8.2.1 Permanent-Magnet DC Motor Fundamentals and Concepts

A permanent-magnet DC motor is like an ordinary shunt motor except that its field is provided by a permanent magnet instead of a salient-pole, wound-field structure. Although there are various types of DC motors, the motor shown in Figure 8.1 is suitable for understanding the basic principles. All kinds of DC motors work on the same principles.

For constructing a DC motor it is essential to establish a magnetic field. The magnetic field is established, obviously, by means of magnet. The magnet can by of many types including an electromagnet or permanent magnet. When a permanent magnet is used to create a magnetic field in a DC motor, the motor is referred to as a permanent-magnet DC or PMDC motor.

As shown in Figure 8.1, the permanent magnets of a PMDC motor are supported by a cylindrical steel stator, which also serves as a return path for the magnetic field. Figure 8.2 shows the field system with a magnetic pole shoe.

Materials used for the permanent magnet are divided into diverse types. The magnet shown in Figure 8.1 has materials with high coercivity and residual flux density.

8.2.2 PMDC Motivations and Applications

The motivations and limitations for using a PMDC motor are shown in Table 8.1.

PMDC motors are used extensively where small DC motors are required and very effective control is not required, such as in automobiles starters, toys, wipers, washers, hot blowers, air conditioners, and computer disc drives.

8.2.3 PMDC Design Models and Materials

Equivalent Circuit of a PMDC Motor

Since in a PMDC motor the field is produced by permanent magnet, there is no need to draw field coils in the equivalent circuit of a PMDC motor Figure 8.3. The supply voltage to the armature will have an drop in its armature resistance and the rest of the supply voltage will be countered by back emf of the motor. Hence the voltage equation of the motor is given as

(8.1)

where I is the armature current, R is the armature resistance of the motor, E_b is the back emf, and V is the supply voltage.

The armature is connected in series with the armature current and resistor R. This yields the back emf as

(8.2)

where k_a is the armature current, ϕ_d is the flux per pole, and ω_m is the flux speed in rad/sec.

Because of friction, the torque generated by the motor may be treated as being made up of frictional torque and usable torque:

(8.3)

where T_d is the total torque of the permanent-magnet motor, T_l is the load torque or useful torque, and T_f is the frictional torque.

Mechanical power is defined as

(8.4)

where ω is the rotational speed (rad/s).

The performance of a PMDC motor with respect to the current–torque relationship and flux-speed–current relationship is given in Figure 8.4.

In a PMDC motor, we have flux per pole defined as

(8.5)

where k = k_a × ϕ_d.

So, back emf

(8.6)

k_a = torque constant

T_d = total torque of permanent magnet motor and is

(8.7)

If V_t terminal voltage of the motor changes at no load, we can do performance analyses such as

(8.8)

where ω_FL is the rotational speed at full load and ω_NL is the rotational speed at no load.

Materials for PMDC Motors

The advances in permanent magnet motors is due to special characteristics of new materials. The highest-performance magnets are brittle ceramics. Other desirable properties include chemical sensitivity, temperature sensitivity, and demagnetizing fields.

Materials used in permanent magnet motors all have B–H loops. Figure 8.5 [1] illustrates a typical B–H characteristic loop. It is divided into four quadrants, namely a magnetization quadrant, a demagnetization quadrant, a residual quadrant, and a coercive region.

The demagnetization quadrant is referred to as the demagnetizing curve.

The worst-case demagnetization shows at point D when flux density is reduced compared to the original line. It is recommended to keep the separating point away from the worst-case demagnetization condition.

The behavior or performance of a PMDC motor depends on the materials used in the permanent magnet. To understand the performance of a PMDC motor, it is important to know the properties of these materials:

1. Alnico magnets have good magnetization/demagnetization properties but very low coercive force and the B–H loop is too square shaped. The permanent demagnetization occurs too easily.
2. Ceramic ferrite magnets use barium and strontium. They are low cost and have a moderately high-service temperature at 400°C. This type of magnet is represented by a straight-line demagnetization curve B_r (residual). Flux density is low and hence the machine volume and size are typically large.
3. Rare-earth magnets, using neodymium-iron-boron (NdFeB), have very good demagnetization properties except when at Curie temperature of 150°C. A range of B–H for each material and ordering measures for useful applications are determined. In general, rare-earth materials are expensive but cost-effective and useful in very small resistors.
4. Samarium-cobalt (Sm₂Co₇) magnets have very good demagnetization properties but are expensive because samarium is rare.

