ruuskanen v. modelling the brushless excitation system for a synchronous machine

Published in IET Electric Power ApplicationsReceived on 3rd April 2008Revised on 3rd November 2008doi: 10.1049/iet-epa.2008.0079

ISSN 1751-8660

Modelling the brushless excitation systemfor a synchronous machineV. Ruuskanen1 M. Niemela1 J. Pyrhonen1 S. Kanerva2

J. Kaukonen3

1Department of Electrical Engineering, Laboratory of electrical drives technology, Lappeenranta University of Technology,PO Box 20, FI-53851 Lappeenranta, Finland2ABB Oy, Machines, PO Box 186, FI-00381 Helsinki, Finland3ABB Oy, MV Drives, PO Box 94, FI-00381 Helsinki, FinlandE-mail: [email protected]

Abstract: The structure and the operation of the model for the brushless excitation system for a synchronousmachine are presented. The nonlinear model including the excitation machine, the AC–AC converter supplyingthe excitation machine and the rectifier diode bridge, mounted on the rotor, is based on a state machine.The states are defined by current commutation in the power electric devices. The operation of the excitationsystem model is verified by measurements with a slip-ring machine imitating the excitation machine. Theexcitation system model is integrated and simulated as a part of a synchronous machine simulator.

1 IntroductionElectrically excited synchronous machines can be divided bytheir excitation systems into brushed and brushless machines.The excitation system depends on the application; a brushedexcitation system is used when high dynamic performance isrequired, whereas the benefit of brushless excitation systemsis their need for lower maintenance. Brushless excitationsystems are commonly used in marine drives, where thedynamical requirements are not too tight, but extremereliability is needed and maintenance is difficult. Asynchronous machine with a brushless excitation system ispresented in Fig. 1.

The target of the excitation current control is to set the powerfactor of the machine to the desired value and keep the machinestable during the transient states. The simulation model createdcan be used to simulate the effects of the excitation systemdynamics on the synchronous machine during the transientstates. The state machine model gives an accurate descriptionof the currents of the excitation machine rotor circuit. Thesimple time constant does not describe the excitation systemwell enough. The time constant for increasing and decreasingthe excitation current differ from each other because of the

free-wheeling state. It is not possible to force the excitationcurrent to fall using negative excitation voltage with a diodebridge.

1.1 Brushless excitation system

A brushless excitation system consists of the excitation machine,that is, a traditional wound-rotor three-phase inductionmachine mounted to the main machine shaft and fed by anAC–AC converter, and a diode rectifier connected to therotor of the excitation machine. The use of the wound-rotorexcitation machine supplied by the AC–AC converter makesit possible to generate the excitation current also at zerospeed, which is a significant benefit compared with the otherkind of excitation method modelled by Cingoski et al. [1] andDarabi and Tindall [2]. The thyristor pair converter isexamined, because it is the traditional converter for theexcitation machine of the large sychronous machine.

The rotor of the excitation machine is joined to the shaft ofthe synchronous machine. The stator flux of the excitationmachine rotates in an opposite direction compared with therotational direction of the synchronous machine. Thereforethe slip of the excitation machine is always greater

IET Electr. Power Appl., 2009, Vol. 3, Iss. 3, pp. 231–239 231doi: 10.1049/iet-epa.2008.0079 & The Institution of Engineering and Technology 2009

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than one. The excitation machine is fed by a thyristor pairpower converter that is connected to the stator connectorsof the synchronous machine or to an external network.The excitation machine takes its power partly from thesupplying network and partly from the axis of thesynchronous machine when operating at a slip greater thanone. There are two extra thyristor pairs for changing therotational direction of the field when the rotationaldirection of the synchronous machine is changed. If thesynchronous machine is used as a generator, one pole paircan be equipped with permanent magnets to enable thebuild-up of the generator also in an isolated networkoperation. The rotor currents of the excitation machine arerectified and fed to the excitation winding of thesynchronous machine with a six-pulse diode bridgerectifier. The complete excitation system configuration ispresented in Fig. 2.

