Electric Motor Control (PDF)

CHAPTER Control of direct current motors 2 Direct current (DC) motors have been widely used in adjustable speed drives or variable torque controls because their torque is easy to control and their range of speed co...

CHAPTER Control of direct current motors 2 Direct current (DC) motors have been widely used in adjustable speed drives or variable torque controls because their torque is easy to control and their range of speed control is wide. However, DC motors have a major drawback in which they need mechanical devices such as commutators and brushes for a continuous rota- tion. These mechanical parts require regular maintenance and hinder high-speed operations. Recently due to a major development in alternating current (AC) motor control technology, power electronics technology, and microprocessors, AC motors have been used instead of DC motors, but DC motors are still used in lower power ratings below several kilowatts. In this chapter, we will study the current and the speed control of DC motors. Understanding the current and speed control techniques for DC motors is very important because these techniques are directly applicable to the AC motor control, which will be discussed in Chapter 6. 2.1 CONFIGURATION OF DIRECT CURRENT MOTORS A DC motor has two windings as shown in Fig. 2.1. One is found in the stator and produces a field flux. This winding is called a field winding. The other one resides in the rotor and is called an armature winding. The armature current inter- acts with the stator field flux to produce torque on the rotor shaft. Small DC motors commonly have a permanent magnet instead of a field winding to produce the stator flux. A DC motor also has a mechanical commutation device consisting of brushes and commutators, which converts the DC current given from the DC power supply to AC current in the armature winding. The DC motors can be mainly classified into two different categories: a separately excited DC motor and a self-excited DC motor. In a separately excited DC motor, the armature winding and field winding each have separate DC power supplies, while in a self-excited DC motor, the two windings share one DC power supply. The self-excited DC motors can also be categorized into a shunt motor, series motor, and compound motor depending on the ways in which the two windings are connected to the DC power supply. Nowadays the DC motors mostly use a Electric Motor Control. DOI: http://dx.doi.org/10.1016/B978-0-12-812138-2.00002-7 © 2017 Elsevier Inc. All rights reserved. 39 40 CHAPTER 2 Control of direct current motors FIGURE 2.1 DC motor configuration. F N N F Alternate S S F – i – i + F + DC power source DC power source FIGURE 2.2 Force on a one-turn coil. permanent magnet to produce the field flux and the DC power supply thus applies only to the armature winding. Now we will learn about the roles of brushes and commutators, which are essential for the continuous rotation of DC motors. First, we will take a look at Fig. 2.2, which shows a one-turn coil in a rotor winding. When the coil is connected to the DC power supply, a current flows in the coil. We will assume that, in the ini- tial stage, the flux distribution and the current flowing in the coil are formed like the figure on the left. In this condition, the force on the conductors on both sides makes the coil rotate in the clockwise direction according to the direction of the force given by Fleming’s left-hand rule. When the coil rotates and comes to the position as in the figure on the right, the force produced on the conductors on both sides will return the coil again to its initial position as shown in the left side of Fig. 2.2. Since the force produced on the coil is not produced continuously in one direction, the  2.1 Configuration of Direct Current Motors 41 F N S i F – + Coil current i 0 t N N Torque Te S S – – + 0 t + F N S Commutator i F – + Brush FIGURE 2.3 Continuous rotation by the use of brushes and commutators. coil also cannot rotate in the same direction continuously. However, when the coil is at the position shown on the right, if the direction of its current is reversed, then the force produced on the conductors on both sides is reversed and thus, the force on the coil will remain in the same direction of clockwise. Thus the coil will rotate continuously in that direction. By the commutation actions of the brush and the commutator, the coil current can be reversed as shown in Fig. 2.3. It should be noted that the current that flows in the coil must be an alternating current in DC motors. The frequency of this alternating current is always the same as the angular velocity of the coil, which indicates the speed of the rotor. In Section 1.2.3, we have already seen that this meets the necessary condition for the continuous torque production in the doubly fed machine. The armature winding of the DC motor has many coils to increase the aver- age torque and reduce torque ripple as shown in Fig. 2.4. Each coil is connected to the DC power supply through a brush and its own isolated commutator. 42 CHAPTER 2 Control of direct current motors Brush Torque ia T θ Coils Commutators FIGURE 2.4 Armature winding. As described earlier, in order for a DC motor to rotate continuously, it must have brushes and commutators in its structure. However, due to the wear and the arcing associated with their brushes and commutators, DC motors require much more maintenance and are limited in high-speed operation. 2.2 MODELING OF DIRECT CURRENT MOTORS A complete dynamic model of the DC motor drive system can be expressed as the following four equations: armature circuit, back-electromotive force (back-EMF), torque, and mechanical load system. 2.2.1 ARMATURE CIRCUIT The voltage equation for the armature winding of a DC motor is given by dia Va 5 Ra ia 1 La 1 ea (2.1) dt where ia is winding current, La is winding inductance, Ra is winding resistance, and ea is back-EMF induced by the rotation of the armature winding in a magnetic field. The equivalent circuit of a DC motor derived from Eq. (2.1) is shown in Fig. 2.5. We need a back-EMF ea to obtain the current flowing through the armature winding for a voltage Va applied to a DC motor from Eq. (2.1). 2.2.2 BACK-ELECTROMOTIVE FORCE As shown in Fig. 2.6, when a conductor of length l meters is moving with a constant velocity v in the presence of a uniform magnetic field B, a voltage e 5 Blv (called back-EMF) will be induced in the conductor. Similarly, in a DC motor, when the conductors of the armature winding with an angular velocity ωm  2.2 Modeling of Direct Current Motors 43 Armature Armature Ra La resistance inductance Ia Va I a Armature Te Field flux current φf Armature E a Back-EMF Va voltage ωm ωm Te FIGURE 2.5 Equivalent circuit of a DC motor. Motion of conductor Magnetic field v B v S B l e N e e=Blv Induced voltage FIGURE 2.6 Induced voltage for a moving conductor. are moving in the presence of the uniform stator magnetic flux φf , the back-EMF in the armature winding will be expressed as e a 5 k e φf ω m ð’e 5 BlvÞ (2.2) where ke is the back-EMF constant (Vs/rad/Wb). Commonly, since the flux remains constant, the back-EMF is proportional to the angular velocity. With the armature current ia calculated from the applied input voltage Va and the back-EMF ea , we can obtain the developed torque of a DC motor as the following. 2.2.3 TORQUE As shown in Fig. 2.7, when a conductor of length l meters carrying a current of i amps is placed in a uniform magnetic field B, the force acting on the conductor 44 CHAPTER 2 Control of direct current motors Force F=Bil F Magnetic field F B S B N l i i Current FIGURE 2.7 Force for a current carrying conductor. will be F 5 Bli. Similarly, in a DC motor, the torque experienced by the conductors of the armature winding carrying a current ia in the uniform stator magnetic flux φf is given by Te 5 kT  φf  ia ð’F 5 BliÞ (2.3) where kT is the torque constant (Nm/Wb/A). The numerical values of kT and ke constants are equal in SI units (the International System of Units), i.e., kT 5 ke. It is interesting to note that the torque of a DC motor can be expressed as a simple arithmetical product of the stator magnetic flux φf and the armature cur- rent ia. This is because the stator magnetic flux φf and the conductors carrying current ia are always perpendicular to each other. Thus the DC motors inherently have a maximum torque per ampere characteristic. The speed of a rotor can be determined by the developed torque as the following. 