Symbols and Abbreviations

A magnetic vector potential A line current density Ar amplitude of the radial displacement a number of parallel current paths of the armature winding of a.c. motors;

number of pairs of parallel current paths of the armature winding of d.c. commutator motors

B vector magnetic flux density B magnetic flux density b instantaneous value of the magnetic flux density; width of slot bbr brush shift bfsk skew of PMs bp pole shoe width bsk skew of stator slots C number of commutator segments; capacitance; cost Cc cost of ferromagnetic core C0 cost of all other components independent of the shape of the machine CPM cost of PMs Csh cost of shaft Cw cost of winding c wave velocity; tooth width cCu cost of copper conductor per kg cE armature constant (EMF constant) cFe cost of ferromagnetic core per kg cPM cost of PMs per kg csteel cost of steel per kg cT torque constant D vector electric flux density D diameter dM external diameter of PM E EMF, rms value; Young modulus Ef EMF per phase induced by the rotor without armature reaction

i Er resultant reactive EMF of self-induction and mutual induction of a

short-circuited coil section during commutation e instantaneous EMF; eccentricity F force; MMF; energy functional; vector-optimization objective function Fexc MMF of the rotor excitation system Fa armature reaction MMF f frequency fc frequency of cogging torque fr natural frequency of the rth order F space and/or time distribution of the MMF G permeance GCD(s1, 2p) greatest common divisor of s1 and 2p g air gap (mechanical clearance) gMy air gap between PM and stator yoke in d.c. machines g

′ equivalent air gap

gi nonlinear inequality constraints H vector magnetic field intensity H magnetic field intensity h height hi equality constraints hM height of the PM I area moment of inertia; electric current; sound intensity Ia armature d.c. or rms current Iash armature current at zero speed (“short circuit” current) i instantaneous value of current or stepping motor current ia instantaneous value of armature current J vector electric current density J moment of inertia Ja current density in the armature winding Kr lumped stiffness k coefficient, general symbol k1R skin effect coefficient for armature conductors kC Carter’s coefficient kad reaction factor in d-axis; coefficient of additional losses in armature

core kaq reaction factor in q-axis kd1 distribution factor for fundamental space harmonic ν = 1 kE EMF constant kE = cEΦf kf form factor of the field excitation kf = Bmg1/Bmg kfault fault-tolerant rating factor kfskμ PM skew factor ki stacking factor of laminations kN coefficient depending on the number of manufactured machines ko1 slot opening factor for fundamental space harmonic ν = 1

ocf ocf maxout kp1 pitch factor for fundamental space harmonic ν = 1 ksat saturation factor of the magnetic circuit due to the main (linkage)

magnetic flux kskk stator slot skew factor referred to the slot (tooth) pitch t1 kskμ stator slot skew factor referred to the pole pitch τ kT torque constant kT = cTΦf kw1 winding factor kw1 = kd1kp1 for fundamental space harmonic ν = 1 L inductance; length LCM(s1, 2p) least common multiple of s1 and 2p Lc axial length of the interpole Li armature stack effective length Lw sound power level l1e length of the one-sided end connection lFe length of ferromagnetic yoke lM axial length of PM M magnetization vector M mutual inductance Mb ballistic coefficient of demagnetization Mr lumped mass m number of phases; mass ma amplitude modulation index mf frequency modulation index N number of turns per phase; number of machines Ncog number of poles-to-GCD(s1, 2p) ratio n rotational speed in rpm; independent variables ncog fundamental cogging torque index ne number of curvilinear squares between adjacent equipotential lines n0 no-load speed nΦ number of curvilinear squares between adjacent flux lines P active power; acoustic power; probability Pelm electromagnetic power ΔP active power losses Δp1/50 specific core loss in W/kg at 1T and 50 Hz p number of pole pairs; sound pressure pr radial force per unit area (magnetic pressure) Q electric charge, reactive power Qen enclosed electric charge R resistance Ra armature winding resistance of d.c. commutator motors R1 armature winding resistance of a.c. motors Rbr resistance of contact layer between brush and commutator Rc resistance of a coil section Re Reynolds number Rint interpole winding resistance

μM Rμg air gap reluctance Rμla external armature leakage reluctance r vibration mode S apparent power; surface SM cross section area of PM SM = wMLM or SM = bpLM s cross section area; displacement s1 number of stator teeth or slots; s2 number of rotor teeth or slots; se estimate of the variance σ T torque Tc cogging torque Td developed torque Tdsyn synchronous or synchronizing torque Tdrel reluctance torque T0 constant of avarage component of the torque Tp period Tr periodic component of the torque Tsh shaft torque (output or load torque) Tm mechanical time constant t time; slot pitch; relative torque tN normalized torque tr torque ripple V electric voltage; volume v instantaneous value of electric voltage; linear velocity vC commutator linear velocity W energy produced in outer space of PM; rate of change of the air gap

energy Wm stored magnetic energy w energy per volume, J/m3

wM width of PM X reactance Xad d-axis armature reaction (mutual) reactance Xaq q-axis armature reaction (mutual) reactance Xdamp reactance of damper Xsd d-axis synchronous reactance X

