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AC CIRCUITS Universitatea Tehnica din Cluj-Napoca, Facultatea de Constructii de Masini 1/34 AC wave forms AC = A lternating C urrent peak peak-to- peak T f = 1/T frequency [Hz] ω = 2πf angular frequency [rad/s] Instantaneous value: i(t) = I m sin(ωt + φ)

# Electrotehnica si masini electrice

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Electrotehnica si masini electrice

### Text of Electrotehnica si masini electrice

• AC wave formsAC = Alternating Currentf = 1/T frequency [Hz] = 2f angular frequency [rad/s] Instantaneous value: i(t) = Im sin(t + )

• Sinus wave Average value: Iav = (1/T)i(t)dt = 0 Root Mean Squared value: I = (1/T)i2(t)dt = Im/2 = 0.707 Im Instantaneous value: i(t) = 2 I sin(t + )I = RMS value2I = peak value (amplitude)f = frequencyT = 1/f = period = 2f = angular frequency = phase u(t) = 325 sin(314t)

• Phase differencetThe two waveforms are /6 (30o) out-of phase.The current lags the voltage by angle .The voltage leads the current by angle .The current cross the horizontal reference axis reaching its peak and zero values after the voltage waveform. u(t) = Um sin(t)

i(t) = Im sin(t - )

= phase difference (phase shift)

• Complex quantities. Phasor diagramthe current phasor lags behind the voltage phasorU = Uej0 I =Ie-j/6A Phasor Diagram can be used to represent two or more stationary sinusoidal quantities: voltage, current, or some other alternating quantity of the same frequency. 2A sin(t + ) A = A ej (j = -1)ej = cos + j sinA = A (cos + j sin)A = A cos + j A sin

• Phasor additionu1(t) = 2U1 sin(314t + 1) u2(t) = 2U2 sin(314t + 2)

u(t) = u1(t) + u2(t)

u (t) = 2U sin(314t + ) U = U1 + U2U2 = U12+U22+2U1U2cos(1-2)tan = (U1sin1+ U2sin2)/(U1cos1+ U2cos2)u1(t) = 230 sin(314t + /3)u2(t) = 220 sin(314t)u(t) = u1(t) + u2(t) = 2U sin(314t + )U = ?, = ? U = 43.6 V, = 23.4

• Impedance, Phase angle, Resistance, Reactance ImpedanceZ = U/I []Phase angle = (u - i) [rad]i(t) = 2(U/Z) sin(t + u - )u(t) = 2100 sin(314t + /3)i(t) = 22 sin(314t + /6)Z = ? = ?Z,

• Impedance, Phase angle, Resistance, Reactance ResistanceR = Z cos []Reactance X = Z sin [] = (u - i) [rad]

Z = (R2 + X2)tan = X/Ru(t) = 2100 sin(314t + /3)i(t) = 22 sin(314t + /6)R = ?X = ?R, X

• Complex ImpedanceZ = Ueju/Ieji Z = (U/I)eju-i Z = ZejZ = Z(cos + j sin )Z = R + j XComplex ImpedanceZ = U/IZ

• Power in AC circuitsInstantaneous powerp(t) = u(t)i(t)p(t) = 2UI sin(t + u) sin(t + i)

• True powerP = (1/T)p(t)dtP = UI cos [W] = u - i Power in AC circuitsU = Z IP = Z I2 cosP = R I2Apparent powerS = U I [VA]U = Z IS = Z I2Power factorkp = P/S = cos Reactive powerQ = UI sin [VAr]U = Z IQ = Z I2 sinQ = X I2S = (P2 + Q2)S = U I* = Ueju Ie-ji = (UI)ej(u-i) = Sej = P + jQ

• Power in AC circuits. Applicationsu(t) = 2100 sin(314t + /3)i(t) = 22 sin(314t + /6)P = ?Q = ?S = ?

• AC Resistor with a sinusoidal supplyu(t) = 2U sin(t) (u= 0)i(t) = u(t)/R i(t) = 2 (U/R) sin(t)I = U/R Z = U/I = Ri = u = u - i = 0 R = Z cos = RX= Z sin = 0 Z = R + jX = R

• AC Resistor with a sinusoidal supplyP = RI2 > 0Q = XI2 = 0 S = ZI2 = RI2 = P (kP = 1)

• AC Inductor with a sinusoidal supplyu(t) = 2U sin(t + u)i(t) = 2I sin(t) ( i = 0)u + e = Ri = 0 u = - eu(t) = d/dt = Ldi/dtu(t) = 2LI cos(t)u(t) = 2 LI sin(t + /2)U = LI Z = U/I = L u = /2 = u - i = /2 R = Z cos = 0X= Z sin = L > 0 Z = R + jX = jL

• AC Inductor with a sinusoidal supplyP = RI2 = 0Q = XI2 = LI2 > 0 S = ZI2 = LI2

• AC Capacitor with a sinusoidal supplyu(t) = 2U sin(t) (u = 0)i(t) = 2I sin(t + i)i(t) = dq/dt = C du/dtu(t) = (1/C)i(t)dtu(t) = -2I(1/C) cos(t + i)u(t) = 2I(1/C) sin(t + i - /2)U = I/(C) Z = U/I = 1/(C) u = i -/2 = 0 i = /2 = u - i = -/2, (kP = 0) R = Z cos = 0X= Z sin = -1/(C) < 0 Z = R + jX = -j/(C)

• AC Capacitor with a sinusoidal supplyP = RI2 = 0Q = XI2 = -1/(C)I2 < 0 S = ZI2 = I2/(C)

