dynamic modelling and simulation of a prototype race car

Dynamic modelling and simulation of aprototype race car in MATLAB/Simulinkapplying different types of electric motors

Guszt�av �Aron Sz�ıki, Attila Sz�ant�op and Tam�as Mankovits

Faculty of Engineering of the University of Debrecen, Ótemet}o u. 2-4, H-4024 Debrecen, Hungary

Received: June 14, 2020 • Accepted: August 31, 2020Published online: December 9, 2020

ABSTRACT

Nowadays, vehicles with alternative drives are playing an increasingly important role in roadtransport. Among the various types of alternative-drives, the most widespread ones are hybrid andelectric ones, thanks for the rapid development of modern batteries, and hybrid and electricalsystems. The above fact establishes the importance of research on various types of electric motors.The Faculty of Engineering of the University of Debrecen has more than a decade of experience indeveloping prototype race cars with pneumatic and electric drive. For the more effective andconscious development and racing a simulation program has been developed in MATLAB/Simu-link environment for the computation of the dynamic functions of a vehicle moving in a linearpath. The program is characterized by a modular structure, so the different structural units (vehiclebody, front and rear wheels, power train, motor) are modelled and simulated separately. In thisstudy we present models and simulation programs for different electric motor types (series-woundDC and brushless DC motors) in MATLAB/Simulink and apply them in our vehicle dynamicssimulation program. From the simulation results the performance of a vehicle – driven by anelectric motor – can be predicted in a race situation, consequently the most suitable motor with theoptimal characteristics can be selected to it.

KEYWORDS

electric motor, vehicle dynamics, modelling and simulation, MATLAB/Simulink

1. INTRODUCTION

Nowadays, vehicles with alternative drives are playing an increasingly important role in roadtransport. The reasons for this are, firstly, regulations and laws on climate change, andsecondly, the rapid development of technology. Among the various types of alternative-drives, the most widespread ones are hybrid and electric drives, thanks to the rapid devel-opment of modern batteries, and hybrid and electrical systems. The range of electric vehiclesthat are commercially available today or will be available in the near future can be more than500 km in many cases, while charging time can be reduced to 10–15 min by using a suitablecharger. Thus, the above vehicles are becoming more usable, which is continuously increasingsales while reducing the costs of production. As a result, one of the different electric motortypes will soon be found in the drive system of almost every vehicle, which establishes theimportance of research on various electric motor types. As a response to the above devel-opment trends the Ministry for Innovation and Technology in Hungary has started theAutomotive Industry Thematic Program at the University of Debrecen in 2019. In theframework of this program our research group has started to develop simulation programsfor electric drives applied in modern road vehicles.

On the other hand, the Faculty of Engineering of the University of Debrecen has morethan a decade of experience in developing prototype race cars with electric and pneumaticdrives. With these cars, student teams of our Faculty take part in different domestic and

International Review ofApplied Sciences andEngineering

12 (2021) 1, 57–63

DOI:10.1556/1848.2020.00145© 2020 The Author(s)

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foreign competitions (MVM Energy Race, Shell ECOMarathon, Pneumobile and Formula Student competitions)announced for university students. At the different compe-titions the students face several different tasks regulated bystrict rules. The race cars have to fulfil the requirements, andadditionally, have to be optimally designed for the givencompetition task. Thus, the designers have to apply differenttechnical solutions and choose the different structural units ofthe cars (e.g. the electric motor) and their technical charac-teristics properly. For the more effective and consciousdevelopment and racing a simulation program has beenrealized in MATLAB/Simulink [1] previously for the calcula-tion of the dynamic functions of a vehicle moving on a lineartrack. The program is characterized by a modular structurethus the different structural units (vehicle body, motor, powertrain, front and rear wheels) are modelled and simulatedseparately. Applying the program and supplementing it withan optimization procedure, the optimal values of technicaldata (e.g. optimal gear ratio in the drive train) can becomputed for a given vehicle dynamics aim (racing task). Thisaim can be e.g. to complete the race in the shortest possibletime or with the lowest energy consumption driving above agiven minimum speed.

In the recent study we present models and simulationprograms for different types of electric motors (series-wound DC (SWDC) and brushless DC (BLDC) motors) andapply them in our vehicle dynamics simulation program.From the simulation results the performance of a car –driven by different types of electric motors – can be pre-dicted in a race situation. Complementing our simulationprogram with an optimization procedure the most suitablemotor with the most advantageous parameters and charac-teristics can be selected for a given racing task.

