N 十 FIGURE 66.2 Left, cross section of a four-pole synchro- FIGURE 66. 3 Cross section of a nous machine. Rotor has a salient pole structure. righ chrono schematic diagram for phase a windings The rotors depicted in Figs. 66. 1 and 66.2 are salient since the poles are protruding from the shaft. Such structures are mechanically weak, since at a high speed(3600 rpm and 1800 rpm, respectively) the centrifugal force becomes a serious problem. Practically, for high-speed turbines, round-rotor(or cylindrical-rotor)struc tures are preferred. The cross section of a two-pole, round-rotor machine is depicted in Fig. 663. Fror ractical viewpoint, salient rotors are easier to build because each pole and its winding can be manufacture separately and then mounted on the rotor shaft. For round rotors, slots need to be reserved in the rotor where the windings can be placed The mathematical model for round-rotor machines is much simpler than that for salient-rotor ones. Th stems from the fact that the rotor body has a permeability much higher than that of the air In a steady state, he stator field and the rotor body are at a standstill relative to each other. (They rotate at the same speed discussed earlier. )If the rotor is salient, it is easier to establish the magnetic flux lines along the direction of the rotor body(when viewed from the cross section). Therefore, for the same set of stator currents, different positions of the rotor alter the stator field in different ways; this implies that the induced emfs are different. If the rotor is round, then the relative position of the rotor structure does not affect the stator field. Hence, the associated mathematical model is simplified. In the following, the steady-state models of the round-rotor and salient-rotor generators are explained. Refer to Fig 66.3 which shows a two-pole round-rotor machine. without loss of generality, one can select phase a(i.e, winding ad) for the development of a mathematical model of the machine. As mentioned previously, the(armature or stator) winding of phase a is exposed to two magnetic fields: rotor field and stator fie 1. Rotor field. Its flux as seen by winding ad varies with the rotor position; the flux linkage is largest when the N-S axis is perpendicular to the winding surface and minimum(zero) when this axis aligns with the surface. Thus, one can express the flux due to the rotor field as seen by winding ad as M,=l(e)I where 0 is to denote the angular position of the n-S axis (of the rotor field) relative to the surface of aa, IF is the rotor current(a dc current, and L is a periodic function of 8. 2. Stator field. Its flux as seen by winding aa is a combination of three individual fields which are due to currents in the stator windings, ia, ib, and ie. This flux can be expressed as M2=L, i+ Imi+ lmie, where the self (mutual) inductance. Because the rotor is round, L, and Lm are not dependent on 8, the relative position of the rotor and the winding. Typically, the sum of the stator currents i+ i,+ i is near zero; thus, one can write n2=(L-Lm)i The total flux seen by winding ad is n =L(O)IF-(L-Lm), where the minus sign in n-n,is due to the fact that the stator field opposes ptor field. The induced emf across the winding ad is dN/dt, the time derivative of n: e 2000 by CRC Press LLC© 2000 by CRC Press LLC The rotors depicted in Figs. 66.1 and 66.2 are salient since the poles are protruding from the shaft. Such structures are mechanically weak, since at a high speed (3600 rpm and 1800 rpm, respectively) the centrifugal force becomes a serious problem. Practically, for high-speed turbines, round-rotor (or cylindrical-rotor) struc￾tures are preferred. The cross section of a two-pole, round-rotor machine is depicted in Fig. 66.3. From a practical viewpoint, salient rotors are easier to build because each pole and its winding can be manufactured separately and then mounted on the rotor shaft. For round rotors, slots need to be reserved in the rotor where the windings can be placed. The mathematical model for round-rotor machines is much simpler than that for salient-rotor ones. This stems from the fact that the rotor body has a permeability much higher than that of the air. In a steady state, the stator field and the rotor body are at a standstill relative to each other. (They rotate at the same speed as discussed earlier.) If the rotor is salient, it is easier to establish the magnetic flux lines along the direction of the rotor body (when viewed from the cross section). Therefore, for the same set of stator currents, different positions of the rotor alter the stator field in different ways; this implies that the induced emf’s are different. If the rotor is round, then the relative position of the rotor structure does not affect the stator field. Hence, the associated mathematical model is simplified. In the following, the steady-state models of the round-rotor and salient-rotor generators are explained. Refer to Fig. 66.3 which shows a two-pole round-rotor machine. Without loss of generality, one can select phase a (i.e., winding aa¢) for the development of a mathematical model of the machine. As mentioned previously, the (armature or stator) winding of phase a is exposed to two magnetic fields: rotor field and stator field. 1. Rotor field. Its flux as seen by winding aa¢ varies with the rotor position; the flux linkage is largest when the N–S axis is perpendicular to the winding surface and minimum (zero) when this axis aligns with the surface. Thus, one can express the flux due to the rotor field as seen by winding aa¢ as l1 = L(q)IF where q is to denote the angular position of the N–S axis (of the rotor field) relative to the surface of aa¢, IF is the rotor current (a dc current), and L is a periodic function of q. 2. Stator field. Its flux as seen by winding aa¢ is a combination of three individual fields which are due to currents in the stator windings, ia, ib, and ic. This flux can be expressed as l2 = Ls ia + Lmib + Lmic, where Ls (Lm) is the self (mutual) inductance. Because the rotor is round, Ls and Lm are not dependent on q, the relative position of the rotor and the winding. Typically, the sum of the stator currents ia + ib + ic is near zero; thus, one can write l2 = (Ls – Lm)i a. The total flux seen by winding aa¢ is l = l1 – l2 = L(q)IF – (Ls – Lm)ia, where the minus sign in l1 – l2 is due to the fact that the stator field opposes the rotor field. The induced emf across the winding aa¢ is dl/dt, the time derivative of l: FIGURE 66.2 Left, cross section of a four-pole synchro￾nous machine. Rotor has a salient pole structure. Right, schematic diagram for phase a windings. FIGURE 66.3 Cross section of a two-pole round-rotor synchronous machine