2.2.4 MECHANICAL LOAD SYSTEM When a DC motor with the output torque in Eq. (2.3) is driving a mechanical load system, the rotor speed can be determined from the equation of motion in Eq. (2.4) as mentioned in Chapter 1. dωm Te 5 J 1 Bωm 1 TL (2.4) dt where ωm is the angular velocity of the rotor, TL is the load torque, J is the moment inertia and B is the friction coefficient of the DC motor drive system, respectively. From the above dynamic model represented by Eqs. (2.1)(2.4), we can under- stand the steady-state as well as the transient characteristics of the current and the speed in the DC motor drive system. First, on the basis of this model, we will exam- ine the speedtorque characteristic of the DC motor in the steady-state condition.  2.3 Steady-State Characteristics of Direct Current Motors 45 2.3 STEADY-STATE CHARACTERISTICS OF DIRECT CURRENT MOTORS The term “steady-state” refers to a condition that exists after all initial transients or fluctuating conditions have damped out. The steady-state behavior of the DC motor is easily determined by making the time derivatives of the variables in Eqs. (2.1) and (2.4) zero, i.e., dia =dt 5 0, dωm =dt 5 0. From the steady-state equations, the speedtorque relation may be written as Va Ra ωm 5 2 TL ðk 5 kT 5 ke Þ (2.5) kφf ðkφf Þ2 Here, the friction coefficient B is neglected. We can see that, from this torquespeed relation, the speed of the DC motor decreases steadily with increasing load torque (or armature current) as shown in Fig. 2.8. Here, N is the speed expressed in terms of revolution per minute (r/min) for angular speed ωm. From the curve, we can find the speed and the current at a certain load. Eq. (2.5) implies that the steady-state speed of a DC motor can be adjusted by the armature voltage Va or the stator field flux φf. Thus there are two speed control methods for the DC motor: armature voltage control and field flux control. In addition to the two methods, we can see from Eq. (2.5) that the speed can be controlled by varying the armature resistance Ra. However, this inefficient method is no longer in use. Speed N No-load speed Current Ia Speed–torque Torque–current curve curve Rated speed Rated current Ra Slope= (Kφ f )2 No-load current Torque TL Rated torque FIGURE 2.8 Speedtorque characteristic of a DC motor in the steady-state. 46 CHAPTER 2 Control of direct current motors Now, we will take a look at the speed control characteristics of the DC motor by armature voltage control and field flux control. 2.3.1 ARMATURE VOLTAGE CONTROL With a constant stator field flux, Eq. (2.5) expressing the torquespeed characteristic can be reduced to Va Ra ωm 5 2 TL 5 K1 Va 2 K2 TL (2.6) kφf ðkφf Þ2 where K1 5 Va =kφf and K2 5 Ra =ðkφf Þ2 From this expression, we can see that the speed ωm of the DC motor changes linearly with the armature voltage Va. Thus, for a constant torque load such as elevators and cranes, the speed of the load can be controlled linearly by adjusting the armature voltage Va of the DC motor as shown in Fig. 2.9. To employ the armature voltage control, variable DC power supplies such as phase-controlled rectifiers or choppers are needed. Because a voltage higher than the rated voltage should not be applied to the motor, this armature voltage control method can provide a speed control range only up to the rated speed. To operate in a speed range above the rated speed, the DC motor has to adopt the field flux control, which will be described later. Permanent magnet DC motors, however, cannot use the field flux control method, so they are incapable of an operation above the rated speed. N Applied voltage No-load speed = rated voltage under rated voltage Torque for rated current Rated speed Ia Speed Applied increase φf + voltage Voltage Ea Va increase – Operating ωm points TL Full torque FIGURE 2.9 Armature voltage control.  2.3 Steady-State Characteristics of Direct Current Motors 47 2.3.2 FIELD FLUX CONTROL Under a constant armature voltage, Eq. (2.5) can be rewritten as a function of field flux: Va Ra K1 K2 ωm 5 2 TL 5 2 2 TL (2.7) kφf ðKφf Þ2 φf φf where K1 5 Va =k and K2 5 Ra =k2. From Eq. (2.7), we can see that the speed of the DC motor varies inversely with the field flux. Fig. 2.10 shows the torquespeed characteristic of the DC motor using the field flux control. When using this field flux control method, the field flux has to be controlled below the rated value to avoid magnetic saturation of iron core because the rated flux is commonly given as a value near the knee of the mag- netization curve as shown in the left side of Fig. 2.10. If the field flux is reduced below the rated value, then the speed can increase above the rated speed. When the speed of the DC motor can be controlled above the rated value by adjusting the field flux control, there are potential problems that can arise. When the field flux is reduced to increase the speed, the developed torque will also be reduced for the same armature current. If the armature current is increased to pro- duce the same torque, then the efficiency will be lowered due to the increased copper losses. The armature current also may not be fully increased due to the thermal limit and the capacity of the DC power supply. Moreover the inductance of the field winding is normally designed to be large to produce the flux effec- tively. Thus the field flux cannot be changed rapidly, and this will result in a slow response of the speed regulation. In addition, in a high-speed range, the speed regulation will be poor for the applied load, and the resultant lower field N Flux decrease Torque for rated current Field current and flux φf Rated flux Torque-speed curve Speed if for rated flux increase Full load torque TL FIGURE 2.10 Field flux control. 48 CHAPTER 2 Control of direct current motors flux will be easily influenced by the armature reaction. Besides these electrical problems, the DC motors also suffer from speed limitations due to mechanical commutation devices. On this account, the achievable speed range of this field flux control is generally limited up to three times the rated speed. 2.3.3 OPERATION REGIONS OF DIRECT CURRENT MOTORS The operating speed range of the DC motor is divided into two regions as follows: constant torque region and constant power region. The operation in these two regions can be achieved through the armature voltage control and the field flux control. 