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Symbols and Constants Q F~ ~ E ~ D Φ V ~ H ~ B Φ ~ S Z P N charge (C) force (N) electric field (V/m) displacement field (C/m2 ) electric flux (C) electric potential (V) magnetic field (A/m) magnetic flux density (T) magnetic flux (Wb) surface vector (m2 ) impedance (Ω) power (W) number of turns (coil) ρV ρS ρL 0 R W I µ0 µR volume charge density (C/m3 ) surface charge density (C/m2 ) linear charge density (C/m) 8.854 × 10−12 F/m dielectric constant energy (J) current (A) 4π × 10−7 Wb/Am relative permeability Y T admittance (S) period (s) Mathematics spheres volume = (4/3)πa3 area = 4πa2 cylinders volume = πa2 L area: (cross section) πa2 , (side wall) 2πaL complex numbers ejθ = cos θ + j sin θ for z = α + jβ |z| = (α2 + β 2 )1/2 and 6 θ = tan−1 (β/α) Electric fields Coulomb’s law Q1 Q2 F~2 = R̂12 4π0 R2 point charge ~ = E electric flux ~ ·S ~ Φ=D linear dielectric ~ = 0 R E ~ D charge density Q = ρV × volume Guass’s law for a closed surface Φ = Qencl ~ to V relating E Q r̂ 4π0 r2 VAB = − ~ F~ = QE V = Q 4π0 r ~ (for flat surface and constant D) Z A B Q = ρS × area ~ ~ in the same direction as x) |E|dx (for E C= 0 R A (parallel plate cap.) d capacitance C = Q/V energy 1~ ~ ~ WE = E · D × volume (for constant E) 2 W = QV Magnetic fields long straight wire ~ = I φ̂ H 2πr solenoid ~ = nI ẑ H Faraday’s law emf = − self-inductance L = N Φ/I (from linked flux) mutual-inductance M = N2 Φ12 /I1 (from linked flux) linear materials ~ = µ0 µR H ~ B energy 1~ ~ ~ WH = H · B × volume (for constant H) 2 Lorentz force ~ F = q~v × B (n is the number of turns per meter) dΦ dt ~ force on a straight wire F~ = I~l × B sliding bar ~ V = l~v × B where Φ is magnetic flux L= 2WH (from energy) I2 ~ constant along wire) (l is the length; B ~ constant over loop) (l is the length and ~v the velocity; B Electric circuits dV (capacitor) dt I=C stored energy 1 1 W = CV 2 (capacitor) W = LI 2 (inductor) 2 2 dissipated power P = I 2 R = V 2 /R = IV RL and RC transients V (t) = V1 + V2 e−t/τ (constant source) V =L dI (inductor) V = IR (resistor) dt fundamental equations I(t) = I1 + I2 e−t/τ τ= 1 L or RC R Kirchoff’s laws sum of the voltage drops around a loop equals zero sum of currents into a node equals zero Electric AC circuits frequency ω = 2πf f= Ohm’s law V = IZ impedance Z= 1 (capacitor) jωC admittance Y = 1 Z combining impedance series Z = Z1 + Z2 1 T Z = R (resistor) Z = jωL (inductor) parallel Z = Z1−1 + Z2−1 −1 !1/2 1ZT 2 V (t) dt (T is the period) T 0 RMS VRMS = instantaneous power P = Re(V )Re(I) actual average power P̄ = phasor notation zejωt ⇒ |z|6 θ 2 VRMS cos θ where Z = |Z|ejθ |Z| power factor = cos θ Magnetic circuits coil mmf = N I material section mmf = Hl flux Φ = BA field to flux density obtain from the magnetization curve for a material B = µ0 µR H (µR = 1 for air) reluctance R= Ohm’s law mmf = ΦR (l is the length) (A is the cross-sectional area) l µ0 µR A (l is the length and A is the cross-sectional area)