The combination of magnets with ferro-magnet materials is used to achieve high performance, high torque, and high efficiency at low voltage levels, and at a low cost.

A summary of the four magnet types is shown in Table 8.2.

Permanent Magnet	Br (T)	Hc (kA/M)	Cost	Resistivity (μ Ω − cm)	Max. Working Temperature	Curie Temperature (°C)
Alnico	1.3	60	–	47	>500	–
Ceramic ferrite magnets	0.4	300	Low	>10,000	250	450
NdFeB (sintered)	1.1	850	Medium	150	80–200	310–350
Sm2Co7 (sintered)	1.0	750	Higher than NdFeB	86	250–350	700–800

Example 1

Consider a PMDC motor, where R_a is 1.2 Ω, V_c is 60 V, and ω is 1,950 rev/m. If I_a = 1.5 A at no load find:

1. torque constant
2. no-load rotational losses
3. P_out in HP if speed changes to 1,500 rpm from a 50 V supply

Solution:

1. Torque constant k = k_a * ϕ_d

   if

   
   

   where

   

   thus

   

   But

   
2. At no load, rotational losses
   

   if

   

   then

   
3. 

Problem 2

A motor with k = 5.89 oz ⋅ in/A and a coil resistance of 1.76 Ω is driven with a supply voltage of 12 V. If the motor's friction torque is 1.2 oz ⋅ in

Find:

1. maximum torque available for driving a load
2. how much current is flowing under these conditions

Solution:

1. Using Equations 8.3 and 8.6, we can find the torque available at the output shaft
   
   
2. Using Ohm's law to substitute for current
   
   
   

Problem 3

A permanent-magnet motor with a terminal resistance of 0.316 Ω and KT = 30.2 Nm/A is powered by a 12 V supply. Measurements show that the operating rotational speed is 3,616 rpm with a current flow of 1.79A. How much power does the motor generate under these conditions?

Solution:

Using Equations 8.3 and 8.4, full expression for motor-power output becomes

Here, T_f is assumed to be negligible, and simplifies the expression

Rotational speed in radians/sec instead of rpm is obtained as:

Substituting Equation 8.7 back to expression of power:

8.3.1 Permanent-Magnet Synchronous Motor Fundamentals and Concepts

A permanent-magnet synchronous motor (PMSM) or permanent-magnet motor (PMM) is a synchronous motor that uses permanent magnets rather than windings in the rotor.

The frequency of the excitation dictates rotor speed. The angular position of the PMM rotates at an angle ω₀, which is equal to angle ω_s of the synchronous machine and thus can produce AC currents. For a typical AC PMM, when energized, the machine produces torque with Pω₀ = T.

First, we present the rotor construction of PMMs. In permanent magnet motors, there is no field-winding sector. The structure is smooth and cylindrical in shape. PMSMs can be built with different rotor configurations (as shown in Figure 8.6). Some PMSM types based on the arrangement of magnets on the rotor (shown in Figure 8.7 [2]) are as follows:

• surface-mounted, where magnets are mounted on the surface of the outer periphery of rotor lamination (Figure 8.7(a))
• surface-insert, where magnets are placed on the grooves of the outer periphery of the rotor lamination (Figure 8.7(b))
• interior transverse (Figure 8.7(c))
• salient poles with laminated-pole shoes and a cage-winding (Figure 8.6(a))
• interior-magnet rotors (Figure 8.6(b))
• surface-magnet rotors (Figure 8.6(c))
• inset-magnet rotors (Figure 8.6(d))
• rotor with buried magnets symmetrically distributed (Figure 8.6(e))
• rotor with buried magnets asymmetrically distributed (Figure 8.6(e))

In PMSMs, the stator contains 3Φ distribution windings, which is similar to the induction machine.

The arrangements for each magnet construction for permanent magnets in synchronous motors are shown in Figures 8.7 and 8.8 [3].

