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Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek
377 10.1. INTRODUCTION The direct-current ( dc ) machine is not as widely used today as it was in the past. For the most part, the dc generator has been replaced by solid-state rectifi ers. Nevertheless, it is still desirable to devote some time to the dc machine since it is still used as a drive motor, especially at the low-power level. Numerous textbooks have been written over the last century on the design, theory, and operation of dc machines. One can add little to the analytical approach that has been used for years. In this chapter, the well- established theory of dc machines is set forth, and the dynamic characteristics of the shunt and permanent-magnet machines are illustrated. The time-domain block diagrams and state equations are then developed for these two types of motors. 10.2. ELEMENTARY DC MACHINE It is instructive to discuss the elementary machine shown in Figure 10.2-1 prior to a formal analysis of the performance of a practical dc machine. The two-pole elementary machine is equipped with a fi eld winding wound on the stator poles, a rotor coil ( a − a ′ ), Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk, Scott Sudhoff, and Steven Pekarek. © 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc. DC MACHINES AND DRIVES 10 378 DC MACHINES AND DRIVES and a commutator. The commutator is made up of two semicircular copper segments mounted on the shaft at the end of the rotor and insulated from one another as well as from the iron of the rotor. Each terminal of the rotor coil is connected to a copper segment. Stationary carbon brushes ride upon the copper segments whereby the rotor coil is connected to a stationary circuit. The voltage equations for the fi eld winding and rotor coil are vri d dt fff f =+ λ (10.2-1) vri d dt aa aaa aa − ′ − ′ − ′ =+ λ (10.2-2) Figure 10.2-1. Elementary two-pole dc machine. f 1 f 1 ¢ f 2 f 1 f-axis f 2 ¢ f 2 ¢ f 2 f 1 ¢ ¢ Brush a a Insulation Copper segment i a v a i a v a i f v f + + – – – q c + ELEMENTARY DC MACHINE 379 where r f and r a are the resistance of the fi eld winding and armature coil, respectively. The rotor of a dc machine is commonly referred to as the armature ; rotor and armature will be used interchangeably. At this point in the analysis, it is suffi cient to express the fl ux linkages as λ fffffaaa Li Li=+ − ′ (10.2-3) λ a a af f aa a a Li Li − ′ − ′ =+ (10.2-4) As a fi rst approximation, the mutual inductance between the fi eld winding and an armature coil may be expressed as a sinusoidal function of θ r as LL L af fa r ==−cos θ (10.2-5) where L is a constant. As the rotor revolves, the action of the commutator is to switch the stationary terminals from one terminal of the rotor coil to the other. For the confi gu- ration shown in Figure 10.2-1 , this switching or commutation occurs at θ r = 0, π , 2 π , . . . . At the instant of switching, each brush is in contact with both copper segments, whereupon the rotor coil is short-circuited. It is desirable to commutate (short-circuit) the rotor coil at the instant the induced voltage is a minimum. The waveform of the voltage induced in the open-circuited armature coil during constant-speed operation with a constant fi eld winding current may be determined by setting i aa− ′ = 0 and i f equal to a constant. Substituting (10.2-4) and (10.2-5) into (10.2-2) yields the following expression for the open-circuit voltage of coil a − a ′ with the fi eld current i f a constant: vLI aa r f r− ′ = ωθ sin (10.2-6) where ω r = d θ r / dt is the rotor speed. The open-circuit coil voltage v aa− ′ is zero at θ r = 0, π , 2 π , . . . , which is the rotor position during commutation. Commutation is illustrated in Figure 10.2-2 . The open-circuit terminal voltage, ν a , corresponding to the rotor posi- tions denoted as θ ra , θ rb ( θ rb = 0), and θ rc are indicated. It is important to note that during one revolution of the rotor, the assumed positive direction of armature current i a is down coil side a and out coil side a ′ for 0 < θ r < π . For π < θ r < 2 π , positive current is down coil side a ′ and out of coil side a . Previously, we let positive current fl ow into the winding denoted without a prime and out the winding denoted with a prime. We will not be able to adhere to this relationship in the case of the armature windings of a dc machine since commutation is involved. The machine shown in Figure 10.2-1 is not a practicable machine. Although it could be operated as a generator supplying a resistive load, it could not be operated effectively as a motor supplied from a voltage source owing to the short-circuiting of the