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ECE 8830 - Electric Drives Topic 4: Modeling of Induction Motor using qd0 Transformations Spring 2004 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 (cont’d) 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 Equations Stator Voltage Equations: d as vas ias rs dt d bs vbs ibs rs dt d cs vcs ics rs dt Voltage Equations (cont’d) Rotor Voltage Equations: d ar var iar rr dt d br vbr ibr rr dt d cr vcr icr rr dt Flux Linkage Equations Model of Induction Motor To build up our simulation equation, we could just differentiate each expression for , e.g. d as d [first row of matrix] vas dt dt But since Lsr depends on position, which will generally be a function of time, the trig. terms will lead to a mess! Park’s transform to the rescue! Park’s Transformation The Park’s transformation is a three-phase to two-phase transformation for synchronous machine analysis. It is used to transform the stator variables of a 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-axis is defined as leading the +ve q-axis by /2. (see Fig. 5.16c Ong on next slide). Park’s Transformation (cont’d) 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. Park’s Transformation (cont’d) The Park’s transformation equation is of the form: fq f a f T f d qd 0 b f0 fc where f can be i, v, or . Park’s Transformation (cont’d) cos q 2 Tqd 0 ( q ) sin q 3 1 2 2 2 cos q cos q 3 3 2 2 sin q sin q 3 3 1 1 2 2 Park’s Transformation (cont’d) The inverse transform is given by: cos q 1 2 Tqd 0 ( q ) cos q 3 2 cos q 3 sin q 2 sin q 3 2 sin q 3 Of course, [T][T]-1=[I] 1 1 1 Park’s Transformation (cont’d) Thus, vq va v T v d qd 0 b v0 vc and iq ia i T i d qd 0 b i0 ic Induction Motor Model in qd0 Acknowledgement: 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 (cont’d) 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, λ qd 0 Ts λ abc on the stator, where [Ts]= [T()], and λ dq 0r [Tr ]λ abcr ( to be defined) on the rotor, where [Tr]= [T()], ( to be defined) Induction Motor Model in qd0 (cont’d) STATOR: “abc”: abcs = Ls iabcs + Lsr iabcr “qd0”: qd0s= Ts abcs= Ts Ls Ts-1 iqd0s +Ts Lsr Ts-1 iqd0r ROTOR: qd0r= Tr abcr= Tr LsrT Ts-1 iqd0s +Tr Lr Tr-1 iqd0r Induction Motor Model in qd0 (cont’d) After some algebra, we find: Lar Tr LrTr1 0 0 and similarly for 0 Lar 0 0 0 where Lar= Lr-Lab Lar 1 s s . Ts L T But what about the cross terms? They depend on the choice of and . Let = - r , where r is the rotor position. Induction Motor Model in qd0 (cont’d) Now: 3 Lm 2 Tr LTsrTs1 Ts LsrTr1 0 0 0 3 L 2 m 0 0 0 0 Just constants!! Our double reference frame transformation eliminates the trig. terms found in our original equations. Induction Motor Model in qd0 (cont’d) We know what and r must be to make the transformation work but we still have not determined what to set to. We’ll come back to this but let us first look at our new qd0 constitutive law and work out simulation equations. vqd 0 s d T s vabcs T s Riabcs T s abcs dt d 1 1 T s RT s iqd 0 s T s T s qd 0 s dt d 1 Riqd 0 s T s T s qd 0 s dt Induction Motor Model in qd0 (cont’d) Using the differentiation product rule: vqd 0 s d d 1 Riqd 0 s qd 0 s T s T s qd 0 s dt dt Riqd 0 s d qd 0 s dt 0 d dt 0 d dt 0 0 0 0 qd 0 s 0 Induction Motor Model in qd0 (cont’d) For the stator this matrix is: 0 0 0 0 0 0 0 For the rotor the terminal equation is essentially identical but the matrix is: 0 ( r ) 0 ( ) 0 0 r 0 0 0 Induction Motor Model in qd0 (cont’d) Simulation model; Stator Equations: vqs iqs rs ds d qs dt d ds vds ids rs qs dt d 0 s v0 s i0 s rs dt Induction Motor Model in qd0 (cont’d) Simulation model; Rotor Equations: vqr iqr rr ( r )dr d qr dt d dr vdr idr rr ( r )qr dt d 0 r v0 r i0 r rr dt Induction Motor Model in qd0 (cont’d) 