SECTION - I PULL-IN-OPERATION Op SYNCHRONOUS MOTORS
14 S 1.1 INTRODUCTION The starting of synchronous, reluctance and permanent magnet synchronous motors is normally carried out by damper winding. The starting characteristics of these motors are therefore dominated by the induction motor action in the first phase and by the synchronous motor action in the later phase near synchronous speed. As the motor initially at rest is switched on to the supply, it will have first electrical transients. These electrical transients will die out quickly due to low electrical time-constant. The rotor gains acceleration and picks up speed -under the influence of average torque of.induction motor action. The time required to attain a particular speed depends upon the developed torque, connected load and inertia. The rotor attains a speed slightly below synchronous speed and would have stably operated at this speed had the reluctance torque been absent. The anisotrophy. in the rotor gives rise to pulsating torque pulsating at double the slip frequency. This pulsating torque is due to the difference in the electric and magnetic parameters of the rotor in the 3- and <3- axfs. Though the average value of this pulsating torque is zero, it does cause fluctuations in speed. These fluctuations increase in magnitude, but its frequency decreases as the synchronous speed is approached. If the synchronising (including reluctance) torque is
15 sufficiently high, the rotor along with its load is accelerated upto synchronous speed from the pseudo-equilibrium speed of induction motor action, during the positive half cycle of the synchronising torque. Due to inertia, the rotor sweeps past the synchronous speed giving rise to a retarding torque. The retardation is further increased due to negative half-cycle of the. synchronising torque. Thus the rotor hunts for sometime before settling down to synchronous speed* Above discussion reveals that the starting characteristic of the motor can be divided into three.different zones. These are s (i) the period during which the rotor speed is negligibly small, but electrical transients are large; (ii) the period of acceleration of the rotor from a very low speed to the pseudo equilibrium speed as an induction motor; (iii) pull-in due to synchronising torque i,e., due to both the exciting field and saliency. In this section, the performance of a synchronous machine with constant excitation in the third zone, and those of reluctance and permanent-magnet motors in the second and the third zones are studied. The differential equations of motion during the synchronising process are non-linear; as such the phase-plane method, the numerical integration and differenceequation techniques are employed using a digital computer. In case of salient-pole synchronous motor the field winding is initially short circuited through a resistance and as the motor reaches a speed very near to its synchronous speed,
16 direct voltage is applied to the field winding to pull the motor into synchronism. The synchronising phenomenon is of nature marked by following important factors s (a) Initially under asynchronous condition the salient-pole motor does not run at a constant speed due to non-uhiform reluctance of the gap and unbalanced nature of the damper winding. (b) The torque required to accelerate the motor from asynchronous to synchronous speed operation is dependent on the moment of inertia of the rotor and its associated mass. Presence of damper winding modifies the effective inertia constant to certain extent as shown by Tolmach(^)«(c) With the application of field excitation, voltage in the armature rises to a steady value, the time taken for such a rise depending mainly on the field circuit time-constant and also the inertia of the rotor and its associated mass. (d) The maximum value of the synchronising torque for a particular excitation depends approximately on the steady value of the field current attained, and as such the available torque during the process of synchronisation varies continuously, as the field current rises. In chapters I and II the synchronising phenomenon in salient-pole synchronous motor is studied, keeping in mind the above factors. The phase-plane techniqueand the Runge-Kutta^'*^ method are used in Chapter I; the influence of idle currents and torques are neglected and only electro-