2.3.3.1 Constant torque region ( 0 # ωm # base speed ωb ) In the operating region below the rated speed, the field flux is maintained at a constant and the torque production is controlled only by the armature current Ia. This region is called the constant torque region because the same torque can be developed by the same armature current at any speed within this region. As the operating speed is increased, the back-EMF is increased. Thus the applied motor voltage should also be increased to maintain the same armature current, i.e., the same torque. Like this, in this region, the terminal voltage of the DC motor varies with its speed. Because the voltage applied to the motor should not exceed the rated value, the range of this region is normally limited to the speed at which the terminal voltage of the DC motor reaches the rated value. This speed may be different from the rated speed depending on the operating conditions. Thus it is usually referred to as the base speed ωb. 2.3.3.2 Constant power region (ωm $ base speed): field-weakening region Once the motor voltage reaches the rated value, the speed can no longer be controlled by the voltage. At speeds higher than the base speed, the applied voltage remains constant. Thus, instead of the voltage control, the speed can be increased by the reduction of the field flux (is called field-weakening operation). The reduction of the field flux can make the back-EMF constant regardless of the speed increase. This allows the armature current, which produces a torque, to flow. However, even though the same current is flowing, the output torque will be decreased with the operating speed due to the reduced field flux. Because the terminal voltage and the current remain constant in this region, the motor operates in a constant power, P 5 ðVa UIa 5 Te Uωm Þ. Thus this speed region is called the constant power region. Fig. 2.11 shows the speedtorque characteristic of the DC motor, which is commonly called the capability curve. The capability curve demonstrates the achievable speed range and the output torque of a motor under the available voltage and current. For DC motor drives with the four-quadrant operation modes as described in Section 1.3.2, this speedtorque characteristic appears in all four quadrants. Considering the voltage drop in armature resistance and losses, the out- put torque in Quadrants 2 and 4 are somewhat larger than those in Quadrants 1  2.3 Steady-State Characteristics of Direct Current Motors 49 Voltage control Field flux control Constant torque region Constant power region Torque Power Voltage Speed limit Current Flux ωm 0 ω rated ωmax = 3ωrated FIGURE 2.11 Capability curve of the DC motor. and 3. For the four-quadrant operation, the direction of the output torque needs to be reversed to change the direction of the rotor rotation. In this case, changing the polarity of the armature current will be more effective. Even though the direction of the torque may be changed by reversing the polarity of field flux, its response is slow due to a large inductance of the field winding, and thus the stability of the drive system may be deteriorated. EXAMPLE 1 The rated values for a separately excited DC motor are 5 kW, 140 V, 1800 r/min and the efficiency η is 91% (the losses such as mechanical losses and iron losses are ignored). The armature resistance is 0.25 Ω. Evaluate the following questions. 1. the armature current and the rated torque 2. the speed at full-load when the armature voltage is reduced by 10% 3. the speed at full-load current when the flux is reduced by 10% and the output torque available by armature current at this speed Solution 1. From the relation of the input and output powers as Pin Va UIa Pout 5 5 : η η the armature current is Pout 5 kW Ia 5 5 5 44:96 A: Va Uη 125 VU0:91 From the output power Pout 5 Te Uωm , the rated torque is 50 CHAPTER 2 Control of direct current motors Pout 3150 W Te 5 5
 5 26:5 N m: ωm 3000  2π 60 2. The armature voltage reduced by 10%, Vnew , is 112.5V. At this voltage, the speed is Ea Vnew 2 Ra Ia 101:25 ωm 5 5 5 kφf kφf 0:60344 : 5 167:8 rad=s ð 5 1602:2 r=minÞ Here, kφf 5 Ea =ωm 5 ðVnew 2 Ra Ia Þ=ωm 5 0:60344. 3. The speed at the flux reduced by 10% is Ea ðVnew 2 Ra Ia Þ ωm 5 5 0:9kφf 0:9kφf 113:75 5 5 209:44 rad=sð 5 2000 r=minÞ ð0:9  0:60344Þ At this speed, the output torque available by armature current is Te@2000 5 0:9kφf Ia 5 23:87N m: 2.4 TRANSIENT RESPONSE CHARACTERISTICS OF DIRECT CURRENT MOTORS In Section 2.3, we studied the steady-state characteristics of DC motors, which demonstrate the speed control by the armature voltage or the field flux. The steady-state characteristic shows what the final speed is, whereas the transient response shows how to reach the final speed. In addition to the steady-state character- istic, the transient characteristic is very important for high-performance motor drives. This is because the transient response characterizes the dynamics of the drive system. As an example, transient responses to a step speed command are shown in Fig. 2.12. Even though all three responses reach the same final speed, their response times and moving trajectories to reach the final speed are different. Although an oscillatory or slow response can ultimately achieve the speed command, it has no practical use. Therefore we need to take into account the transient characteristic as well as the steady-state characteristic for high-performance motor drives. Now, let us take a look at the transient response of reaching the final speed by the applied armature voltage for a DC motor. An easy way to examine the transient response is to use the transfer function of the system. We can obtain the  2.4 Transient Response Characteristics of Direct Current Motors 51 Speed Oscillatory response Command ωm Adequate response Slow response Time FIGURE 2.12 Transient response to a step speed command. TL − Va + 1 Ia Te + 1 ωm kT φ f La s + Ra Js + B − Ea Load DC motor ke φ f FIGURE 2.13 Block diagram for a DC motor. transfer function for the DC motor drive system by taking the Laplace transform of the differential equations of Eq. (2.1) and (2.4) as follows. dia va 5 Ra ia 1 La 1 ea - Va ðsÞ 5 ðRa 1 sLa ÞIa ðsÞ 1 Ea ðsÞ dt (2.8) 5 ðRa 1 sLa ÞIa ðsÞ 1 ke φf ωm ðsÞ dωm Te 5 J 1 Bωm 1 TL - Te ðsÞ 5 ðJs 1 BÞωm ðsÞ 1 TL 5 kT φf Ia ðsÞ (2.9) dt Here, s denotes the Laplace operator. Eqs. (2.8) and (2.9) together can be represented by the closed-loop block diagram as shown in Fig. 2.13. From the block diagram in Fig. 2.13, the transfer function from the input armature voltage Va ðsÞ to the output angular velocity ωm ðsÞ is given by 1 1  ke φf  ωm ðsÞ La s 1 Ra Js 1 B 5 ! Va ðsÞ 1 1 11  k e φf  kT φf La s 1 Ra Js 1 B (2.10) k JLa 5

 ðk 5 kT φf 5 ke φf Þ Ra B Ra B k2 s2 1 1 s1 1 La J La J JLa 52 CHAPTER 2 Control of direct current motors Stable region Im(s) Unstable region Command Underdamping Critical damping Overdamping Re(s) FIGURE 2.14 System pole locations and the corresponding transient responses. This transfer function is a second-order system including two poles, which are important factors in determining the transient response of the system. Fig. 2.14 shows the pole locations of a system in the s-plane and the corre- sponding transient responses. The imaginary axis location of the pole deter- mines the frequency of oscillations in the response. Thus, if the poles are complex roots, the response is oscillatory. The real axis location of the pole determines the speed of the exponential decay of the transient situation. Therefore, the farther the distance of the pole from the origin is to the left in the s-plane, the faster the transient situation disappears, i.e., the system reaches the final value faster. CLOSED-LOOP TRANSFER FUNCTION A system with a feedback loop has the following transfer function: YðsÞ GðsÞ 5 RðsÞ 1 1 GðsÞHðsÞ R(s) Y(s) G(s) + – H(s)  2.4 Transient Response Characteristics of Direct Current Motors 53 From the locations of the two poles in Eq. (2.10), we can predict the transient response of the speed to the applied armature voltage. Thus the transient response depends heavily on the system parameters such as motor parameters Ra and La and mechanical parameters J and B, as shown in the following. Assuming that B 5 0 in Eq. (2.10) to analyze the system easily, we can obtain a simplified transfer function as
 k 1 Ra k2 U ωm JL k L JR 5
a
25
a
a  (2.11) Va Ra k Ra Ra k2 s 1 2 s1 s 1 2 s1 U La JLa La La JRa Letting the electromechanical time constant Tm 5 JRa =k2 and the electrical time constant Ta 5 La =Ra , Eq. (2.11) can be rewritten as