The thyristor power converter in the stator circuit and thediode rectifier in the rotor circuit cause strong nonlinearitiesinto the excitation system. Currents and voltages are notsinusoidal, which makes modelling quite difficult.

2 State machine modelBecause of the strong nonlinearities in the excitation circuit,the conventional flux vector model cannot be applied tomodel the excitation machine. The dynamics of theexcitation system changes constantly by the currentcommutations in the power electronic devices. Zahawi et al.[3] have introduced a state-space model for a Kramer drivethat also includes nonlinearities in the rotor circuit. Therectifier model presented by Akpinar is also based ondifferent commutation states [4, 5]. The modelling of thewhole excitation system by a state machine model is quite alaborious task; first of all, the description of all thedirection combinations of the stator and rotor currents

would require a large number of states. Further, a majorproblem would be finding a stable method to commutatebetween the states. Therefore only the rotor circuit ismodelled with a state machine. The block diagram of themodel is presented in Fig. 3.

The stator voltage is generated by a PI controller from thedifference between the desired and actual excitation currents.The stator circuit is modelled with a sinusoidally fed single-phase equivalent circuit. The phase voltages for the rotor statemachine are generated by the rotor frequency and themagnetising voltage given by the equivalent circuit.

The method based on different commutation modes for asinusoidally fed diode rectifier bridge is presented for instancein [6]. The method gives average values of the excitationmachine rotor circuit currents. The mode selection is basedon the commutation overlapping angle, that is, defined bythe excitation current and the exciter flux linkage. Themethod was not used for some practical reasons. The fluxlinkage of the excitation machine is changing continuously,and the rectifier mode searching conditions are alsochanging, which increases the amount of calculation.Instead of the averaging model, the waveforms of theexcitation machine rotor currents were desired.

2.1 Single-phase equivalent circuitof the stator

A single-phase equivalent circuit can be used to modelthe stator circuit regardless of the diode rectifier andthe excitation winding in the rotor circuit, when theresistance of the excitation winding is modest comparedwith the resistances of the excitation machine. The highinductance of the excitation winding can be neglected, whenthe excitation current is assumed to be a smooth DCcurrent. According to the measurements, the short-circuiteddiode bridge corresponds to the short-circuited rotorwithout a diode bridge. The measurements were made usinga slip-ring machine as an excitation machine. Theparameters of the slip-ring machine are presented in Table 1.

A six-pulse diode bridge was connected to the slip-rings.The load was varied by connecting loads with differentresistances and inductances to the DC buses of the diodebridge. The slip was varied by rotating the rotor of the slip-ring machine with a DC machine. The slip was changedfrom the rotor rotating at the synchronous speed in thedirection same as that of the stator field (s ¼ 0) in thelocked-rotor situation (s ¼ 1), and further, the rotorrotating at a synchronous speed but in a direction opposite

Figure 2 Brushless excitation system configuration

Figure 1 Synchronous machine with a brushless excitationsystem

The rotor circuit is indicated with a lighter line

Figure 3 Block diagram of the excitation system model

232 IET Electr. Power Appl., 2009, Vol. 3, Iss. 3, pp. 231–239

& The Institution of Engineering and Technology 2009 doi: 10.1049/iet-epa.2008.0079

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to that of the stator field (s ¼ 2). The stator voltage of theslip-ring machine was kept constant. The measuredamplitudes of the fundamental harmonics of the rotorphase current at different loads connected to the diodebridge are presented in Fig. 4 as a function of slip. Also,the case with the rotor windings short-circuited without adiode bridge is presented.

In real excitation machines, the inductance of theexcitation winding is very large compared with that ofthe excitation machine. Because of the high slip of theexcitation machine, the equivalent circuit can be assumedto be short-circuited with a locked rotor. For the excitationmachine, based on Fig. 4, it makes no difference if the slipis one or more. The single-phase equivalent circuit of aninduction machine is presented in Fig. 5 [7].

