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An Investigation into the Induction Motor of Tesla Model S Vehicle

Conference Paper · June 2018

DOI: 10.1109/ISEM.2018.8442648

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IEEE TRANSACTIONS ON EDUCATION,, VOL. XX, NO. X, ...... 2017 1

An Investigation into the Induction Motor

of Tesla Model S Vehicles
Grzegorz Sieklucki

Abstract—There are no scientific publications on a electric

motor in Tesla cars, so let’s try to deduce something. Tesla’s
induction motor is very enigmatic so the paper tries to introduce a
basic model. This secrecy could be interesting for the engineering
and physics students. Multidisciplinary problem is considered:
kinematics, mechanics, electric motors, numerical methods, con-
trol of electric drives. Identification based on three points in
the steady-state torque-speed curve of the induction motor is
presented. The field weakening mode of operation of the motor
is analyzed. The Kloss’ formula is obtained. The main aim of
the article is determination of a mathematical description of the
torque vs. speed curve of induction motor and its application for
vehicle motion modeling. Additionally, the moment of inertia of
the motor rotor and the electric vehicle mass are considered in
one equation as electromechanical system. A mathematical model
of the mechanical gear is considered. Presented approach may
seem like speculation, but it allows to understand the problem Fig. 1. Torque-speed and power-speed curves of the car
of a vehicle motion. The article composition is different from
classical approach – studying should be intriguing.
Index Terms—Electric vehicles, engineering education, elec-
tromechanical systems, field weakening, induction motors, kine- Field weakening region
matics, Kloss’ formula, mathematical models, mechanical gears, constant slip frequency
motor torque, Tesla Model S, traction motors. curve (critical)

Mode 1 Mode 2 Mode 3

I. M OTIVATION

T ESLA Model S is a modern electric vehicle (EV), so

its simple mathematical model (dynamics) should be
interesting for students. Induction motors in Tesla cars are
enigmatic, but a few facts are known: copper rotor, four poles
(two pole pairs pb = 2).
The main aim of the article is determination of a mathe-
matical description of the torque vs. speed curve (fig. 1) of
Fig. 2. Induction motor operating conditions
induction motor and its application for vehicle motion mod-
eling. Moreover, it is an interesting example for engineering
and physics students: kinematics, mechanics, electric motors,
numerical methods, control of electric drives. • Mode 3: costant power*rotor angular velocity region, slip
Control of the traction motor has been well known since the frequency is constant and equals ω2K (critical value, pull-
70s of the 20th century, eg. [1]. The analysis of the electric out frequency). This region is similar to DC series motor.
motors for EV applications is presented in [2]–[6].
Fig. 2 shows 3 modes of operation for the induction motor Mode 3 is the most interesting, because this curve contains
(Me motor torque, ψS stator linkage flux, ωm rotor angular information about some parameters of the induction motor.
velocity, ω1 , ω2 stator and slip frequency, US stator voltage): Thus, critical curve is the relationship between critical (pull-
• Mode 1: constant torque region, out) torque MeK and ωm where US = USN (subscript N
• Mode 2: constant power region, denotes nominal (rated) value). Me vs. ωm in mode 3 is
applied to simple identification in further part of the paper.
G. Sieklucki is with the AGH University of Science and Technology, Fac- The inertia of a motor is usually omitted in popular litera-
ulty of Electrical Engineering, Automatics, Computer Science and Biomedical ture, e.g. [6]–[8] but it is energy storage. Hence, motor moment
Engineering, Department of Power Electronics and Energy Control Systems,
Krakow 30-059, Poland (e-mail: sieklo@agh.edu.pl). of inertia is included in the presented model. Furthermore, the
Manuscript received March 19, 2017; revised ...... . friction in a gear could be modeled as constant force [9].
IEEE TRANSACTIONS ON EDUCATION,, VOL. XX, NO. X, ...... 2017 2