 1 1 ωm k Tm Ta 5 (2.12) Va 1 1 s2 1 s 1 Ta Tm Ta The poles can be obtained from the roots of the denominator (also called the characteristic equation) of Eq. (2.12) as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 Ta s1;2 5 2 6 2 (2.13) 2Ta Ta 4 Tm If Tm $ 4Ta , then the characteristic equation has two real roots. A DC motor drive system with a large J or Ra corresponds to this case, and its response is either ➀ or ➁ in Fig. 2.14. If Tm , 4Ta , then the characteristic equation has two complex roots. In this case, the system response is oscillatory such as ➂ in Fig. 2.14. We can also find out the response characteristic from the damping ratio for this second-order system. To do so, we compare this equation with the following prototype of the second-order system. ω2n GðsÞ 5 (2.14) s2 1 2ζωn s 1 ω2n Given Eq. (2.14), the undamped natural frequency ωn and the damping ratio ζ for the DC motor drive system are given by 1 k ωn 5 pffiffiffiffiffiffiffiffiffiffi 5 pffiffiffiffiffiffiffi (2.15) Ta Tm JLa rffiffiffiffiffiffi rffiffiffiffiffi 1 Tm 1 Ra J ζ5 5 (2.16) 2 Ta 2 k La 54 CHAPTER 2 Control of direct current motors Speed Speed Speed Command Under damping Critical damping Over damping t t t (A) Underdamping (ζ 1) FIGURE 2.15 System responses according to the damping ratio. (A) Underdamping (ζ , 1), (B) critical damping (ζ 5 1), and (C) overdamping (ζ. 1). There are three typical responses of the system according to the value of the damping ratio ζ: underdamping (0,ζ ,1), critical damping (ζ 5 1), or overdamp- ing (ζ.1). As shown in Fig. 2.15, in the case of underdamping (0,ζ ,1), the system comes rapidly to equilibrium but has a slight oscillatory response. As ζ is increased, the oscillation is reduced. In the case of overdamping (ζ.1), the sys- tem is not oscillatory but has a very slow response. A critically damped system response (ζ 51), which comes to equilibrium as fast as possible without oscillat- ing, is the most desirable. Eq. (2.16) implies that the damping ratio ζ, which implies the dynamic perfor- mance of the speed to the applied armature voltage, depends on the electrome- chanical time constant value Tm and the electrical time constant value Ta. If Tm ,4Ta , then ζ ,1 and its speed response will be oscillatory. By contrast, in the case of Tm $4Ta , the system will have a stable speed response without oscillation. Normally, small motors have Tm.4Ta , while servo motors or high power motors have Tm ,4Ta , resulting in an oscillatory response. Even though the system response is inherently oscillatory, we can achieve the desired response by adopting an adequate controller, which will be described later. In most motor drive systems, the time response of an electric system is faster than that of a mechanical system. In that case, we can ignore the transient of the electric circuit so that La 5 0. Thus Eq. (2.11) can be written as
 ωm k 1 ωc k2 5 5 ω c 5 (2.17) Va Ra Js 1 K 2 k s 1 ωc Ra J This equation implies that, in a DC motor, the relationship between the speed ωm and the armature voltage Va can be considered as a first-order, low-pass filter with a cut-off frequency ωc. This means that the speed ωm is directly proportional to the armature voltage Va with its changing rate below the ωc , though there is a time delay between the two. Generally it can be assumed that the rotational speed of a DC motor is proportional to the voltage applied to it.  2.4 Transient Response Characteristics of Direct Current Motors 55 EXAMPLE 2 Consider a separately excited DC motor with the following parameters. The armature resistance is 0.28 Ω, the armature inductance is 1.7 mH, the inertia of moment J of this motor is 0.00252, and kφf 5 0:4078. 1. Calculate the locations of the two poles and predict the transient response of the speed to an armature voltage at no-load. 2. Predict the transient response when this motor is driving the load with the inertia of moment of 6J. Solution 1. From Eq. (2.13), the two poles are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 Ta s1;2 5 2 6 2 5 2 76:47 6 j181:57: 2Ta Ta 4 Tm Since the system has two complex poles, we can expect an oscillatory system response. Also, from Eq. (2.16), the damping ratio is rffiffiffiffiffiffi rffiffiffiffiffi 1 Tm 1 Ra J ζ5 5 5 0:3881: 2 Ta 2 k La Because ζ , 1, we can expect the system to have an under-damped response and an overshoot. From Figure A, we can see the oscillatory speed response of this system to the step applied voltages. 2. When this motor is driving the load with the inertia of moment of 6J, the system has two real poles of 2152 and 20.0002. Thus the system response is not oscillatory. Also, since ζ 5 1.0269, we can expect the critically damped system response to be as Figure B. A B 4000 4000 3000 3000 Speed (r/min) 2000 2000 1000 1000 0 0 –500 –500 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Time (s) Time (s) 56 CHAPTER 2 Control of direct current motors 2.5 MOTOR CONTROL SYSTEM An electric motor is an electromechanical device that converts electrical energy into mechanical torque. As a torque-producing device, the electric motor needs to have its torque controlled as a priority. When the torque produced by the motor is applied to a mechanical load, the torque will change the speed ωm of the driven load and, in turn, change its position θm as shown in Fig. 2.16. Therefore, in motor drives, the most effective way for controlling the position or speed of the load connected to the motor is to control the torque of the motor directly. However, the torque control is expensive due to the high cost of the torque sensor for motor torque feedback. We can see from Eq. (2.3) that the torque is directly proportional to the current under a constant field flux. This tells us that we can control the torque by using the current instead of the torque itself. Thus the current control is important for the speed or position control of a motor. 2.5.1 CONFIGURATION OF CONTROL SYSTEM Fig. 2.17 illustrates the typical configuration of controllers in the motor drive systems, in which the controllers required for position, speed and current control are usually connected in a cascade. Among these, the current controller must be placed at the most inner loop and has the widest bandwidth. As mentioned above, since the speed and position can be controlled through the current control, the response time for the current control must be the fastest among the three. Likewise, the response for the control of the inner loop should be sufficiently faster than that of the outer loop to improve the response performance and the stability of the outer control loop. For example, the bandwidth of the current loop needed to be at least five times wider than that of the speed loop to ensure that the current control will not have any influence on the speed control performance. Voltage Torque Speed Position Load ωm θm Motor Te ∝ φ i a θm = ∫ ωm dt dωm Te = ( J M + J L ) + TL dt FIGURE 2.16 The motor drive system.  2.5 Motor Control System 57 Control system Power source Speed Current Position Voltage command command command command Position Speed Current Converter + controller + controller + controller – – – Current Current sensor Speed Position Position sensor Motor Load FIGURE 2.17 Configuration of the motor control system. Reference System input System output R(s) U(s) Y(s) G p−1 ( s ) Gs (s) System FIGURE 2.18 Open-loop control. 2.5.2 TYPES OF CONTROL The aim of control is to adjust the control input so that the state or output of the system reaches the required goal. In the motor drive systems, the state or output may be the current, flux, speed, position, or torque, while the control input is only the input voltage. For control performance, the ability of reference tracking, i.e., the system output follows the reference input as accurately as possible, is the most important factor. The control system is also needed to have the disturbance rejection capability while tracking the reference input. This means that no distur- bance can affect the system tracking the reference input as much. There are two basic types of control: open-loop control and closed-loop control, also known as nonfeedback and feedback, respectively. 2.5.2.1 Open-loop control (or nonfeedback control) Open-loop control is a type of control that determines the control input UðsÞ of a system by using the system model Gs ðsÞ as shown in Fig. 2.18. A drawback of the open-loop control is that it requires perfect knowledge of the system. Since the open-loop control does not use the feedback of the system output YðsÞ, the error between the reference RðsÞ and the output YðsÞ cannot be reflected in determining its control input UðsÞ. Therefore its output may be 58 CHAPTER 2 Control of direct current motors different from the desired goal due to the change in system parameters or disturbances. Because of its simplicity and low cost, the open-loop control is often used in systems where it has the well-defined relationship between input and output and are not influenced by disturbances. However, this open-loop control is inadequate for the high-performance motor drive systems, which are easily affected by parameter variations and disturbances such as the back-EMF and load torque. 