2. THE VEHICLE DYNAMICS MODEL ANDSIMULATION PROGRAM

Applying the developed dynamics model, and the simulationprogram based on it [1], the dynamic functions of a vehicle,

moving on a linear path, can be calculated from its technicalcharacteristics and data. Complementing the program withan optimization procedure, the optimal values of technicaldata can be computed for a given driving dynamics aim [2].

The program takes into consideration almost all thefactors that influence the motion of the vehicle. Includingthe electromagnetic and dynamic characteristics of themotor, air and rolling resistance, moment of inertia of therotating machine parts, position of the centre of gravity ofthe vehicle, and the coefficient of friction between the roadand the tyres as a function of tyre slip [1].

Beyond the usual driving dynamics functions (accelera-tion, velocity and position-time functions), the program iscapable of calculating the forces on the front and rear wheelsand loads on the front and rear axles versus time intangential and normal direction. Other calculated functionsare: rolling and bearing resistance torques, air resistanceforce, tyre slip, angular speed and torque of the motor versustime.

For modular development and greater clarity, the mainvehicle components are organized into separate blocks,which are:

� vehicle body� motor� powertrain� front wheels� rear wheels

The block diagram of the simulation program is shownin Fig. 1.

The detailed description of the simulation program,together with the input technical characteristics and data thatare necessary for running it, can be found in reference [1].

3. MODELING AND SIMULATION OF THEELECTRIC MOTOR

A main part of the vehicle dynamics simulation program isthe “motor simulation block” which is responsible for the

Fig. 1. Block diagram of the vehicle dynamic simulation program [1]

58 International Review of Applied Sciences and Engineering 12 (2021) 1, 57–63


simulation of the electric motor. There are different types ofmotors (Brushed DC motor (BDC), Brushless DC motor(BLDC), Induction (Asynchronous) motor (IM), Synchro-nous motor (SM), Switched Reluctance motor (SRM)) thatare commercially available for driving electric vehicles.Regarding the student competitions, for the wide range ofcompetition tasks, different types of electric motors withdifferent technical characteristics and parameters can beoptimal. Thus, for an optimal choice, the vehicle dynamicssimulation program must be run with all the possibledifferent motor types. As a result of this, a decision wasmade by the authors to model and simulate two of thefrequently used electric motor types first, and then theremaining ones in the near future.

3.1. Modeling and simulation of series-wound DCmotors

In Fig. 2 the applied model for series-wound DC motors ispresented.

The model [3] is based on the following differentialequations [4, 5]:

Ubatt � ðRs þ Rr þ Rbatt þ RwireÞ$I� ðLsðIÞ þ LrðIÞÞ$dIdt � LsrðIÞ$ur$I ¼ 0:

(1)

MeðIÞ �MresðurÞ �MloadðurÞ ¼ Jr$dur

dt(2)

where the electromagnetic torque of the motor can becalculated as:

MeðIÞ ¼ LsrðIÞ$I2 (3)

In the equations Ubatt [V] is the electromotive force of thebattery, Rs, Rr [Ω] are the electric resistances of the statorand the rotor windings of the motor, while Rbatt and Rwire

[Ω] are the internal resistance of the accumulator and theresultant resistance of the wires connecting the motor to thebattery. Characteristics Lr(I), Ls(I) and Lsr(I) [H] are the self-dynamic inductances of the rotor and stator windings, andthe mutual dynamic inductance, respectively. The abovecharacteristics depend on the intensity of the current flowingthrough the motor. Characteristic Mres(ur) [Nm] is thebearing resistance and brush friction torques together, whileMload(ur) [Nm] is the loading torque on the rotor of themotor and Jr [kgm2] is the rotor’s moment of inertia.Characteristics Lr(I), Ls(I), Lsr(I) and Mres(ur) have to bemeasured [3], while the value of Jr sometimes can be foundin the catalogue of the motor. Mres(ur) is also frequentlyapproximated as:

MresðurÞ ¼ B$ur (4)

where ur [rad/s] is the angular velocity of the rotor and B[Nms] is a constant. The detailed description of the simu-lation program based on this model can be found in refer-ence [3], together with the detailed description of themeasurement set-up and procedures for the determinationof the technical characteristics of the motor, that arenecessary as input data for the simulation. In reference [3]

all the necessary input data are given for a 4 kW SWDCmotor. We use these data when simulating the SWDC motorin the prototype race car in Section 4.

3.2. Modeling and simulation of brushless DC motors

In Fig. 3 the applied model for brushless DC (BLDC) motorwith star-connected strator windings is presented.

