Single-Phase AC Circuits
in a DC Circuit In a DC circuit, we deal with one type of power. P = I I W = t2 t 1 Pdt = P(t 2 t 1 ) = P t (J) DC CIRCUIT
in an AC Circuit Instantaneous : p(t) v(t)i(t) i(t)=i max cos( ω t θ i ) v(t)= max cos( ω t θ v ) AC CIRCUIT Trigonometric Identity 1 cosxcosy = [cos(x y) cos(x y)] 2 p(t) = maxi maxcos(ωtθ v)cos(ωtθ i ) p(t) = 1 2 maximaxcos(θv θ i) 1 2 maximaxcos(2ωtθv θ i)
in an AC Circuit p(t) = 1 2 maximaxcos(θv θ i) 1 2 maximaxcos(2ωt2θv (θv θ i)) = 1 2 maximaxcos(θv θ i) 1 2 maximaxcos(θv θ i) cos(2ωt2θ v) }{{}}{{} P (W) P (W) 1 2 maximaxsin(θv θ i) sin(2ωt2θ v) }{{} Q ( Ar) Trigonometric Identity cos(x y) = cosxcosy sinxsiny
p(t) = 1 2 maximaxcos(θv θ i) 1 2 maximaxcos(2ωt2θ i (θ v θ i )) = 1 2 maximaxcos(θv θ i) 1 2 maximaxcos(θv θ i) cos(2ωt2θ i ) }{{}}{{} P (W) P (W) 1 2 maximaxsin(θv θ i) sin(2ωt2θ i ) }{{} Q ( Ar) Trigonometric Identity cos(x y) = cosxcosy sinxsiny
Interpretation of Real or Active p(t) v(t)i(t) = P Scos(2ωt θ v θ i ) = P P cos(2ωt2θ v)qsin(2ωt2θ v) = P P cos(2ωt2θ i )Qsin(2ωt2θ i ) P avg = 1 T T 0 p(t)dt = P
Triangle P = 1 2 maxi max cos(θ v θ i ) Q = 1 2 maxi max sin(θ v θ i ) S = 1 2 maxi max S = P 2 Q 2 S φ=θ θ v P i Triangle Q
for Parallel Loads p 1 (t) = P 1 P 1 cos(2ωt 2θv) Q 1 sin(2ωt 2θv) p 2 (t) = P 2 P 2 cos(2ωt 2θv) Q 2 sin(2ωt 2θv) p(t) v(t)i(t) = v(t)[i 1 (t)i 2 (t)] = p 1 (t)p 2 (t) P,Q v(t)= max cos( ω t θ v ) P 1,Q 1 p(t) = (P 1 P 2 ) (P 1 P 2 ) cos(2ωt2θ v)(q 1 Q 2 ) sin(2ωt 2θ v) }{{}}{{}}{{} P P Q i(t) i 1 (t) P 2,Q 2 i 2 (t) Therefore, KPL: P = P 1 P 2 KQL: Q = Q 1 Q 2
for Series Loads p 1 (t) = P 1 P 1 cos(2ωt 2θ i ) Q 1 sin(2ωt 2θ i ) p 2 (t) = P 2 P 2 cos(2ωt 2θ i ) Q 2 sin(2ωt 2θ i ) p(t) v(t)i(t) = i(t)[v 1 (t)v 2 (t)] = p 1 (t)p 2 (t) P 1,Q 1 v (t) 1 v 2 (t) P 2,Q 2 P(t) = (P 1 P 2 ) (P 1 P 2 ) cos(2ωt2θ i )(Q 1 Q 2 ) cos(2ωt2θ i ) }{{}}{{}}{{} P P Q P,Q i(t)=i max cos( ω t θ i ) v(t) Therefore, KPL: P = P 1 P 2 KQL: Q = Q 1 Q 2
p(t) v(t)i(t) = P Scos(2ωt θ v θ i) = P P cos(2ωt 2θ v) Qsin(2ωt 2θ v) = P P cos(2ωt 2θ i) Qsin(2ωt 2θ i) i(t)=i max cos( ω t θ i ) v(t)= max cos( ω t θ v ) AC CIRCUIT
Definitions Real or Active : P = 1 2 maximaxcos(θv θ i) = rmsi rmscos(θ v θ i ) Reactive or Imaginary Q = 1 2 maximaxsin(θv θ i) = rmsi rmssin(θ v θ i ) Apparent S = 1 maximax = rmsirms 2
