Journalof sciendo

sciendo

Journal of ELECTRICAL ENGINEERING, VOL 72(2021), NO3, 176–183

PAPERS

Electrical equivalent circuit for modellingpermanent magnet synchronous motors

Esra Kandemir Beser1

In permanent magnet synchronous motor (PMSM) models, only the stator part is given as an electrical circuit andmechanical equations are used for modelling the mechanical part of the machine. In this study, electrical equivalents ofmechanical equations are also obtained and mechanical parameters of a PMSM are expressed as an electrical circuit element.In this way, an exact electrical equivalent circuit is proposed in which both the stator and the mechanical part can be modelledas an electrical circuit for the PMSMs dynamic model. Although PMSM model includes mechanical parameters and variables,the complete model is expressed only in electrical elements and variables. The proposed PMSM circuit was simulated fordifferent load torques in the circuit simulation program. Simulation results show that the proposed circuit operates like aPMSM. Simulation results were verified by another method in the form of solution of the differential equations that constitutethe mathematical model of PMSM. Due to the proposed circuit that enables the conversion of mechanical parameters intoelectrical parameters, PMSM can be modelled and simulated as an electrical circuit with completely electrical elements in acircuit simulation program.

K e y w o r d s: equivalent circuits, circuit simulation, ac machines, permanent magnet motor, modelling, synchronousmotor

1 Introduction

In recent days, permanent magnet synchronous motors(PMSMs) are very popular since there is continuous im-provement and price decrease in permanent magnet ma-terials and PMSMs have a lot of advantages over manymotor types [1]. They have a widespread use in homeappliances and industrial applications [2]. PMSMs con-tain permanent magnets on the rotor; therefore, they donot need the magnetizing current provided by the stator[3]. This allows PMSMs to operate at high power factorand efficiency values [4, 5]. In addition, there is no cop-per loss in the rotor due to the absence of windings [4].The PMSMs have often a flattened air gap flux densitydistribution and the shape of the flux linkage and theinduced voltage strongly depends on the configuration ofthe winding. However, they are generally designed to gen-erate a sinusoidal electromotive force (EMF) and in orderto generate constant torque, sinusoidal current must beapplied to stator windings. PMSMs can be classified assurface mounted and interior types according to the waythe magnets are placed on the rotor [6].

The fact that PMSM has advantages over many ma-chines has increased the studies on this subject. In the lit-erature, there are many studies related to PMSM such asvector control, direct torque control, speed control, cur-rent control, etc. The simulation model of PMSM andits system is of great importance for verification of con-trol algorithms and optimization of the system. For thisreason, in most studies, mathematical model is createdfor PMSM to analyse the operation of the system. The

machine model consists of electrical equations containingvoltage equations and mechanical equations containingtorque produced in response to mechanical load.

Among the studies including machine model, the d−qequations are used in [4] and [7-9]. An extended model isproposed in [10]. PMSM equations are constituted in thestator flux reference frame for the direct torque controlfor PMSM in [11]. Work [12] proposes an advanced be-haviour motor model based on the stator flux linkage ind−q reference frame. A direct flux vector control strategyis proposed for PMSM drives in [13] and machine model isconstituted in d−q reference frame. The stationary α−βframe is preferred for the mathematical model of PMSMin order to realize model predictive current control in [14].A model predictive control is proposed for PMSM andthe machine mathematical model is obtained with d−qreference frame theory in [15]. A simplified d−q equiva-lent circuit model of a PMSM is proposed for analysingeffect of inter-turn fault in [16-18]. A real-time motor sim-ulator is designed in MATLAB/Simulink environment byusing the mathematical model of PMSM in d−q rotatingcoordinate system in [19]. A PMSM model using vectorcontrol algorithm is built and simulated in [20]. The mo-tor model is first obtained in a,b,c phase system thenthe equations are converted to d−q rotating coordinatesystem. In [21], the equations of PMSM model are ob-tained for field-oriented control by using synchronous ref-erence frame transformations. [22] gives steady-state andgeneral PMSM equivalent circuits in d−q coordinate sys-tem. Parameter estimation is made in [23] and the surfacemounted PMSM model is constituted in d−q rotor refer-

1Kocaeli University, Electrical Engineering Department, Umuttepe Campus, 41001, Izmit Kocaeli, [email protected]

https://doi.org/10.2478/jee-2021-0024, Print (till 2015) ISSN 1335-3632, On-line ISSN 1339-309Xc©This is an open access article licensed under the Creative Commons Attribution-NonCommercial-NoDerivs License

(http: //creativecommons.org/licenses/by-nc-nd/3.0/).