II. W HICH M ODEL S IS IT ? Hence, the critical curve in fig. 2 is described as:
The red curve in fig. 3 is probably steady-state motor
 1 1
MeK (ωm ) = mk 2 = mk (6)

characteristic of Tesla Model S P85 (2012) [10]. That vehicle US =USN ω1 (pb ωm + ω2K )2
has the following performance: 3p U 2 (1−σ)
where mk = b 2σLSN
.
• 0-60mph (0-96,6km/h): 4,6s, S
The parameters MeK , mk , SK , ω2K , ω1N are obtained in
• 1/4-mile (402,5m): 13,3s and speed: 104 mph
subsection V-A.
(167,5km/h),
• top speed: 134 mph (215,7km/h) which leads to vmax =
V. S USPECT
59, 9m/s.
The torque-speed and power-speed curves of Tesla traction
Approximated and assumed specification:
motor (fig. 3) are available on the following websites (actual
• mass (car+driver) m2 ≈ 2100 + 100 = 2200kg,
2
source is unknown):
• moment of inertia (electric motor rotor) J1 ≈ 0, 06kgm
http://electronics.stackexchange.com/questions/271674/
(assumption: radius of the rotor about 0,1m, mass of rotor tesla-car-maximum-torque-at-0-rpm-is-this-correct,
about 12kg), https://geektimes.ru/post/279758/
• aerodynamic drag coefficient Cd = 0.24, frontal area of
http://teslatrails.blogspot.com/2015/03/
the vehicle A = 2, 4m2 , electric-motors-are-incredible.html
• gear ratio i = 9, 73 with efficiency η ≈ 0.97 (assumed in
[11] for Tesla Roadster).
In gears one observes dry or Coulomb friction, which is
nearly independent of speed [9], so gear produces load
torque for the motor:
4M = Mmax (1 − η) · sign (ωm ) = 18 sign (ωm ) (1)
where ωm is angular velocity of a motor.
• tires: P245/45R19 which leads to wheel diameter dw =
0.703m.
Two kinetic energy storages are found in every ground vehicle:
rotor of a motor (moment of inertia J1 ) and a vehicle mass
(m2 ), so in dynamics the total moment of inertia J = J1 + J2
or the total mass m = m1 + m2 can be used:
4i2 d2w
m1 = J1 = 46kg, J2 = m2 = 2, 87kgm2 (2)
d2w 4i2
The values m1 , J1 are much smaller than m2 , J2 but let us
not forget about kinetic energy.

III. I NVESTIGATION Fig. 3. Torque-speed and power-speed curves of the car (v speed, n rotations
per minute of rotor, fw frequency of the wheels)
This article seeks an answer to the following question: ”How
the mechanical characteristics of the Tesla traction motor is
described?” Analysis of the red line (Mmax = 600Nm, Pmax = 310kW)
is presented in the paper. The figure is described by the author
IV. WANTED – K LOSS ’ FORMULA for the educational aspect and the information comes from
sections II and VI.
Simplified Kloss’ formula [9], [12] is described as (φ =
ω1
ω1N , κ = UUSN
S
are per unit values):
A. Interpolation
 2
κ The motor torque curve in mode 3: Me (ωm ) =
2MKN mk
φ (pb ωm +ω2K )2 can be interpolated, where two parameters are
Me = (3) unknown (mk , ω2K ), and two points in fig. 3 are known
S SK
+ (ωm1 , MeK1 ) and (ωm2 , MeK2 ). Hence, interpolation (numer-
SK S
ical methods) leads to an exercise with a high school difficulty
where critical (pull-out) motor torque and slip are defined as
level: function with two parameters crosses two points. Thus,
(σ is total leakage factor of the motor):
the equations for analyzed induction motor have the following
 2
US 1−σ form: mk
MeK = 3pb (4)
 
ω1 2LS σ  MeK1 = (p ω − ω )2 
 
b m1 2K
mk (7)
RR RR ω2K ω2K  MeK2 =
SK = = = = (5)
 

φω1N LR σ ω1 σLR ω1 ω2K + pb ωm (pb ωm2 − ω2K )2
IEEE TRANSACTIONS ON EDUCATION,, VOL. XX, NO. X, ...... 2017 3

The geometrical interpretation of considered problem is shown

in fig. 4.

