Modelling of IM Using Dq Transformation

Topic 4: Modeling of Induction Motor using qd0 TransformationsSpring 2004ECE 8830 - Electric Drives Introduction Steady state model developed in previous topic neglects electrical transients due to load changes and stator frequency variations. Such variations arise in applications involving variable-speed drives. Variable-speed drives are converter-fed from finite sources, which unlike the utility supply, are limited by switch ratings and filter sizes, i.e. they cannot supply large transient power. Introduction (contd)Thus, we need to evaluate dynamics of converter-fed variable-speed drives to assess the adequacy of the converter switches and the converters for a given motor and their interaction to determine the excursions of currents and torque in the converter and motor. Thus, the dynamic model considers the instantaneous effects of varying voltages/currents, stator frequency and torque disturbance. Circuit Model of a Three-Phase Induction Machine (State-Space Approach) Voltage EquationsStator Voltage Equations:asas as sdv i rdt +bsbs bs sdv i rdt +cscs cs sdv i rdt + Voltage Equations (contd)Rotor Voltage Equations:arar ar rdv i rdt +brbr br rdv i rdt +crcr cr rdv i rdt + Flux Linkage Equations Model of Induction Motor To build up our simulation equation,we could just differentiate each expression for , e.g. But since Lsr depends on position, which will generally be a function of time, the trig. terms will lead to a mess! Parks transform to the rescue!asasd dvdt dt [first row of matrix] Parks Transformation The Parks transformation is a three-phaseto two-phase transformation for synchronous machine analysis. It is used to transform the stator variables ofa synchronous machine onto a dq reference frame that is fixed to the rotor. The +ve q-axis is aligned with the magnetic axis of the field winding and the +ve d-axisis defined as leading the +ve q-axis by /2. (see Fig. 5.16c Ong on next slide).

Parks Transformation (contd)

The result of this transformation is that all time-varying inductances in the voltage equations of an induction machine due to electric circuits in relative motion can be eliminated. Parks Transformation (contd)The Parks transformation equation is of the form: where f can be i, v, or .00q ad qd bc ]] ]] ] ] ]] ]] ]]f ff T ff f Parks Transformation (contd)02 2cos cos cos3 32 2 2( ) sin sin sin3 3 31 1 12 2 2q q qqd q q q q ]| ` | ` + ]. , . , ] ]| ` | ` ] + ] ]. , . , ] ] ] ]T Parks Transformation (contd)The inverse transform is given by:Of course, [T][T]-1=[I]10cos sin 12 2( ) cos sin 13 32 2cos sin 13 3q qqd q q qq q ] ] ] ]| ` | ` ] ] ]. , . , ] ]| ` | `+ + ] . , . , ]T Parks Transformation (contd)Thus,and 00q ad qd bcv vv T vv v ]] ]] ] ] ]] ]] ]]00q ad qd bci ii T ii i ]] ]] ] ] ]] ]] ]] Induction Motor Model in qd0Acknowledgement: The following notes covering the induction motor modeling in qd0 space are mostly courtesy of Dr. Steven Leeb of MIT. Induction Motor Model in qd0 (contd) This transform lets us define new qd0 variables. Our induction motor has two subsystems - the rotor and the stator - to transform to our orthogonal coordinates: So,on the stator,

where [Ts]= [T()], ( to be defined) andon the rotor, where [Tr]= [T()], ( to be defined)[ ]0 qd s abc T0[ ]dq r r abcr T Induction Motor Model in qd0 (contd)STATOR:abc:abcs = Ls iabcs + Lsr iabcrqd0: qd0s= Ts abcs= Ts Ls Ts-1 iqd0s +Ts Lsr Ts-1 iqd0rROTOR:qd0r= Tr abcr= Tr LsrT Ts-1 iqd0s +Tr Lr Tr-1 iqd0r Induction Motor Model in qd0 (contd)After some algebra, we find:where Lar= Lr-Laband similarly for.