Journal of ELECTRICAL ENGINEERING 72(2021), NO3 177

ence frame. In [24], dq0 and embedded model of PMSMwas built for real-time simulation. [25] realizes equivalentcircuit analysis for interior PMSM and d and q axis equiv-alent circuits are obtained based on a synchronous d−qreference frame.

In the existing studies, electrical equivalent circuit ofthe stator part is given while modelling PMSM. The me-chanical part is expressed by only mechanical equations.In this study, different from machine models made forPMSM, electrical equivalents of mechanical equations arealso obtained. Mechanical parameters are expressed asan electrical circuit element in the equivalent circuit, andtherefore, the exact electrical equivalent circuit of the mo-tor is obtained for both the stator and the mechanicalpart. Thus, the complete PMSM model is expressed byelectrical elements and variables. A similar approach hasbeen made for separately excited direct current (dc) mo-tor [26] and brushless dc motor [27] as well. However, inthis study, mechanical equations are expressed with a dif-ferent electrical circuit model than [26] and [27]. PMSMcan now be modelled in any program (PSPICE, LTspice,etc) that simulates an electrical circuit.

In this study, the mathematical model of PMSM isbuilt in a,b,c phase system and an exact electrical equiv-alent circuit including mechanical components in themodel is created. Simulations of the obtained electricalcircuit are made in the circuit simulation program andthe results are given. In order to prove the accuracy ofthe obtained circuit scheme, another simulation study hasbeen carried out. For this, the differential equations in thestate space model that form the mathematical model ofPMSM are solved numerically and their results are in-cluded.

2 Mathematical model of PMSM

Mathematical model of a PMSM can be constituted bya, b, c phase variables. The following assumptions havebeen made to simplify the equations.

• Saturation, hysteresis and eddy current loses are ne-glected.

• The induced EMF has a sinusoidal shape.

• Stator winding resistances are equal and constant.

• A surface mounted PMSM is considered in the model.Therefore, self and mutual inductance values of thestator windings are equal and constant.

The voltage equations of PMSM can be given in matrixform

Va

Vb

Vc

=

Ra 0 00 Rb 00 0 Rc

IaIbIb

+d

dt

λa

λb

λc

, (1)

where, Vi - are the phase-neutral voltages, Ii - are thephase currents, Ri are the winding resistances, and λi -are the total flux linkages, while i = a,b,c. The total fluxlinkage in each phase can be expressed in detail as,

λa

λb

λc

=

Laa Lab Lac

Lba Lbb Lbc

Lca Lcb Lcc

IaIbIc

+

λma

λmb

λmc

, (2)

where, Lii - are the self-inductances, Lik - are the mutualinductances, λmi are total magnet fluxes in the phasewindings, and k = a,b,c. Generally, winding inductancedoes not change in surface mounted PMSMs dependingon the rotor position or there is little change. Therefore,it can be said that the self and mutual inductances areequal. Self-inductance values Lii can be taken as L inthe inductance matrix and mutual inductance values Lij

as M . Thus, the inductance matrix can be rearranged toobtain

λa

λb

λc

=

L M MM L MM M L

IaIbIc

+

λma

λmb

λmc

. (3)

According to these assumptions, the derivative of (3)can be placed (1), to give

Va

Vb

Vc

=

Ra 0 00 Rb 00 0 Rc

IaIbIc

+

+d

dt

L M MM L MM M L

IaIbIc

+

λma

λmb

λmc

.

(4)

In case of using Y-connection in stator winding, Ia +Ib + Ic = 0, therefore

Va

Vb

Vc

=

Ra 0 00 Rb 00 0 Rc

IaIbIc

+

+

L−M 0 00 L−M 00 0 L−M

d

dt

IaIbIc

+

+∂

∂θe

λma

λmb

λmc

dθedt

.

