EE201 Lecture 38 P 1 Complex Power and its Components In SSS the complex power absorbed by a twoterminal device is a complex number defined by S VeffIeff Ieff is the complex conjugate of Ieff Example Calculate complex power if i t I m cos t i v t Vm cos t v I m 2 I eff Vm 2Veff Average power Pave Veff I eff cos v i and EE201 Lecture 38 P 2 Complex power S V eff I eff Veff e j v I eff e j i S Veff I eff e j v i Re S Veff I eff cos v i Pave Characteristics of complex power S P Qj S S Veff I eff Complex form P 2 Q 2 Apparent Power Units VA Q reactive power Im S Units VAR Volt Amp Reactive EE201 Lecture 38 P 3 Calculation of power components for inductors iL t Suppose iL t 2 I L sin t vL t iL t I m cos t 90 I L jI L I L 90 VL ZLIL j LIL LIL VL 0o Complex power SL VLIL jVLIL PL jQL Average power PL 0 VLILcos 0o 90o Reactive power QL VLIL EE201 Lecture 38 Instantaneous power pL t pL t vL t iL t L 2ILcos t x 2 ILsin t pL t VLIL sin 2 t Stored energy WL t 1 WL t LiL2 t LI L2 sin 2 t 2 1 2 WL t LI L 1 cos 2 t 2 1 cos 2 t WL t LI L2 2 1 cos 2 t WL t VL I L 2 1 cos 2 t WL t QL 2 QL WL t MAX P 4 EE201 Lecture 38 P 5 Calculation of power components for capacitors iC t vC t Suppose vC t 2 Vcsin t vC t Vm cos t 90o VC jVC VC 90 IC YCVC j CVC CVC IC 0o Complex power SC VCIC jVCIC PC jQC Average power PC 0 VCICcos 90o 0o Reactive power QC VCIC EE201 Lecture 38 P 6 Instantaneous power pC t pC t vC t iC t pC t C 2Vccos t 2 Vcsin t pC t VCICsin 2 t Stored energy WC t 1 WC t Cvc2 t CVc2 sin 2 t 2 1 WC t CVc2 1 cos 2 t 2 1 cos 2 t WC t CVc2 2 1 cos 2 t WC t Vc I c 2 QC Wc t MAX EE201 Lecture 38 P 7 Example Consider the circuit in the figure where vin t 100 2 cos 2000 t V The quantity 100 2 is a maximum value Find the complex average reactive and apparent power absorbed by the load iin t Load vin t 100 10k 16nF Zin j Solution Step 1 Compute Zin j Zin j2000 100 1 10 4 j2000 16 10 9 5074 j5000 EE201 Lecture 38 P 8 Step 2 Compute Iin Vin Iin 10 j9 85 mA Zin Step 3 Compute complex power absorbed by load S VeffI eff 100 10 j9 85 10 3 1 j0 985 VA Step 4 Compute average reactive and apparent power Pav Re S 1 W Average Q Im S 0 985 var Reactive S 1 404 VA Apparent EE201 Lecture 38 P 9 Conservation principles General Principle of Conservation of Power In all circuits instantaneous power is conserved Sum of absorbed powers of all circuit elements is zero Principle of Conservation of Complex Power in AC circuits In AC circuits operating in steady state complex power is conserved Sum of all absorbed complex powers for all elements is zero Note conservation principle does not hold for apparent power EE201 Lecture 38 P 10 Example Calculate power delivered by the voltage source and the phasor current Iin if Iin 100V S1 S2 S5 S1 360 j160VA S3 S2 360 j120VA S4 S3 420 j540VA S4 130 j80VA S5 40 j100VA SS 1310 j560 VA 100 Iin Pave 1310 W Qs 560 VAR Ss 1425 VA Iin 13 1 j5 6 A EE201 Lecture 38 P 11 Example Consider a circuit which depicts a motor connected to a commercial power source The motor absorbs 50 kW of average power and 37 5 kvar of reactive power and has a terminal voltage Vm 230 V Find Is the complex power delivered by the source Ss and Vs Rline 0 5 Vs Is Vm Motor 50kW Solution Step 1 Find the apparent power Sm absorbed by the motor Since Sm Pm jQm 50 j37 5 kVA Sm 62 5 kVA EE201 Lecture 38 P 12 Step 2 Find Is Sm VmI s 230 Is Is 271 7 A Step 3 Compute the line loss Pline Rline Is 2 36 92 kW Step 4 Compute the complex power delivered by the source From the conservation of power Ss Sm Sline Sm Pline 50 j37 5 36 92 86 92 j37 5 kVA Step 5 Compute Vs Vs Ss I s Ss Is 348 4 V
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