Measured Values

AC currents and voltages

Frequency

f = 50 Hz ; 60 Hz ; 16 2/3 Hz (trains)

Complex Values (Phasors) ⇔ Sine Curves

U = U · e ^{j φ_u} ⇔ u ( t ) = √2 · U · cos ( ω t + φ_u ) = √2 · Re{ U · e ^{j ω t} }

I = I · e ^{j φ_i} ⇔ i ( t ) = √2 · I · cos ( ω t + φ_i ) = √2 · Re{ I · e ^{j ω t} }

U , I : effective values φ_u , φ_i : phase angles ω = 2 π f : angular frequency

with e ^{j α} · e ^{j β} = e ^{j ( α + β )} ; e ^{j φ} = cos φ + j sin φ

complex phasor U = U · e ^{j φ_u} ⇔ u ( t ) = √2 · U · cos ( ω t + φ_u ) Browser unterstützt kein Java

Phase angle φ_(L) between voltage phasor U and current phasor I

Load Impedance

Z_L = U / I = U / I · e ^{j ( φ_u - φ_i )} = Z_L · e ^{j φ_L} = R + j X

with e ^{j α} / e ^{j β} = e ^{j ( α − β )}

φ_L = φ_u - φ_i ; cos φ_L : power factor

pure resistive :	φ_i = φ_u	=> φ_L = 0	(current in phase with voltage)
inductive :	φ_i < φ_u	=> φ_L > 0	(lagging current)
capacitive :	φ_i > φ_u	=> φ_L < 0	(leading current)

Power (single phase)

S = U · I^* = U e ^{j φ_u} · I e ^{-j φ_i} = U · I e ^{j ( φ_u - φ_i )} = S e ^{j φ_L} = P + j Q

with ( e ^{j φ} )^* = e ^{-j φ} ; e ^{j α} · e ^{-j β} = e ^{j ( α − β )}

P = S · cos φ_L = U · I · cos ( φ_u - φ_i ) : active power ( kW, MW)

Q = S · sin φ_L = U · I · sin ( φ_u - φ_i ) : reactive power ( kvar, Mvar)

S = √( P² + Q² ) : apparent power ( kVA, MVA)

cos φ_L = P / S : power factor (ind. or cap.)

p ( t ) = u ( t ) · i ( t ) = 2 · U · I · cos ( ω t + φ_u ) · cos ( ω t + φ_i )

   = U · I · cos ( φ_u - φ_i ) + U · I · cos ( 2 ω t + φ_u + φ_i )

   = P + S · cos ( 2 ω t + φ_u + φ_i )

   with 2 · cos α · cos β = cos ( α − β ) + cos ( α + β )

Demonstration

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three-phase systems

- power (three-phase systems)

S = U₁· I₁^* + U₂· I₂^* + U₃· I₃^*

- symmetrical voltages

U₂ = U₁ e ^{-j 120 °} ; U₃ = U₁ e ^{-j 240 °} : phase to ground voltages

U₁₂ = U₁ − U₂ = √ 3 U₁ e ^{j 30 °} ;

U₂₃ = U₂ − U₃ ; U₃₁ = U₃ − U₁ : phase to phase voltages

- symmetrical currents

I₂ = I₁ e ^{-j 120 °} ; I₃ = I₁ e ^{-j 240 °} : phase currents

- power (symmetrical)

S = 3 · U₁· I₁^* ; U₁ : phase to ground voltage

- rated (nominal) voltage of voltage level and three-phase equipment

e.g. U_N = 400 kV : phase to phase voltage

U₁ = U_N / √ 3 ≈ 231 kV : phase to ground voltage

U_{phase to phase} = √ 3 · U_{phase to ground}

- rated power (rated data of three-phase equipment)

S_N = √ 3 · U_N · I_N : total three-phase apparent power

U_N : phase to phase voltage ; I_N : phase current (of one phase)

P_N = √ 3 · U_N · I_N · cos φ_N : total three-phase active power

Q_N = √ 3 · U_N · I_N · sin φ_N : total three-phase reactive power