armature coil at each commutation. A practicable dc machine, with the rotor equipped with an a winding and an A winding, is shown schematically in Figure 10.2-3 . At the rotor position depicted, coils aa 44 − ′ and AA 44 − ′ are being commutated. The bottom brush short-circuits the aa 44 − ′ coil while the top brush short-circuits the AA 44 − ′ coil. Figure 10.2-3 illustrates the instant when the assumed direction of positive current 380 DC MACHINES AND DRIVES is into the paper in coil sides a 1 , A 1 ; a 2 , A 2 ; . . . , and out in coil sides ′ a 1 , ′ A 1 ; ′ a 2 , ′ A 2 ; . . . It is instructive to follow the path of current through one of the parallel paths from one brush to the other. For the angular position shown in Figure 10.2-3 , positive cur- rents enter the top brush and fl ow down the rotor via a 1 and back through ′ a 1 ; down a 2 and back through ′ a 2 ; down a 3 and back through ′ a 3 to the bottom brush. A parallel current path exists through AA 33 − ′ , AA 22 − ′ , and AA 11 − ′ . The open-circuit or induced armature voltage is also shown in Figure 10.2-3 ; however, these idealized waveforms require additional explanation. As the rotor advances in the counterclockwise direction, the segment connected to a 1 and A 4 moves from under the top brush, as shown in Figure 10.2-4 . The top brush then rides only on the segment connecting A 3 and ′ A 4 . At the same time, the bottom brush is riding on the segment connecting a 4 and ′ a 3 . With the rotor so positioned, current fl ows in A 3 and ′ A 4 and out a 4 and ′ a 3 . In other words, current fl ows down the coil sides in the upper one half of the rotor and out of the coil sides in the bottom one half. Let us follow the current fl ow through the parallel paths of the armature windings shown in Figure 10.2-4 . Current now fl ows through the top brush into ′ A 4 out A 4 , into a 1 out ′ a 1 , into a 2 , out ′ a 2 , into a 3 out ′ a 3 to the bottom brush. The Figure 10.2-2. Commutation of the elementary dc machine. i a + – a a a ¢ a ¢ a a ¢ q ra v a i a + – q rc q rc q ra q rb v a v a i a + – q rb = 0 v a ELEMENTARY DC MACHINE 381 Figure 10.2-3. A dc machine with parallel armature windings. f 1 A 2 A 1 f 1 ¢ f 2 f-axis f 2 ¢ a 2 A 3 a 3 A 4 a 4 a 3 -a 3 A 1 a 1 a 1 Rotation ¢ ¢ A 2 a 2 ¢ ¢ A 3 a 3 ¢ ¢ ¢ a 2 -a 2 ¢ a 4 -a 4 ¢ A 4 -A 4 ¢ A 3 -A 3 ¢ A 2 -A 2 ¢ A 1 -A 1 ¢ A 1 -A 1 ¢ a 1 -a 1 ¢ A 4 a 4 ¢ ¢ i a v a v a + – t Rotor position shown above 382 DC MACHINES AND DRIVES Figure 10.2-4. Same as Figure 10.2-3 , with rotor advanced approximately 22.5° counterclockwise. t f 1 f 1 ¢ f 2 f-axis f 2 ¢ A 2 A 1 a 2 a 1 A 3 a 3 A 4 a 4 A 1 a 1 ¢ ¢ A 2 a 2 ¢ ¢ A 3 a 3 ¢ ¢ a 3 -a 3 ¢ a 2 -a 2 ¢ a 4 -a 4 ¢ A 4 -A 4 ¢ A 3 -A 3 ¢ A 2 -A 2 ¢ A 1 -A 1 ¢ A 1 -A 1 ¢ a 1 -a 1 ¢ A 4 a 4 ¢ ¢ i a v a v a + – Rotor position shown above p 2pq r q r w r 0 ELEMENTARY DC MACHINE 383 parallel path beginning at the top brush is AA 33 − ′ , AA 22 − ′ , and AA 11 − ′ , and ′ −aa 44 to the bottom brush. The voltage induced in the coils is shown in Figure 10.2-3 and Figure 10.2-4 for the fi rst parallel path described. It is noted that the induced voltage is plotted only when the coil is in this parallel path. In Figure 10.2-3 and Figure 10.2-4 , the parallel windings consist of only four coils. Usually, the number of rotor coils is substantially more than four, thereby reducing the harmonic content of the open-circuit armature voltage. In this case, the rotor coils may be approximated as a uniformly distributed winding, as illustrated in Figure 10.2-5 . Therein, the rotor winding is considered as current sheets that are fi xed in space due to the action of the commutator and which establish a magnetic axis positioned orthogonal to the magnetic axis of the fi eld winding. The brushes are shown positioned on the current sheet for the purpose of depicting commutation. The small angular Figure 10.2-5. Idealized dc machine with uniformly distributed rotor winding. Current into paper 2g Short-circuited coils Rotation a-axis – Magnetic axis of equivalent armature winding f-axis Current out of paper i a i a i f v f N f v a + _ v a r a r f + _ 384 DC MACHINES AND DRIVES displacement, denoted by 2 γ , designates the region of commutation wherein the coils are short-circuited. However, commutation cannot be visualized from Figure 10.2-5 ; one must refer to Figure 10.2-3 and Figure 10.2-4 . In our discussion of commutation, it was assumed that the armature current was zero. With this constraint, the sinusoidal voltage induced in each armature coil crosses through zero when the coil is orthogonal to the fi eld fl ux. Hence, the commutator was arranged so that the commutation would occur when an armature coil was orthogonal