, the voltages induced depend on the slip speed and overcome the IR voltage drop in the rotor bar resistances. The rotor bar resistances, the voltage drop, and the power losses in them are represented in the per-phase equivalent circuit of Figure 11.11a by a voltage drop across an equivalent resistance . Using Kirchhoff’s voltage law in Figure 11.11a,

(11.27)

Hence,

(11.28)

Using Equations 11.23, 11.25, and 11.28, the combined torque of all three phases is

(11.29)

where in the above equation is a machine constant that shows that the torque produced is linearly proportional to the slip speed. Using Equations 11.19 and 11.29,

(11.30)

11.2.3.1 Requirements Imposed by Induction Machines on the PPU

Based on Equation 11.30, the torque-speed characteristics are shown in Figure 11.12 for various applied frequencies to the stator. Based on Equation 11.21, the stator voltage as a function of frequency is shown in Figure 11.13 to maintain the peak of the flux-density distribution at its rated value. The relationship between the voltage and the synchronous speed or shown in Figure 11.13, is approximate. A substantially higher voltage than indicated (shown dotted) is needed at very low frequencies where the voltage drop across the stator winding resistance becomes substantial at higher torque loading and hence cannot be neglected.

In summary, in induction motor drives, similar to DC and PMAC drives, the voltage rating of the power-processing unit is dictated by the maximum speed, and the current rating is dictated by the maximum load torque being supplied.

Mechanical Systems

1. 11.1 A constant torque of 5 Nm is applied to an unloaded motor at rest at time t = 0. The motor reaches a speed of 1800 rpm in 5 s. Assuming the damping to be negligible, calculate the motor inertia.
2. 11.2 In an electric motor drive, the combined motor-load inertia is The load torque is . Plot the electromagnetic torque required from the motor to bring the system linearly from rest to a speed of 100 rad/s in 5 s and then maintain that speed.
3. 11.3 Calculate the power required in Problem 11.2.