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ABB_11_E_02_druck

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2 51 2 General Electrotechnical Formulae 2.1 Electrotechnical symbols as per ISO 31 and IEC 60027 Table 2-1 Mathematical symbols for electrical quantities (general) Symbol Quantity Sl unit Q quantity of electricity, electric charge C E electric field strength V/m D electric flux density, electric displacement C/m 2 U electric potential difference V ϕ electric potential V ε permittivity, dielectric constant F/m ε o electric field constant, ε o = 0.885419 · 10 –11 F/m F/m ε r relative permittivit
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         2 51 2General Electrotechnical Formulae 2.1Electrotechnical symbols as per ISO 31 and IEC 60027 Table 2-1 Mathematical symbols for electrical quantities (general)SymbolQuantitySl unit Q quantity of electricity, electric chargeC E  electric field strengthV/m D electric flux density, electric displacementC/m 2 U  electric potential differenceV ϕ  electric potentialV ε  permittivity, dielectric constantF/m ε  o electric field constant, ε  o = 0.885419 · 10 –11 F/mF/m ε  r relative permittivity1 C electric capacitanceF I electric current intensityA   J  electric current densityA/m 2  x , γ  , σ  specific electric conductivityS/m ρ  specific electric resistance  Ω   · m G electric conductanceS R electric resistance  Ω θ  electromotive forceA  Table 2-2 Mathematical symbols for magnetic quantities (general)SymbolQuantity -Sl unit Φ  magnetic fluxWb B magnetic flux densityT H magnetic field strengthA/m V  magnetomotive forceA  µ  permeabilityH/m µ  o absolute permeability, µ  o = 4 π  · 10 –7 · H/mH/m µ  r relative permeability1 L inductanceH L mn mutual inductanceH  ABB_11_E_02 13.08.2007 8:17 Uhr Seite 51   Die ABB AG übernimmt keinerlei Verantwortung für eventuelle Fehler oder Unvollständigkeiten in diesem Dokument. Vervielfältigung - auch von Teilen - ist ohne vorherige schriftliche Zustimmung durch die ABB AG verboten. Copyright © 2007 by ABB AG, Mannheim  Alle Rechte vorbehalten.  52 Table 2-3 Mathematical symbols for alternating-current quantities and network quantitiesSymbolQuantitySl unit S apparent powerW, (VA) P active powerW Q reactive powerW, (var) ϕ  phase displacementrad ϑ load anglerad λ  power factor, λ  = P/S, λ  cos ϕ 1) 1 δ  loss anglerad d  loss factor, d = tan δ  1) 1  Z  impedance  Ω Y  admittanceS R resistance  Ω G conductanceS  X  reactance  Ω B susceptanceS γ  impedance angle, γ  = arctan  X/R rad Table 2-4 Numerical and proportional relationshipsSymbolQuantitySl unit η efficiency1  s slip1  p number of pole-pairs1 w, N  number of turns1  n tr  (t) transformation ratio1  m number of phases and conductors1  k  overvoltage factor1  n ordinal number of a periodic component1  g fundamental wave content1 d  harmonic content, distortion factor1  k   r  resistance factor due to skin effect,1 1)  Valid only for sinusoidal voltage and current. 2.2 Alternating-current quantities With an alternating current, the instantaneous value of the current changes itsdirection as a function of time i = f(t). If this process takes place periodically with aperiod of duration T, this is a periodic alternating current. If the variation of the currentwith respect to time is then sinusoidal, one speaks of a sinusoidal alternating current.  ABB_11_E_02 13.08.2007 8:17 Uhr Seite 52         2 53The frequency f  and the angular frequency ω  are calculated from the periodic time T  with12 π  f = – and ω  = 2 π  f  = — . TT  The equivalent d. c. value of an alternating current is the average, taken over oneperiod, of the value:This occurs in rectifier circuits and is indicated by a moving-coil instrument, for example.The root-mean-square value (rms value) of an alternating current is the square root ofthe average of the square of the value of the function with respect to time. As regards the generation of heat, the root-mean-square value of the current in aresistance achieves the same effect as a direct current of the same magnitude.The root-mean-square value can be measured not only with moving-coil instruments,but also with hot-wire instruments, thermal converters and electrostatic voltmeters. A non-sinusoidal current can be resolved into the fundamental oscillation with thefundamental frequency f  and into harmonics having whole-numbered multiples of thefundamental frequency. If I 1 is the rms value of the fundamental oscillation of analternating current, and I 2 , I 3 etc. are the rms values of the harmonics havingfrequencies 2 f  , 3 f  , etc., the rms value of the alternating current is I =     I 21 + I 22 + I 23 + …If the alternating current also includes a direct-current component  i  –, this is termed anundulatory current. The rms value of the undulatory current is I =     I 2– + I 21 + I 22 + I 23 + …The fundamental oscillation content  g is the ratio of the rms value of the fundamentaloscillation to the rms value of the alternating current I 1 g = –– . I The harmonic content d (distortion factor) is the ratio of the rms value of the harmonicsto the rms value of the alternating current.   I 22 + I 23 + … d  = —————— =   1 –  g 2 I The fundamental oscillation content and the harmonic content cannot exceed 1.In the case of a sinusoidal oscillation the fundamental oscillation content  g = 1,the harmonic content d  = 0. I ii  = 1d12d 2020 T tt  T2  ⋅ ∫  =  ⋅ ∫  π ω  π  . ⏐⏐   ⏐⏐ ⏐⏐ i T itit  T  = 1d12d 0 02  ∫  =  ∫  π ω  π  .  ABB_11_E_02 13.08.2007 8:17 Uhr Seite 53  54 Forms of power in an alternating-current circuit  The following terms and definitions are in accordance with DIN 40110 for thesinusoidal wave-forms of voltage and current in an alternating-current circuit. apparent power S = UI =      P 2 + Q 2 ,active power P = UI cos ϕ  = S cos ϕ  ,reactive power Q = UI sin ϕ  = S sin ϕ  , P power factor cos ϕ  = – SQ reactive factorsin ϕ  = – S When a three-phase system is loaded symmetrically, the apparent power is S = 3 U  1 I 1 =   3 UI 1 where I 1 is the rms phase current, U  1 the rms value of the phase to neutral voltage and U  the rms value of the phase to phase voltage. Also active power P = 3 U  1 I 1 cos ϕ    =   3 UI 1 cos ϕ  reactive power Q = 3 U  1 I 1 sin ϕ    =   3 UI 1 sin ϕ  The unit for all forms of power is the watt (W). The unit watt is also termed volt-ampere(symbol VA) when stating electric apparent power, and Var (symbol var) when statingelectric reactive power. Resistances and conductances in an alternating-current circuit US impedance  Z  = – = – =   R 2 +  X  2 II 2 U cos  ϕ  P resistance R = ——–— = — =  Z  cos ϕ  =    Z  2 –  X  2 II 2 U sin  ϕ  Q reactance  X  = ——–— = — =  Z  sin ϕ  =    Z  2 – R 2 II 2 inductive reactance  X  i = ω  L 1capacitive reactance  X  c = —– ω  CIS1 admittance Y  = – = — =  G 2 + B 2 = – UU  2  Z I cos ϕ  PR conductance G = ——— = — = Y  cos ϕ  =     Y  2 – B 2 = — UU  2  Z  2 I sin ϕ  QX  conductance B = ——— = — = Y  sin ϕ  =  Y  2 – G 2 = — UU  2  Z  2 1inductive susceptance B i = —– ω  L capacitive susceptance B c = ω  C  ABB_11_E_02 13.08.2007 8:17 Uhr Seite 54

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