to fi eld fl ux. When current fl ows in the armature winding, the fl ux established therefrom is in an axis orthogonal to the fi eld fl ux. Thus, a voltage will be induced in the armature coil that is being commutated as a result of “cutting” the fl ux established by the current fl owing in the other armature coils. Arcing at the brushes will occur, and the brushes and copper segments may be damaged with even a relatively small armature current. Although the design of dc machines is not a subject of this text, it is important to mention that brush arcing may be substantially reduced by mechanically shifting the position of the brushes as a function of armature current or by means of interpoles. Interpoles or commutating poles are small stator poles placed over the coil sides of the winding being commutated, midway between the main poles of large horsepower machines. The action of the interpole is to oppose the fl ux produced by the armature current in the region of the short-circuited coil. Since the fl ux produced in this region is a function of the armature current, it is desirable to make the fl ux produced by the interpole a function of the armature current. This is accomplished by winding the interpole with a few turns of the conductor carrying the armature current. Electrically, the interpole winding is between the brush and the terminal. It may be approximated in the voltage equations by increasing slightly the armature resistance and inductance ( r a and L AA ). 10.3. VOLTAGE AND TORQUE EQUATIONS Although rigorous derivation of the voltage and torque equations is possible, it is rather lengthy and little is gained since these relationships may be deduced. The armature coils revolve in a magnetic fi eld established by a current fl owing in the fi eld winding. We have established that voltage is induced in these coils by virtue of this rotation. However, the action of the commutator causes the armature coils to appear as a station- ary winding with its magnetic axis orthogonal to the magnetic axis of the fi eld winding. Consequently, voltages are not induced in one winding due to the time rate of change of the current fl owing in the other (transformer action). Mindful of these conditions, we can write the fi eld and armature voltage equations in matrix form as v v rpL LrpL i i f a fFF rAF a AA f a ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0 ω (10.3-1) where L FF and L AA are the self-inductances of the fi eld and armature windings, respec- tively, and p is the short-hand notation for the operator d/dt . The rotor speed is denoted as ω r , and L AF is the mutual inductance between the fi eld and the rotating armature VOLTAGE AND TORQUE EQUATIONS 385 coils. The above equation suggests the equivalent circuit shown in Figure 10.3-1 . The voltage induced in the armature circuit, ω r L AF i f , is commonly referred to as the counter or back emf. It also represents the open-circuit armature voltage. A substitute variable often used is kLi vAFf = (10.3-2) We will fi nd this substitute variable is particularly convenient and frequently used. Even though a permanent-magnet dc machine has no fi eld circuit, the constant fi eld fl ux produced by the permanent magnet is analogous to a dc machine with a constant k v . For a dc machine with a fi eld winding, the electromagnetic torque can be expressed TLii eAFfa = (10.3-3) Here again the variable k v is often substituted for L AF i f . In some instances, k v is multi- plied by a factor less than unity when substituted into (10.3-5) so as to approximate the effects of rotational losses. It is interesting that the fi eld winding produces a station- ary MMF and, owing to commutation, the armature winding also produces a stationary MMF that is displaced (1/2) π electrical degrees from the MMF produced by the fi eld winding. It follows then that the interaction of these two MMF ’ s produces the electro- magnetic torque. The torque and rotor speed are related by TJ d dt BT e r mr L =++ ω ω (10.3-4) where J is the inertia of the rotor and, in some cases, the connected mechanical load. The units of the inertia are kg·m 2 or J·s 2 . A positive electromagnetic torque T e acts to turn the rotor in the direction of increasing θ r . The load torque T L is positive for a torque, on the shaft of the rotor, which opposes a positive electromagnetic torque T e . The constant B m is a damping coeffi cient associated with the mechanical rotational system of the machine. It has the units of N·m·s and it is generally small and often neglected. Figure 10.3-1. Equivalent circuit of dc machine. + – –– ++ r f r a i f i a v a v f L FF L AF w r i f L AA 386 DC MACHINES AND DRIVES 10.4. BASIC TYPES OF DC