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 (cont’d) We can also write down the flux linkages: qs Las 0 ds 0 s 0 qr 3 2 Lsr dr 0 0 r 0 0 Las 0 0 3 2 Lsr 0 0 0 Las 0 0 0 0 3 2 Lsr 0 0 Lar 0 0 0 3 2 Lsr 0 0 Lar 0 0 iqs 0 ids 0 i0 s 0 iqr 0 idr Lar 0 i0 r Induction Motor Model in qd0 (cont’d) How do we pick ? One good choice is: d e dt where e is synchronous frequency. Remember that this choice makes a balanced 3 voltage set applied to the stator look like a constant. Induction Motor Model in qd0 (cont’d) The torque of the motor in qd0 space is given by: 3 P m qr idr dr iqr 2 2 where P= # of poles F=ma, so: d r J ( m l ) dt where l = load torque Induction Motor Model in qd0 (cont’d) Example: The equations for a balanced 3, squirrel cage, 2-pole rotor induction motor: Constitutive Laws: 3 m qr idr dr iqr 2 qs Las 0 ds qr 3 2 Lsr dr 0 0 Las 0 3 2 Lsr 0 Lar 3 2 Lsr 0 0 iqs 3 2 Lsr ids 0 iqr Lar idr Induction Motor Model in qd0 (cont’d) State equations: d qs rs iqs ds vqs dt d ds rs ids qs vds dt d qr rr iqr ( r )dr dt d dr rr idr ( r )qr dt d r ( m l ) dt J r= rotor speed = frame speed J= shaft inertia l = load torque 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 (cont’d) qd0 Induction Motor Model in Stationary Reference Frame (cont’d) Stator and Rotor Voltage Equations: vqs vds vqr vdr d rs iqs qs dt d rs ids ds dt d rr iqr qr r dr dt d rr idr dr r qr dt d v0 s rs i0 s 0 s dt d 0 r v0 r rr i0 r dt qd0 Induction Motor Model in Stationary Reference Frame (cont’d) Flux Linkage Equations: qs xls xm 0 ds 0 s 0 qr xm dr 0 0 r 0 0 xls xm 0 0 xm 0 0 0 xls 0 0 0 xm 0 0 xlr xm 0 0 0 xm 0 0 xlr xm 0 0 iqs 0 ids 0 i0 s 0 iqr 0 idr xlr i0 r qd0 Induction Motor Model in Stationary Reference Frame (cont’d) Torque Equation: 3P Tem (qr idr dr iqr ) 22 3P (ds iqs qs ids ) 22 3P xm (idr iqs iqr ids ) 22 Induction Motor Model in qd0 Example Example 5.3 Krishnan 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 (cont’d) qd0 Induction Motor Model in Synchronous Reference Frame (cont’d) Stator and Rotor Voltage Equations: d qs vqs iqs rs e ds dt d ds vds ids rs e qs dt d 0 s v0 s i0 s rs dt vqr iqr rr (e r )dr d qr dt d dr vdr idr rr ( e r )qr dt d 0 r v0 r i0 r rr dt qd0 Induction Motor Model in Synchronous Reference Frame (cont’d) Flux Linkage Equations: qs xls xm 0 ds 0 s 0 qr xm dr 0 0 r 0 0 xls xm 0 0 xm 0 0 0 xls 0 0 0 xm 0 0 xlr xm 0 0 0 xm 0 0 xlr xm 0 0 iqs 0 ids 0 i0 s 0 iqr 0 idr xlr i0 r qd0 Induction Motor Model in Synchronous Reference Frame (cont’d) Torque Equation: 3P Tem (qr idr dr iqr ) 22 3P (ds iqs qs ids ) 22 3P xm (idr iqs iqr ids ) 22 Induction Motor Model in Synchronous Reference Frame Example Example 5.5 Krishnan Steady State Model of Induction Motor The stator voltages and currents for an induction machine at steady state with balanced 3 phase operation are given by: vas Vms cos(et ) ias I ms cos(et s ) 2 vbs Vms cos( et ) 3 4 vcs Vms cos( et ) 3 2 ibs I ms cos( et s ) 3 4 ics I ms cos( et s ) 3 Steady State Model of Induction Motor (cont’d) Similarly, the rotor voltages and currents with the rotor rotating at a slip s are given by: var Vmr cos(set r (0) ) iar I mr cos(set r (0) r ) 2 vbr Vmr cos( s et r (0) ) ibr I mr cos( s et 2 r (0) r ) 3 3 4 4 i I cos( s t r (0) r ) vcr Vmr cos( s et r (0) ) cr mr e 3 3 Steady State Model of Induction Motor (cont’d) Transforming these stator and rotor abc variables to the qd0 reference with the q-axis aligned with the a-axis of the stator gives: v s v jv Vms e s qs s ds i s i ji I ms e s qs s ds jet j e jet v r (vqrr jvdrr )e jr (t ) (Vmr e j ( set r (0) ) )e jr (t ) i r (i ji )e r qr r dr j r ( t ) ( I mr e j ( set r (0) ) )e j r ( t ) where s and r= qd0 components in stationary frame and rotating ref. frames, respectively. Steady State