But what about the cross terms? They depend on the choice of and . Let = - r , where r is the rotor position.10 00 00 0arr r r ararLT L T LL ] ] ] ] ]1s s sT L T Induction Motor Model in qd0 (contd) Now:

Just constants!! Our double reference frame transformation eliminates the trig. terms found in our original equations. 1 130 0230 020 0 0mTr sr s s sr r mLT L T T L T L ] ] ] ] ] ] ] Induction Motor Model in qd0 (contd) We know what and r must be to make the transformation work but we still have not determined what to set to. Well come back to this but let us first look at our new qd0 constitutive law and work out simulation equations.( )0 qd s abcs abcs abcss s sdv T v T Ri Tdt +1 10 0 qd s qd ss s s sdT RT i T Tdt +10 0 qd s qd ss sdRi T Tdt + Induction Motor Model in qd0 (contd)Using the differentiation product rule:( )10 0 0 0 qd s qd s qd s qd ss sd dv Ri T Tdt dt ] + + ] ]( ) 0 0 00 00 00 0 0qd s qd s qd sddtd dRidt dt ] ] ] ] + + ] ] ] ] ] Induction Motor Model in qd0 (contd)For the stator this matrix is:For the rotor the terminal equation is essentially identical but the matrix is:0 00 00 0 0 ] ] ] ] ]0 ( ) 0( ) 0 00 0 0rr ] ] ] ] ] Induction Motor Model in qd0 (contd)Simulation model; Stator Equations:dsds ds s qsdv i rdt +qsqs qs s dsdv i rdt + +00 0ss s sdv i rdt + Induction Motor Model in qd0 (contd)Simulation model; Rotor Equations:( )drdr dr r r qrdv i rdt +( )qrqr qr r r drdv i rdt + +00 0rr r rdv irdt + Induction Motor Model in qd0 (contd) Zero-sequence equations (v0s and v0r) may be ignored for balanced operation. For a squirrel cage rotor machine, vdr=vqr=0. Induction Motor Model in qd0 (contd)We can also write down the flux linkages:0 0 00 0 00 0 3 2 0 00 0 0 3 2 00 0 0 0 03 2 0 0 0 00 3 2 0 0 00 0 0 0 0qs as sr qsds as sr dss as sqr sr ar qrdr sr ar drr ar rL L iL L iL iL L iL L iL i ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] Induction Motor Model in qd0 (contd)How do we pick ?One good choice is:where e is synchronous frequency. Remember that this choice makes a balanced 3 voltage set applied to the stator look like a constant. eddt Induction Motor Model in qd0 (contd)The torque of the motor in qd0 space is given by:where P= # of polesF=ma, so: where= load torque ( )32 2m qr dr dr qrPi i | ` . ,( )rm ldJdt l Example: The equations for a balanced 3, squirrel cage, 2-pole rotor induction motor: Constitutive Laws:Induction Motor Model in qd0 (contd)( )32m qr dr dr qri i 0 3 2 00 0 3 23 2 0 00 3 2 0qs as sr qsds as sr dsqr sr ar qrdr sr ar drL L iL L iL L iL L i ]]] ]]] ]]] ]]] ]]] ]]] Induction Motor Model in qd0 (contd)State equations:r= rotor speed = frame speedJ= shaft inertia l = load torqueds s ds qs dsdr i vdt qs s qs ds qsdr i vdt + ( )dr r dr r qrdr idt ( )qr r qr r drdr idt + ( )m l rddt J qd0 Induction Motor Model in Stationary Reference Frame The qd0 induction motor model in the stationary reference frame can be obtained by setting =0. This model is known as the Stanley model and the equivalent circuits are given on the next slide. qd0 Induction Motor Model in Stationary Reference Frame (contd) qd0 Induction Motor Model in Stationary Reference Frame (contd)Stator and Rotor Voltage Equations:qs s qs qsdv r idt +ds s ds dsdv r idt +qr r qr qr r drdv r idt + dr r dr dr r qrdv r idt + +0 0 0 s s s sdv r idt +00 0rr r rdv r idt + qd0 Induction Motor Model in Stationary Reference Frame (contd)Flux Linkage Equations:0 00 00 0 0 00 0 0 00 0 0 0 00 0 0 00 0 0 00 0 0 0 0qs ls m m qsds ls m m dss ls sqr m lr m qrdr m lr m drr lr rx x x ix x x ix ix x x ix x x ix i+ ]]] ]]]+ ]]] ]]] ]]]+ ]]] ]]]+ ]]] ]]] ]]] qd0 Induction Motor Model in Stationary Reference Frame (contd)Torque Equation:3( )2 2em qr dr dr qrPT i i 3( )2 2ds qs qs dsPi i 3( )2 2m dr qs qr dsPx i i i i Example 5.3 KrishnanInduction Motor Model in qd0 Example qd0 Induction Motor Model in Synchronous Reference Frame The qd0 induction motor model in the synchronous reference frame can be obtained by setting = e . This model is known as the Kron model and the equivalent circuits are given on the next slide. qd0 Induction Motor Model in Synchronous Reference Frame (contd) qd0 Induction Motor Model in Synchronous Reference Frame (contd)Stator and Rotor Voltage Equations:qsqs qs s e dsdv i rdt + +dsds