Me Nm
1070.5 MKN mk
Me (ωm ) =
(pb ωm + ω2K )2

Fig. 5. Arbitrary forces acting on a vehicle

Me

600

VI. L ONGITUDINAL V EHICLE M ODEL

MeK1
343
A. Forces
Most of the significant forces (road loads and traction force)
MeK2
110 acting on the vehicle are shown in fig. 5 e.g. [7].
0 Tractive force – force from motor which is function of ωm
0 516 904 1609
ωm or v (fig. 3):
2i
Fig. 4. Steady-state curves of induction motor FT = (Me (ωm ) − 4M ) (11)
dw
Total rolling resistance Fr is the sum of the resistances from
The solution is a quadratic equation: all wheels [7] (cos α is neglected, because for 10% road grade
2
(MeK1 − MeK2 )ω2K + 2pb (ωm1 MeK1 − ωm2 MeK2 )ω2K is cos( 0,1π
4 ) = 0, 9969):
2 2 2
+pb (ωm1 MeK1 − ωm2 MeK2 ) = 0  v 
Fr = fr m2 g, fr = f0 1 + (12)
0 00 161
thus, zeros are: ω2K = 33, 1rad/s and ω2K = −2318rad/s, so
00 where v is vehicle speed in km/h and the rolling coefficient
ω2K is ignored and critical slip frequency is:
is f0 = 0, 01 ÷ 0, 02 for concrete surface. In literature there
0 are a lot of the empirical relationships which can be used as
ω2K = ω2K = 33, 1 rad/s
exercises for students.
and the parameter mk which is used in critical curve (from Aerodynamic drag force:
(6) and (7)) equals: 1
FDF = % Cd A(v − vw )2 (13)
2
mk = 1, 1627 · 109
where ρ = 1.225kg/m3 is air density at sea level, at 15o C and
The calculation of MKN , and ω1N bases on mk , ω2K and vw speed of wind.
the point (ωmN , Mmax ) in fig. 3. Then simplified Kloss’ Grade resistance (gravitational force acting on the vehicle) –
formula (3) with (5) can be presented in the following form: differentiation between the front and rear axle loads is omitted:
2mk ω2K (ω1N − pb ωmN ) Fg = m2 g sin(α) (14)
Mmax = 2 2 ω2 (8)
ω1N (ω1N − pb ωmN )2 + ω1N 2K Application of Newton’s second law leads to:
which leads to equation: dv(t)
m = FT (t) − (FDF (t) + Fr (t) + Fg (t)) (15)
4
ω1N − 3
2pb ωmN ω1N + (p2b ωmN
2 2
+ ω2K )ω1N dt } | {z }
(9) 0
| {z
−2 Mmmax
k
ω2K ω1N + mk
2 Mmax ω2K pb ωmN = 0 a(t) F (t)

Two zeros are complex (not considered) and two are positive The tractive force (11) and the aerodynamic drag force (13)
real values: are nonlinear functions of vehicle speed (v), so the ordinary
0 00
differential equation (15) is nonlinear, too. Thus, the numerical
ω1N = 1128 rad/s, ω1N = 1050 rad/s solution of ODE is necessary.

but the first value does not satisfy the condition ω1N −
pb ωmN < ω2K (stable part of mechanical characteristic of B. Torques
induction motor), thus: The forces applied to the wheels and the torques on induc-
ω2K tion motor shaft are shown in fig. 6. The conversion of the
ω1N = 1042rad/s ⇒ f1N = 165, 8Hz ⇒ SK = = 0, 32 road loads F 0 to torques on the motor shaft is:
ω1N
(10) M0 F 0 dw
and nominal critical torque equals MKN = 1070Nm. Mm = 4M + , M0 = (16)
i 2
IEEE TRANSACTIONS ON EDUCATION,, VOL. XX, NO. X, ...... 2017 4

B. Simulation results
IM The above equations (19) and (20) are the basis for nu-
merical computations and they allow us to calculate the
performance of the EV. The slip between tires and surface is
neglected in computations and so is the deflection of the tire.
For the performance analysis the gear (1) simplifies to the
Fig. 6. Simplified powertrain
form 4M = 18 Nm. The integral interval equals Ts = 0, 05s.
The Matlab script used is included in the appendix.
The dynamics of the analyzed model is presented in fig. 8
and 9 where three modes of operation of the induction motor
Thus, the Newton’s second law for the rotational motion is:
control can be observed.
dωm (t)
J = Me (t) − Mm (t) (17)
dt
600 215
The equations (15) and (17) describe the same electromechan- 200
ical system in different reference frames.

150
C. Model 400

343
Previous equations lead to the diagram in fig. 7 which can

v/kmh
Me
be helpful for understanding vehicle motion. 96.6

200
Motor
+
Control

0 0
0 5 10 15 20 25 30
t

Fig. 7. Powertrain as signal processing Fig. 8. Induction motor torque and vehicle speed

16000
VII. V ERIFICATION FT
14000 FDF
A. Numerical integration – Euler’s Method
Fr
Considering the natural problem where the reference system 12000
FDF +F r
is the Earth (15), the vehicle acceleration and speed are:
10000
FT (t) − F 0 (t) dv(t)
Forces

a(t) = , = a(t) (18) 8000

m dt
and applying the Euler’s method (Ts is integral interval or 6000

integration step): 4000

dv(t) v(t + Ts ) − v(t)

