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Symbols and Abbreviations Symbols and Abbreviations A magnetic vector potential A line current density; cross-section area a number of parallel current paths of the stator (armature) winding B vector magnetic flux density B magnetic flux density; damping of the system b instantaneous value of the magnetic flux density; width of slot bp pole shoe width Cf cost of frame Cins cost of insulation C0 cost of all other components independent of the shape of the machine CPM cost of PMs Crc cost of the rotor core Csh cost of shaft Cw cost of winding cCu cost of copper conductor per kg cE armature constant (EMF constant) cF e cost of ferromagnetic core per kg cins cost of insulation per kg cp specific heat at constant pressure cPM cost of PMs per kg csteel cost of steel per kg cv heat capacity D diameter; duty cycle of power semiconductor switches Din inner diameter of PMs equal to the inner diameter of stator bars Dout outer diameter of PMs equal to the outer diameter of stator bars E EMF (rms value); electric field intensity Ef EMF per phase induced by the rotor without armature reaction Ei internal EMF per phase e instantaneous EMF; eccentricity F force; number of winding sections F12 shape factor of two surfaces involved in radiation 328 Symbols and Abbreviations F space and/or time distribution of the MMF Fa armature reaction MMF Fexc MMF of the rotor excitation system f frequency; friction factor G permeance; gap ratio g/R g air gap (mechanical clearance); gravitational acceleration Gr Grashof number 0 g equivalent air gap H vector magnetic field intensity H magnetic field intensity h height; heat transfer coefficient hM height of the PM I electric current Ia stator (armature) current i instantaneous value of current; enthalpy J vector electric current density J moment of inertia Ja current density in the stator (armature) winding Kc current regulator gain Ki inverter gain k coefficient, general symbol; thermal conductivity k1R skin effect coefficient for the stator conductor resistance kad reaction factor in d-axis kaq reaction factor in q-axis kC Carter’s coefficient kd inner-to-outer diameter ratio kd = Din/Dout kd1 distribution factor for fundamental kE EMF constant kE = cEΦf kf form factor of the field excitation kf = Bmg1/Bmg ki stacking factor of laminations kocf overload capacity factor kocf = Tmax/Tshr kp1 pitch factor for fundamental ksat saturation factor of the magnetic circuit due to the main (linkage) magnetic flux kT torque constant kT = cT Φf kw1 winding factor kw1 = kd1kp1 for fundamental L inductance; length l1e length of the one-sided end connection Li armature stack effective length lM axial length of PM M mutual inductance Mo momentum m number of phases; mass m˙ mass flow rate ma amplitude modulation ratio Symbols and Abbreviations 329 mf frequency modulation ratio N number of turns per phase; number of machines Nu Nusselt number n rotational speed in rpm; independent variables n0 no-load speed P active power Pelm electromagnetic power Pin input power Pout output power P r Prandtl number ∆P active power losses ∆P1F e stator core losses ∆P1w stator winding losses ∆P2F e rotor core losses ∆Pe eddy current losses in stator conductors ∆Pfr friction losses ∆PPM losses in PMs ∆Prot rotational (mechanical) losses ∆Pwind windage losses ∆p specific core loss p number of pole pairs; pressure pr radial force per unit area ℘ wetted perimeter Q electric charge; reactive power; volumetric flow rate Q1 number of slots per pole Qc number of stator coils Qen enclosed electric charge q1 number of slots per pole per phase R radius; resistance R1 armature winding resistance of a.c. motors Rin inner radius of PMs equal to the inner radius of stator bars Rout outer radius of PMs equal to the outer radius of stator bars Re Reynolds number <µg air gap reluctance <µla external armature leakage reluctance <µM permanent magnet reluctance S apparent power; surface SM cross section area of PM; SM = wM LM or SM = bpLM s cross section area of stator conductor s1 number of stator slots equal to the number of stator teeth T torque Td electromagnetic torque developed by the machine Tdrel reluctance torque Tdsyn synchronous or synchronizing torque Tm mechanical time constant 330 Symbols and Abbreviations Tsh shaft torque (output or load torque) t time; slot pitch U internal energy u tangential velocity V electric voltage; volume v instantaneous value of electric voltage; 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; radial velocity wM width of PM X reactance X1 stator winding leakage reactance Xad d-axis armature reaction (mutual) reactance Xaq q-axis armature reaction (mutual) reactance Xsd d-axis synchronous reactance; Xsd = X1 + Xad Xsq q-axis synchronous reactance; Xsq =√X1 + Xaq Z impedance Z = R + jX; | Z |= Z = R2 + X2 z number of coils in a phase group α complex attenuation constant of electromagnetic field αi effective pole arc coefficient αi = bp/τ γ form factor of demagnetization curve of PM material δ power (load) angle δi inner torque angle eccentricity ε emissivity; surface spectral property η efficiency γ equivalent sand grain roughness θ rotor angular position for brushless motors θm slot pitch angle θre electrical angle corresponding to the coil layer width ϑ temperature; angle between Ia and Iad λ coefficient of leakage permeance (specific leakage permeance) λT turbulent parameter µ dynamic viscosity −6 µo magnetic permeability of free space µo = 0.4π × 10 H/m µr relative magnetic permeability µrec recoil magnetic permeability µrrec relative recoil permeability µrrec = µrec/µo ν number of the stator νth harmonic; kinematic viscosity ξ reduced height of the stator conductor ρ specific mass density σ electric conductivity; Stefan-Boltzmann constant σf form factor to include the saturation effect σp output coefficient Symbols and Abbreviations 331 σr radiation factor τ average pole pitch; thermal time constant Φ magnetic flux Φf excitation magnetic flux Φl leakage flux φ power factor angle Ψ flux linkage Ψ = NΦ; angle between Ia and Ef ψ flux linkage Ω angular speed Ω = 2πn ω angular frequency ω = 2πf Subscripts a armature (stator) avg average c conduction cv control volume Cu copper d direct axis; differential; developed e end connection; eddy-current elm electromagnetic eq equivalent exc excitation ext external F e ferromagnetic f field; forced fr friction; free g air gap h hydraulic; hysteresis in inner l leakage M magnet m peak value (amplitude) n, t normal and tangential components out output, outer q quadrature axis r rated; remanent; radiation; rotor r, θ, z cylindrical coordinate system rel reluctance rot rotational s slot; synchronous; system; stator sat saturation sh shaft st starting 332 Symbols and Abbreviations syn synchronous or synchronizing t teeth; total u useful v convection vent ventilation wind windage y yoke x, y, z cartesian coordinate system 1 stator; fundamental harmonic; inlet 2 rotor; exit Superscripts inc incremental (sq) square wave (tr) trapezoidal Abbreviations A/D analog to digital AFPM axial flux permanent magnet AIFI American Iron and Steel Industry a.c. alternating current AWG American Wire Gauge BPF band pass filtering BSCCO bismuth-strontium-calcium-copper-oxygen CAD computer-aided design CPU central processor unit DSP digital signal processor d.c. direct current EDM electro-discharge machining EMALS electro-magnetic aircraft launch system EMF electromotive force EMI electromagnetic interference EV electric vehicle FCEV fuel cell electric vehicle FDB fluid dynamic bearing FEM finite element method FPGA field programmable gate array FRP fiberglass reinforced plastic GCD greatest common divisor HDD hard disk drive HEV hybrid electric vehicle HTS high temperature superconducting Symbols and Abbreviations 333 IC integrated circuit IGBT insulated-gate bipolar transistor ISG integrated starter-generator LPF low pass filter LVAD left ventricular assist device MMF magnetomotive force MMT moving magnet technologies MOSFET metal oxide semiconductor (MOS) field effect transistor MTBF mean time between failures MVD magnetic voltage drop NdFeB neodymium iron boron PFM pulse frequency modulation PLD programmable logic device PM permanent magnet PWM pulse width modulation RFI radio frequency interference RFPM radial flux permanent magnet SEMA segmented electro-magnetic array SMC soft magnetic composite SmCo samarium cobalt SSC solid state converter YBCO yttrium-barium-copper oxide (YBa2Cu3O7−x) References 1. Abdel-Razek A.A., Coulomb J.L., Feliachi M., and Sabonnadiere J.C. (1981). The calculation of electromagnetic torque in saturated electric machines within combined numerical and analytical solutions of the field equations, IEEE Trans. MAG-17(6):3250–3252. 2. Abdel-Razek A.A., Coulomb J.L., Feliachi M., and Sabonnadiere J.C. (1982). Conception of an air-gap element for the dynamic analysis of the electromag- netic field in electric machines, IEEE Trans. MAG-18(2):655–659. 3. Acarnley P.P., Mecrow B.C., Burdess J.S., Fawcett J.N., Kelly J.G., and Dickinson P.G. (1996). Design principles for a flywheel energy store for road vehicles, IEEE Trans. IA-32(6):1402–1408. 4. Accucore, TSC Ferrite International, Wadsworth, IL, U.S.A., (2000), www.tscinternational.com 5. Afonin A.A., and Cierznewski P. (1999). Electronically commutated disc- type permanent magnet motors (in Russian), Int. Conf. on Unconventional Electromechanical and Electr Systems UEES’99. St Petersburg, Russia, pp. 271–276. 6. Afonin A.A., Kramarz W., and Cierzniewski P. (2000) Electromechanical En- ergy Converters with Electronic Commutation (in Polish). Szczecin: Wyd Ucz PS. 7. Ahmed A.B., and de Cachan L.E. (1994). Comparison of two multidisc con- figurations of PM synchronous machines using an elementary approach, Int. Conf. on Electr. Machines ICEM’94, vol. 1, Paris, France, pp. 175–180.
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