This paper shows a universal method for composing a simula- tion model for any ... in a PM brush-less motor or PM stepping motor, by currents through rotor ...
193
A Universal Method for Modelling Electrical Machines A. Veltman
P.P.J. van den Bosch
Abstract This paper shows a universal method for composing a simulation model for any type of electrical rotating machine. Models for Asynchronous machines, Permanent Magnet- or exited Synchronous machines, brush-less and brush DC, Stepping- and Variable Reluctance motors are presented. A simulation model is needed when detailed interactions between driver circuitry, machine, load and controller are t o be studied. In such cases it is usually not possible to transform the system into a solvable matrix differential equation. A general applicable transient machine model, interacting easily with existing models of its surroundings, the inverter and the mechanical load, is to be preferred.
1
General machine characteristics
In literature different transient machine models are used. For large machines 121, the shaft speed is considered piece wise constant in order t o solve the machines fourth or higher order differential equations. An other method considers a transformation to flux coordinates ]1],[4]. Both mentioned methods do not permit straight forward implementation of auxiliary effects like stator tooth effects, salient rotor poles and skin effects. Further more, switching from voltage input t o current input is not easy. In this paper a universal method for modelling electrical machines, that does permit easy addition of these auxiliary effects and optional voltage or impressed current application, is presented. One common characteristic of all rotating electrical machines is that it consists of a rotating part, the rotor, and a static non moving part, the stator. Multiphase machines can be reduced t o a n equivalent model with just two orthogonal windings in stator and rotor, as depicted in figure 1. In this paper, all electrical quantities like currents, voltages and fluxes are considered as space vectors. These vectors are represented in different orthogonal coordinate systems. An equivalent circuit or machine model contains two identical per phase circuits, one for each space axis. A vector named V that applies to node pair N represented in coordinate system S is written as:
%
=
(Z)
The indices a,@ refer to standstill stator coordinates, while a, b refer to coordinates fixed to the rotor. The rotor coordinate system is turned by shaft angle 0 relative to the stator coordinates. The asynchronous machine model is used as the basic electrical machine in this paper. The electrical properties or the windings as indicated in figure 1, representing resistance R, a mutual main inductance L,, and a leakage inducta.nce Lo,
Figure 1: Defining the stator (a,@)and rotor (a,b) relerence frames.
conclude to the general linear steady state machine model for an asynchronous machine with shorted rotor windings and slip frequency slip, depicted in figure 2
$=----?3
Gi
1,
E¶ dip
Figure 2: General linear steady state per phase asynchronous machine model, supplied by a sinusoidal voltage source.
The machine can produce torque T., based on the Lorenlz = 1 . f x l?, the force on a conductor with IengtJ Force: 1 and current f i n a space with magnetic field flux density R . Lorents’s law acts on both rotor and stator windings. Windings are applied to generate a current distribution around the circumference of the stator and/or the rotor. The magnetic flux density B can be generated by a permanent magnet like in a P M brush-less motor or PM stepping motor, by currents through rotor windings as in,an exited synchronous machipe or induced as in asynchronous machines.
-
The eleLtrical machine is an electro mechanical transducer and can be decomposed in three fundamental sections: the stator section, the rotor section and the mechanical section. Energy and power are transferred between these three sections. Each section contains a t least one energy bufIer. Magnetic energy is stored in inductors in stator, rotor and the air gap. Mechanical energy is stored in the inertia of the rotating parts. For example in an induction machine, electrical energy can be transferred from the stator t o the rotor as in an ordinary bansformer by means of the mutual inductance of both windings. Electrical energy is first transferred into magnetic energy in the machine’s core and air gap, and is succeedingly transferred back into electrical energy in the secondary (rotor) winding. These basic ways of energy exchange are depicted in figure 3.
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i i s supposc~l,caused by an arl>il,raryrornplex irnpcrlanrr ZR nu I.hp rnt.or side of the IRTF. The energy in a single inductor is the p r o d u c t of flux and currrnt: EL = W L I L . T h e energy E in a system with vector qnantitirs, seen from the stator or the rotor is the inner product of both flux and current. wctors. Torque can be expressed by means of t l ~ epartial derivative Te = 'l'liis torqrip is direrted into the IRl'Fsee ( R ) , llicrefore a minus sign is added to get, a n expression for tlie developed torque to t l w nrotor shaft..
g.