(5)

Here, θe indicates rotor electrical position. The mag-net flux varies sinusoidally in PMSMs depending on the

rotor position, hence∂λmi

∂θecan be expressed by a sinus

function for each phase, as

∂

∂θe

λma

λmb

λmc

=

Kf sin(θe)

Kf sin(θe −2π3)

Kf sin(θe +2π3)

=

Kfa

Kfb

Kfc

. (6)

Since the electrical angular velocity ωe =dθedt

, we can

write

Va

Vb

Vc

=

Ra 0 00 Rb 00 0 Rc

IaIbIc

+

+

L−M 0 00 L−M 00 0 L−M

d

dt

IaIbIc

+

eaebec

,(7)

178 E. K. Beser: ELECTRICAL EQUIVALENT CIRCUIT FOR MODELLING PERMANENT MAGNET SYNCHRONOUS MOTORS

Fig. 1. Stator electrical equivalent circuit of PMSM

Fig. 2. Series RL circuit

where ei is the back electromotive force (EMF) voltage

and depends on the variation of the magnet flux depend-

ing on the rotor position(

∂λmi/∂θe)

and the electricalangular velocity ωe = pωm .

Back EMF voltages are written depending on Kfi and

rotor electrical (ωe) or mechanical (ωm) angular veloci-

ties,

eaebec

=

Kfa

Kfb

Kfc

ωe =

Kfa

Kfb

Kfc

pωm, (8)

where p is the number of pole pairs. According to voltage

equations, the stator equivalent circuit of the PMSM is

illustrated in Fig. 1.

To complete the model of PMSM, it is necessary to

express the mechanical equation

Te − TL = Jdωm

dt+Bωm, (9)

where, Te - is the electromagnetic torque, TL - is the load

torque, J - is the inertia torque, and B - is the viscous

friction coefficient.

Electromagnetic torque is also given by electricalquantities

Te =p

IaIbIc

⊤

∂

∂θe

λma

λmb

λmc

=

=p

IaIbIc

⊤

Kfa

Kfb

Kfc

.

(10)

Electromagnetic torque and mechanical equations to-gether with voltage equations constitute the mathemati-cal model of PMSM.

By placing the back EMF (8) into (7) and electromag-netic torque (10) into (9), the voltage and mechanicalequations can be converted into a form

Va

Vb

Vc

=

Ra 0 00 Rb 00 0 Rc

IaIbIc

+

+

L−M 0 00 L−M 00 0 L−M

d

dt

IaIbIc

+

+

Kfa

Kfb

Kfc

pωm,

(11)

p

IaIbIc

⊤

Kfa

Kfb

Kfc

− TL = Jdωm

dt+Bωm , (12)

dθedt

= ωe = pωm . (13)

Thus the state-space model of the motor can be con-stituted as

X = [Ia Ib Ic ωm θe]⊤,

V = [Va Vb Vc TL]⊤,

A =

−

Ra

L−M0 0 −

pKfa

L−M0

0 −

Rb

L−M0 −

pKfb

L−M0

0 0 −

Rc

L−M−

pKfc

L−M0

pKfa

J

pKfb

J

pKfc

J−

B

J0

0 0 0 p 0

,

C =

1

L−M0 0 0

01

L−M0 0

0 01

L−M0

0 0 0−1

J0 0 0 0

,


Table 1. The electrical equivalents of the mechanical parametersaccording to (9) and (15)

Parameters

mechanical electrical

Te Ve

TL VL

J LJ

ωm ILJ

B RB

Fig. 3. Integrator circuits: (a) – without a reset option, and (b) –with the reset option

dX

dt= AX +CV . (14)

Dynamic model of PMSM includes mechanical as wellas electrical parameters. In order to obtain the electricalequivalent circuit, mechanical parameters must also bereplaced by their electrical equivalents. To accomplishthis, an approach can be made (9) and a series RL circuitin Fig. 2 can be proposed to do this.

The mechanical expression (9) is electrically analogousto the expression of the series RL circuit

Ve − VL = VLJ+ VRB

, (15)

Ve − VL = LJ

dILJ

dt+RBILJ

. (16)

There are also studies where this equation is modelledwith a parallel RC circuit [26, 27]. Torques are expressedby current sources and J and B are expressed by aparallel RC circuit [26,27]. But in the proposed circuit,torques are expressed by voltage sources, and J and Bare expressed by a series RL circuit.