8.3.2 PMSM Design Motivations and Applications

The motivations and limitations for using a PMSM are shown in Table 8.3.

PMSMs are used in air conditioners, heaters, blowers, windshield wipers, washers, power seats radios, etc. They are used at homes to operate electric stoves, electric toothbrushes, vacuum cleaners, electric knives, power tools, lawn mowers, etc. They are also used as starter motors for outdoor purposes, and in computers as tape drivers.

8.3.3 PMSM Design Models and Materials

Using the diagrams in Figures 8.7 and 8.8, we can derive the model equation for a permanent-magnet AC motor.

First let us discuss the permanent-magnet materials in the linear region of demagnetization. It has B_r (residual), with B_m > B_r (critical view), so magnetic flux density

(8.9)

where

We need to develop the PMM,

(8.10)

(8.11)

This gives

(8.12)

using ampere circuital law, where, l_m is the toroidal radial length and g_e is the effective length of air gap.

If the flux is continuous, B_m = B_g, then combining Equations 8.7 and 8.10 gives

(8.13)

Assume

to get,

If μ_r = steel, by equation produce 0.9 Tesla.

For different materials, with a neodymium magnet with r = 5 mm and air gap 1 m, we get B = 0.9T. If, B_r is from a samarium–cobalt magnet, B_r = 0.857T. If the material is ferrite, the air-gap flux density ≤ 0.3T.

To build a circuit model in general we have from the recoil line, Figure 8.9

(8.14)

and range-slope equation

(8.15)

If ΔF tends to zero (0),

(8.16)

so the equation of the circuit model is

(8.17)

Equation 8.17b suggests the equivalent circuit for a permanent magnet as shown in Figure 8.10.

It can easily be shown that source flux Φ is retarded or opposed by the reluctance in the magnetic circuit just as the electrical-equivalent circuits can be used to represent the magnetic circuit (mmf representing emf, flux representing current, and reluctance representing resistance), e.g., in source transformation shown in Figure 8.11.

The equivalent-circuit model of PMM is developed, and hence for each part of the machine it goes specifically as follows:

For the rotor with magnetic length l_m we derive,

(8.18)

If the width of the magnet is 2γ of the magnet, then,

(8.19)

Thus, the current value is determined from

(8.20)

where

For different materials, different values of β are determined, which give different values of L_m.

For the case of μ_r = 1

(8.21)

where N_se is the number of turns, l is the axial length, g_e is the gap length, l_m is the rotor-magnetic length, r is the radius of the outer coil, L_m is the magnetic inductance, and l is typically small, 0.25 to 0.4, based on the machine rating of two to five for induction machines.

From the B–H loop, the region of magnetic saturation is frequently of negligible effect. Thus, the mmf required for large machines is frequently of negligible effect.

Note that L_ls and L_m are constants. We may need to now provide the equivalent circuit of permanent magnet motor as shown in Figure 8.12.

(8.22)

we can rewrite the equivalent diagram.

The inductive element function of current can be given as

(8.23)

(8.24)

which is the new system as shown in Figure 8.13

where i_f is the source current and i′_f is the output current.

From the equivalent circuit, as the PMM rotates we have ω_s = ω₀ at demagnetization region. The flux-density value B_d from the B–H loop = ± B₀.

For different materials, e.g., ferrite, the value decreases with temperature. For samarium–cobalt (Sm-Co), B_d becomes negative:

(8.25)

If

(8.26)

You can convert to jω domain and show i_s at different frequencies. Using jω transformation

(8.27)

Now jω domain dominates.

We obtain

(8.28)

The overall equivalent-circuit model of PMM is shown in Figure 8.14 [2]. The inductance is lumped with the mutual inductance of the stator and damping windings. The combined mutual inductance is given by L_md. The magnetizing current gives the equivalent magnetizing current for the PMM at the stator side.

The constant of the equivalent-circuit model for the PMM is given in Figure 8.14, defined into a direct quadrature axes reference frame.

Figure 8.15 [3] shows a B–H curve with different operating points of a PMSM machine. At a no-load separating point B′ there is demagnetization. The shape is smaller with a larger air gap.

With current in the state, there is further demagnetization of the permanent magnet, which results in a different operation point at C′ at full load.

