m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Electric car and motor design

Connect road loads to battery energy, inverter power, motor torque, regeneration and thermal management.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Battery and vehicle energy

Definitions & inputs. Eusable battery energy,e consumption per distance,η charging efficiency,Vpack voltage,Ah charge capacity.

  1. Volts times ampere-hours gives watt-hours, then apply the usable fraction.

    Enom≃VpackCAh,Eusable=fusableEnomE_{nom}\simeq V_{pack}C_{Ah},\quad E_{usable}=f_{usable}E_{nom}

  2. Divide available energy by a defined driving-cycle consumption.

    d≃Eusable/ed\simeq E_{usable}/e

  3. Distinguish grid energy from energy stored in the pack.

    Egrid=Estored/ηchargeE_{grid}=E_{stored}/\eta_{charge}

Interpretation. Range is not a fixed battery property; speed, weather, HVAC, tires and terrain change consumption.

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2. Motor torque and field interaction

Definitions & inputs. B flux density,I current,l conductor length,kt torque constant,ke back EMF constant,ω speed.

  1. Lorentz force on a perpendicular conductor is the elementary torque mechanism.

    F=BIlF=BIl

  2. Winding voltage pays for resistance, inductive change and motion-induced EMF.

    T=ktI,V=RI+L dI/dt+keωT=k_tI,\quad V=RI+L\,dI/dt+k_e\omega

  3. Separate shaft power from copper loss.

    Pmech=Tω,PCu=I2RP_{mech}=T\omega,\quad P_{Cu}=I^2R

Interpretation. Permanent-magnet synchronous, induction and reluctance machines have different field production and control; this equivalent model is not universal.

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3. Synchronous-machine dq model

Definitions & inputs. p pole pairs,ψf magnet flux linkage,id andiq currents,Ld andLq inductances,ωe electrical speed.

  1. Magnet and reluctance contributions combine in electromagnetic torque.

    T=32p[ψfiq+(Ld−Lq)idiq]T=\tfrac32p[\psi_fi_q+(L_d-L_q)i_di_q]

  2. Rotation couples the d-axis voltage to q-axis flux.

    vd=Rid+Ldi˙d−ωeLqiqv_d=Ri_d+L_d\dot i_d-\omega_eL_qi_q

  3. BackEMF and cross-coupling consume inverter voltage headroom.

    vq=Riq+Lqi˙q+ωe(Ldid+ψf)v_q=Ri_q+L_q\dot i_q+\omega_e(L_di_d+\psi_f)

Interpretation. Field weakening, current limits and voltage ellipse constrain high speed operation; control loop delay and rotor position accuracy matter.

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4. Regeneration and thermal limits

Definitions & inputs. K vehicle kinetic energy,ηregen net recovery fraction,Rth thermal resistance,Cth thermal capacitance.

  1. Recover only part of the reduction in kinetic energy, subject to power and traction limits.

    Erec≤ηregen12m(V12−V22)E_{rec}\le\eta_{regen}\tfrac12m(V_1^2-V_2^2)

  2. Stored heat and cooling determine component temperature.

    CthT˙=Ploss−(T−Tc)/RthC_{th}\dot T=P_{loss}-(T-T_c)/R_{th}

  3. Steady rise and transient timescale follow from the thermal ODE.

    ΔTsteady=PlossRth,τth=RthCth\Delta T_{steady}=P_{loss}R_{th},\quad\tau_{th}=R_{th}C_{th}

Interpretation. Inverter loss, iron loss, bearing loss, pack heat and coolant power are additional parts of vehicle efficiency.

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Graphical worked example

DC-equivalent linear motor with kt=0.5 N m/A; saturation and thermal limits excluded. X axis: Current (A). Y axis: Fixed-flux motor torque (N m).
DC-equivalent linear motor with kt=0.5 N m/A; saturation and thermal limits excluded. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Nominal pack energy

Definitions & inputs. V400V,capacity150Ah.