~(6,j,q5)= wonm
Tc
The Ideal Rotating TransFormer
Two elFrtrical ports and one mechaniral port.. Two phase orthogonal windings on primary arid serondary side. No winding resistance on pither side. All flux is mutual, no leakage flux. Winding ratio is 1 : I unless otherwise sperified. Magnetizing current is zero, its main inductance L, =
7;
-
~
is
=
lGsll.sl
sin 4
x $ ~
(7)
(8)
t 'Pglz = I : / - '@If :
(9)
IJecause t,he defined orientation of the rotor current is opposite to the stator current, the torque in (8) expressed in quantities on the rotor side have t,he opposite in sign, and T. does not depend on the shaft angle 0. It is not surprising to find great similarity between ( 8 ) and Lorentz's force in section 1. N o rest.rictions to the momentary phase angle 4 were made, therefore ( 8 ) ran be used in general.
2.2
00.
= I ~ s ~ ~ 1 3 ~ c o s(6) ~
~_ ( $i,_ , 4) -a_
Equation (8) shows that torque is determined by the outer product of f'and 6 , (8) ran also lie written as:
T. = -*:I; I n figure 3, the Ideal Rotattrig TransFormer, IRTF [G]thkes care nf the energy exchaiigP trrtween stator, rolnr and the mechanical section. The IRTF is an ideal transformer with:
@.is
84
Figure 3: IJasic ways oj energy ezchange in a general electrical machine
2
=
Virtual impedances
Suppose an IRTF with a rotor circuit that just contains a resistor Rn switched in series with an inductor L R . Under steady state conditions and sinusoidal voltage and currents, the rotor impedance is:
The angle between t,he primary referenre frame and the secondary reference frame is equal to the shall, displacement angle 0. When the shaft angle 0 does not change, the IRTF is equal to an ordinary transformer model with fixed windings.
pn Z R = RR + J W R L R= -1R
Tlie IRTF is ail ideal r~itilti-p~rt, translorrner wit.11 no losses and no eni'rgy storage. Therefore, tlie total input power of the I R 7 F , the sun1 of stator power Ps, rotor power PR and mechanical power P M , is zero. Mechanical power is the product of shaft speed wW and torqne 2;.
(3) A n ideal transformer transforms current and voltage of any In frequency perfectly by means of the mutual main flux I. order t o maintain a flux in an IRTF, no magnetizing current is needed, so the current on the rotor side of the IRTF is calculated from the current on the stator side or visa versa. Only one electrical quantity can be input on either rotor or stator side so that if is input on the rotor side, 6 : must be inpnt on the stator side, as indicated by the following expressions.
(10)
The rotor flux is tlip time integral of impedance times current: ,jjn
~
RR + i ~ ~ J , n j ~ , ( ~ ~ t + ~ ) iwn
(11)
T h e impedance that, accordingly apprars a t the stator winding of tlie IRTF will depend on t.he shaft speed. T h e stator flux 3" a n d stator current is ran be calculated by ( 4 ) . By means of the stator voltage, the time derivative of the stator flux, the virtual impedance Zs on the stator is found:
From (13) follows rhat, an indnct,or on the rotor side of the IRTF appears at, the stator side as if it was connected to the stator side itself. T h e only difrerence is that it's equivalent on the stator side experiences the stator frequency u s 1 while the inductor i n the rotor circuit experiences rotor frequency W R . Freqnenry dependent inductors can therefore not be moved t o the opposite side of the IRTF. The resistor Rn is scaled by a factor * = which corresponds t o the apparent rotor resiswn tanre change in the general steady state asynchronous machine model in figure 2 151. A n easy way to illustrate the fact that a lincar inductor can be moved across an IRTF is t h e fixed relation between flux and current in an induct,or: 0, = L I . Flux amplitude and current amplitude are equal on both sides of an
&,
Equation 4 shows that current and flnx on the rotor side are equal to the quantities on the stator side, turned over angle 0 ( t ) . So (I) yields:
IRlF.