2.5.2.2 Closed-loop control (or feedback control) Closed-loop control is a type of control that adjusts the control input UðsÞ by the feedback of the output YðsÞ, as shown in Fig. 2.19. The closed-loop control is generally used to control the position, speed, current, or flux in the motor drive systems. The controller Gc ðsÞ adjusts its control input UðsÞ to reduce the error between the output and the desired goal. Thus the role of the controller Gc ðsÞ in the closed- loop control is very critical to the system’s performance. There are several types of widely used types of the feedback controller in the motor drive systems as follows. 2.5.2.2.1 Feedback control Proportional controller (P controller) In this type of controller, which is illustrated in Fig. 2.20, the controller output is proportional to the present error eðtÞð 5 rðtÞ 2 yðtÞÞ as given by UðtÞ 5 Kp eðtÞ (2.18) where Kp represents the proportional gain. Reference Error System input System output R(s) E(s) U(s) Y(s) Gc (s) Gs (s) + − Controller System Sensor Measured output FIGURE 2.19 Closed-loop control. R(s) E(s) U(s) Y(s) Kp Gs (s) + − P controller System FIGURE 2.20 Proportional control.  2.5 Motor Control System 59 An advantage of the proportional control is its immediate response to errors. The larger the proportional gain is, the faster the response is. Thus it should be noted that the proportional gain determines the control bandwidth. However, there is a limit to increasing the proportional gain because a large gain can lead to system instability or may require unrealizable control input. A drawback of the proportional control is that the error can never be zero. In other words, there exists a steady-state error in the P control. This is because when the error comes to zero, the output of the P controller becomes zero. Thus the system will have an error again. We can evaluate the steady-state error of the P controller from its close-loop transfer function of the block diagram in Fig. 2.20 as, RðsÞ Go ðsÞ Kp Gp ðsÞ 5 5 (2.19) YðsÞ 1 1 Go ðsÞ 1 1 Kp Gp ðsÞ where Go ðsÞð 5 Kp Gp ðsÞÞ is the open-loop transfer function of the system including the proportional controller. From Eq. (2.19), if the proportional gain Kp is very large, then RðsÞ=YðsÞ will be close to one and the error will be very small, but it will never reach zero. Integral controller (I controller) In this type of controller, the controller output is the integral of the error as shown in Fig. 2.21, so the past errors as well as the present error have an effect on the control input as given by ð ð Kp UðtÞ 5 Ki eðtÞdt 5 eðtÞdt (2.20) Tpi where Ki represents the integral gain and Tpi ð 5 Kp =Ki Þ represents the integral time constant. The integral controller can remove the steady-state error completely for the non-time varying reference RðsÞ. This is because it continuously produces nonzero control input from the accumulated past errors even though the present error is zero. However, due to the accumulated errors in the integral term, it responds slowly to change in errors. For this reason, the integral controller is not normally used alone but is combined with the P controller. Proportionalintegral controller A widely used controller in motor drive systems is the proportionalintegral controller (PI controller), which is a combination of a proportional controller and an integral controller to achieve both a fast response and a zero steady-state error. The block diagram of a proportionalintegral controller is shown in Fig. 2.22. R(s) E(s) Ki U(s) Y(s) Gs (s) + − s I controller System FIGURE 2.21 Integral control. 60 CHAPTER 2 Control of direct current motors R(s) E(s) + U(s) Y(s) Kp Gs (s) + + − System Ki PI controller s Gc (s) FIGURE 2.22 Proportionalintegral control. Error e(t) t Control input PI controller u(t) ∫ I controller K i e(t ) dt P controller K p e(t) t FIGURE 2.23 Control inputs for an error. Fig. 2.23 compares the control inputs for an error in the three controllers described earlier. 2.5.2.3 Feedforward control Disturbances are always present in the control process. In motor drive systems, e.g., the back-EMF acts as a disturbance on the current control, and the load torque acts as a disturbance on the speed and position control. We can see from the Section 2.5.2.2.1 that the feedback control operates to reduce the error between the output and the desired goal. Thus the feedback control can react to the disturbance only after the disturbance has an effect on the output. Accordingly, the feedback control cannot achieve a fast response to the disturbance. Feedforward control is an effective way to rapidly eliminate the effect of the disturbance on the output. The feedforward control predicts and compensates for the disturbance by adding it to the control input in advance.  2.5 Motor Control System 61 Feedforward controller D(s) Disturbance U ff Ufb + − R(s) Feedback + U(s) + Y(s) G(s) + controller − System FIGURE 2.24 Feedforward control. The feedforward control is always used along with the feedback control. Fig. 2.24 shows the block diagram of a typical system with the feedback control and the feedforward control. This control system can reduce the effect of the disturbance DðsÞ on the output YðsÞ more than the system with the feedback control alone. For the feedforward control, accurate system parameters are required to predict the disturbance correctly. In motor current control systems, the performance of the current control could be significantly improved by the feedforward control, which compensates for the back-EMF that acts as a disturbance on the current control. This feedforward control for current control of DC motors and AC motors will be described in detail later. For motor speed control, the performance of the speed control could also be significantly improved by feedforward control for the load torque. 2.5.3 DESIGN CONSIDERATION OF CONTROL SYSTEM The PI controller has been widely used in motor drive systems as the controller for the current, speed, or flux linkage control because it could give a satisfactory performance. The performance of the PI controller depends heavily on its selected gains. Thus it is very important to select appropriate gains of the PI controller for obtaining a satisfactory performance in a given system. Now, we will discuss the gain selection method for the current and speed PI controllers. Through the frequency response of the whole system including the PI controller, we will determine the P and I gains for achieving the desired transient and steady-state characteristics. WHAT IS FREQUENCY RESPONSE? The output of a linear system to a sinusoidal input is a sinusoid of the same frequency but with a different amplitude and phase. The frequency response refers to a system’s open-loop responses at the steady state to sinusoidal inputs at varying frequencies as shown in the following figure. (Continued) 62 CHAPTER 2 Control of direct current motors WHAT IS FREQUENCY RESPONSE? (CONTINUED) System Sinusoidal input Steady-state output G(s) r (t ) = A sin ω t c(t ) = B sin(ω t + φ ) t φ The frequency response is used to characterize the dynamics of a system and measure the ability of the system following an input reference. The frequency response is characterized by the magnitude and phase of the system’s response as a function of frequency, which are calculated from the transfer function GðsÞ of the system as  CðsÞ     5 GðsÞs5jω 5 Gð jωÞ+φð jωÞ RðsÞ s5jω   where GðjωÞ and +φðjωÞ represent the amplitude and phase angle between the input and output signals, respectively. We can easily see the frequency response of a dynamic system through a Bode plot, which displays the magnitude (in dB) and the phase (in degrees) of the system response as a function of frequency. When designing a control system, we should consider the three specifications: stability, response time, and steady-state error. We will begin with briefly reviewing these specifications. 