  1. Choose the governing model and isolate the requested quantity.

    E=VCE=VC

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    400(150)/1000400(150)/1000

  3. Evaluate the expression; the result uses the units shown.

    Result=60 kWh\mathrm{Result}=60\ \mathrm{kWh}

Interpretation. Nominal voltage approximation.

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Example 02. Usable energy

Definitions & inputs. Nominal60kWh,f.9.

  1. Choose the governing model and isolate the requested quantity.

    Eu=fEE_u=fE

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .9(60).9(60)

  3. Evaluate the expression; the result uses the units shown.

    Result=54 kWh\mathrm{Result}=54\ \mathrm{kWh}

Interpretation. Excludes reserve by definition.

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Example 03. Cycle range

Definitions & inputs. Eu54kWh,consumption.18kWh/km.

  1. Choose the governing model and isolate the requested quantity.

    d=Eu/ed=E_u/e

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    54/.1854/.18

  3. Evaluate the expression; the result uses the units shown.

    Result=300 km\mathrm{Result}=300\ \mathrm{km}

Interpretation. Defined cycle only.

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Example 04. Charging input

Definitions & inputs. Store54kWh,η.9.

  1. Choose the governing model and isolate the requested quantity.

    Eg=Es/ηE_g=E_s/\eta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    54/.954/.9

  3. Evaluate the expression; the result uses the units shown.

    Result=60 kWh\mathrm{Result}=60\ \mathrm{kWh}

Interpretation. Includes assumed charge conversion loss.

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Example 05. Ideal constant-power charge time

Definitions & inputs. Energy30kWh,power50kW.

  1. Choose the governing model and isolate the requested quantity.

    t=E/Pt=E/P

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    30/5030/50

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6 h\mathrm{Result}=0.6\ \mathrm h

Interpretation. Real charging typically tapers.

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Example 06. Pack current

Definitions & inputs. DCpower40kW,V400V.

  1. Choose the governing model and isolate the requested quantity.

    I=P/VI=P/V

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    40000/40040000/400

  3. Evaluate the expression; the result uses the units shown.

    Result=100 A\mathrm{Result}=100\ \mathrm A

Interpretation. Ignore voltage sag in this step.

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Example 07. Series cells

Definitions & inputs. Cell nominal3.7V,target approximately400V,108cells.

  1. Choose the governing model and isolate the requested quantity.

    V=NVcV=NV_c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    108(3.7)108(3.7)

  3. Evaluate the expression; the result uses the units shown.

    Result=399.6 V\mathrm{Result}=399.6\ \mathrm V

Interpretation. Integer cell count gives nominal pack voltage.

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Example 08. Parallel capacity

Definitions & inputs. Cell5Ah,30parallel.

  1. Choose the governing model and isolate the requested quantity.

    C=NCcC=NC_c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    30(5)30(5)

  3. Evaluate the expression; the result uses the units shown.

    Result=150 Ah\mathrm{Result}=150\ \mathrm{Ah}

Interpretation. Matched cells and appropriate balancing.

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Example 09. DC-equivalent torque

Definitions & inputs. kt.5Nm/A,I100A.

  1. Choose the governing model and isolate the requested quantity.

    T=ktIT=k_tI

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(100).5(100)

  3. Evaluate the expression; the result uses the units shown.

    Result=50 N m\mathrm{Result}=50\ \mathrm{N\,m}

Interpretation. Fixed flux and linear regime.

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Example 10. Back EMF

Definitions & inputs. ke.5Vs/rad,ω300rad/s.

  1. Choose the governing model and isolate the requested quantity.

    E=keωE=k_e\omega

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(300).5(300)

  3. Evaluate the expression; the result uses the units shown.

    Result=150 V\mathrm{Result}=150\ \mathrm V

Interpretation. Consistent SI constant.

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Example 11. Copper loss

Definitions & inputs. I100A,R.05Ω.

  1. Choose the governing model and isolate the requested quantity.

    P=I2RP=I^2R

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1002(.05)100^2(.05)

  3. Evaluate the expression; the result uses the units shown.

    Result=500 W\mathrm{Result}=500\ \mathrm W

Interpretation. Equivalent winding resistance convention.