≈ 2000
dt Ts
leads to difference equation in the following form: 0 10 20 30 40 50 60
t
v(t + Ts ) = v(t) + Ts a(t)
Fig. 9. Forces acting on vehicle, introduced in fig. 5
or
v(k + 1) = v(k) + Ts a(k) (19)
The fig. 10 shows the simulation results for performance
where k is the iteration step. analysis.
Z t
Comparison of the simulation results with performance data
The calculation of the vehicle position spos (t) = v(τ )dτ (sec. II) indicates the following errors:
0
needs numerical integration, too. Thus, the Euler method can • acceleration 0-60 mph: δ =
4,6−4,08
· 100% = 11, 3%
4,6
be applied: 13,3−12,75
• 1/4 mile: time: δ = 13,3 · 100% = 4, 1%, speed:
|167,4−171.6|
spos (k + 1) = v(k) + Ts spos (k) (20) δ= 167,4 · 100% = 2, 5%.
IEEE TRANSACTIONS ON EDUCATION,, VOL. XX, NO. X, ...... 2017 5

150 Stator voltage is a function of frequency US (f1 ) in the

v/kmh AC motors, so stator frequency is omitted. The steady-state
100
equation is assumed as:
50 U R
ωm = − 2 Me , U = US (21)
ψe ψe
0 2 4 6 8 10 12
t and it is complemented by Newton’s second law (17). If stator
400
voltage is known, e.g. UN = 400V then generalized flux
ψe and generalized resistance R can be calculated as (for
300
considered problem MeN = Mmax = 600 Nm):
spos/m

200
UN ψeN (UN − ψeN ωmN )
100 ψeN = , R= (22)
ωm0 MeN
0
0 2 4 6 8 10 12 Thus, ψeN = 0, 768 Vs and R = 0, 005 Ω.
t The block diagram of the generalized traction motor is
Fig. 10. Acceleration 0-60 mph: time 4,08 s, 1/4 mile Race: time 12,75 s
presented in fig. 12.
(speed 171,6 km/h or 106,8 mph. Driver mass equals 100kg

The error for acceleration is not satisfying. Perhaps the vehicle

analyzed it is not Tesla model P85? Answer: the P85+ Model
S has the same torque and power, but better performance:
• 0-60mph: 4,0s,
• 1/4-mile (402,5m): 12,7s and speed: 107.8 mph (173,6 Fig. 12. Generalized model of a traction motor
km/h).
The tires of P85+: front P245/35/R21; rear 265/35/R21 (dw = The model can be used in torque controller optimization
0, 719m). After simulation (uncomment line 06 in Matlab and designing the cruise-control of the vehicle or the traction
script – appendix) the error levels are reduced to: control (slip-speed ratio control). Presented model allows
• acceleration 0-60 mph: δ = |4,0−4,14|
4,0 · 100% = 3, 5% students to understand necessity of the field weakening region
|12,7−12,77| in every traction motor control.
• 1/4 mile: time: δ = 12,7 · 100% = 0, 55%, speed:
|173,6−172,5|
δ= 173,6 · 100% = 0, 63%.
IX. J UDGMENT
The obtained results are better. However, in the simulations
Finally it can be said ”The P85+ Model S should blame its
the mass of the driver was assumed to be 100kg. If reduced
high performance on the red curve in fig. 3 (control of the
to 50kg, the simulation results will be the best.
induction motor) and the equation (3) with parameters (10)
describes motor mechanical characteristics”, but the question
VIII. BASIC M ODEL OF T RACTION M OTOR still remains: ”Who is the author of the curves in fig. 3?”.

Generalized mathematical model of the motor should in- X. C ONCLUSION

clude signals: motor Me and load Mm torques, back electro- The paper allows to understand some problems of a vehicle
motive force (emf) E and angular velocity ωm . motion and the Tesla model S is only a pretext.
This model applies a simplified equation of a DC motor The next step in students education is the area for:
where inductance is neglected. Generalized model of the
• control of electrical drives:
traction motor bases on linearization which is shown in fig. 11.
– controller optimization for more complex motor
models [13], [9], [14], [15].
– final realization of the control system: inverter,
ωm0
520
Clarke-Park transformations, measurement systems,
ωmN
observers, vector control method, etc.
ωm

510 • kinematics, mechanics:

– simulation with speed of wind and grade, how the

500 car performance will change?
0 200 400 600 800 1000 1200 – optimization of the gear for best acceleration or
Me
maximum speed of vehicle.
Fig. 11. Linearization
– consideration rotating masses between gear and
ground (wheels, brake discs, drive shafts).
IEEE TRANSACTIONS ON EDUCATION,, VOL. XX, NO. X, ...... 2017 6

Presented problem includes issues of kinematics, mechan- 52: s_pos=zeros(1,length(t));

53: for k=2:length(t) %replace: s_pos=t_step*cumsum(v_ms);
ics, electric motors, numerical methods, control of electrical 54: s_pos(k)=t_step*v_ms(k)+s_pos(k-1);
drives. The Matlab/Octave script allows to simulate any vehi- 55: end;
56: %%---- VISUALIZATION------
cle with any propulsion: electric, hybrid, internal combustion 57: figure(2)
engine (the steady-state curve Me (ωm ) in line 40 can be 58: v_kmh=v_ms*3.6; % convert m/s to kmh
59: %p1=plot(t,F_T,t,F_DF,t,F_r,t,F_DF+F_r);
changed). This script or the diagram in fig. 7 can be translated 60 % yl=ylabel(’Forces’);
into Simulink or Xcos (Scilab) graphical form, then additional 61: %leg1=legend(’F_T’,’F_{DF}’,’F_r’,’F_{DF}+F_r’);
62: %set(leg1,’FontSize’,15)
exercises for students could be realized. In further exercises 63: %p1=plot(t,a);
the block diagram (fig. 7) should be complemented by motor 64: %p1=plot(t,wm);
65: p1=plot(t,v_kmh,’k’,[t(1) max(t)],[96.6 96.6],’r’);
model and torque controller, et cetera. 66: yl=ylabel(’v/kmh’);
67: %p1=plot(t,s_pos,’k’,[t(1) max(t)],[402.5 402.5],’r’);
68: %yl=ylabel(’s_{pos}/m’);
A PPENDIX A 69: grid; axis tight
T HE M ATLAB /O CTAVE SCRIPT 70: set(p1,’LineWidth’,1.5)
71: xl=xlabel(’t’);
In the following lines of the script there are: 72: set(xl,’FontSize’,15);set(yl,’FontSize’,15);
73: set(gca,’FontSize’,12);
• 1÷ 15 – vehicle parameters and 3 modes of operation of
the induction motor,
• 17÷ 30 – steady-state curve of the motor,
R EFERENCES
• 32÷ 55 – Euler’s method, where t_step= Ts and [1] A. B. Plunkett and T. A. Lipo, “New methods of induction motor torque
regulation,” IEEE Trans. Ind. Appl., vol. IA-12, no. 1, pp. 47–55, Jan
t_end is simulation time, 1976.
• 57÷ 73 – visualization of the simulation results, the [2] M. Ehsani, Y. Gao, and S. Gay, “Characterization of electric motor drives
performances are marked with the red line. for traction applications,” in Ind. Electron. Soc., 2003. IECON ’03. The
29th Annu. Conf. of the IEEE, vol. 1, Nov 2003, pp. 891–896 vol.1.
01: clear [3] J. Goss, M. Popescu, and D. Staton, “A comparison of an interior
02: g=9.81; %gravitational acceleration permanent magnet and copper rotor induction motor in a hybrid electric
03: g_i=9.73;%gear ratio
04: mass=2245;m2=mass-45; % mass=car+driver+m_1
vehicle application,” in 2013 Int. Elec. Mach. Drives Conf., May 2013,
05: d_w=(19*25.4+2*245*0.45)/1000; %Diameter of wheels P85 pp. 220–225.
06: %d_w=(21*25.4+2*265*0.35)/1000; %Diameter of wheels P85+ [4] M. Barcaro, N. Bianchi, and F. Magnussen, “PM motors for hybrid
07: Pn=310000; Mmax=600;pb=2; electric vehicles,” in 2008 43rd Int. Universities Power Eng. Conf., Sept
08: wI=516; %end of mode I 2008, pp. 1–5.
09: wII=904; % end of mode II [5] M. Karamuk, “A survey on electric vehicle powertrain systems,” in Int.
10: wIII=1920; %max. of IM Aegean Conf. on Elec. Mach. and Power Electron. and Electromotion,
11: w_2k=33.1; m_k=1.1627e+09; Joint Conf., Sept 2011, pp. 315–324.