The magnetising voltage can be calculated by the statorvoltage and the stator current

um ¼ us � Rsis � jLssvsis (1)

where Rs is the stator winding resistance, Lss the stator strayinductance and vs the stator flux angular speed. The stator

current can be calculated by the stator voltage andimpedances as

is ¼us

Zs þ ((ZmZr)=(Zm þ Zr))(2)

where Zs is the stator impedance, Zm the magnetisingimpedance and Zr the rotor impedance. The impedancesare defined as

Zs ¼ Rs þ jvsLss (3)

Zm ¼ jvsLm (4)

Zr ¼R0rsþ

R00Fsþ jvsL

0rs (5)

where R0r is the rotor winding resistance referred to the stator,s the slip, R00F the excitation winding three-phase equivalentresistance referred to the stator and L0rs the rotor strayinductance referred to the stator. The rotor electromotiveforce is determined by the magnetising voltage um, the slips and the reduction factor n between the stator and the rotor

ur ¼ umsn (6)

Figure 5 Single-phase equivalent circuit for the excitationmachine

Table 1 Parameters of the slip-ring machine

Parameter Symbol Value

nominal power Pn 1.8 kW

nominal voltage Un 380 V Y

nominal current In 4.5 A

nominal power factor cos(f) 0.8

nominal speed nn 1400 r/min

nominal frequency fn 50 Hz

stator resistance Rs 2.2 V

rotor resistance (referred to thestator)

Rr0 5.4 V

rotor resistance Rr 1.0 V

magnetising inductance Lm 0.271 H

stator stray inductance Lss 12 mH

rotor stray inductance (referred tothe stator)

Lrs0 27 mH

rotor stray inductance Lrs 5.0 mH

excitation winding inductance(referred to the stator)

LF0 0.353 mH

excitation winding inductance LF 65 mH

reduction factor between thestator and the rotor

n 0.43

Figure 4 Rotor phase current fundamental harmonicamplitudes of the slip-ring machine with a differentlyloaded diode bridge connected to the slip-rings as afunction of slip

The stator voltage of the slip-ring machine is constant


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The phase voltages for the state machine can be generatedwith the rotor voltage amplitude and the rotor frequency.

2.2 Rotor state machine

There are 12 different phase current direction combinationsin the rotor circuit. If only two phases are conducting,there are six different current combinations, which areselected as odd states. Between the odd states, all threephases are conducting atleast for a short commutating timeinterval; the current combinations, in which all the phasesare conducting, are selected as even states. The double-ended arrow illustrates the ongoing commutation. Allpossible rotor states are presented in Table 2. The positivedirection of the current is chosen from the rotor winding tothe diode bridge. The first letter indicates the positivecurrent and the second the negative current.

2.2.1 Voltage equations: The directions of the rotorcircuit currents in the first state cb are presented in Fig. 6.When the diode resistances and the threshold voltages areneglected, the voltage equation for the first state is given by

uc � ub

� �¼ 2Rr þ RF

� �ic

� �

þ 2Lrs þ LF

� � dic

dt

� � (7)

where Rr is the rotor phase resistance. The voltage equationsfor all the odd states have the same gain matrices; only theconducting phases change.

During the commutation, all the rotor phases areconducting. The directions of the currents in the secondstate are illustrated in Fig. 7. In the even states, two rotorcurrents must be solved. The voltage equations for the

second state are

uc � ub

uc � ua

� �¼�(Rr þ RF) Rr

Rr 2Rr

� �ib

ic

� �

þ�(Lrs þ LF) Lrs

Lrs 2Lrs

� � dib

dtdic

dt

264

375 (8)

The voltage equations for the rest of the states are formed inthe same way. Both the gain matrices and the phases change.The voltage equations for the even rotor states are presentedin the Appendix.