2.5.3.1 Stability First of all, the whole system, including the controller, must be stable. Therefore the stability is the most important requirement for the design of a control system. The response of an unstable system to a bounded input signal will continue to oscillate, but the magnitude of this oscillation will never diminish or get larger. The stability of linear systems can be assessed by checking the poles of the transfer function of the system, i.e., the roots of the characteristic equation in the s-plane as we already saw in Fig. 2.14. A linear system is stable if all the poles lie in the left-half s-plane. However, merely knowing whether a system is stable or unstable is not sufficient. We need to find out how stable the system is. We can measure the relative stability of a system by examining the gain margin and the phase margin obtained from the Bode plot of the frequency response. A brief description of the gain margin and the phase margin is presented below.  2.5 Motor Control System 63 2.5.3.1.1 Gain margin and phase margin Phase margin and gain margin are the measures of closed-loop control systems stability. They are illustrated in Fig. 2.25. 1. Gain margin Gain margin is defined as the amount of change in open-loop gain needed to make a closed-loop system unstable. The gain margin is the difference between 0 dB and the gain at the phase  cross-over  frequency that gives a phase of 2180. If the gain GHðjωÞ at the frequency of +GHðjωÞ 5 2 180 is greater than 0 dB as shown in the left of Fig. 2.25, meaning a positive gain margin, then the closed-loop system is stable. 2. Phase margin Phase margin is defined as the amount of change in open-loop phase needed to make a closed-loop system unstable. The phase margin is the dif- ference in phase between 2180 and the phase at the gain cross-over fre- quency  that gives a gain of 0 dB. If the phase +GHðjωÞ at the frequency of GHðjωÞ 5 1 is greater than 2180 as shown in the left of Fig. 2.25, meaning a positive phase margin, the closed-loop system is stable. In general, as the gain of a system increases, the system becomes less stable. The gain margin and the phase margin indicate how much the gain increases until the system becomes unstable. Let us take a look at the relationship between the gain and phase margins and the stability of a system. For example, assume that the phase  margin  of a system is zero degree, i.e., +GHðjωÞ 5 180 at a frequency ω of GðjωÞ 5 1 dB. In this case, as shown in Fig. 2.26, the phase difference between YðsÞ and RðsÞ is 180 , Stable system Unstable system |G|(dB) |G|(dB) Positive Negative gain margin gain margin 0 ω 0 ω ∠G ∠G –90° –90° –180° ω –180° ω Positive Negative phase margin phase margin FIGURE 2.25 Gain margin and phase margin. 64 CHAPTER 2 Control of direct current motors Y(s) R(s) G(s) + – FIGURE 2.26 Example of an unstable system. (A) (B) |G|(dB) Large bandwidth Fast response 0 ω –3dB Slow response Small bandwidth t FIGURE 2.27 Speed of response versus system bandwidths. (A) Frequency responses and (B) step responses. and this closed-loop system thus becomes a positive feedback system. This sys- tem will continue to amplify its output and become unstable. If a closed-loop system is stable, both the gain margin and the phase margin need to be positive. In general, the phase margin of 3060 degrees and the gain margin of 210 dB are desirable in the closed-loop system design. A system with a large gain margin and phase margin is stable but has a sluggish response, while the one with a small gain margin and phase margin has a less sluggish response but is oscillatory. 2.5.3.2 Response time/speed of response How quickly a system responds to the change in input is an important measure of the system’s dynamic performance. The speed of response of a closed-loop system may be proportional to its control bandwidth. Thus the control bandwidth is a measure of the speed of response. We can see the system responds according to its control bandwidth from Fig. 2.27, which is showing the responses to a step command versus system band- widths. Generally, the wider the system’s control bandwidth, the faster the speed of response is. However, if the control bandwidth of a system is too large, the system may be sensitive to noise and become unstable.  2.5 Motor Control System 65 This control bandwidth is mainly dependent on the proportional gain of the controller. Thus we need to adjust the P gain to obtain the desired response speed. WHAT IS BANDWIDTH? Bandwidth describes how well a system follows a sinusoidal input. We can easily find the bandwidth from the system’s closed-loop frequency response. As in the following figure, the bandwidth is the range of frequencies for which the magnitude jGj of the frequency response is greater than 3 dB, i.e., 0 # ω # ωb. The bandwidth is a measure of the feedback system’s ability to follow the input signal. Sinusoidal inputs with a frequency less than the bandwidth frequency ωb are followed reasonably well by the system, whereas sinusoidal inputs with a frequency greater than the bandwidth frequency are attenuated in magnitude by a factor of 1/10 or greater (and are also shifted in phase). The concept of the bandwidth is very important in control systems because it describes how quickly a system follows its command. |G|(dB) ωb 0 ω –3dB Bandwidth 2.5.3.3 Steady-state error A steady-state error is defined as the difference between the desired value and the actual value of a system when the response has reached the steady state. We can calculate the steady-state error of the system using the final value theorem. This theorem is given for the unit feedback system in Fig. 2.28 as s Final value theorem: eN 5 lim eðtÞ 5 lim sEðsÞ 5 lim RðsÞ (2.21) t-N s-0 s-0 1 1 GðsÞ
 RðsÞ Error: EðsÞ 5 RðsÞ 2 YðsÞ 5 1 1 GðsÞ As an example, let us evaluate the steady-state error for a step input command in the control system for a DC motor, where the speed or current command is R(s) E(s) Y(s) G(s) + – Command Controller + system FIGURE 2.28 Unit feedback system. 66 CHAPTER 2 Control of direct current motors generally given as a step signal. Assume that the control system for the DC motor is a unit feedback system as in Fig. 2.28, and the transfer function of the current control system is GðsÞ. For the unit step input ð rðtÞ 5 1-RðsÞ 5 1=sÞ to this system, the steady-state error is given from the final value theorem by s s 1 1 eN 5 lim RðsÞ 5 lim 5 lim (2.22) s-0 1 1 GðsÞ s-0 1 1 GðsÞ s s-0 1 1 GðsÞ The steady-state error depends on the system type (0, I, or II). First, suppose a control system of type 0 expressed as GðsÞ 5 K=ðTs 1 1Þ. In this system the steady-state error is given as 1 1 eN 5 5 (2.23) 1 1 Gð0Þ 1 1 K The magnitude of the error depends on the system gain K and will be never zero. Next, suppose a type I system of GðsÞ 5 K=sðTs 1 1Þ. This control system has no steady-state error as 1 1 eN 5 5 50 (2.24) 1 1 Gð0Þ 1 1 N From this, it can be seen that if a control system includes an integrator 1/s, then the steady-state error to a step reference will be zero. 2.6 CURRENT CONTROLLER DESIGN The PI controller is the most commonly used in the DC motor drive systems for current control. In this section we will introduce how to design a current control- ler, i.e., how to select the P and I gains of a PI current controller for the desired performance. The PI controller consists of a proportional term, which depends on the pres- ent error, and an integral term, which depends on the accumulation of past and present errors as follows: Ki Gpi ðsÞ 5 Kp 1 (2.25) s where Kp and Ki are the proportional and integral gain, respectively. Fig. 2.29 shows the frequency response of the PI controller. The PI controller has an infinite DC gain (i.e., gain when ω 5 0) due to an integrator. The transfer function of Eq. (2.25) can be rewritten as