2.1
Torque computation
In order to calculate t.he torque, produced by an IRTF as a funrlion of its ii1put.s 6,l'and 0, a phase angle q5 bel.weeri and
A n IRTP' inrorporaks I.he fundamental relations between two relative moving winding systrnis, and can be seen as the hart of all rotating i~IecI.ricaImachines. T h e next step in synthesizing a machine model is t o add the other machine sections from ligure 3, the stator and rotor and the mechanical section.
195
3 3.1
Model construction Leakage inductances
By applying Thevenin's equivalent theory 121, 161,the main inductance L, (figure 2) can be moved parallel to rotor resistance R R , or t o the left side of L,s. Both rotor leakage and stator leakage inductances can be replaced by one ellective total leakage iuduclance Lo,,, if all leakage is concentrated in the stator. In case all leakage inductance is supposed in the rotor, LOtzapplies. In this way the three dependent inductances are replaced by just two independent ones. t
%
a
1
2
8
,
-
,...._._____........~~.......~~~~~~~~~-.--.-.
Figure 5: General simulation model structure Jor a separately eztled machine.
' Eq.l '
'
'
' Eq.11
Figure 4: Both possible simulation models, all leakage inductance i n the stator or i n the rotor. The IRTF can be inserted a t three dilTerent places f ,2,3 in both circuits. Dashed lines indicate the possible points of insertion. Both figures 4 q . I , E q . I 1 are equivalent in terms of representing the circuit in figure 2. Their simulation models are equivalent but not identical. A major dilTerence is that, in case the IRTF is inserted between the two inductors (option 2). Eq.1,2 produces rotor flux, while Eq.11,2 produces rotor current. Both equivalents in figure 4 are starting points for more elaborate machine models. The transformed parameters in both equivalents are related to the general parameters in the steady state machine model from figure 2.
shaft angle (21) and J, the total inertia of all rotating parts on the shaft:
e
=
Illlo,
(21)
The number of pole pairs acts as a mechanical gearbox with transmission ratio f (20),(21). The electrical torque 2'. is a n output of the IRTF and determines with given load (all other loads, including lriction) the actual acceleration of the total inertia J . on the machine-shaft. The block diagram for simulation is straight lorward (figure 6).
Figure 6 Mechanical section: simulation model m'th
pole
pairs.
Unless otherwise specified, the number of poles is considered to he two. The Lransformation ratio is then equal to one.
3.2
3.4
Inductor model
Because numerical integration is accurate and dilTerentiation is inaccurate, a computational model for an inductor should preferably consist of an integrator.
Integrators in the following block diagrams are indicated by $, with p the Laplace operator ( p = Using (19), the general per phase induction machine model from figure 4&.1,2can be translated into figure 5. Figure 5 shows the two circuits in a,o and a , b coordinates connected by an IRTF.
5).
The IHTF also links the mechanical section. As was indicated before, the IRTF could just as well be put on the left of or right of main inductance without changing the models characteristics.
&,
3.3
2
mechanical section
The mechanical section can he represented by the following differential equation with np the number of poles, 8, the physical
Computational causality
In physical reality, there is no distinction between cause and elTect 131. One can say that voltage across a resistor causes a current to flow, but one can say just as well that a current Row through a resistor causes a voltage drop across the device. A resistor simulation model therefore has a n arbitrary causality. An inductor according to (19). has voltage causality. Inductor current in this way is a function of voltage. It should be noted that if not voltage, but flux is used, a linear inductor model concludes to a multiplier. Both flux and current causality are then possible. The I R l F a l s o inhabits a computational causality. As shown in (4), shalt angle 0 is an input and torque II:, is an output of the IRTF. Its causality on the electrical ports is two fold. Current in and flux out on the rotor side, or flux in and current out on the rotor side.