2.2.2 Commutation: The commutation between states isbased on the voltages and currents. The state machine movesfrom an odd state to an even state, when the voltage of thenon-conducting phase reaches the value of the conductingphase with the same polarity. For example, when rotatingto the positive direction, the step from the first (cb) to thesecond state (cb$ ab) takes place when the voltage of thephase a reaches the value of the phase c.

The state change from an even to an odd state takes placewhen one of the phases stops conducting as it reaches thezero current. For example, when rotating to the negative

Table 2 States of the rotor state machine

State # Conducting phases

1 cb

2 cb$ ab

3 ab

4 ab$ ac

5 ac

6 ac$ bc

7 bc

8 bc$ ba

9 ba

10 ba$ ca

11 ca

12 ca$ cb

Figure 6 Directions of the rotor circuit currents in the firststate cb

Figure 7 Directions of the rotor circuit currents in thesecond state cb$ ab



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direction, the state of the state machine changes from thesecond state to the first state when the current of the phase adrops to zero.

2.2.3 Free-wheeling state: As can be seen in Fig. 2,the excitation winding current can flow freely through thediode bridge without passing through the rotor winding ofthe excitation machine; this is known as a free-wheelingstate. In the free-wheeling state, the rotor current candecrease freely while the excitation current keeps passingthrough the diode bridge, damping out because of theresistive losses in the diodes and the excitation winding.

In the free-wheeling state, the excitation current consists ofthe rotor current and the free-wheeling current. The free-wheeling effect doubles the number of the rotor states. Thefree-wheeling states are equivalent to the states presentedabove, but the resistance and inductance of the excitationwinding are neglected. In that case, the rotor state machinegenerates only the rotor phase currents. The excitationwinding current must be calculated separately with a modelfor the damping current. If the excitation machine rotorcurrents are assumed to be small, the voltage equation forthe excitation current is written as

0 ¼ RFiF þ LF

diF

dt(9)

3 Simulation and measurementsThe excitation system model was constructed and simulatedwith Matlab Simulink. The results were compared with themeasured values. The measured machine is a small 1.8 kWslip-ring induction machine fed by a variable voltagetransformer. A six-pulse diode bridge was connected to theslip-rings of the rotor. The diode bridge was loaded withan RL branch to emulate the excitation winding. Thestator voltages of the simulated and the measured slip-ringmachine were set equal, and the rotor currents werecompared with each other.

3.1 Zero speed

The system was simulated and measured at zero speed, whichmeans that the slip is equal to one. The simulated andmeasured currents are presented in Fig. 8.

The amplitudes of the simulated and measured currents arealmost equal. The small difference is a consequence of theinaccuracy of the model and the parameters for the slip-ringmachine. The measured current is in a steady state all thetime. In the simulated current, it is possible to detect the timeconstant of the excitation system. The frequency of the rotorcurrents is equal to the stator frequency, because the rotor isstationary. The shapes of the measured currents in the upperfigure are congruent with the simulated ones given below.There is only a small ripple in the excitation DC current.

The trapetzoidal shape of the phase currents can be explainedby the current commutations. The free-wheeling-state is notclearly visible at zero speed.

3.2 Reverse speed

When the rotor is rotating in the direction opposite to that ofthe stator flux, the slip is more than one and the rotorfrequency is higher than the stator frequency. Themeasured and simulated currents at the slip equal to twoare presented in Fig. 9.

The rotor frequency is double compared with the locked-rotor situation. Hence, also the electromotive force of the

Figure 8 One rotor phase current and the excitationwinding current at zero speed

Figure 9 Excitation current and one rotor phase currentmeasured and simulated at a slip equal to two


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rotor is twice the value in the locked-rotor situation based on(6). The amplitudes of the rotor current are only slightlyhigher than above because of the increasing rotorimpedance. This justifies the use of the single-phaseequivalent circuit. The time constant of the rotor circuit isnow smaller than at zero speed. The effect of the excitationwinding impedance diminishes, because both theelectromotive force and the impedance of the rotor circuitare increased while the slip is increased. The currents arepresented at a shorter interval in Fig. 10 to observe theshape of the currents.

