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E E 2415 Lecture 9 Phasor Circuit Analysis, Effective Value and Complex Power: Watts, VAR’s and Volt-Amperes Effective Value of a Sinusoid (1/3) i(t) Average Power: 10 Effective Value of a Sinusoid (2/3) Effective Value of a Sinusoid (3/3) In our example: Also: The effective value is also called the Root Mean Square value or rms value. R-C Circuit Example (1/6) 40 vs(t) + vR - + i vc - 88.42 F Capacitive Reactance R-C Circuit Example (2/6) 40 120 ~ + VR - + ~ ~ I VC - Using rms phasor for voltage source. -j30 R-C Circuit Example (3/6) Calculate Real Power: And Reactive Power: Apparent power is the product of voltage and current of the source. Also: R-C Circuit Example (4/6) Power Factor is the ratio of real power to apparent power: Power Factor is also the Cosine of the angle between the load voltage and the load current: If the load current leads the load voltage, the power factor is leading; if it lags the load voltage, the power factor is lagging. R-C Circuit Example (5/6) 2.4A ~ I 36.87o 120V ~ V Phasor Diagram of Voltage and Current Current leads voltage. R-C Circuit Example (6/6) imag 230.4 W 288.0 VA real -172.8 VAR The Power Triangle showing leading power factor. Calculating Complex Power (1/2) ~ I jX R ~ V + S P jQ I ~ I c jd 2 - R jX I c jd , 2 I I I I c jd * then I I c d I S P jQ I I * * * 2 R jX 2 2 Calculating Complex Power (2/2) V R jX I S P jQ V I * From now on, we use the above method to calculate complex power. Lagging Power Factor Example (1/4) 5 v(t) i 22.97 mH Lagging Power Factor Example (2/4) 5 1200° ~ I j8.66 Calculate complex power directly: Lagging Power Factor Example (3/4) Power Factor: Power Factor is Lagging imag 1200° V 60o real 12-60° A Phasor Diagram of Voltage and Current Lagging Power Factor Example (4/4) 14 40 VA imag 1247 VAR's real 720 W Power Triangle for lagging Power Factor Phasor Power Example (1/4) 0.5 ~ I j2 + 40 ~ V 480 V(rms) -j150 j30 - (40 j30)( j150) Zp (56.25 j18.75) 40 j 30 j150 4800 V I 7.94 20.08 A (56.75 j 20.75) Phasor Power Example (2/4) V I (56.25 j18.75) 471 1.65 S RLC V I * (471V 1.65)(7.94 A20.08 ) 3.55 kW j1.18 kVAR PF 3.55 0.949 or 94.9% Lagging 3.74 Phasor Power Example (3/4) I RL 471V 1.65 9.42 A 38.52 (40 j30) * S RL V I RL (471V 1.65)(9.42 A38.52) 3.55 kW j 2.66 kVAR Capacitor VAR’s: 4712 V 2 Qc 1.48 kVAR 150 Phasor Power Example (4/4) 2.66 kVAR 1.18 kVAR Q 44 . 4 A V k }1.48 kVAR VA k 3.74 3.55 kW P Impedance and Admittance Z R jX Impedance 1 Y G jB Admittance Z R Conductance G 2 R X2 X B 2 R X2 Susceptance