Factor Angle Factor φ = θ v θ i pf = cosφ Definitions: pf is said to be lagging if 0 φ 90 o (Inductive Load) pf is said to be leading if 90 o φ 0 (Capacitive Load)
Resistor v(t) = max cos(ωt θ v) i(t) = I max cos(ωt θ i) i(t) R Ohm s Law: v(t) = Ri(t) maxcos(ωtθ v) = RI maxcos(ωtθ i) { max = RI max θ v = θ i rms v(t) R φ = θ v θ i = 0 (v(t) and i(t) are in phase) P = 1 2 maximax = 1 2 RI2 max = 1 2 2 max R rms = rmsi rms = RI 2 rms = 2 rms R Q = 0 and S = 1 maximax = rmsirms = P 2
Inductor v(t) = max cos(ωt θ v) i(t) = I max cos(ωt θ i) i(t) L Ohm s Law: v(t) = L di dt max cos(ωt θ v) = ωli max sin(ωt θ i) = ωli max cos(ωt θ i 90 o ) = ωli max cos(ωt θ i 90 o ) max = ωli max θ v = θ i 90 o rms φ = θ v θ i = 90 o v(t) j ω L Q = 1 2 maximax = 1 2 ωli2 max = 1 2 = rmsi rms = ωli 2 rms = 2 rms ωl 2 max ωl rms P = cos90 o = 0 S = 1 maximax = rmsirms = Q 2
Capacitor v(t) = max cos(ωt θ v) i(t) = I max cos(ωt θ i) i(t) C Ohm s Law: i(t) = C dt I max cos(ωt θ i) = ωc max sin(ωt θ v) = ωc max cos(ωt θ v 90 o ) = ωc max cos(ωt θ v 90 o ) I max = ωc max θ i = θ v 90 o φ = θ v θ i = 90 o < 0 Q = 1 2 maximax = 1 2 ωc 2 max = 1 2 = rmsi rms = ωc 2 rms = I2 rms ωc P = S cos90 o = 0 S = 1 maximax = rmsirms = Q 2 I 2 max ωc rms v(t) 1 j ω C rms
a Resistor v(t) = max cos(ωt θ v) i(t) = I max cos(ωt θ i) Ohm s Law: v(t) = Ri(t) 2rms cos(ωt θ v) = 2RI rms cos(ωt θ i ) ω 2 rms sin(ωt θ v) = ω 2RI rms sin(ωt θ i ) ω 2j rms sin(ωt θ v) = ωj 2RI rms sin(ωt θ i ) rms [cos(ωt θ v) j sin(ωt θ v)] = RI rms [cos(ωt θ i ) j sin(ωt θ i )] rmse j(ωtθv) = RI rmse j(ωtθ i ) I R rmse jθv = RI rmse jθ i Ohm s Law: Ṽ rms = RĨrms
an Inductor v(t) = max cos(ωt θ v) i(t) = I max cos(ωt θ i) Ohm s Law: v(t) = L di dt v(t) = L di dt 2 cos(ωt θ v) = 2ωLI sin(ωt θ i) ωj 2 sin(ωt θ v) = jω 2 2LI cos(ωt θ i) I jx=j ω L [cos(ωt θ v) j sin(ωt θ v)] = jωli [cos(ωt θ i) j sin(ωt θ i)] e j(ωtθv) = jωlie j(ωtθ i ) e jθv Ṽ = jωlie jθ i = jωlĩ
a Capacitance v(t) = 2 cos(ωt θ v) i(t) = 2I cos(ωt θ i) Ohm s Law: v(t) = C dv dt 2I cos(ωt θi) = 2ωC sin(ωt θ v) I 1 1 jx= =j j ω C ω C = 2ωC cos(ωt θ v 90 o ) = 2ωC cos(ωt θ v 90 o ) Ie jθ i = jωc e jθ Ĩ = jωcṽ
