**4. Basic permanent magnet motor design process**

Design procedure for any PM servomotor is shown in Fig. 10. This process comprises three main steps: Electromagnetic, structural and thermal designs. Electromagnetic design starts with magnetic circuit modeling and parameter optimization with a given set of design specifications. A series of optimizations such as pole number, loading, current density, dimensional limits etc. have to be performed to find the optimum parameters of the motor before proceeding further. When a design is obtained that meets the technical spec, a quick motor simulation and the influence of parameter variation must be carried out using simulation software such as SPEED (PC-BDC Manual, 2002). A detailed electromagnetic finite element analysis (FEA) either in 2D or 3D is the next step to verify that the design meets the specified torque-speed characteristics and performance. After an electromagnetic design is finalized, structural and thermal analyses (MotorCAD Manual 2004) have to be completed. It should be pointed out that structural analysis is not a necessity at low speed servomotor designs. If the motor does not meet the structural or thermal tests, then the electromagnetic design study should be repeated for a better design. A motor design has to be finalized after a design passes all of the main steps (Aydin et al. 2006).

Fig. 10. PM servomotor design process

### **5. Dynamic model of PM servomotors**

Following assumptions are made for the analysis of PM servomotors: The inverter is ideal with no losses; no DC voltage ripple exists in the DC link; the supply current is sinusoidal and no saturation is considered; eddy current and hysteresis loses are negligible; and all motor parameters are constant. Based on these assumptions, permanent magnet servomotor dynamic equations in the synchronous rotating reference frame are written as

$$
\sigma\_q = r\_s \dot{\mathbf{i}}\_q + \mathbf{L}\_q \frac{d\dot{\mathbf{i}}\_q}{dt} - \alpha\_e \mathbf{L}\_d \dot{\mathbf{i}}\_d + \alpha\_e \mathbf{\mathcal{A}}\_f \tag{1}
$$

Brushless Permanent Magnet Servomotors 285

The finite element method (FEM) is a numerical method for solving the complex electromagnetic field problems and circuit parameters. It is specifically convenient for problems with non-linear material characteristics where mathematical modeling of the system would be difficult. This method involves dividing the servomotor cross section or volume into smaller areas or volumes. It could be 2D objects in the case of 2D FEM analysis or 3D objects in the case of 3D analysis. The variation of the magnetic potential throughout the motor is expressed by non-linear differential equations in finite element analysis. These differential equations are derived from Maxwell equations and written in terms of vector potential where the important field quantities such as flux, flux direction and flux density

The FEM can accurately analyze the magnetic systems which involve permanent magnets of any shape and material. There is no need to calculate the inductances, reluctances and torque values using circuit type analytical methods because these values can simply be extracted from the finite element analysis. Another important advantage of using FEM over analytical approach is the ability to calculate the torque variations or torque components such as cogging torque, ripple torque, pulsating torque and average torque accurately

There are various FEA packages used for motor analysis. FEA packages have 3 main mechanisms which are pre-processor, field solver and post-processor. Model creation, material assignments and boundary condition set-up are all completed in the pre-processor part of the software. Field solver part has 4 main steps to solve the numerical problems. After the pre-process, the software generates the mesh, which is the most important part of getting accurate results. User's experience in generating the mesh has also an important effect on the accuracy of the results. Then, the FEA package computes the magnetic field, performs some analysis such as flux, torque, force and inductance, and checks if the error criteria have been met. If not, it refines the mesh and follows the same steps based on the user's inputs until it reaches the specified error limit. This procedure is shown in Fig. 12. In the post-processor, magnetic field quantities are displayed and some quantities such as

Permanent magnet servomotors are widely used in many industrial applications for their small size, higher efficiency, noise-free operation, high speed range and better control. This makes quality of their torque an important issue in wide range of applications including servo applications. For example, servomotors used in defense applications, robotics, servo

One of the most important issues in PM servomotors is the pulsating torque component which is inherent in motor design. If a quality work is not completed during the design stage, this component can lead to mechanical vibrations, acoustic noise, shorter life and drive system problems. In addition, if precautions are not taken, it can lead to serious control issues especially at low speeds. Minimization of the pulsating torque components is

In general, calculation of torque quality is a demanding task since the torque quality calculation does not only consider the torque density of the motor but also consider the

**6. Use of finite element analysis in PM servomotors** 

can be determined.

without too much effort.