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Example 12. Mechanical power

Definitions & inputs. T50Nm,ω300rad/s.

  1. Choose the governing model and isolate the requested quantity.

    P=TωP=T\omega

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    50(300)50(300)

  3. Evaluate the expression; the result uses the units shown.

    Result=15000 W\mathrm{Result}=15000\ \mathrm W

Interpretation. Shaft torque definition.

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Example 13. Motor efficiency

Definitions & inputs. Pout15kW,loss1kW.

  1. Choose the governing model and isolate the requested quantity.

    η=Pout/(Pout+Ploss)\eta=P_{out}/(P_{out}+P_{loss})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    15/1615/16

  3. Evaluate the expression; the result uses the units shown.

    Result=0.9375 \mathrm{Result}=0.9375\ {}

Interpretation. All relevant losses assumed included.

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Example 14. Electrical frequency

Definitions & inputs. p4pairs,shaft3000rpm.

  1. Choose the governing model and isolate the requested quantity.

    fe=pn/60f_e=pn/60

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4(3000)/604(3000)/60

  3. Evaluate the expression; the result uses the units shown.

    Result=200 Hz\mathrm{Result}=200\ \mathrm{Hz}

Interpretation. Synchronous machine.

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Example 15. Surface-PM torque

Definitions & inputs. p4,ψf.1Wb,iq100A,Ld=Lq.

  1. Choose the governing model and isolate the requested quantity.

    T=32pψfiqT=\tfrac32p\psi_fi_q

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.5(4)(.1)(100)1.5(4)(.1)(100)

  3. Evaluate the expression; the result uses the units shown.

    Result=60 N m\mathrm{Result}=60\ \mathrm{N\,m}

Interpretation. Specified peak dq convention.

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Example 16. Regenerated energy

Definitions & inputs. m1800kg,V20→0m/s,η.7.

  1. Choose the governing model and isolate the requested quantity.

    E=.7mV2/2E=.7mV^2/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .7(.5)(1800)(400).7(.5)(1800)(400)

  3. Evaluate the expression; the result uses the units shown.

    Result=252000 J\mathrm{Result}=252000\ \mathrm J

Interpretation. Brake power and battery acceptance ignored.

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Example 17. Recovered watt-hours

Definitions & inputs. E252000J.

  1. Choose the governing model and isolate the requested quantity.

    EWh=EJ/3600E_{Wh}=E_J/3600

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    252000/3600252000/3600

  3. Evaluate the expression; the result uses the units shown.

    Result=70 Wh\mathrm{Result}=70\ \mathrm{Wh}

Interpretation. Small compared with pack capacity.

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Example 18. Steady motor temperature rise

Definitions & inputs. Ploss1000W,Rth.03K/W.

  1. Choose the governing model and isolate the requested quantity.

    ΔT=PRth\Delta T=PR_{th}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1000(.03)1000(.03)

  3. Evaluate the expression; the result uses the units shown.

    Result=30 K\mathrm{Result}=30\ \mathrm K

Interpretation. Linear constant coolant temperature.

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Example 19. Thermal time constant

Definitions & inputs. Rth.03K/W,Cth10000J/K.

  1. Choose the governing model and isolate the requested quantity.

    τ=RthCth\tau=R_{th}C_{th}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .03(10000).03(10000)

  3. Evaluate the expression; the result uses the units shown.

    Result=300 s\mathrm{Result}=300\ \mathrm s

Interpretation. Single node approximation.

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Example 20. Road-to-pack power

Definitions & inputs. Wheel20kW,motorη.94,inverterη.97.

  1. Choose the governing model and isolate the requested quantity.

    Ppack=Pw/(ηmηi)P_{pack}=P_w/(\eta_m\eta_i)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    20/(.94(.97))20/(.94(.97))

  3. Evaluate the expression; the result uses the units shown.

    Result=21.93463 kW\mathrm{Result}=21.93463\ \mathrm{kW}

Interpretation. Gear and accessory losses not included.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.