 1
 s1 1 Tpi Gpi ðsÞ 5 Kp 1 1 5 Kp (2.26) Tpi s s  2.6 Current Controller Design 67 | Gpi (s) | PI corner ∠Gpi (s) 1 frequency Tpi 0° ω KP – 45° ω 1 – 90° Tpi FIGURE 2.29 Frequency response of the PI controller. Ea DC motor _ Ia + PI current Va + 1 I a* _ controller La s + Ra FIGURE 2.30 Block diagram of the DC motor including a current controller. where Tpi 5 Kp =Ki is the integral time constant of the PI controller and 1=Tpi is called the corner frequency of the PI controller. From Eq. (2.26), we can see that the PI controller can add one pole at s 5 0 and one zero at s 5 2 Ki =Kp to the transfer function of the system. The location of this zero has a strong influence on the system performance, which will be described in detail later. If the zero is correctly placed, then the damping of the system can be improved. As mentioned earlier, the pole at s 5 0 of the PI control- ler causes the system type to be increased by adding an integrator 1=s to the transfer function, improving the steady-state error. However, the increased system type may make the system unstable when gains are large. In general, the PI controller can improve the steady-state error but at the expense of stability. In addition, the PI controller can reduce the maximum overshoot and improve the gain and phase margins as well as the resonant peak, but it reduces the bandwidth and extends the rise time. The PI controller also has a characteristic of a low-pass filter that reduces the noise. Now, let us evaluate the performance of the PI controller for current control of a DC motor. The transfer function of a DC motor system that includes a PI current controller as shown in Fig. 2.30 is given as Kpc s 1 Kic  s IðsÞ 5 I ðsÞ 2 EðsÞ (2.27) La s2 1 ðRa 1 Kpc Þs 1 Kic La s2 1 ðRa 1 Kpc Þs 1 Kic where Kpc and Kic represent the proportional and integral gains of the current controller, respectively. We can see from Eq. (2.27) that the back-EMF EðsÞ of the DC motor will act as a disturbance to the current control system. If the current reference of the 68 CHAPTER 2 Control of direct current motors DC motor remains a constant value (i.e., s 5 0), then EðsÞ will have no influence  on the current control performance. However, when the current reference I ðsÞ is changed, EðsÞ may have an influence on the dynamic performance of the actual current IðsÞ following its reference. For a large system inertia, the effect of EðsÞ on the current control can be ignored since the variation of EðsÞ due to the speed variation is very small. However, for servo motors designed with a small inertia for a fast speed response, the effect of EðsÞ cannot be ignored. To obtain a good performance of the current control by eliminating this undesirable effect of the back-EMF, we can adopt the feedforward control as described in the Section 2.5.2.3. Since the back-EMF of a DC motor is ea 5 ke φf ωm , we can easily estimate it from the speed information. When the disturbance EðsÞ is compensated by the feedfor- ward control as in Fig. 2.31A, the DC motor will be simplified as a R-L circuit shown in Fig. 2.31B. For the case of Fig. 2.31B, the current control will be easier and its performance will be improved. In this case, Eq. (2.27) becomes IðsÞ Kpc s 1 Kic 5 (2.28) I  ðsÞ La s2 1 ðRa 1 Kpc Þs 1 Kic From Eq. (2.28), it can be seen that the current control characteristic depends on the performance of the current controller, which will be determined by the P and I gains selected for the PI controller. Now, we will look at how the proper gains are selected to achieve the desired performance of the current control. Here, we will introduce a gain selection method based on the frequency response of the system. Back-EMF Cancel K compensation êa ea + I* + Current V* - 1 i T 1 ωm controller K + sL + R Js + B – Motor + load (A) I* + Current V* 1 i T 1 controller K ωm sL + R Js + B – (B) FIGURE 2.31 Current control system. (A) Feedforward control of the back-EMF and (B) system after compensating the back-EMF.  2.6 Current Controller Design 69 2.6.1 PROPORTIONALINTEGRAL CURRENT CONTROLLER A current control system containing a PI controller is shown in Fig. 2.32. Here, assume that the feedforward control compensates the back-EMF of the DC motor perfectly. In this PI current controller, the relationship between the current error and its output voltage is given as
 1   Va 5 Kpc 1 1 Ia 2 Ia (2.29) Tpi s where Tpi ð 5 Kpc =Kic Þ is the integral time constant of the PI current controller and Kic is the integral gain. There are several PI gain tuning rules such as the famous ZieglerNichols method. Here, we will adopt the technique called the pole-zero cancellation method. By using the pole-zero cancellation, we can remove the current control characteristic of the DC motor itself so that the PI controller may determine the performance of the current control. The open-loop transfer function Goc ðsÞ of this system is given by 0 1
 1 Kic 1 s1 s1 B T C 1 K L Goc ðsÞ 5 Kpc B C 
a  pi pc @ s A  La s 1 Ra 5 Kpc s Ra (2.30) s1 La If the zero ð2 Kic =Kpc Þ of the PI controller is designed to cancel the pole ð2 Ra =La Þ of the DC motor by the pole-zero cancellation method, i.e. 1 Kic Ra 5 5 (2.31) Tpi Kpc La then, Eq. (2.30) becomes 1 Goc ðsÞ 5
 (2.32) La s Kpc Fig. 2.33 shows the bode plot of the open-loop frequency response from Goc ðsÞ. The phase  is 290 at the gain cross-over frequency ωcc where the magni-   tude Gc ðjωÞ 5 0 dB. Thus the gain margin is positive and the system will be o * + K Va* = Va 1 Ia I K pc + ic a s La s + Ra – PI controller Motor FIGURE 2.32 PI current control system. 70 CHAPTER 2 Control of direct current motors K pc Gain (dB) ωcc = Cross-over frequency La ω (Log) 0 Phase (°) ω (Log) 0° –90° FIGURE 2.33 Open-loop frequency response. stable. Also, in this case, the transfer function of Eq. (2.32) becomes the system of type 5 1 and, thus, we can expect the steady-state error to be zero as 1 1 eN 5 lim 5 50 (2.33) s-0 1 1 Go c ðsÞ 11N From Eq. (2.32), the gain cross-over frequency ωcc can be determined by  o  G ð jωcc Þ 5  1  5 1 - ωcc 5 Kpc (2.34) c  La  La   K jωcc  pc The gain cross-over frequency ωcc of the open-loop frequency response is equal to the gain cut-off frequency of the close-loop frequency response, which indicates the bandwidth of the current control system. Let us check the bandwidth of this current control system. The closed-loop transfer function of this system is given by IðsÞ Goc ðsÞ 1 ωcc  5 Gcc ðsÞ 5 5
 5 (2.35) I ðsÞ 1 1 Gc ðsÞ o La s 1 ωcc s11 Kpc Its frequency response is shown in Fig.  2.34.  We can see pffiffithat ffi the bandwidth of this system is given as ωcc by letting Gcc ð jωÞ equal to 1= 2(523 dB) as    c    G ð jωÞ 5  ωcc  5 p1ffiffiffi (2.36) c  jω 1 ωcc  2 Thus the bandwidth is equal to the gain cross-over frequency of the open-loop frequency response.  2.6 Current Controller Design 71 K pc Bandwidth ωcc = La 0 Gain (dB) ω (Log) –3 Phase (°) ω (Log) 0° – 45° – 90° FIGURE 2.34 Closed-loop frequency response. Like this, if the zero of the PI controller is designed to cancel the pole of the system, the current control system is expressed simply as a first-order system with a cut-off frequency ωcc , which is stable. In this case, there will be no over- shoot in the response and the time required for the response to reach the final value will be about four times the time constant of the system. Based on the explanation given above, the proportional and integral gains of the current controller needed to achieve the required bandwidth can be determined as follows. If the required control bandwidth is ωcc , then the proportional gain Kpc can be obtained from Eq. (2.34) and the integral gain Kic can be obtained from Eq. (2.31) as Proportional gain: Kpc 5 La  ωcc (2.37) Ra Integral gain: Kic 5 Kpc 5 Ra  ωcc (2.38) La It is important to note that the gains of the controller depend on the motor parameters La and Ra. Therefore the gain values to achieve the same current control performance may differ from motor to motor. For example, a motor with a large value of winding inductance, whose current is difficult to change, needs a larger proportional gain to obtain an equal bandwidth of the current control than that with a small value of winding inductance, whose current is easy to change. Therefore, accurate information about the motor parameters is essential to achieve the required current control performance. 