Refer to Section 8.1.3 for materials of a PMSM machine.

PMSMs provide a wide range of speed by using inverters on good magnetic types of materials such as neodymium–iron–boron or samarium–cobalt. Note that these materials with the same sizes may differ in efficiency under different temperatures. The permanent-magnet materials used in this motor are same as that of a PMDC motor.

Example 1

The parameters of a star-connected, six-pole, 1.5 kW, 9.2 A, 1,500 rpm, 9.55 Nm, three-phase PMSM are:

• Rs = 0.513 Ω, Ld = 4.74 mH, Lq = 9.51 mH.
• Inverter input voltage = 285 V.

Find:

1. normalized stator resistance
2. normalized direct-axis inductance
3. normalized quadrature-axis inductance

Solution:

Choose the base value as follows:

Base voltage,

Base speed,

Base impedance,

Base impedance,

Normalized stator resistance,

Stator phasor voltage V_s is calculated as:

Normalized value,

Normalized direct axis inductance,

Normalized quadrature axis inductance,

8.4.1 Trapezoidal Surface-Magnet Motors

A trapezoidal surface PMM is very similar to a sinusoidal PMM. The difference is that the trapezoidal surface-magnet motor has its 3Φ winding made of concentrated, full-pitch distribution instead of a sinusoidal distribution. Figure 8.16 represents the field winding and rotor.

The two-pole motor in Figure 8.16 [3] consists of a gap in the rotor magnets to reduce flux (magnetic losses), and has four slots per phase winding per pole. When the machine rotates, the flux linkage varies linearly at the demagnetizing region when it passes the magnetic gap. If the machine is driven by a prime mover, the stator three-phase voltages will have a trapezoidal wave shape as shown in Figure 8.17 [4].

Furthermore, an electronic inverter is required to establish a six-step current wave, which generates torque. A trapezoidal surface-magnet motor can serve as a brushless DC motor if it makes use of the inverter and an absolute-position sensor is mounted on its shaft. The sinusoidal-surface PMM can also serve as a brushless DC motor but the trapezoidal surface PMM gives closer DC-machine-like performance.

8.4.2 Synchronous Reluctance Motors

A synchronous reluctance motor has the same structure as that of a salient-pole synchronous motor except that it does not have a field winding on the rotor. Figures 8.18 and 8.19 [5] show the cross-section and torque-angle curve, respectively, of a synchronous reluctance motor.

The stator has a symmetrical winding, which creates a sinusoidal rotating field in the air gap. There is magnetic field induced in the rotor, causing it to align with the stator field in a minimum reluctance, which creates a reluctance torque on the rotor. The torque developed in this motor is given by

(8.29)

where T_e is the torque developed in the motor,  P is the pair of poles or pole pair, Ψ is the flux linkage, L_ds is the direct-axis inductance, L_qs is the quadrature-axis inductance, and δ is the rotor angle.

8.4.3 Variable Reluctance Motors

A variable reluctance motor has double saliency, i.e., both the rotor and stator have saliency. This motor type includes the stepper motor and the switched reluctance motor.

Stepper Motor

A stepper motor is a type of synchronous motor built to rotate a specific number of degrees in response to a digital input in the form of a pulse. This means that a digital signal is used to drive the motor, and every time it receives a digital pulse it rotates a specific number of degrees in rotation.

Stepper motors are excellent devices for accurate speed control or precise position control without any feedback. In such usage, the axis of the motor's magnetic field steps around the air gap at a speed that is based on the frequency of pulses. The rotor inclines to align itself with the axis of the magnetic field, which means that the rotor steps in sync with the motion of the magnetic field, thus it is called a stepper motor. Figure 8.20 shows operation of a 30° per step stepper motor.

Stepper motors are not suitable for variable-speed drives.

Switched Reluctance Motor

The structure of the switched reluctance motor, as shown in Figure 8.21 and Figure 8.22, has four stator-pole pairs and three rotor-pole pairs. The rotor has neither windings nor permanent magnets.

The stator poles have concentrated winding rather than sinusoidal winding. Each stator-pole pair is aligned and is then de-energized. The stator-pole pairs are sequentially excited using a rotor position encoder for timing.