B.  Synchronous machine

The theory of the synchronous machine is well known, so only the basic model characteristics will be described here.  A cross sectional view of a 3-phase, 2-pole, salient-pole synchronous machine is shown in Fig. SM-1.  The stator phases are represented by three sinusoidally distributed windings, displaced by 120 degrees.  The rotor contains one field winding (fd) (shown under the poles of the rotor), one damper winding in the same magnetic axis as the field (kd) (shown at the center of the rotor), and two damper windings in an axis  ahead of the  magnetic axis of the field (kq1 and kq2) (shown in the polefaces).  The damper windings are short circuited windings which represent paths for induced rotor currents.   Since most synchronous machines are operated as generators, it is assumed that positive stator current flows out of the machine. Therein positive electromagnetic torque corresponds to generator operation and negative electromagnetic torque corresponds to motor operation. From Fig. SM-1, it can be seen that the self-inductances of the stator windings as well as the mutual inductances between the stator and rotor windings are functions of the rotor angular position.  In the model used herein, Park’s transformation is applied to the stator variables (voltage, current, flux linkage), which replaces the variables associated with the stator windings  with variables associated with fictitious windings that rotate with the rotor.  This change of variables has the effect of eliminating the angular position-dependence of the inductances.

Fig. Sm-1:  Cross-sectional view of a two-pole, salient-rotor, three-phase synchronous machine

The change of variables which constitutes Park’s transformation may be expressed symbolically as


and f  may be voltage, current, or flux linkage.

For state-variable based simulation, the equations of the synchronous machine are conveniently modeled by expressing the flux linkages per second (state variables) in terms of the voltages applied to the machine.

where the superscript r  represents that the equations are transformed to the rotor frame of reference. The subscript “s” indicates stator quantities. The rotor voltage equations are not transformed; however, they are expressed in terms of the
transformed stator variables.  In particular,
It is assumed that rotor variables are referred to stator windings by the appropriate turns ratio [10]

The quadrature and direct magnetizing flux linkages are expressed as

The expressions for the stator (in rotor reference frame) and rotor currents can be found using equations SM-14 to SM-20

Equations SM-4 to SM-20 provide electrical quantities of the machine.  The synchronous machine is an electromechanical device, and thus also requires expressions for the electromagnetic torque and the speed of the machine. Equation SM-21 expresses the electromagnetic torque in terms of the flux linkages, and equation SM-22 determine the rotational speed from the electromagnetic torque, load torque, and moment of inertia.  In both equations, P represents the number of poles.  It should be noted that the model neglects core loss as well as friction and windage loss.

It is often convenient for analysis and physical interpretation to relate the position of the rotor of a synchronous machine to a voltage or the rotor of another machine.  In this model, the rotor angle is referenced to the maximum positive value of the fundamental component of the terminal voltage of phase a.  The change in rotor angle is expressed in terms of the rotor angular velocity and the electrical angular velocity of the phase voltages as
For constant electrical angular velocity, the rotor angle can be used in relating the torque and rotor speed

Fig. SM-2 shows the Graphic Modeller simulation of the synchronous machine.  As noted above, the model for the synchronous machine requires voltages as inputs.  Thus one block consists of a three-phase source that provides a balanced set of three-phase voltages.  The synchronous machine block is a compound block that contains another level, and will be described in the next paragraph. The final block in the model is the mechanical torque applied to the shaft of the synchronous machine.  In the model developed, currents were assumed positive out of the machine, indicating positive torque for generator operation, negative torque for motor operation.  The mechanical torque represents an input torque for generator operation  and a load torque for motor operation.  For convenience there are also two strip plot recorders that plot the stator phase currents and rotor angle.  Double clicking them, after a simulation run, will plot the respective variables.


Fig. SM-2:  Graphic Modeller simulation of the synchronous machine

Double clicking on the synchronous machine block reveals the next level of detail as shown in Fig. SM-3. Compound blocks can be used to allow multiple levels in a model.  In this model compound blocks were used so the model could be used as a tool for undergraudates who are not concerned with the simulation equations.  More advanced undergraduate and graduate students, on the other hand can go down a level to understand the theory behind the simulation.

Fig. SM-3:  Details of compound block representing the synchronous machine

Since the machine model is based on Park’s equivalent circuits, the input phase voltages and ouput currents must be transformed to and from the rotor reference frame, respectively.  Thus the leftmost green block contains the equations to transform the phase voltages to the rotor reference frame, and the q-d-0 rotor reference frame currents to phase currents.  The center red block contains ACSL code representing equations SM-4 to SM-21, and is thus the actual simulation of the electrical portion of the synchronous machine.  This block also contains constants for the parameters of the machine, which can be changed by the user to represent other machines.  The yellow box contains the code for equations SM-22 to SM-24 and determines the speed and rotor angle as a function of time.

The model of the synchronous machine is versatile and can be used to support lectures, laboratory experiments, and homework assignments.  One example where the model provides an excellent accessory to education is in explaining various operating characteristics of the machine. Several texts, particularly in power system analysis, develop analytical methods to approximate the transient dynamic characteristics of the synchronous machine. These include developing transient torque-angle curves along with determining transient stability using analytical tools such as the equal area criteria.  Although useful, these analytical techniques require approximations which lead to significant inaccuracies and do not account for actual machine behavior.  In contrast, using the detailed synchronous machine model to explain the dynamics, a more complete behavior of the system is obtained, without requiring any increase in student prerequisite knowledge.