12: %-----drag force
13: rho=1.225; % Density of air kg/m3
[6] K. Nam, AC Motor Control and Electrical Vehicle Applications. CRC
14: Cd=0.24; % drag coefficient Press, 2010.
15: A=2.4; % the reference area. [7] T. Gillespie, Fundamentals of Vehicle Dynamics. Society of Automotive
16: %%---- IM curve Engineers, 1992.
17: wm_vect=0:0.5:1920; [8] F. Giri, AC Electric Motors Control: Advanced Design Techniques and
18: for i=1:length(wm_vect) Applications. Wiley, 2013.
19: Me(i)=(wm_vect(i)<=wI)*Mmax... [9] W. Leonhard, Control of Electrical Drives. Berlin, Springer-Verlag,
+(wm_vect(i)>wI&&wm_vect(i)<=wII)*Pn/wm_vect(i)... 2001.
+(wm_vect(i)>wII)*m_k/(pb*wm_vect(i)+w_2k)ˆ2;
20: end
[10] D. Sherman, “Drag queens. five slippery cars enter a wind tunnel;
21: P=Me.*wm_vect; one slinks out a winner.” https://www.tesla.com/sites/default/files/blog
22: fig1=figure(1); hold off attachments/the-slipperiest-car-on-the-road.pdf.
23: plot(wm_vect,Me,’k’,’LineWidth’,1.5); hold on [11] J. G. Hayes, R. P. R. de Oliveira, S. Vaughan, and M. G. Egan,
24: plot(wm_vect,P/1000,’k--’,’LineWidth’,1.5); “Simplified electric vehicle power train models and range estimation,”
25: xlabel(’$\omega_m$’,’FontSize’,15,’Interpreter’,’latex’) in 2011 IEEE Vehicle Power and Propulsion Conf., Sept 2011, pp. 1–5.
26: ylabel(’$M_e, P$’,’FontSize’,15,’Interpreter’,’latex’) [12] I. Boldea and S. Nasar, Electric Drives. Boca Raton, CRC Press, 1999.
27: leg1=legend(’$M_e$ Nm’,’$P$ kW’); [13] R. Krishnan, Electric Motor Drives. Modelling, Analysis and Control.
28: set(leg1,’FontSize’,15,’Interpreter’,’latex’);
29: set(gca,’FontSize’,12);
Upper Saddle River, Prentice Hall, 2001.
30: grid on [14] R. Klempka and B. Filipowicz, “Comparison of using the genetic algo-
31: %%------ Numerical methods rithm and cuckoo search for multicriteria optimisation with limitation,”
32: t_step=0.05; t_end=33; %interval and simulation time Turkish Journal of Elec. Eng. & Comp. Sci., vol. 25, no. 2, pp. 1300–
33: t=0:t_step:t_end; 1310, 2017.
34: wm=zeros(1,length(t))+0.01; %initial value of wm [15] G. Sieklucki, “Analysis of the transfer-function models of electric drives
35: v_ms=zeros(1,length(t))+0.0001; %initial value of v with voltage controlled source,” Przeglad Elektrotechniczny, vol. 88,
36: F_g=0*wm; % grade resistance no. 7A, pp. 250–255, 2012.
37: clear Me;
38: dM=18; %load torque in gear
39: for k=1:length(t)-1
40: Me(k)=(wm(k)<=wI)*Mmax...
+((wm(k)>wI)&&(wm(k)<=wII))*Pn/wm(k)...
+(wm(k)>wII)*m_k/(pb*wm(k)+w_2k)ˆ2;
41: F_T(k)=2/d_w*g_i*(Me(k)-dM);
42: F_r(k)=m2*g*0.01*(1+v_ms(k)*3.6/161); Grzegorz Sieklucki was born in Cracow, Poland, in 1972. He received the
43: F_DF(k)=1/2*rho*Cd*A*v_ms(k)ˆ2;
M.S. degree in automatics and robotics and Ph.D. in electrical engineering
44: a(k)=1/mass*(F_T(k)-F_r(k)-F_DF(k)-F_g(k));
45: if (wm(k)>wIII) a(k)=0; end from AGH University of Science and Technology, Cracow in 1997 and 2000,
46: v_ms(k+1)=v_ms(k)+t_step*a(k); % speed m/s respectively. He is currently with Department of Power Electronics and Energy
47: wm(k+1)=2*pi*g_i/(pi*d_w)*v_ms(k); Control Systems, AGH UST. His research interests include control theory and
48: end electrical drives.
50: Me(k+1)=Me(k); a(k+1)=a(k); %complement the vectors
51: F_T(k+1)=F_T(k);F_DF(k+1)=F_DF(k);F_r(k+1)=F_r(k);

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