**7. Torque quality** 

force, torque, flux, inductance etc. are all calculated.

systems, electric vehicles all require smooth torque operation.

of great importance in the design of permanent magnet servomotors.

$$
\sigma\_d = r\_s \dot{\mathbf{i}}\_d + \mathbf{L}\_d \frac{d\dot{\mathbf{i}}\_d}{dt} - \alpha \mathbf{e}\_e \mathbf{L}\_q \dot{\mathbf{i}}\_q \tag{2}
$$

$$T\_m = \frac{\Re}{2} \frac{P}{2} \left[ \mathcal{A}\_f \dot{i}\_q + \left( L\_d - L\_q \right) \dot{i}\_d \dot{i}\_q \right] \tag{3}$$

where *vd* and *vq* are d and q axis voltages, *id* and *iq* are d and q-axis currents, *rs* is stator resistance, *Ld* and *Lq* are d-q axis inductances, ωe is synchronous speed, λ*f* is magnet flux linkage, *P* is pole number and *Tm* is the electromagnetic torque of the motor. During constant flux operation, *id* becomes zero and the torque equation becomes

$$T\_m = \frac{\mathfrak{Z}}{2} \frac{P}{2} \mathcal{A}\_f i\_q = \mathcal{K}\_T i\_q \tag{4}$$

where *KT* is the torque constant of the motor. This equation becomes similar to standard DC motor and therefore provides ease of control. The torque dynamic equation is

$$T\_m = T\_L + Bo\_m + f \frac{do\_m}{dt} \tag{5}$$

equivalent circuit of the steady-state operation of the PM servomotors in d-q reference model is shown in Fig. 11. Independent control of both q-axis and d-axis components of the currents is possible with the vector controlled PM servomotors. Both voltage controlled and current controlled inverters are possible to drive the motor.

Fig. 11. Equivalent circuit of PM servomotors (a) d-axis equivalent circuit and (b) q-axis equivalent circuit

Following assumptions are made for the analysis of PM servomotors: The inverter is ideal with no losses; no DC voltage ripple exists in the DC link; the supply current is sinusoidal and no saturation is considered; eddy current and hysteresis loses are negligible; and all motor parameters are constant. Based on these assumptions, permanent magnet servomotor

> *q q s q q e dd e f di v ri L L i dt*

 <sup>3</sup> 2 2 *m f q d qd <sup>q</sup> <sup>P</sup> T i L L ii* 

where *vd* and *vq* are d and q axis voltages, *id* and *iq* are d and q-axis currents, *rs* is stator resistance, *Ld* and *Lq* are d-q axis inductances, ωe is synchronous speed, λ*f* is magnet flux linkage, *P* is pole number and *Tm* is the electromagnetic torque of the motor. During

> 2 2 *m fq Tq <sup>P</sup> T i Ki*

where *KT* is the torque constant of the motor. This equation becomes similar to standard DC

*<sup>m</sup> mL m <sup>d</sup> T TB J dt*

equivalent circuit of the steady-state operation of the PM servomotors in d-q reference model is shown in Fig. 11. Independent control of both q-axis and d-axis components of the currents is possible with the vector controlled PM servomotors. Both voltage controlled and

(a) (b)

Fig. 11. Equivalent circuit of PM servomotors (a) d-axis equivalent circuit and (b) q-axis

 

*d d sd d e q q di v ri L L i dt* 

  (1)

(2)

(4)

(5)

(3)

dynamic equations in the synchronous rotating reference frame are written as

constant flux operation, *id* becomes zero and the torque equation becomes

motor and therefore provides ease of control. The torque dynamic equation is

current controlled inverters are possible to drive the motor.

equivalent circuit

3

**5. Dynamic model of PM servomotors** 