17 magnetic moments as functions of 5 and $ are considered and not the equations derived by Park^. In case of "conventional" synchronous motor,-the synchronising torque is predominently due to the exciting current. But in case of a reluctance machine, the synchronising torque is developed due to saliency alone. If the synchronising torque due to saliency is not sufficient, the motor may fail to reach the synchronous speed and will run stably at a speed below synchronous speed.with fluctuations in torque a^d speed. In ca.se of permanent-magnet synchronous motor, in the period during which the rotor speed is negligibly'small but electrical transients are large, the presence of permanent magnetism in the rotor provides braking action. During the period of acceleration of the rotor from very low speed to about 90$ of synchronous speed as an induction motor, the presence of saliency and permanent-magnetism in the rotor provides a pulsating torque resulting in continuous pulsation of speed. During pull-in, due to synchronising torque,, the presence of permanent-magnetism and saliency in the rotor provides necessary acceleration under certain conditions. S 1.2. ASSUMPTIONS AND LIMITATIONS : 1. Simplified equations ^ of a synchronous machine with two damper coils are used. As such assumptions involved in the formulation of equations based on Park s 'Ideal* synchronous machine are also tenable here
18 2* The synchronisation phenomenon is studied from a speed about 9o to 95$ of synchronous speed*, 3. Following similar lines, as Kingsley^4^ the effect of saturation is taken into account by modifying the parameter Xj and Xr»A corresponding to the rated voltage by means of a saturation factor. 4. Quadrature axis saturation affects are ignored. S 1.3 EQUATIONS FOR SYNCHRONISING PH3K0MSSIA s When a synchronous, reluctance or permanent-magnet motor is running at a speed, very near synchronous speed, the position of the reference axis, at any instant is given by - «l-...(si-1) With the varying speed, the rotor position is ^ ^ ~ ^...(SI-2) Differentiating both sides of equation (SI-2) with respect to i and noting M to bs ^ ^ = co -...(SI-3) or ^ =...(SI-4) as ^ is the fractional slip. The phase voltage can be expressed as s i
19 S- try, CO L~ 1. *, C& + 2) * ^<~% C*>6 + Cot...(SI-5) Comparing the R. H* S«of equation (SI.5) with Park* s transformation^^ equation s J = m 1 (..(SI»6) Using equation (SI.6) and writing Park* voltage equations : ev, ^ s -- :«W -...(SI.7) E T S -- - <^d + (.-f + t=s " (8I,8) where co -h, -= Cb) Ki * & C^~) es? (SI.9) co '4' <V '<jr i, (SI.10) X<=v. Cb) - C i * V' /»J) C I + f>)..(si.11) ^ ^ ( > + Tel'b) C' + r*i I*) v Xc/ C<V * ---------------- ------------XJ...(SI,12) C + j>) (t+ TjJ'b) (For conventional and permanent magnet synchronous motors)
20 ' C l + T<sJ I» ^ O) - --------r-=~, ( i+ WfO...(SI*13) (For a reluctance motor as the field exciting winding is absent) & 6 60 C ' + TkJ /O -- (l + r/~yjx^^ f>) XiwJ f...(si.14) (For ordinary synchronous motor) <ZCk>)...(SI.15) (For reluctance Motor) <S 00 *f CO...(SI.16) (For a permanent-ihagnet synchronous motor) where to I is the direct axis flux-linkage due to presence of permanent-magnetism in the rotor alone. This unidirectional flux is similar to the field flux In a conventional synchronous machine, as defined by Walshaw and Lynn (43) and Is given by where EJ XI...(S 1.17) B0 is the useful flux density at no load i.e., FJ reduced by an amount corresponding to leakage (figure S I.l) a is the cross-sectional area of the magnets. & N is the equivalent turns of the direct axis armature winding.
21 1 S 1 ROJRE'SI.I 're=manl'\t - MAGNET OPERATING DIAGRAM. m
22 From equation (S 1*3), the acceleration is Km) = - K* Hence the torque equation according to Adkins (1) o is...(s 1.18) - \ri + -Te * -Tt 03..(S 1.19) where -r. = ^ ^ ' *' - ^ ' */ )..(S 1.20) Combining equations (S 1.19) and (S 1.20) th Km)- * CO *>* 03 (K, K^'d) ***^s I#21* Equations (S 1.7), (S 1.8), <Sl.?),(S I.10) and (S 1.21) are the complete set of differential equations for all these types of synchronous motor during pull-in-operation. These equations are non-linear, due to presence of multiples of two variables aamely j=>s, axis currents and axis flux-linkages. The analytical solution for them does not exist and hence the use of analogue computer or numerical methods with the help of digital computer becomes inevitable. For studying the pull-in operation for synchronous motor using phase-plane method, simpler expressions of in equation (S 1.19) is chosen as statad in details in Chapter I (article 1.3).