It is seen that there is a structural similarity between(9) and (15). Here Te is defined as a controlled voltage

source since it depends on current and Kfi parameters.The electrical equivalents of the mechanical parametersaccording to (9) and (15) are given in Tab. 1.

In the mechanical expression (9), there is Kfi param-eter in expression for Te according to (10). This, Kfi de-pends on θe , so that in order to obtain the exact electricalequivalent circuit, θe must be

θe = p

∫

ωmdt , (17)

which is structurally similar to the current expression

ILp=

1

Lp

∫

VILJdt . (18)

The VILJsource is a current controlled voltage source

and the produced voltage is equal to the ILJ. The elec-

trical equivalents of the mechanical parameters accordingto (17) and (18) are given in Tab. 2.

Table 2. The electrical equivalents of the mechanical parametersaccording to (17) and (18)

Parameters

mechanical electrical

θe ILp

p 1/Lp

ωm VILJ

In Fig. 3(a), ILpcurrent value corresponding to θe in-

creases and goes towards an infinite value. In some appli-cations, absolute rotor position value may be required. Inthis case, θe needs to be reset at 360 degrees. Therefore,the revision seen in Fig. 3 (b) is made on the integratorcircuit. In this circuit, when the θe value reaches at 360degrees, SR flip flop goes into reset state. When the flipflop is in the reset state, the switching element S turns off.The current in the Lp coil is discharged through the resis-tance Rd and diode D. When the current is zero, the SRflip flop goes into the set state and the switching elementS turns on. In this way, the current value in the coil is resetevery 360 degrees and the absolute θe value is obtained.If the Rd value is chosen much greater than the Lp , theLp/Rd time constant in the circuit will be a small value.Thus, the delay in the circuit will be very low. Hence, theabsolute position θe can be obtained more accurately. Incases where absolute θe is required, the circuit given inFig. 3(b) can be used for rotor position data.

When the electrical equivalent circuit in Fig. 1 is com-bined with the circuits in Fig. 2 and Fig. 3, where me-chanical quantities are converted into electrical format,the exact electrical equivalent circuit is obtained as inFig. 4. It can be seen from Fig. 4 that the circuit con-sists of electrical components completely. Motor analysiscan be realized entirely with electrical variables. In ad-dition, PMSM analysis can be performed in a computerenvironment using any circuit software.


Fig. 4. Exact electrical equivalent circuit of PMSM

3 Simulation study

In order to confirm the validity of the proposed exactelectrical equivalent circuit of PMSM, simulations werecarried out. Typical data and the parameters of PMSMused in the simulation study are given in Tab. 3 andTab. 4, respectively. The electrical equivalent circuit wasbuilt using the PMSM parameters given in Tab. 4. As de-scribed in Section 2, the values of the circuit elements inthe electrical circuit are adjusted to provide these param-eters. In simulations, the frequency value was set to 50Hz and the peak value of the supply voltages was set to300 V. In simulation studies, the PMSM equivalent cir-cuit is connected to a 3-phase alternating current sourceand operated as line-start.

Table 3. Typical data of PMSM

Motor Type Surface mounted PMSM

Rated Power 1.5 kW

Rated Voltage 220 V

Rated Current 6 A

Rated Speed 1500 rpm

Rated Torque 25 Nm

Motor Insulation Class Class F

Protection Class IP65

Table 4. PMSM parameters

Parameter Symbol (Unit) Value

Stator resistance R(Ω) 0.68

Stator self inductance L (mH) 5.10

Stator mutual inductance M (mH) 1.00

Inertia J (kgm2) 0.0060

Friction B (Nms) 0.0041

Number of poles 2p 4

Max PM Flux Value Kf(Wb) 0.96

First, simulation of the no-load condition of the motorwas performed. In order to constitute the no-load condi-

tion (TL = 0), the value of the independent dc source(VL) in the exact equivalent circuit was set to zero.

Phase A, and phase currents in this case are givenin Fig. 5 (a) and (b). Motor angular speed (ωm) andproduced electromagnetic torque (Te) are also given inFig. 5 (c) and (d). Shown ωm and Te values correspond toILJ

and Ve in the proposed electrical equivalent circuit.