For example, Fig. SM-4 shows a detailed dynamic response of a 835 MVA steam turbine generator to a 3-phase fault at the stator terminals of the machine.  Initially, the machine is operated at 50% of rated conditions (Ti = 1.11e+6 Nm, Exfd = 52 kV). With the machine operating in steady state, a 3-phase fault is simulated by setting the phase voltages to zero at  t = 6.0 sec. The fault is then removed by reapplying the voltages at  t = 6.2 sec.  From Fig. SM-4 it is seen that the fault causes an immediate offset in the stator phase currents, represented by Ibs. This offset is present because flux contained in the machine cannot change instantaneously; therefore, a dc offset in the phase currents occurs to maintain the flux at pre-fault values.  The dc offset in phase currents is reflected as oscillations in the transformed stator currents and the electromagnetic torque. Most undergraduate students will not have knowledge of Park’s transformation, or the respective transformed variables; however the compound blocks provide a means to remove that level of detail. In this example, the initial swing of the electromagnetic torque causes the rotor to decelerate.  This, in turn, causes the rotor angle to decrease, in what is commonly referred to as a ‘back swing’.  Back swings are not accounted for in analytical methods of analysis, and are a source of error in approximating transient dynamic behavior. Because the phase voltages are zero, the machine has no means of transmitting power to the system, and therefore the rotor begins to accelerate. The rotor continues to accelerate until the fault is cleared by reapplying the phase voltages.  Upon reapplication of the phase voltages (clearing of the fault), initial offsets in the phase currents appear and again produce 60 Hz oscillations in the rotor reference variables and the electromagnetic torque.


Fig. SM-4:  Results of application and removal of a three-phase short circuit

In this particular example, the machine eventually returns to the inital pre-fault operating point, as demonstrated by the plot of the power angle, delta.   Repeated fault simulations can be used to determine the critical clearing time, and to demonstrate the inability of the system to obtain pre-fault operating conditions once the critical clearing time is exceeded.

The results of a fault-study in which the fault is cleared after the critical clearing time are shown in Fig. SM-5.  In this case the rotor angle settles to a steady-state, post-fault value of approximately 390 degrees.  In other words, the rotor has advanced a complete revolution ahead of the system (which is referred to as slipping a pole).  Because of the large torques that occur on the turbine-generator during pole slipping, synchronous machines in the United States are designed with protection to prevent its occurance.

Smf41.jpg (29598 bytes)
Fig. SM-5:  Results of application and removal (after the critical clearing time) of a three-phase short circuit

For graduate and more advanced undergraduate students, the linear menu in the ACSL command window provides the user with a host of analysis options.  Included are options to compute the Jacobian, a linearized system model, the eigenvalues of the system, as well as bode, nyquist, and root-locus plots.  These options provide a means to test various control techniques, evaluate operating conditions, determine input/output impedances, and explain system behavior in more detail.  In the synchronous machine stability example described above, the eigenvalues of the system were computed within ACSL at steady state and are shown in Table SM-1.  Using these, the system behavior  can be explained in linear system-theory detail.  In particular, the complex set of eigenvalues with imaginary components approximately equal to the base frequency of the system correspond to the transient stator currents that produce 60 Hz oscillations in the electromagnetic torque.  This complex pair is often referred to as the stator eigenvalues.  The low frequency complex pair corresponds to the principal mode of oscillation of the rotor relative to the electrical angular velocity of the system and are portrayed in the damping and long-term oscillatory behavior observed in the rotor velocity and angle.  The remaining eigenvalues are real and correspond to the rate of decay of the transient currents in the rotor electrical windings.

Table SM-1:  Complex eigenvalues of a large synchronous machine, in ascending order, computed by ACSL 
   1   -0.37690500      
   2   -0.98638200      
   3   -5.95093000      
   4   -1.94917000   +/-9.69484000   9.888840  0.197108
   6   -10.2192000      
   7   -53.7368000      
   8   -4.44606000   +/-376.957000   376.984   
The model is compatible with machines of all ratings. Therefore it can be used in a laboratory environment to demonstrate the expected behavior of actual machines the students are using.  In addition the dynamic performance of various types of machines can be easily compared and the differences explained.  For example, a small horsepower synchronous machine will have a significant difference in dynamic performance compared to a large-scale generator.  This is well illustrated by comparing the fault responses of the 835 MVA generator depicted in Fig.-SM4 with the fault response of a 5-HP machine used in the Emerson Electric Machines Laboratory at Missouri-Rolla.  The 5-HP machine is a 220 volt, three-phase, two-pole generator.  The response of the machine, initially operating at Ti = 8.0 Nm and Exfd = 200 V, to a three-phase fault at the stator terminals is shown in Fig. SM-6.  Comparing Fig. SM-4 and SM-6, it is clear the 5-HP machine exhibits a much more damped response.  In particular, the stator currents and electromagnetic torque exhibit a much more damped response and they do not contain the sustained 60 Hz oscillations exhibited by the large machine.  This is due to the fact that the rotor circuits of the 5HP machine have much larger resistance values than those of the turbine-generator.  Therefore, the rotor circuits have much shorter time constants, and the flux in the 5 HP machine can change much faster than the turbine-generator.  Since the flux can change faster, the DC offset in the stator variables is very short lived.  Thus the 60 HZ oscillations in the torque and the transformed stator variables are barely observed. The difference in responses can also be explained by comparing the eigenvalues of the respective machines.  The eigenvalues of the 5-HP generator are shown in Table-SM-2.
Smf6.jpg (28043 bytes)
Fig. SM-6:  Short Circuit Response of a 5-HP Laboratory Generator
Table SM-2:  Complex eigenvalues of a 5 hp synchronous machine, in ascending order, computed by ACSL 
   1   -3.13336000   +/-14.7266000  15.05620  0.208111
   3   -40.1660000      
   4  -211.383000      
   5  -323.253000      
   6  -130.354000  +/-336.736000   361.0860   0.361005
   8  -4629.69