′ sd d-axis transient synchronous reactance

X ′′ sd d-axis subtransient synchronous reactance

Xsq q-axis synchronous reactance X

′ sq q-axis transient synchronous reactance

X ′′ sq q-axis subtransient synchronous reactance

Z impedance Z = R + jX; | Z |= Z = √R2 +X2 z1 number of teeth on wheel 1 z2 number of teeth on wheel 2

αd angle between d-axis and y-axis αi effective pole arc coefficient αi = bp/τ β overlap angle of pole χ magnetic susceptibility γ mechanical angle; gear ratio; form factor of demagnetization curve of

PM material γs step angle of a rotary stepping motor ΔVbr voltage drop across commutation brushes δ power (load) angle; bias error δi inner torque angle δl learning rate δ random error relative eccentricity η efficiency θ rotor angular position for brushless motors ϑ temperature; angle between Ia and Iad λ coefficient of leakage permeance (specific leakage permeance); intensity

of failures; wavelength μ number of the rotor μth harmonic μdyn dynamic viscosity μ0 magnetic permeability of free space μ0 = 0.4π × 10−6 H/m μr relative magnetic permeability μrec recoil magnetic permeability μrrec relative recoil permeability μrrec = μrec/μo ν number of the stator νth harmonic; relative speed ξ coefficient of utilization; reduced height of the armature conductor ρ specific mass density σ electric conductivity σf form factor to include the saturation effect σp output coefficient σr radiation factor τ pole pitch Φ magnetic flux Φf excitation magnetic flux Φl leakage flux φ power factor angle Ψ flux linkage Ψ = NΦ; angle between Ia and Ef ΨE total electric flux Ψsd total flux linkage in d-axis Ψsq total flux linkage in q-axis Ω angular speed Ω = 2πn ω angular frequency ω = 2πf

a armature av average br brush c commutation cog cogging Cu copper d direct axis; differential dyn dynamic e end connection; eddy-current elm electromagnetic eq equivalent exc excitation ext external Fe ferromagnetic f field fr friction g air gap h hysteresis in inner l leakage l, m, n labels of triangular element M magnet m peak value (amplitude) n, t normal and tangential components out output, outer q quadrature axis r rated; remanent r, θ, x cylindrical coordinate system rel reluctance rhe rheostat rot rotational s slot; synchronous; system sat saturation sh shaft sl sleeve; slot st starting str additional syn synchronous or synchronizing t teeth u useful vent ventilation wind windage y yoke

1 primary; stator; fundamental harmonic 2 secondary; rotor

Superscripts

inc incremental (sq) square wave

Abbreviations

A/D analog to digital ASM additional synchronous motor a.c. alternating current CAD computer-aided design CD compact disk CFRP carbon fiber reinforced polymer CLV constant linear velocity CPU central processor unit CSI current source inverter CVT continuously variable transmission DBO double-beam oscilloscope DFT discrete Fourier transform DSP digital signal processor d.c. direct current EIA Energy Information Administration (Dept. of Energy, U.S.A.) EMF electromotive force EMI electromagnetic interference EV electric vehicle EVOP evolutionary operation FDB fluid dynamic bearing FES flywheel energy storage FEM finite element method FFT fast Fourier transform GA genetic algorithm GCD greatest common divisor GFRP glass fiber reinforced polymer GS generator/starter GTO gate turn-off (thyristor) HDD hard disk drive HEV hybrid electric vehicle HVAC heating, ventilating and air conditioning IC integrated circuit IGBT insulated-gate bipolar transistor ISG integrated starter-generator

LDDCM liquid dielectric d.c. commutator motor LIGA litography (Litographie), electroforming (Galvanoformung) and

molding (Abformung) LVAD left ventricular assist device MEA more electric aircraft MEMS micro electromechanical systems MG motor/starter MMF magnetomotive force MRI magnetic resonance imaging MRSH multiple-restart stochastic hill climbing MTBF mean time between failures MTOE million tons of oil equivalent MTTF mean time to failure MVD magnetic voltage drop OECD Organisation for Economic Co-operation and Development PBIL population-based incremental learning PDF probability density function PFM pulse frequency modulation PLC programmable logic controller PM permanent magnet PSD power split device PWM pulse width modulation RESS rechargeable energy storage system RFI radio frequency interference RSM response surface methodology SA simulated annealing SCARA selective compliance assembly robot arm SRM switched reluctance motor SWL sound power level TDF tip driven fan TFM transverse flux motor TSM tested synchronous motor UPS uninterruptible power supply UV underwater vehicles URV underwater robotic vehicle VCM voice coil motor VSI voltage source inverter VSD variable-speed drive VVVF variable voltage variable frequency