At a higher slip, the free-wheeling state is clearly visible. Ifthe rotor system is assumed to be free-wheeling in the statecb, the diodes 3 and 5 and the free-wheeling diodes areconducting, and the current of the phase a is zero. Thefree-wheeling diodes are the diode pairs that are carryingthe excitation current but not the rotor phase current. Therotor phase currents and different commutation states withfree-wheeling are presented in Fig. 11.

The diode 1 starts to conduct immediately, and the systemis in the state cb$ ab, when the diodes 1, 3 and 5 areconducting. As the absolute value of the current ib reachesthe excitation current, the excitation winding inductancestarts to prevent the absolute value of the current ib fromrising and the free-wheeling state is over. When theabsolute value of the phase current ib tries to becomesmaller than the excitation current iF, the free-wheelingstarts. A part of the excitation current starts to passthrough the diode bridge without flowing through the rotorof the excitation machine. The current of the phase b canchange without being dependent on the excitation current.The current of the phase a rises, and the current of thephase c is decreases until the current ic becomes zero; the

system is in the state ab, and the diodes 1, 5, and the free-wheeling diodes are conducting. The system does not stayin an odd state, because the excitation machine inductancesare too small compared with the excitation machine rotorresistances to keep the rotor current at the value of theexcitation current long enough. Thus, the rotor currents arecontinuous, and the current ic continues decreasing tonegative values immediately after reaching zero.

Fig. 12 illustrates the currents of a real excitation machine.The current commutations are as described above in the caseof the slip-ring machine, but now also the odd states arevisible.

Figure 10 Closer view of the currents of the slip-ringmachine measured and simulated at a slip equal to two

Figure 11 Rotor current commutations and free-wheelingstates at a slip equal to two

The odd state points of time are marked, although all the threephases are conducting continuously

Figure 12 Rotor current commutations of a real excitationmachine measured and simulated at a slip equal to 1.5



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The measured and simulated current waveformscorrespond to each other. The phase currents do notimmediately start to flow to the opposite direction afterreaching zero, and the system stays in an odd state for awhile. The free-wheeling time is very short and cannot beclearly seen. The difference between the slip-ring machineand the real excitation machine is a consequence ofdifferent inductance ratios of the systems. In the slip-ringmachine system, the magnetising inductance of the slip-ring machine is notably higher compared with theexcitation winding inductance than in the case of the realexcitation machine.

4 Integration into thesynchronous machine simulatorThe excitation system model was integrated into thesynchronous machine simulator. The use of the simulatorfor simulating asynchronous and synchronous machines hasbeen presented in [8, 9]. Fig. 13 illustrates the blockdiagram of the developed excitation system model as a partof the synchronous machine simulator.

The inputs of the excitation system model are theexcitation current and its reference, slip, and the statorsupply frequency of the excitation machine. The excitationwinding voltage is the only output.

The stator voltage of the excitation machine model isgenerated with a PI controller, which has the differencebetween the excitation winding current and the referencevalue of the excitation winding current as the input. Therotor electromotive force is calculated with a single-phaseequivalent circuit. The excitation winding current isgenerated with the rotor circuit state machine.

The excitation current is calculated at two places at thesame time: in the synchronous machine simulator and inthe excitation system model. The problem is to fit thesecurrents together. The conventional calculation of theexcitation winding voltage with the voltage equation for theRL branch, uF ¼ RFiF þ LF(diF=dt) does not work in thiscase. The excitation system model calculates the excitationwinding current without coupling with the synchronousmachine model that also generates the excitation currentindependently based on the excitation winding voltage. Theonly coupling between the models is the excitation windinginductance. A stronger coupling between the models wouldrequire to integrate the excitation system model very deepinto the synchronous machine model, which would be very

difficult. In that case, the excitation system could not beseparated as a block of its own as currently. It is notpossible to modify the synchronous machine simulator tohave the excitation current as an input to keep thesimulator modular. The brushless exciter model must beeasily replaced by the model of the brushed excitation. Themodel of the brushed exciter includes just the model of thethyristor bridge with a voltage output.