• ApplicationsZC = 1/(2fC) = 3.18 k IC = 2fCU = 15.7 mA U = 50 Vf = 5 kHzZC = 1/(2fC) = 31.8 IC = 2fCU = 1.57 A ZL = 2fL = 691.15 IL = U/2fL = 72.3 mA U = 50 Vf = 5 kHzZL = 2fL = 69.115 k IL = U/2fL = 0.723 mA

• Parallel RLC circuitu(t) = 2U sin(t) (u = 0)i(t) = 2I sin(t + i)i(t) = iR(t) + iL(t) + iC(t)I = IR + IL + ICIR = U/ZR = U/RIL = U/ZL = U/jL = -jU/LIC = U/ZC = U/(-j/C) =jCU I = U[1/R - j(1/L - C)]1/Z = [1/R - j(1/L - C)]1/Z = [(1/R)2 + (1/L - C)2]tan = (1/L - C)/(1/R)

• Parallel RLC circuitu(t) = 250 sin(628t)R = 50 L = 20 mHC = 5 FZ = ?, IR = ?, IL = ?, IC = ? I = ?, = ?ZL = L = 6280.02 = 12.6 ZC = 1/C = 1/(628510-6)= 318.3 Z = 1/[(1/R)2 + (1/L - C)2] = 12.7 IR = U/R = 1 A, IL = U/ZL = 3.9 A, IC = U/ZC = 0.16 A, I = [IR2 + (IL IC)2] = 3.87 A, = arctg[(1/L - C)/(1/R)] = 75.3

• Parallel resonance circuitResonance : = 0tan = (1/L - C)/(1/R)1/L = C0 = 2f0 = 1/LCf0 = 1/ (2 LC)1/Z0 = (1/R)2Z0 = R = Zmax

• Parallel resonance circuitI0 = Imin = U/R

• Three phase wave formu1(t) = 2U sin(t)u2(t) = 2U sin(t 2/3) u3(t) = 2U sin(t 4/3)u2(t) = 2U sin(t + 4/3) u3(t) = 2U sin(t + 2/3)

• Three phase voltageU1 U2 = U1(1 a2)a2 = ej4/3 = = cos(4/3) + j sin(4/3)(1 - a2) = 3/2 + j3/2 = = 3(3/2 + j1/2) == 3[cos(/6) + j sin(/6)]U1 U2 = 3 U1 ej/6 U1 U2/6

• Three phase connectionsUph = Phase VoltageUl = Line VoltageIph = Phase CurrentIl = Line CurrentIl = IphUl UphIl Iph

• Unbalanced 3-phase load, STAR connectionU1, U2 = a2U1, U3 = aU1U1= U1 U0 U2 = U2 U0 U3 = U3 U0 I1 + I2 + I3 = I0I1 = U1/Z1 = (U1 U0)/Z1I2 = U2/Z2 = (U2 U0)/Z2I3 = U3/Z3 = (U3 U0)/Z3 (U1 U0)/Z1 + (U2 U0)/Z2 + + (U3 U0)/Z3 = U0/Z0 U0 = (U1/Z1 + U2/Z2 + U3/Z3)/(1/Z1 + 1/Z2+ 1/Z3+ 1/Z0)

• Unbalanced 3-phase load, STAR connectionN

• Balanced 3-phase load, STAR connectionZ1 = Z2 = Z3 = Zej U0 = [(U1 + U2 + U3)/Z]/(3/Z + 1/Z0) = 0I1 = U1/Z = U1/(Zej) = (U1/Z)e-jI2 = U2/Z = a2U1/(Zej) = a2I1I3 = U3/Z = aU1/(Zej) = aI1U1, U2 = a2U1, U3 = aU1I1, I2 = a2I1 I3 = aI1Ul =3 Uph

• Balanced 3-phase load, DELTA connectionZ12 = Z23 = Z31 = Zej U12, U23 = a2U12, U31 = aU12I12, I23 = a2I12 I31 = aI12Il =3 IphI12 = U12/Z = U12/(Zej) = (U12/Z)e-jI23 = U23/Z = a2U12/(Zej) = a2I12I31 = U31/Z = aU12/(Zej) = aI12

• Power in 3 - phase circuitsU1 = Uf, U2 = a2Uf, U3 = aUfI1f = U1/Z = Uf/(Zej) = = (Uf/Z)e-j = If e-j I*1f = If ej I2f = U2/Z = a2Uf/(Zej) = a2If e-jI*2f = aIf ejI3f = U3/Z = aUf/(Zej) = aIf e-jI*3f = a2If ejZ1 = Z2 = Z3 = Zej S = U1f I*1f + U2f I*2f + U3f I*3f

• Power in 3 - phase circuitsS = Uf If ej + a2Uf aIf ej + + aUf a2If ej = 3UfIf ej (a3 = e2 = 1)S = P + jQej = cos + j sinS = 3UfIf cos + j 3UfIf sinP = 3UfIf cosQ = 3UfIf sinP = 3UlIl cosQ = 3UlIl sinSTAR Uf = Ul/3, If = IlDELTAUf = Ul, If = Il/3

• Power in 3 - phase circuits. ApplicationZ1 = Z2 = Z3 = R = 100 PY = 3UlIl cos = = 3400(230/100)cos0 = = 1600 WZ1 = Z2 = Z3 = R = 100 P = 3UlIl cos = = 3400(3400/100)cos0 = = 4800 WU1 = U2 = U3 = 230 V U12 = U23 = U31 = 400 V

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