The time-varying quantities are normally sinusoidal, and for practical purposes, can be represented by phasors. Thus the above expression becomes E=E-(L2-Ln)j0og。全E-jxF where Oo is the angular speed (rad/s)of the rotor in a steady state. This equation can be modeled as a voltage burce-Er behind a reactance jX, as shown in Fig. 66.4; this reactance is usually referred to as synchronot reactance. The resistor Ra in the diagram represents the winding resistance, and V, is the voltage measured acro am6m=4++,mmm→人。 are now dependent on the(relative) position of the rotor For example (refer to Fig. 66.1), L, is maximum when the rotor is in a vertical position and minimum when the rotor is 90 away. ea In the derivation of the mathematical/ circuit model for salient-rotor machines, the stator field B, can be resolved into two components when the rotor is viewed from a cross section, one component aligns along the rotor and the other is perpendicular to the rotor(Fig. 66.5) The component Ba, which directly opposes the rotor field, is said to FIGURE 66.4 Per-phase equivalent cir- elong to the direct axis, the other component, Be is weaker and machines -E is the internal voltage belongs to the quadrature axis. The model for a salient-rotor machine consists of two circuits, direct-axis circuit and quadrature-axis circuit, (phasor form) and V, is the terminal vol each similar to Fig. 66.4. Any quantity of interest, such as Ia, the current inding ad, is made up of The round-rotor machine can be viewed as a special case of the salient pole theory where the corresponding parameters of the d-axis and q-axis circuits are equal B Dynamic models. When a power system is in a steady state (ie operated at an equilibrium), the electrical output of each generator is equal to the power applied to the rotor shaft. Various losses have beer a neglected without affecting the essential ideas provided in this discus- sion)Disturbances occur frequently in power systems, however Examples of disturbances are load changes, short circuits, and equip- ment outages. A disturbance results in a mismatch between the power input and output of generators, and therefore the rotors depart from IHITHmTHILIH their synchronous-speed operation. Intuitively, the impact is more FIGURE 66.5 In the salient-pole the- severe for machines closer to the disturbance. When a system is per- ory, the stator field(represented by a turbed, there are several possibilities for its subsequent behavior. If the single vector B, is decomposed into B, disturbance is small, the machines may soon reach a new steady speed, and B. Note that [ BI>IB which is close to or identical to their synchronous speed, in which case the system is said to be stable. It may also happen that some machines speed up while others slow down. In a more complicated situation, a rotor may oscillate about its synchronous speed. This results in an unstable case An unstable situation can result in abnormal changes in system frequency and voltage and, unless properly controlled, may lead to damage to machines (e.g, broken shafts). To study these phenomena, dynamic models are required. Details of a dynamic model depend on a number of factors such as location of disturbance and time duration of interest. An overview of dynamic generator models is given here. In essence, there are two aspects that need be modeled: electromechanical and electromagnetic e 2000 by CRC Press LLC© 2000 by CRC Press LLC The time-varying quantities are normally sinusoidal, and for practical purposes, can be represented by phasors. Thus the above expression becomes: where w0 is the angular speed (rad/s) of the rotor in a steady state. This equation can be modeled as a voltage source–EF behind a reactance jXs , as shown in Fig. 66.4; this reactance is usually referred to as synchronous reactance. The resistor Ra in the diagram represents the winding resistance, and Vt is the voltage measured across the winding. As mentioned, the theory for salient-rotor machines is more com￾plicated. In the equation l2 = Ls ia + Lmib + Lmic, the terms Ls and Lm are now dependent on the (relative) position of the rotor. For example (refer to Fig. 66.1), Ls is maximum when the rotor is in a vertical position and minimum when the rotor is 90° away. In the derivation of the mathematical/circuit model for salient-rotor machines, the stator field B2 can be resolved into two components; when the rotor is viewed from a cross section, one component aligns along the rotor and the other is perpendicular to the rotor (Fig. 66.5). The component Bd , which directly opposes the rotor field, is said to belong to the direct axis; the other component, Bq, is weaker and belongs to the quadrature axis. The model for a salient-rotor machine consists of two circuits, direct-axis circuit and quadrature-axis circuit, each similar to Fig. 66.4.Any quantity of interest, such as Ia, the current in winding aa¢, is made up of two components, one from each circuit. The round-rotor machine can be viewed as a special case of the salient￾pole theory where the corresponding parameters of the d-axis and q-axis circuits are equal. Dynamic models. When a power system is in a steady state (i.e., operated at an equilibrium), the electrical output of each generator is equal to the power applied to the rotor shaft.(Various losses have been neglected without affecting the essential ideas provided in this discus￾sion.) Disturbances occur frequently in power systems, however. Examples of disturbances are load changes, short circuits, and equip￾ment outages. A disturbance results in a mismatch between the power input and output of generators, and therefore the rotors depart from their synchronous-speed operation. Intuitively, the impact is more severe for machines closer to the disturbance. When a system is per￾turbed, there are several possibilities for its subsequent behavior. If the disturbance is small, the machines may soon reach a new steady speed, which is close to or identical to their synchronous speed, in which case the system is said to be stable. It may also happen that some machines speed up while others slow down. In a more complicated situation, a rotor may oscillate about its synchronous speed. This results in an unstable case. An unstable situation can result in abnormal changes in system frequency and voltage and, unless properly controlled, may lead to damage to machines (e.g., broken shafts). To study these phenomena, dynamic models are required. Details of a dynamic model depend on a number of factors such as location of disturbance and time duration of interest. An overview of dynamic generator models is given here. In essence, there are two aspects that need be modeled: electromechanical and electromagnetic. e d dt dL dt I L L di dt e L L di dt a F s m a F s m a = = = l – ( – ) – ( – ) D E E L L j I E jX I a = - F s m a = F s a ( – ) w0 D – FIGURE 66.4 Per-phase equivalent cir￾cuit of round-rotor synchronous machines. –EF is the internal voltage (phasor form) and Vt is the terminal volt￾FIGURE 66.5 In the salient-pole the￾ory, the stator field (represented by a single vector B2 ) is decomposed into Bd and Bq . Note that *Bd* > *Bq*