2.6.1.1 Selection of the bandwidth for current control The value of the proportional gain Kpc determines how fast the system responds, whereas the value of the integral gain Kic determines how fast the steady-state 72 CHAPTER 2 Control of direct current motors error is eliminated. When the value of these gains is larger, the control perfor- mance is better. However, large gains may lead to an oscillatory response and result in an unstable system. As can be seen in Eq. (2.37) and (2.38), since the gains of a current controller are determined by its bandwidth ωcc , we first need to select an appropriate bandwidth ωcc for the target current control system. A wider bandwidth will lead to a faster response but an unstable system. The bandwidth ωcc of a current controller is limited by the following two factors: the switching frequency of the power electronic converter realizing the output of the current controller and the sampling period for detecting an actual current for feedback in the digital controller. Since a motor current cannot change faster than the switching frequency of the power electronic converter, the switch- ing frequency limits the bandwidth of the current control. If the current is sampled twice every switching period, as a rule of thumb, the maximum available bandwidth can be up to 1/10 of the switching frequency. On the other hand, if it is sampled once every switching period, the maximum available bandwidth can be up to 1/20 of the switching frequency. For this case, it is desirable to restrict the bandwidth to 1/25 of the sampling frequency [1,2]. As an example, suppose that the switching frequency of a chopper, which provides the voltage applied to a DC motor, is 5 kHz. In this case, the maximum available bandwidth of the current controller can be up to at most 1/10 of the switching frequency, i.e., 5 kHz/10 5 500 Hz(  3100 rad/s). However, for a stable current control or if the current is sampled once every switching period, it is better to restrict to 1/20 of the switching frequency, i.e., 5 kHz/20 5 250 Hz (  1550 rad/s). Fig. 2.35 compares the current control performances according to the gains, which are determined by the gain selection method as described earlier. We can identify the different responses according to the control bandwidth. 25 25 ωcc=500 Hz ω cc=1000 Hz 20 20 i* i* 15 i 15 i (A) (B) 10 10 5 5 0 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0 0.002 0.004 0.006 0.008 0.01 0.012 Time (s) Time (s) FIGURE 2.35 Current control performance according to the control bandwidth.  2.6 Current Controller Design 73 2.6.2 ANTI-WINDUP CONTROLLER As stated before, we can see that the integral controller can effectively eliminate the steady-state error. This is because of the nature of the integral controller producing its output from the accumulated past errors. However, this nature is often the cause of control performance degradation in cases where the output of the controller is limited. The output of the PI current controller, which indicates the voltage reference applied to a DC motor, should be limited to a feasible value by the following reasons. First, a voltage exceeding its rated value should not be applied to a motor. Furthermore, power electronic converters, which produce the voltage applied to the motor, normally have a restricted output voltage due to the limited availability of the input voltage and the voltage rating of switching devices. Once the output of the PI controller exceeds its limit due to sustained error signals for a significant period of time, the output will be saturated, but the integrator in the controller may have a large value by its continual integral action. This phenomenon is known as integral windup. When the windup occurs, the controller is unable to immediately respond to the changes in the error due to a large accumulated value inside the integrator. It is required that the error have an opposite sign for a long time until the controller returns to the normal state. This system becomes an open-loop system because the output remains at its limit irrespective of the error. As a result, the system will exhibit a large overshoot and a long setting time. To avoid such integrator windup, the magnitude of the integral term should be kept at a proper value when the saturation occurs, so that the controller can resume the action as soon as the control error changes. There are several methods to avoid the integrator windup such as back calcu- lation, conditional integration, and limited integration [3,4]. The anti-windup scheme of back calculation as shown in Fig. 2.36 is widely used because of its satisfactory dynamic characteristic. Anti-windup controller Ka + – Current – error + KI + + Va* e Va*−ref s + + Limiter KP Output limit Êa FIGURE 2.36 Anti-windup control by the back calculation method. 74 CHAPTER 2 Control of direct current motors (A) (B) i* i* i i Value of integrator Limit Value of integrator value 0 Time 0 Time FIGURE 2.37 Anti-windup control. (A) Without anti-windup control and (B) with anti-windup control. In this scheme, when the output is saturated, the difference between the controller output and the actual output is fed back to the input of the integrator with a gain of Ka so that the accumulated value of the integrator can be kept at a proper value. The gain of an anti-windup controller is normally selected as Ka 5 1=Kp to eliminate the dynamics of the limited voltage. Fig. 2.37 shows the phenomenon of integrator windup for a PI current control- ler, which is generated by a large change in the reference value. Fig. 2.37A shows the performance of a current controller without an anti-windup control. Due to its saturated output voltage, the actual current exhibits a large overshoot and a long setting time. On the other hand, Fig. 2.37B shows a current controller with an anti-windup control. When the output is saturated, the accumulated value of the integrator can be kept at a proper value, resulting in an improved performance. 2.6.2.1 Gains selection procedure of the proportionalintegral current controller 1. Find the switching frequency fsw of the system. 2. Select the control bandwidth ωcc of the current controller to be within 1/101/ 20 of the switching frequency fsw and below 1/25 of the sampling frequency. 3. Calculate the proportional and integral gains from the selected bandwidth ωcc as Proportional gain: Kpc 5 La  ωcc Integral gain: Kic 5 Ra  ωcc 4. Select the gain of the anti-windup controller as K a 5 1=Kpc. The procedures 1 and 2 are interchangeable with each other, i.e., the switching frequency can be determined by the required bandwidth ωcc for current control.  2.7 Speed Controller Design 75 2.7 SPEED CONTROLLER DESIGN When controlling the speed of a motor, the speed controller must be placed in the outer loop of the current controller as shown in Fig. 2.38. In this case, if the bandwidth of current control is chosen to be sufficiently wider than that of speed control, then the current control will not have any influence on the speed control performance, and the stability of the speed control will be improved. Like the current controller, a PI controller is widely used for speed control. Now, we will explain how the PI speed controller is designed, i.e., how to select the proper gains to achieve the desired speed control performance. 2.7.1 PROPORTIONALINTEGRAL SPEED CONTROLLER The transfer function of a PI speed controller is given by Kis Gpi ðsÞ 5 Kps 1 (2.39) s where Kps and Kis are the proportional and integral gains of a PI speed controller, respectively. From the bloc