The problem is solved by coupling the excitation currentstogether with a PI controller. The PI controller generates theexcitation winding voltage such that the difference betweenthe excitation currents will disappear. The time constant ofthe PI controller has to be much smaller than the timeconstant of the excitation system. The excitation windingvoltage is not a physical but a virtual value.

The synchronous machine is rotating at its nominal speedwith no load when a load torque step to the nominal load isadded. Later, the load is decreased to zero again. Theexcitation winding current and the reference value of theexcitation current are shown in Fig. 14.

The excitation current follows the reference value well atthe end of the steady state when the excitation current isrising. The control of the excitation current operates asdesired.

When the load ceases to be effective, the excitation currentstays higher than the reference value. The excitation current isin a free-wheeling state, and it damps because of the resistivelosses in the diode bridge and the excitation winding. Withbrushless excitation, it is not possible to decrease theexcitation current faster by controlling the excitationwinding voltage negative. At the end of the falling edge,there is a clear undershoot. The undershoot is a

Figure 13 Excitation system model as a part of thesynchronous machine simulator

Figure 14 Excitation current and the reference value of theexcitation current at the torque steps to nominal andzero loads


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consequence of the slow dynamics of the PI-controlledexcitation system.

The excitation current and the rotor phase currents duringthe excitation start-up and the load step are illustrated inFig. 15. The currents are presented at a shorter interval in thesteady state in order to observe the shape of the rotor currents.

The rotor phase currents have the same form as describedabove. There are strong DC components in the rotor currentsduring the free-wheeling state. The DC components are aconsequence of the fast-decreasing amplitude of the phasecurrents. The commutation works in spite of the DCcurrents, because the high-pass filtered rotor currents areused for commutation during the free-wheeling state. Thephase currents return to the zero average when the free-wheeling state ends and the amplitude of the rotor currentrises.

The waveforms of the rotor currents are not the same as inthe case of the measured slip-ring machine. Now, there is aremarkably larger excitation winding impedance that triesto keep the excitation current constant. Because of the highexcitation inductance, the rotor circuit is in the free-wheeling state for most of the time. The variation of theamplitude of the rotor currents is explained by the lowinductances and resistances in the excitation machine rotorwindings.

To observe the operation of the PI controller coupling theexcitation model with the synchronous machine simulator,the excitation currents calculated by the excitation modeland the synchronous machine model are presented in Fig. 16.

The excitation current generated by the synchronousmachine model is on an average equal to the currentgenerated by the excitation system model. The PI controlleroperates well in coupling the models together if the timeconstant of the controller is small enough. It is worthremembering that in this case, the excitation windingvoltage is only a virtual value.

5 ConclusionOnly the excitation model with the state machine for the rotorcircuit illustrates the excitation system with sufficient accuracy.Based on the nearly equal measured and simulated rotorcurrent waveforms, the single-phase equivalent circuitsuffices to illustrate the stator. However, the non-sinusoidalstator voltage supply may change the situation. To verify theexcitation system model for real excitation machines,laboratory measurements are needed. Of the greatestimportance is the need to measure the stator and rotorcurrents and their waveforms at the same time. Adding astator circuit to the state machine drastically increases thenumber of states. Consequently, the stable commutationmethod would require more research in the future.