MACHINES The fi eld and armature windings may be excited from separate sources or from the same source with the windings connected differently to form the basic types of dc machines, such as the shunt-connected, the series-connected, and the compound- connected dc machines. The equivalent circuits for each of these machines are given in this section along with an analysis and discussion of their steady-state operating characteristics. Separate Winding Excitation When the fi eld and armature windings are supplied from separate voltage sources, the device may operate as either a motor or a generator; it is a motor if it is driving a torque load and a generator if it is being driven by some type of prime mover. The equivalent circuit for this type of machine is shown in Figure 10.4-1 . It differs from that shown in Figure 10.3-1 in that external resistance r fx is connected in series with the fi eld winding. This resistance, which is often referred to as a fi eld rheostat , is used to adjust the fi eld current if the fi eld voltage is supplied from a constant source. The voltage equations that describe the steady-state performance of this device may be written directly from (10.3-1) by setting the operator p to zero ( p = d/dt ), whereupon VRI fff = (10.4-1) VrI LI aaa rAFf =+ ω (10.4-2) where R f = r fx + r f and capital letters are used to denote steady-state voltages and cur- rents. We know from the torque relationship given by (10.3-6) that during steady-state operation T e = T L if B m is assumed to be zero. Analysis of steady-state performance is straightforward. A permanent-magnet dc machine fi ts into this class of dc machines. As we have mentioned, the fi eld fl ux is established in these devices by a permanent magnet. The voltage equation for the fi eld winding is eliminated, and L AF i f is replaced by a constant k v , which can be measured if it is not given by the manufacturer. Most small, hand-held, fractional-horsepower dc motors are of this type, and speed control is achieved by controlling the amplitude of the applied armature voltage. Figure 10.4-1. Equivalent circuit for separate fi eld and armature excitation. + – – – ++ r fx r f r a i f i a v a v f L FF L AF w r i f L AA [...]... (10B-5) and solving for Ia yields Ia = = Va + (kv / Bm )TL 2 ra + (kv / Bm ) 6 + [(1.41 × 10 −2 ) / (6.04 × 10 −6 )](3.53 × 10 −3 ) 3 = 0.357 A 7 + (1.41 × 10 −2 )2 / (6.04 × 10 −6 ) (10B-6) From (10B-4), 1.41 × 10 −2 1 0.357 − (3.53 × 10 −3 ) 6.04 × 10 −6 6.04 × 10 −6 = 249 rad/s ωr = (10B-7) The power input is Pin = Va I a = (6)(0.357) = 2.14 W The power output is (10B-8) 394 DC MACHINES AND DRIVES. .. × 10 2 V s/rad and Bm to be 6.04 × 10 6 N·m·s During steady-state operation, (10. 3-6) becomes Te = Bmω r + TL (10B-2) From (10. 3-5), with LAFif replaced by kv, the steady-state electromagnetic torque is Te = kv I a (10B-3) Substituting (10B-3) into (10B-2) and solving for ωr yields ωr = kv 1 Ia − TL Bm Bm (10B-4) From (10. 4-2) with LAFif = kv, Va = ra I a + kvω r (10B-5) Substituting (10B-4) into (10B-5)... va = ∫ ∫ ∫ (10. 7-44) 412 DC MACHINES AND DRIVES va kivs ki+1vs ti+1 + ki+1T ti+1 + kiT ˆ Figure 10. 7-7 v a versus t for va shown in Figure 10. 7-5 and Figure 10. 7-6 ˆ Figure 10. 7-8 Comparison of v a and v a for k = 0.5 + 0.5 sin[r(t)t] After we have gone through the dynamic averaging process, we realize that we could have written the results given in (10. 7-42) and in (10. 7-44) by inspection and concern... All of these dc/ dc converters will be considered later in this chapter 10. 7 ONE-QUADRANT DC/ DC CONVERTER DRIVE In this section, we will analyze the operation and establish the average-value model for a one-quadrant chopper drive A brief word regarding nomenclature: dc/ dc converter and chopper will be used interchangeable throughout the text One-Quadrant dc/ dc Converter A one-quadrant dc/ dc converter... Methods of 398 DC MACHINES AND DRIVES solving equations of the fundamental form given by (10. 5-15) are well known Consequently, it is used extensively in control system analysis [1] 10. 6 SOLID-STATE CONVERTERS FOR DC DRIVE SYSTEMS Numerous types of ac /dc and dc/ dc converters are used in variable-speed drive systems to supply an adjustable dc voltage to the dc drive machine In the case of ac /dc converters,... τ a ) ra (10. 7-33) If T « τa, (10. 7-33) may be approximated as t ⎞ kω ⎛ ia ≈ I 2 ⎜ 1 − ⎟ − v r t ⎝ τ a ⎠ L AA for T . Inc. Published 2013 by John Wiley & Sons, Inc. DC MACHINES AND DRIVES 10 378 DC MACHINES AND DRIVES and a commutator. The commutator is made up of two. 1 41 10 6 04 10 3 53 310 7 1 41 10 6 04 10 0 357 3 22 6 × +× × = − −− ) (. ) /(. ) .A (10B-6) From (10B-4) , ω r = × × − × × = − −− − 141 10 604 10 0