The PMSM was loaded with different load torque (TL)values of 10 and 25 Nm by adjusting the dc source (VL).Phase A current and phase currents for these loads aregiven in Fig. 6. Motor angular speed ωm is given inFig. 6(c). Load torque (TL) and produced electromag-netic torque (Te) are also given in Fig. 6(d).

The PMSM used in the simulation study has 4 poles.It was observed that the motor rotates at 157.07 rad/swhen fed with 50 Hz frequency voltage. The speed of themachine did not change as it was loaded, and the statorcurrents varied depending on the load. It is seen from thesimulation results that the proposed circuit achieves tomodel a PMSM.

In order to verify the obtained results from the electri-cal equivalent circuit, another simulation study was alsocarried out. In this simulation study, the equations instate-space model in (14), which form the dynamic modelof the PMSM were solved by writing scripts in MAT-LAB M-File. While solving the differential equations, theresults were obtained by making iterations with ode 45numerical method. In the simulations, the no-load andloaded conditions of PMSM were examined similar toelectrical equivalent circuit simulations. The simulationresults for the no-load condition are given in Fig. 7. Theresults for the loaded condition are given in Fig. 8.

In the first simulation study, the dynamic model ofPMSM was constituted using the proposed electrical cir-cuit and the results of the circuit were obtained. In thelatter study, the equations forming the PMSM modelwere solved in the simulation program and the resultsof the model were obtained. When the simulation resultsare examined, it is seen that all the results are exactlythe same. So, it can be said that the proposed equivalentcircuit fully meets the dynamic model of the PMSM quitesuccessfully.


Fig. 5. Simulation results of the proposed electrical equivalent circuit for no-load condition (TL = VL = 0Nm) (a) – phase A current,(b) – phase currents, (c) – motor speed (ωm = ILJ

), (d) – produced electromagnetic torque (Te = Ve)

Fig. 6. Simulation results of the proposed electrical equivalent circuit for loaded condition TL = VL=10,25 Nm: (a) – phase A current(b) – phase currents (c) – motor speed (ωm = ILJ

(d) – load torque (TL = VL) and produced electromagnetic torque (Te = Ve)

4 Conclusions

In this study, an exact electrical equivalent circuit is

proposed for a PMSM. The electrical equivalent circuit

of PMSM is expressed only as a stator equivalent cir-

cuit using voltage equations and the mechanical side that

completes the model of the machine is expressed only

with differential equations in the existing studies. The

significant difference of the proposed circuit from the lit-

erature is that the mechanical parameters and variables

in the PMSMs mathematical model are also expressed

with electrical components and variables. Therefore, the

complete PMSMmodel is expressed by electrical elements

and variables and the PMSM model is completely trans-


Fig. 7. Simulation results of the solution of differential equations for no-load condition (TL = 0Nm) (a) – phase A current, (b) – phasecurrents, (c) – motor speed (ωm) , (d) – produced electromagnetic torque (Te)

Fig. 8. Simulation results of the solution of differential equations for loaded condition TL = 10Nm, 25Nm: (a) – phase A current, (b) –phase currents, (c) – motor speed (ωm) , (d) – load torque (TL) and produced electromagnetic torque (Te)

formed into an electrical circuit. This allows PMSM to

be simulated as an electrical circuit in circuit simulation

programs.

The proposed PMSM circuit was simulated in the cir-

cuit simulation program. Simulation results show that the

circuit provides PMSM model quite well. The differential

equations of PMSM model were also solved in another

simulation study. The results are compatible with the pre-

vious simulation results. In the future studies, this model,

which is built with a, b, c phase variables can be devel-

oped for the machine types in which the inductance varies

depending on the rotor position. In this way, the proposed


electrical equivalent circuit can be generalized by consid-ering the mechanical parameters as an electrical circuitelement. In addition, in the Section 2, some assumptionswere made and the model was built. A new model canbe made by including the neglected components in thecircuit model in the future work.

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Received 26 May 2021

Esra Kandemir Beser received her BS, MS and PhDdegrees in electrical engineering from Kocaeli University, Ko-caeli, Turkey, in 2002, 2004 and 2013, respectively. Currently,she is an Assistant Professor with the Department of Elec-trical Engineering, Kocaeli University. Her research interestsinclude design and analysis of electrical machines, perma-nent magnet synchronous motors, brushless dc motors anddrives, motor control, power electronics and multilevel invert-ers.

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