6 References

[1] CINGOSKI V., MIKAMI M., YAMASHITA H., INOUE K.: ‘Computersimulation of a three-phase brushless self-excitedsynchronous generator’, IEEE Trans. Mag., 1999, 35, (3),pp. 1251–1254

[2] DARABI A., TINDALL C.: ‘Brushless exciter modeling forsmall salient pole alternators using finite elements’, IEEETrans. Energy Convers., 2002, 17, (3), pp. 306–312

Figure 15 Rotor currents of the excitation machine and theexcitation current during the torque steps

Given below, are the currents at a shorter interval in the steadystate. At the time 0.2 s the nominal load is added. The excitatoncurrent reaches its nominal value at 0.35 s. At the time 0.7 sthe load is subtracted and the system migrates to the free-wheeling state. The excitation current reaches its referencevalue and the free-wheeling state ends at the time 0.95 s

Figure 16 Excitation currents generated with the excitationsystem state machine model and the synchronous machinesimulator



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[3] ZAHAWI B.A.T., JONES B.L., DRURY W.: ‘Analysis and simulationof static kramer drive under steady-state conditions’, IEEProc., 1989, 136, (6), pp. 281–292

[4] AKPINAR E., PILLAY P.: ‘Modeling and performance of slipenergy recovery induction motor drives’, IEEE Trans.Energy Conver., 1990, 5, (1), pp. 203–210

[5] AKPINAR E., PILLAY P., ERSAK A.: ‘Calculation of the overlapangle in slip energy recovery drives using a d,q/abc model’,IEEE Trans. Energy Conver., 1993, 8, (2), pp. 229–235

[6] ALIPRANTIS D.C., SUDHOFF S.D., KUHN B.T.: ‘A brushless excitermodel incorporating multiple rectifier modes andpreisach’s hysteresis theory’, IEEE Trans. Energy Conver.,2006, 21, (1), pp. 136–147

[7] KRON G.: ‘Steady-state equivalent circuits ofsynchronous and induction machines’, AIEE Trans., 1948,67, pp. 175–181

[8] KANERVA S., STULZ C., GERHARD B., BURZANOWSKA H., JARVINEN J.,SEMAN S.: ‘Coupled fem and system simulator in thesimulation of asynchronous machine drive with directtorque control’. 6th Int. Conf. Electrical Machines(ICEM04), Cracov, Poland, September 2004

[9] BURZANOWSKA H., SARIO P., STULZ C., JOERG P.: ‘Redundantdrive with direct torque control (dtc) and dual-starmachine, simulation and verification’. 12th EuropeanConf. Power Electronics and Applications (EPE 2007),Aalborg, Denmark, September 2007

7 Appendix7.1 Commutating states of the rotorcircuit state machine

cb$ ab

uc � ub

uc � ua

� �¼�(Rr þ RF) Rr

Rr 2Rr

� �ib

ic

� �

þ�(Lrs þ LF) Lrs

Lrs 2Lrs

� � dib

dtdic

dt

264

375

(10)

ab$ ac

ua � ub

ub � uc

� �¼�(2Rr þ RF) �(Rr þ RF)

Rr �Rr

� �ib

ic

� �

þ�(2Lrs þ LF) �(Lrs þ LF)

Lrs �Lrs

� � dib

dtdic

dt

264

375

(11)

ac$ bc

ua � uc

ua � ub

� �¼�Rr �(2Rr þ RF)

�2Rr �Rr

� �ib

ic

� �

þ�Lrs �(2Lrs þ LF)

�2Lrs �Lrs

� � dib

dtdic

dt

264

375

(12)

bc$ ba

ub � uc

uc � ua

� �¼

Rr þ RF �Rr

Rr 2Rr

� �ib

ic

� �

þLrs þ LF �Lrs

Lrs 2Lrs

� � dib

dtdic

dt

264

375

(13)

ba$ ca

ub � ua

ub � uc

� �¼

2Rr þ RF Rr þ RF

Rr �Rr

� �ib

ic

� �

þ2Lrs þ LF Lrs þ LF

Lrs �Lrs

� � dib

dtdic

dt

264

375

(14)

ca$ cb

uc � ua

ua � ub

� �¼

Rr 2Rr þ RF

�2Rr �Rr

� �ib

ic

� �

þLrs 2Lrs þ LF

�2Lrs �Lrs

� � dib

dtdic

dt

264

375

(15)


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ruuskanen v. modelling the brushless excitation system for a synchronous machine

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