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List of Symbols

A: magnetic vector potential (definition: B = ∇ × A or B = curl A or B = rot A), instantaneous value; A: modulus (Wb/m)

A: surface area, dA: vector of area element perpendicular to the surface (m²)

A₁: specific electric loading = linear current density, instantaneous value; A_m1: maximum value (A/m)

a₁, a₂: width of HV and LV transformer winding, respectively (m)

B: magnetic flux density vector, instantaneous value; B: modulus of vector; ${\underline{B}}_{m} = B_{m} e^{j ω t}$ ${\underline{B}}_{m} = B_{m} e^{j ω t}$ , where B_m = |B_m|e ^jψ: complex value; B*: conjugate value

B_m: maximum value in time,

B_avr,m: average value in space of maximum flux density in time;

B_t: tangential component; B_n: normal component; B_s: surface value;

B_sat: saturation flux density (T = Wb/m²)

b, a: linear dimensions (m)

C: capacitance of capacitor (F); constant

c: specific heat [J/(kg ⋅ ^oK)]

c_o: speed of light (m/s)

D: vector of electric flux density, electric displacement (C/m²);

D: internal stator diameter (m)

d: thickness of sheet or screen (m)

e: instantaneous value of electromotive force EMF (V); base of a natural logarithm (e = 2.718); e = 1.60219 × 10⁻¹⁹ C electron charge

E: vector of electric field intensity, instantaneous value; E_m: maximum value (V/m); other symbols as in case of B; E_ext: external (e.g., thermoelectric) field intensity

E_u: electromotive force EMF, rms value; E_um: maximum value (V); F: force vector (N)

F: current flow, magnetomotive force MMF, ampere-turns; F_m: maximum flow (A)

f: frequency (Hz)

H: vector of magnetic field intensity, instantaneous value; H_m: maximum value (A/m); Ĥ_m: maximum in time and space; other symbols as in case of B

h: linear dimension, height (m)

i: current, instantaneous value; I: rms value; I_m: maximum value (A); i = ∫ J ⋅ ds

i, j, k or 1_x, 1_y, 1_z: unit vectors in rectangular (Cartesian) coordinates

J: vector of electric current density, instantaneous value; J_m: maximum value (A/m²); other symbols as in case of B;

$j = \sqrt{- 1}$ $j = \sqrt{- 1}$ : imaginary (complex) unit, j² = –1

$k = \sqrt{\frac{ω μ σ}{2}}$ $k = \sqrt{\frac{ω μ σ}{2}}$ : attenuation constant of electromagnetic wave in metal (1/m)

l: length; dl: vector of length element (m)

L, M: self-inductance and mutual inductance, respectively (H)

M, m: coefficients of mirror images

n: rotational (angular) speed, n_s = (f/2p)(revolutions/s = 1/s); n_s = (3000/p)(rev/min)-synchronous speed

N: number of turns

P: active power (W); P_s = P + jQ: apparent power (V ⋅ A); P₁: value per unit surface (W/m²)

p: screening factor; number of poles pairs

Q: reactive power (var); Q₁: per unit surface, length or volume

Q: electric charge (C); q: volumetric electric charge density (C/m³); q_s: surface charge density (C/m²)

R: electric resistance (Ω)

R_m: reluctance (magnetic resistance), ${\underline{R}}_{m} = R_{m} + R_{mr}$ ${\underline{R}}_{m} = R_{m} + R_{mr}$ : complex reluctance (magnetic impedance) (1/H)

r: radius vector, r: radius (m); 1_r: unit vector in radial direction

S = E × H: Poynting’s vector; ${\underline{S}}_{s} = S_{p} + j S_{q}$ ${\underline{S}}_{s} = S_{p} + j S_{q}$ : vector modulus of complex Poynting’s vector; S_p: active component (W/m²); S_q: reactive component (var/m²)

s_i = is_xi + js_yi + ks_zi: vector in direction of wave propagation

T: period (s); thermodynamic temperature (K)

t: time (s); temperature (^oC)

u or v: voltage, instantaneous value; U or V: rms value; V_m: maximum value (V); V_m = ∫ E ⋅ dl

v: linear velocity vector (m/s)

W: energy (J); w: energy per unit volume (J/m³)

X: reactance; X_L: leakage reactance (Ω)

Z = R + jX: complex impedance; Z* = R – jX; $| Z | = \sqrt{R^{2} + X^{2}} (Ω)$ $| Z | = \sqrt{R^{2} + X^{2}} (Ω)$ ; Z₁, Z₂, Z₃: wave impedances of media 1, 2, and 3 (Z_air = 377 Ω ≫ |Z_Steel| ≈ 2.4 × 10⁻⁴ Ω ≫ |Z_Cu| = 2.7 × 10⁻⁶ Ω)

α = (1 + j)k: propagation constant in space, for solid metal (1/m); α* = (1 − j)k; $| α | = \sqrt{2} k = \sqrt{ω μ σ}$ $| α | = \sqrt{2} k = \sqrt{ω μ σ}$ ; α: thermal coefficient of resistivity (1/K)

α′: coefficient of heat dissipation (W/m²)

$Γ = \sqrt{j ω μ (σ + j ω ϵ)} = α ″ + j β ″$ $Γ = \sqrt{j ω μ (σ + j ω ϵ)} = α ″ + j β ″$ : propagation constant in space

α″: attenuation constant; β″: phase constant

$δ = (1 / k) = \sqrt{2 / ω μ σ}$ $δ = (1 / k) = \sqrt{2 / ω μ σ}$ : equivalent depth of field penetration into conducting half-space (m)

δ_air: interwinding insulation gap in transformer; δ′ = δ_air + (a₁ + a₂/3): equivalent leakage gap

ε = ε_r ε_o: dielectric permittivity; ε_r: relative permittivity

ε_o ≅ 8.854 × 10⁻¹² F/m: absolute dielectric permittivity = electric constant (permittivity of free space)

Θ = t − t_ambient: temperature rise (−^oK)

Λ: permeance: magnetic conductivity (H)

λ: length of wave (m); thermal conductivity [W/(m ⋅ ^oK)]

μ = μ_r μ_o: magnetic permeability; μ_r: relative permeability; μ_o = 4π × 10⁻⁷ H/m: absolute magnetic permeability = magnetic constant (permeability of free space)

ν = 1/μ: reluctivity (m/H)

ρ = 1/σ : resistivity = specific resistance (Ω/m)

σ = 1/ρ: conductivity = specific conductance (S/m)

ρ: volume density of space charge (C/m³)

ρ_m: mass density (kg/m³)

$τ = \frac{π D}{2 p}$ $τ = \frac{π D}{2 p}$ : pole pitch of electric machine (m)

Φ = ∫B ⋅ ds: magnetic flux, instantaneous value; Φ_m: maximum value (Wb)

Φ_L: leakage flux; φ: angle of phase shift (rad)

χ_m: magnetic susceptibility, μ_r + χ_m (pure number) = a measure of magnetic polarization

Ψ = ∫D ⋅ ds: electric flux (C)

ω = 2πf: pulsation, angular frequency (rad/s)

W = A ⋅ B = AB cos α = A_xB_x + A_yB_y + A_zB_z: scalar product; div A (or ≡ ∇ ⋅ A)

F = A × B = n AB sin α = i(A_yB_z − A_zB_y) + j(A_zB_x − A_xB_z) + k(A_xB_y − A_yB_x): vector product; curl A ≡ rot A ≡ ∇ × A

∇² A ≡ Lap A ≡ ΔA

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