4
Synchronous machines
Synclironous machine rotors usually contain two distinct rotor windings, an excited rotor winding to generate the rotor
196
flux, and a short,-circuit damping winding. 1101.11 rotor win& ings have a high mutual flux linkage. Mosl. of this niutiial flux is also linked with the stator wiiidiiig. 111 order t.0 coiisI,rurI a simulation model for two parallel windings, current should be calculated from voltage or flux. Ilencc the leakage irirlurlanre 01 each rotor winding should be on the rot,or side of the main inductance. Uased on figure 4Eq.11, I , a sirnulation model lor a machine with double rotor rircuit IZotZ, Rnz and second rotor winding L , , Z , R k z is shown in figure 7 .
nn I I I P r o l n r side, shows f l n x - r a r d i t , y on thr slnl.or sidr of I.he 1117'1.'. Flux causality n l t h p lR7'F on t h e sl.al.or sidr is nwded i f a multi-wiiirling rot,or is simulated or when skin e k l . i n l h e rotor cage is implemented in thr simulation model. In t,hese rases irnpressrd stator currrnt.s cannot bc simulated without solving art algebraic loop. Th? morlrl output, 4'2 can be used MECHANICAL
Te
- V,"
%I," a
Figure 8: General simulation model /or a machzne with impressed stator current. t o cslirnate the input voltage as a result of impressed stator 1,SRs. current by means of a derivator: G: = d(';Lt$Ls''l
+
Figure 7 : General simulation model for a machine with two rotor windings, one damping cage. Models for P M synchronous machines require a model for a permanent magnet. A permanent magnet can be considered as a current-source that magnetizes the main inductance LmI or L,? and builds a flux GWn = Lnfmp accordingly. If one of the two rotor 'windings' in figure 7 is a permanent magnet, a current source can be added t o the current adding point on the right side of the IRTF.
Adding a permanent magnet t o the rircuit in figure 5 is also possible, because I,, is a n independent quantity. This, as shown in figure 8 results in:
Many synchronous machines contain rotors with salient poles. A salient pole rotor is asymmetrical and inhabits therefore different values for the main inductance L,z and the leakage illductance LOl2in the direct ( a ) and perpendicular ( b ) axes. This asymmetry is fixed t o the rotor, and should therefore be rcpresented in rotor coordinates. T h a t is why, in case of rotor asymmetry, both L, and Lot have t o be on the rotor side of the IRTF (options Eq.l,l; Eq.lI,f) in the simulation model, as shown in figure 7.
5
6 6.1
Auxiliary phenomena Tooth effects
All previous machine modrls were models for smooth air gap machines. An additional romponent that incorporates torque and inductance ellerts lor toothed stators and rotors is presented, The main quantity that. changes with shaft position is the main inductance in both axis: l,,,,ls(0), L,,,lb(0). Because the niairi indurtance determines the reriprocal relation i t is easier t o directly consider this rebetween and lation. Suppose the most siniplc_ronfiguration 01 a two pole rot,or, with magnetizing current I,,,,, and a stator with four arbitrary teeth, one each 90",as depicted in figure 9. T h e resulting flux vector under static conditions in the stator core has a preference for the teeth positions, and will therefore not always be aligned with the current vector in the rotor. This non alignment causes a torque, just as in the IRTF (8).
2,
Voltage or current input
The machine model in figure 5 has stator- and rotor-voltage as an input and a n IRTF with current-causality on the stator side. If machines with impressed stator currents have t o b e simulated, a model with current,-causality on the stator side is needed, so the model in figure 8 , the circuit in figure I E q . 1 is suitable for impressed stator currpnts. The circuit i n figure 4Eq.I1,1 (model in figure 7 ) , with all leakagr iiidiir-tance
Figure 9: Idealized fluz and current vector relations in a uartable reluctance machine: toothed stator.
71,
For 0 t [O,:, n, the flux is high, the alignment is correct: I.hese are stable poqitions. The positions in between, -:,
1-9,
197
$,?I
Impressed stator currents: Eq.I,2,3
show a low flux and correct alignment: these posit.ions are unstable. If the relative amonnt of inductance modulation is I