ds s e qsdv i rdt +00 0ss s sdv i rdt +( )drdr dr r e r qrdv i rdt +( )qrqr qr r e r drdv i rdt + +00 0rr r rdv irdt + qd0 Induction Motor Model in Synchronous Reference Frame (contd)Flux Linkage Equations:0 00 00 0 0 00 0 0 00 0 0 0 00 0 0 00 0 0 00 0 0 0 0qs ls m m qsds ls m m dss ls sqr m lr m qrdr m lr m drr lr rx x x ix x x ix ix x x ix x x ix i+ ]]] ]]]+ ]]] ]]] ]]]+ ]]] ]]]+ ]]] ]]] ]]] qd0 Induction Motor Model in Synchronous Reference Frame (contd)Torque Equation:3( )2 2em qr dr dr qrPT i i 3( )2 2ds qs qs dsPi i 3( )2 2m dr qs qr dsPx i i i i Induction Motor Model in Synchronous Reference Frame ExampleExample 5.5 Krishnan Steady State Model of Induction MotorThe stator voltages and currents for an induction machine at steady state with balanced 3 phase operation are given by:cos( )as ms ev V t 2cos( )3bs ms ev V t 4cos( )3cs ms ev V t cos( )as ms e si I t 2cos( )3bs ms e si I t 4cos( )3cs ms e si I t Steady State Model of Induction Motor (contd) Similarly, the rotor voltages and currents withthe rotor rotating at a slip s are given by:cos( (0) )ar mr e rv V s t cos( (0) )ar mr e r ri I s t 2cos( (0) )3br mr e rv V s t 4cos( (0) )3cr mr e rv V s t 2cos( (0) )3br mr e r ri I s t 4cos( (0) )3cr mr e r ri I s t Steady State Model of Induction Motor (contd)Transforming these stator and rotor abcvariables to the qd0 reference with the q-axisaligned with the a-axis of the stator gives: where s and r= qd0 components in stationary frame and rotating ref. frames, respectively.ej t s ssqs ds msv jv V e vvej t s s jsqs ds msi ji I e e iv( (0) ) ( ) ( )( ) ( )e r r rj s t j t j t r rrqr dr mrv jv e V e e vv( (0) ) ( ) ( )( ) ( )e r r rj s t j t j t r rrqr dr mri ji e I e e iv Steady State Model of Induction Motor (contd) In steady state operation with the rotor rotating at a constant speed of e(1-s), This equation can be used to simplify the rotor voltage and current space vectors which become:( ) (1 ) (0)r e rt s t +ej t s s jrqr dr mrv jv V e e vv( )e rj t j s srqr dr mri ji I e e + iv Steady State Model of Induction Motor (contd) Use phasors to perform steady state analysis. Notation: A - rms values of space vectors- rms time phasors Thus,B02jmsasVe V2sjmsasIe I%2jmrarVe V( )2rjmrarIe + I% Steady State Model of Induction Motor (contd)and2es ss sqs ds j tqs ds asv jvj e V V Vur ur2es ss sqr dr j tqr dr arv jvj e V V Vur ur2es ss sqs ds j tqs ds asi jij e I I Ir r%2es ss sqr dr j tqr dr ari jij e I I Ir r% Steady State Model of Induction Motor (contd) Referring the rotor voltages and currents to the stator side gives: where the primed quantities indicate rotor quantities referred to the stator side. ' ' 'e es sj t j tsqr dr ar arrNj e eN | ` . ,V V V Vur ur' ' 'e es sj t j trqr dr ar arsNj e eN | ` . ,I I I Ir r% % Steady State Model of Induction Motor (contd) In the stationary reference frame, the qd0 voltage and flux linkage equations can be rewritten in terms of the complex rms space voltage vectors as follows:[ ( )]( ) ( ' ')s s s ss sqs ds qs dss e ls m e m qr drj r j L L j j L j + + + V V I I I Iur ur r r r r' ' ( ) ( )s s ssqs ds qrdr e r mj j L j V V I Iuur ur r r' '[ ( )( )( ' ')s sr e r lr m qr drr j L L j + + + I Ir r Steady State Model of Induction Motor (contd) Using the relationships between the rms space vectors and rms time phasors provided earlier, and re-writing (e-r) byse, and dropping the common ejtterm, we get:s =>( ) ( ')as as ass e ls e m arr j L j L + + + V I I I% %% ' ( ' ' ) ' ( ')as arr e lr ar e m arr js L js L + + + V I I I% %%' '( ' ) ' ( ')arrase lr ar e m arrj L j Ls s + + +VI I I% %% Steady State Model of Induction Motor (contd)The relations on the previous slide can be rewritten as:where b is the base or rated angular freq.given bywhere frated =rated frequency in Hz of the machine.( ) ( ')e eas as ass ls m arb br j x j x + + + V I I I% %%' '( ' ) ' ( ')are e raslr ar m arb brj x j xs s + + +VI I I% %%2b ratedf Steady State Model of Induction Motor (contd) A phasor diagram of the stator and rotor variables with is shown below together with an equivalent circuit diagram.'m asar + I I I%%% Steady State Model of Induction Motor (contd) By adding and subtracting rr and regrouping terms, we get the alternative equivalent circuit representation shown below:

e Steady State Model of Induction Motor (contd)The rr (1-s)/s resistance term is associated with the mechanical power developed. The rr/s resistance term is associated with the power through the air gap.

Steady State Model of Induction Motor (contd) If our main interest is on the torque developed, the stator side can be replaced by the Thevenin equivalent circuit shown below: Steady State Model of Induction Motor (contd)In steady state:The average power developed is given by:The average torque developed is given by:' 2 '13em ar rsP I rs| ` . ,' 2 '' 2 '3 (1 )3(1 )mech ar rem ar rrm sm smP Ir sT Irs s s Steady State Model of Induction Motor (contd)The operating characteristics are quite different if the induction motor is operated at constant voltage or constant current. Constant voltage -> stator series impedance drop is small => airgap voltage close to supply voltage over wide range of loading. Constant current -> terminal and airgap voltage could vary significantly. Steady State Model of Induction Motor- Constant Voltage Supply Shorting the rotor windings and operating the stator windings with a constant voltage supply leads to the below Thevenin equivalent circuit. Steady State Model of Induction Motor- Constant Voltage SupplyThe Thevenin circuit parameters are: ( )mth ass ls mjxr j x x+ +V V( )( )m s lsth th ths ls mjx r jxr jxr j x x+ + + +Z Steady State Model of Induction Motor- Constant Voltage Supply The average torque developed for a P-pole machine with constant voltage supply is given by: We can use this equation to generate the torque-slip characteristics of an induction motor driven by constant voltage supply.2 '' 2 ' 2( /) 32 ( /) ( )th reme th r th lrV r s PTr r s x x + + + Steady State Model of Induction Motor- Stator Input ImpedanceThe stator input impedance is given by:The stator input current and complex power are given by:' '' '( / )/ ( )m r lrin s lsr lr mjx r s jxr jxr s j x x+ + ++ +Z asasin VIZ% *3as asin in inP jQ + S V % Steady State Model of Induction Motor- Constant Current Supply With a constant current supply, the stator current is held fixed and the stator voltage varies with the input impedance given on the previous slide. The rotor current Iar can be used to determine the torque and is given by:2 2' 2' 2 ' 2( /) ( )masarr lr mx IIr s x x+ + Comparison of Constant Voltage vs. Constant Current Operation Consider a 20 hp, 60Hz, 220V 3 induction motor with the following equivalent circuit parameters:rs = 0.1062 xls = 0.2145 rr = 0.0764 xlr = 0.2145 xm = 5.834 Jrotor= 2.8 kgm2 A comparison of the performance under constant voltage and constant current is shown in the accompanying handout.