Chapter 3 Generator and Transformer Models The Per Unit System 3 1 Introduction 1 Use of the per phase basis to represent a three phase balanced system 2 Use the model to describe transmission lines see Chapter 5 3 Simple models for generators and transformers to study steady state balanced operation 4 One line diagrams to represent a three phase system 5 The per unit system and the impedance diagram on a common MVA base 3 2 Synchronous Generators Note the coils aa bb and cc are 120o apart Axis of a coil is the x axis as shown These are Fr concentrated full pitch windings In r real machines the a Fsr windings are n distributed among many slots and often c b are not full pitch t We assume the r r windings produce a sinusoidal mmf around the rotor periphery angle F around the airgap s c b with respect to winding axis Assume the rotor is m a excited with DC current I f producing a flux which rotates with the rotor at speed At time t the rotor would have moved an angle t The flux linkage for an N turn a winding will be maximum N at t 0 The flux linkage with the a winding will go to 0 at t 2 The flux linkage will go to N at t and back to 0 at t 3 2 When t 2 the flux linkage will Rotor position shown at w t p 2 be back to N Thus a whole cycle This repeats every revolution of the rotor The flux linkage with winding a is thus given by a N cos t Using Faraday s law the voltage induced in phase a is given by d N sin t dt Emax sin t ea Emax cos t 2 Where Emax N 2 fN since 2 2 4 44 the rms value of the generated voltage is given by E 4 44 fN Fr a r Fsr n c b r t r The frequency is a function of speed and the number of poles thus P n f where n is 2 60 the speed in rpm the synchronous speed and P is the number of poles always an even number If the phase a is connected to a load then Fs b c a current ia will flow Depending on the load this current will have a m a phase angle say see figure lagging the generated voltage ea which is along the xRotor position shown at w t p 2 axis Again this is shown in the figure as the line mn N B The line mn is attached to and rotates with the rotor Be careful also that the Flux vectors are spatial while the voltage and current vectors are phasors in time The same is true for phases b and c but they will lag the voltage in phase a by 120 and 240 degrees respectively 2 Since ea sin t then we have ia Imax sin t ib I max sin t 120o ic I max sin t 240o Since mmf is proportional to the current we then have Fa Fm sin t Fb Fm sin t 120o Fc Fm sin t 240o We now take components of these phasors along the line mn and in quadrature with it Along mn we have F1 Fm sin t cos t Fm sin t 120o cos t 120o Fm sin t 240o cos t 240o 1 Using the identity sin cos sin 2 the above equation becomes 2 F1 Fm sin 2 t sin 2 t 120o sin2 t 240o 2 It is noted that this expression is the sum of three balanced phasors hence is equal to zero Next we consider the components of the mmf perpendicular to mn F2 Fm sin t sin t Fm sin t 120 o sin t 120o Fm sin t 240 o sin t 240o Using the identity sin 2 F2 1 1 cos2 we have 2 Fm 3 cos2 t cos2 t 120o cos2 t 240o 2 The three cosine terms add to zero balanced phasors thus we have 3 FS 3 Fm 2 Thus the result of having three pulsating single phase fluxes produce when applied symmetrically a constant flux perpendicular to line mn and rotates at the same speed n as the rotor This flux is called the armature reaction to the field of the rotor Fr the various fields are shown for one phase say phase a in the diagram below Note that Fs is perpendicular to line mn and rotates with it at the same speed The fact that Fs which is proportional to Ia is perpendicular to line mn indicates that that we can theorize that the armature current Ia is producing a reactive voltage drop parallel to line mn due to inductance in the machine Run the command rotfield in a Matlab window to see a demo of the rotating field Fr Fsr n Ear Esr Fs jX ar Ia V R a Ia m E jX l Ia Ia Figure 3 2 Page 53 is the angle between V and I a it is the power factor angle is the angle between E and V it is the power angle First the rotor field Fr produces the no load generated voltage E at zero armature current Note that E lags Fr by 90 degrees E is called the excitation voltage It is directly proportional to the field current The voltage current phasors for phase a are shown above lagging the flux diagram by 90 deg Note the above diagram is a hybrid combining spatial and temporal vectors 4 Second assume the armature carries a current to a load now the armature reaction flux Fs is produced This is perpendicular to line mn The two fluxes due to rotor and armature combine together to form the resultant flux Fsr The resultant flux induces the generated on load emf Esr The armature mmf Fs induces the voltage Ear known as the armature reaction voltage In all cases each mmf produces a voltage lagging the mmf by 90 degrees Note that the voltage Ea r leads Fs hence Ia by 90 degrees Thus we can theorize an inductor model for this relationship with reactance Xar i e Ear jX ar I a Xar is known as the reactance of armature reaction Thus we now have the circuit equation E Esr jX ar I a The terminal voltage V is found by considering the armature resistance and leakage reactance thus E V Ra j X l X ar I a Which can be simplified to E V Ra jX s I a where X s X l X ar is known as the synchronous reactance The angle between V and Ia is the power factor angle The power angle is the angle between E and V called The circuit model of the machine is as shown below The resistance Ra is much smaller than the synchronous reactance Xs and is often neglected This is shown in the simplified one line diagram below where the machine is shown connected …