Model of Induction Motor (cont’d) In steady state operation with the rotor rotating at a constant speed of e(1-s), r (t ) e (1 s)t r (0) This equation can be used to simplify the rotor voltage and current space vectors which become: v r vqrs jvdrs Vmr e j e jet i r i ji I mr e s qr s dr j ( r ) e jet Steady State Model of Induction Motor (cont’d) Use phasors to perform steady state analysis. Notation: A - rms values of space vectors B - rms time phasors Thus, Vms j 0 V as e 2 I ms js I as e 2 Vmr j V ar e 2 I mr j ( r ) I ar e 2 Steady State Model of Induction Motor (cont’d) and s qs s ds V jV s qs s ds I jI s qr s dr s dr I jI 2 iqss jidss 2 V jV s qr vqss jvdss I as e jet vqrs jvdrs 2 iqrs jidrs 2 V as e jet V ar e jet I ar e jet Steady State Model of Induction Motor (cont’d) Referring the rotor voltages and currents to the stator side gives: Ns V jV Nr 's qr 's dr Nr I jI Ns 's qr 's dr ' jet jet V e V e ar ar ' jet jet I e I e ar ar where the primed quantities indicate rotor quantities referred to the stator side. Steady State Model of Induction Motor (cont’d) 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 qs s ds s qs s ds s s V j V [rs je ( Lls Lm )](I j I ) je Lm (I 'qr j I 'dr ) s qr s qs s ds V ' j V ' j (e r ) Lm (I j I ) s dr s s [rr' j ( e r )( L'lr Lm )(I 'qr j I 'dr ) Steady State Model of Induction Motor (cont’d) Using the relationships between the rms space vectors and rms time phasors provided earlier, and re-writing (e-r) by se, and dropping the common ejt term, we get: Vas (rs je Lls )I as je Lm (I as I 'ar ) V 'ar (r 'r jse L 'lr )I 'ar jse Lm (I as I 'ar ) r 'r V 'ar ( j e L 'lr )I 'ar j e Lm (I as I 'ar ) s => s s Steady State Model of Induction Motor (cont’d) The relations on the previous slide can be rewritten as: e e V as (rs j xls )I as j xm (I as I 'ar ) b b e e r 'r V 'ar ( j x 'lr )I 'ar j xm (I as I 'ar ) s s b b where b is the base or rated angular freq. given by b 2 f rated where frated =rated frequency in Hz of the machine. Steady State Model of Induction Motor (cont’d) A phasor diagram of the stator and rotor variables with I m I as I ' is shown below ar together with an equivalent circuit diagram. Steady State Model of Induction Motor (cont’d) By adding and subtracting rr’ and regrouping terms, we get the alternative equivalent circuit representation shown below: e Steady State Model of Induction Motor (cont’d) 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 (cont’d) 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 (cont’d) In steady state: The average power developed is given by: 1 s ' Pem 3I rr s '2 ar The average torque developed is given by: Tem Pmech rm 3I r (1 s) 3I r s sm (1 s) s sm '2 ' ar r '2 ' ar r Steady State Model of Induction Motor (cont’d) 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 Supply The Thevenin circuit parameters are: jxm V th V as rs j ( xls xm ) jxm (rs jxls ) Zth rth jxth rs j ( xls xm ) 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: Vth2 (rr' / s) 3P Tem 2 e (rth rr' / s) 2 ( xth xlr' ) 2 We can use this equation to generate the torque-slip characteristics of an induction motor driven by constant voltage supply. Steady State Model of Induction Motor- Stator Input Impedance The stator input impedance is given by: jxm (rr' / s jxlr' ) Zin rs jxls ' rr / s j ( xlr' xm ) The stator input current and complex power are given by: V as I as Zin * as as Sin Pin jQin 3V Ι 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 m as ' lr x I I ' 2 2 (rr / s) ( x xm ) '2 ar 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 rr’ = 0.0764 xm = 5.834 xls = 0.2145 xlr’ = 0.2145 Jrotor= 2.8 kgm2 A comparison of the performance under constant voltage and constant current is shown in the accompanying handout.