The model is based on the following differential equa-tions [6, 7]:

Ua ¼ Ra$Ia$La$dIadt

þ Lab$dIbdt

$Lac$dIcdt

þ ea (5)

Ub ¼ Rb$Ib þ Lba$dIadt

þ Lb$dIbdt

þ Lbc$dIcdt

þ eb (6)

Uc ¼ Rc$Ic þ Lca$dIadt

þ Lcb$dIbdt

þ Lc$dIcdt

þ ec (7)

where Ua, Ub and Uc [V] are output voltages from theinverter, Ra, Rb, Rc, [Ω]; La, Lb and Lc [H] are the electricalresistances and the self-inductances of the stator windings,Lab, Lac, Lba, Lbc, Lca and Lcb [H] are the mutual inductancesbetween the stator winding pairs, ea, eb and ec [V] are the

Rwire

La

Rr

L r

Rs Ls dI/dtI

Mres

ILsr2

-

+

Rbatt

Vbatt Mload

Fig. 2. The applied model for the simulation of series-wound DCmotors [3]

Fig. 3. The applied model for the simulation of BLDC motors

International Review of Applied Sciences and Engineering 12 (2021) 1, 57–63 59


Back EMF values generated in the stator windings by therotating permanent magnet rotor. The output voltages fromthe inverter are given as:

Ua ¼ UðqeÞ ¼

8>>>>>><>>>>>>:

�1; if4p3≤ qe<2p

0; if 0≤ qe<p

3or p≤ qe<

4p3

1; ifp

3≤ qe<p

9>>>>>>=>>>>>>;

(8)

Ub ¼ U

�qe � 2p

3

�(9)

Uc ¼ U

�qe � 4p

3

�(10)

In the above equations the electrical angle can be calcu-lated as qe ¼ p

2$qr [rad], where qr [rad] is the angular posi-tion of the rotor, and p

2 is the number of permanent magnetpole-pairs in the rotor.

The arrangement of windings is symmetrical and the selfand mutual inductances can be assumed as constants, thus:

R ¼ Ra ¼ Rb ¼ Rc (11)

L ¼ La ¼ Lb ¼ Lc (12)

M ¼ Lab ¼ Lba ¼ Lac ¼ Lca ¼ Lbc ¼ Lcb (13)

Consequently:

Ua ¼ Ra$Ia þ L$dIadt

þM$dIbdt

þM$dIcdt

þ ea: (14)

Ub ¼ Rb$Ib þM$dIadt

þ L$dIbdt

þM$dIcdt

þ eb: (15)

Uc ¼ Rc$Ic þM$dIadt

þM$dIbdt

þL$dIcdt

þ ec : (16)

For the Y connected stator windings applying Kirchhoff’snodal rule:

Ia þ Ib þ Ic ¼ 0 ⇒dIadt

þ dIbdt

þ dIcdt

¼ 0 ⇒ e:g:

:dIbdt

þ dIcdt

¼ −dIadt

(17)

Thus:

Ua ¼ R$Ia þ ðL�MÞ$dIadt

þ ea (18)

Ub ¼ R$Ib þ ðL�MÞ$dIbdt

þ eb (19)

Fig. 4. The block diagram of the simulation program for BLDC motors

Fig. 5. The internal structure of the program block for the calculation of the phase current intensities



Uc ¼ R$Ic þ ðL�MÞ$dIcdt

þ ec (20)

The Back EMF values can be calculated as:

ea ¼ −ke$ur$FðqeÞ (21)

eb ¼ −ke$ur$F

�qe � 2p

3

�(22)

ec ¼ −ke$ur$F

�qe � 4p

3

�(23)

In the above equations, ur [rad/s] is the angular velocity ofthe rotor, ke [V/(rad/s)] is the Back EMF constant and F(qe)describes the trapezoidal waveform of Back EMF:

FðqeÞ ¼

8>>>>>>>>>><>>>>>>>>>>:

6p$qe � 1; if 0≤ qe<

p

3

1; ifp

3≤ qe<p

�6p$qe þ 7; if p≤ qe<

4p3

�1; if4p3≤ qe<2p

9>>>>>>>>>>=>>>>>>>>>>;

(24)

The equation of motion for the rotor is the same as inSection 3.1 (Eq. 2).

The electromagnetic torque here can be calculated as:

Me ¼ −ke$

�FðqeÞ$Ia þ F

�qe � 2p

3

�$Ib þ F

�qe � 4p

3

�$IcÞ(25)

The block diagrams of the simulation program, based onEqs. (5–25), are shown in Figs. 4–6.