AC Loads in Time and Phasor s i(t)= 2Icos( ω t θ i ) v(t)= 2cos( ω t θ v ) Time Circuit P(t) = v(t)i(t) = P S cos(2ωt θ v θ i ) = P P cos(2ωt 2θ v) Qsin(2ωt 2θ v) = P P cos(2ωt 2θ v) Qsin(2ωt 2θ i ) = I=I θ v θ i _ Z= 1 _ Y Phasor Circuit P = I cos(θ v θ i) Q = I sin(θ v θ i) S = I = p 2 Q 2
in Phasor I S = P jq S=PjQ = I cos(θ v θ i ) j I sin(θ v θ i ) = I [cos(θ v θ i ) j sin(θ v θ i )] = ( e jθv )(Ie jθ i) S = ( e jθv )(Ie jθ i) = Ṽ Ĩ = P jq S=I= P 2 Q 2 φ P Q Triangle φ = θ v θ i S = P 2 Q 2 P = S cosφ = I cosφ Q = S sinφ = I sinφ S P jq = Ṽ Ĩ = S φ tanφ = Q P
Impedance Triangle Z Ṽ Ĩ = θ v I θ i where: Z : R : X : = I (θ v θ i ) = Z φ RjX Impedance (Ω) Resistance (Ω) Reactance (Ω) I Z=RjX Z φ R X
Admittance Triangle I G Y=GjB Y φ B Ȳ = Ĩ Ṽ = I θ i = I θ v (θ i θ v ) = Y φ Ȳ = GjB Y : Admittance (S) G : Conductance (S) B : Susceptance (S)
in a Resistor Also, S = ṼĨ = RĨĨ = }{{} R Ĩ 2 j0 }{{} P R Q R S = ṼĨ = Ṽ { P = R Ĩ 2 = Ṽ 2 R Q = 0 ) (Ṽ = R Ṽ Ṽ R = Ṽ 2 R }{{} P R j0 }{{} Q R
in an Inductor P = 0 Q = X Ĩ 2 = Ṽ 2 X > 0 Q = ωl Ĩ 2 = Ṽ 2 ωl > 0
in a Capacitor P = 0 Q = X Ĩ 2 = Ṽ 2 X < 0 Q = Ĩ 2 ωc = ωc Ṽ 2 < 0 Hence, Q = B Ṽ 2 < 0
in a Series Load { S = ṼĨ { = (R s jx s )ĨĨ = R s }{{ Ĩ 2 jx s Ĩs 2 }}{{} P Q X s = ωl > 0 X s = 1 ωc < 0 if X s > 0 0 o φ 90 o if X s < 0 90 o φ o o Z = R s jx s = Z φ Ṽ = Ĩ = Ṽ Z ZĨ = Ṽ = Z Ĩ I R s jx s
RL Circuit I = I θ i z = RjωL = Z φ φ = tan 1 ωl R > 0 Ĩ = Ĩ θ i = Ṽ Z = Ṽ 0o Z φ Ĩ = Ĩ φ = Ĩ θ i R j ω L = 0 o θ = φ i I = I θ i 0 o φ = tan 1 ω L 90 o R
RC Circuit I = I θ i z = Rj 1 ωc = Z φ φ = tan 1 1 ωcr < 0 Ĩ = Ĩ θ i = Ṽ Z = Ṽ 0o Z φ Ĩ = Ĩ φ = Ĩ θ i R 1 j ωc I = I θ i θ = φ i = 0 o 90 o 1 φ = tan 1 0 ω CR
Algebra z = z 1 z 2 z = z 1 z 2 Proof: Application: z = z 1 z 2 z = z 1 z 2 z = ( z 1 θ 1 )( z 2 θ 2 ) z θ = z 1 z 2 (θ 1 θ 2 ) Z = R s jx s Z = R s X s Ṽ = ZĪ Ṽ = Z Ĩ
in a Parallel Load 1 Z = 1 1 = 1 j 1 R p jx p R p X p 1 Z = 1 j 1 R p X p ) (Ṽ S = ṼĨ = Ṽ = Ṽ 2 Z Z = ( 1 R p j 1 X p ) Ṽ 2 = Ṽ 2 R p }{{} P j Ṽ 2 X p }{{} Q I R p jx p
and Reactive Factors Factor: I S=PjQ pf = cosφ = P S = Reactive Factor: rf = sinφ = Q S = S=I= P 2 Q 2 φ P P P 2 Q 2 P P 2 Q 2 pf is said to be { lagging if rf = sinφ > 0 φ > 0 Q > 0 leading if rf = sinφ < 0 φ < 0 Q < 0 Q