The specifications of the simulated BLDC motor can befound in reference [8]. We also use these data when simu-lating the BLDC motor in the prototype race car in Section4. The used input data for the simulation are: rated power:1.4 kW; rated DC voltage: 100 V, phase resistance: 0.28 Ω,phase inductance: 0.24 mH, number of pole-pairs: 2; line-to-line back EMF constant: 0.0033 V/(r/min); moment ofinertia of the rotor: 0.002139 kgm2, constant B: 0.001 Nms,loading moment of inertia: 0.0688 kgm2 (The loadingmoment of inertia can be modelled as a homogenous circular

Fig. 6. The internal structure of the program block for the calcu-lation of the electromagnetic torque of the motor

Fig. 7. The intensity of current vs. time in phase a without (a) andwith (b) loading moment of inertia on the rotor

Fig. 8. RPM and torque of the motor vs. time (M5Me-Mres).



disk applied on the rotor, the value of Mload is selected to bezero.) The results of the simulation are shown in Figs. 7–9.

4. SIMULATION OF THE PROTOTYPE RACE CAR

The motion of a prototype race car, accelerating from rest ona linear track, was simulated. The car was designed andconstructed at the Faculty of Engineering of the Universityof Debrecen. The detailed technical description of the car

can be found in Reference [1]. The vehicle dynamicssimulation program [1] was applied for the simulation bothwith the SWDC and BLDC motors driving the car. All thevehicle’s technical data were the same in both cases (see thelist of data in Reference [1]) except for the gear ratio in thechain drive, which was 4 when applying the SWDC and 14when the BLDC motor. At the above gear ratios, the time thecar needed to travel 100 m starting from rest, has a mini-mum. Fig. 10 shows the obtained velocity-time functions ofthe vehicle applying the two different motors.

Although the nominal power of the SWDC motor is 4kW, while the one of the BLDC motor is 1.4 kW, the time thecar needed for travelling 100 m, starting from rest, is onlyslightly different in the two cases (9.64 and 9.76 s, respec-tively). Nevertheless, it must be emphasized that the BLDCmotor is significantly overloaded during the acceleration.Fig. 11 shows the torque and RPM of the two motors vs. time.

5. CONCLUSION

The motion of a prototype race car, accelerating from rest ona linear track, was simulated applying our simulation pro-gram that was previously developed in MATLAB/Simulinkenvironment. The program is capable of calculating thedynamics functions of a vehicle, moving on a linear track,from the vehicle’s technical data. Complementing the pro-gram with an optimization procedure, the optimal values oftechnical data can be computed for a given driving dynamicsaim. Simulations were performed both with SWDC andBLDC motors driving the car. The model and simulationprogram for the SWDC motor were developed previously,the detailed modelling and simulation procedure of theBLDC motor is presented here. The obtained results showthat the vehicle dynamics simulation program, and also thesimulation programs for the electric motors work properly.The accuracy of the simulation program for the SWDCmotor was checked by test measurements previously. Themeasured and simulated results are in good agreement. Inthe near future we also intend to perform test measurementson the BLDC motor and compare the results with thesimulated ones. To be able to do that, first of all, we have tomeasure its electromagnetic and dynamic characteristics. Inthe long term we also intend to perform telemetry mea-surements on the prototype race car to check the accuracy ofour vehicle dynamics simulation program.

-350

-250

-150

-50

50

150

250

350

0,02 0,025 0,03 0,035 0,04 0,045

Supplyvoltage[V]Currentintensity[A]Back EMF*5 [V]

t [s]

Fig. 9. Supply voltage, current intensity and Back EMF for phase avs. time

Fig. 10. Velocity-time functions of the prototype race car applyingtwo different types of electric motor

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10

]mN[

M

t [s]

BLDCmotorSWDCmotor

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 2 4 6 8 10

]nim/1[

MPR

t [s]

BLDCmotor

SWDCmotor

Fig. 11. Torque and RPM of the SWDC and BLDC motors vs. time while the car is accelerating from rest (M 5 Me � Mres)



ACKNOWLEDGEMENT

“The research was financed by the Thematic ExcellenceProgramme of the Ministry for Innovation and Technologyin Hungary (ED_18-1-2019-0028), within the framework ofthe (Automotive Industry) thematic programme of theUniversity of Debrecen.”

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dynamic modelling and simulation of a prototype race car

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