1. Power and RMS Values

1. 1. Power and RMS Values

2. + − Circuit in a box, two wires + − Circuit in a box, three wires + − Instantaneous power p(t) flowing into the box Any wire can be the voltage reference Works for any circuit, as long as all N wires are accounted for. There must be (N – 1) voltage measurements, and (N – 1) current measurements.

3. Average value ofperiodic instantaneous power p(t)

4. Two-wire sinusoidal case zero average Power factor Average power

5. compare Root-mean squared value of a periodic waveform with period T Compare to the average power expression The average value of the squared voltage Apply v(t) to a resistor rms is based on a power concept, describing the equivalent voltage that will produce a given average power to a resistor

6. Root-mean squared value of a periodic waveform with period T For the sinusoidal case

7. Given single-phase v(t) and i(t) waveforms for a load • Determine their magnitudes and phase angles • Determine the average power • Determine the impedance of the load • Using a series RL or RC equivalent, determine the R and L or C

8. Determine voltage and current magnitudes and phase angles Using a cosine reference, Voltage cosine has peak = 100V, phase angle = -90º Current cosine has peak = 50A, phase angle = -135º Phasors

9. The average power is

10. Voltage – Current Relationships

11. Thanks to Charles Steinmetz, Steady-State AC Problems are Greatly Simplified with Phasor Analysis (no differential equations are needed) Time Domain Frequency Domain Resistor voltage leads current Inductor current leads voltage Capacitor

12. Problem 10.17

15. Complex power S Projection of S on the imaginary axis Q P Projection of S on the real axis is the power factor Active and Reactive Power Form a Power Triangle

16. Consider a node, with voltage (to any reference), and three currents IA IB IC Question: Why is there conservation of P and Q in a circuit? Answer: Because of KCL, power cannot simply vanish but must be accounted for

17. Voltage and Currentin phase Q = 0 Voltage leads Current by 90° Q > 0 Current leads Voltage by 90° Q < 0 Voltage and Current Phasors for R’s, L’s, C’s Resistor Inductor Capacitor

18. Complex power S Projection of S on the imaginary axis Q P Projection of S on the real axis

19. Resistor also so Use rms V, I ,

20. Inductor also so Use rms V, I ,

21. Capacitor also so , Use rms V, I

22. Active and Reactive Power for R’s, L’s, C’s (a positive value is consumed, a negative value is produced) Active Power P Reactive Power Q Resistor Inductor Capacitor source of reactive power

23. Now, demonstrate Excel spreadsheet EE411_Voltage_Current_Power.xls to show the relationship between v(t), i(t), p(t), P, and Q

24. A Single-Phase Power Example

25. 0.05 + j0.15 pu ohms PL + jQL PR + jQR /0 ° VR = 1.010 / - 1 0 ° VL = 1.020 IS IcapL IcapR j0.20 pu mhos j0.20 pu mhos A Transmission Line Example Calculate the P and Q flows (in per unit) for the loadflow situation shown below, and also check conservation of P and Q.

28. 0.05 + j0.15 pu ohms PL + jQL PR + jQR /0 ° VR = 1.010 / - 1 0 ° VL = 1.020 IS IcapL IcapR j0.20 pu mhos j0.20 pu mhos

29. V 0 0 < D < 1 DT T RMS of some common periodic waveforms Duty cycle controller By inspection, this is the average value of the squared waveform

30. RMS of common periodic waveforms, cont. Sawtooth V 0 T

31. V 0 V 0 V 0 V 0 V 0 V 0 0 -V RMS of common periodic waveforms, cont. Using the power concept, it is easy to reason that the following waveforms would all produce the same average power to a resistor, and thus their rms values are identical and equal to the previous example

32. 2. Three-Phase Circuits

33. a b c n Three-phase, four wire system Reference Three Important Properties of Three-Phase Balanced Systems • Because they form a balanced set, the a-b-c currents sum to zero. Thus, there is no return current through the neutral or ground, which reduces wiring losses. • A N-wire system needs (N – 1) meters. A three-phase, four-wire system needs three meters. A three-phase, three-wire system needs only two meters. • The instantaneous power is constant

34. Observe Constant Three-Phase P and Q in Excel spreadsheet 1_Single_Phase_Three_Phase_Instantaneous_Power.xls

40. 3 Z l ine c c I c 3Z 3Z load load a a b b Z l ine I 3Z a load – V + ab Z l ine I b Balanced three - phase systems, no matter if they are delta connected, wye connected, or a mix, are easy to solve if you follow these steps : Z l ine 1. Convert the entire circuit to an equivalent wye with a a a I a ground ed neutral . 2. Draw the one - line diagram for phase a , recognizing that phase a has one third of the P and Q . 3. Solve th e one - line diagram for line - to - neutral voltages and + line currents . The “One - Line” Z load Van 4. If needed, compute l ine - to - neutral voltages and line currents Diagram – for phases b and c using the ±120° relationships. 5. If needed, compute l ine - to - line voltages and delta currents n using the and ± 30 ° relationships. n

41. Now Work a Three-Phase Motor Power Factor Correction Example • A three-phase, 460V motor draws 5kW with a power factor of 0.80 lagging. Assuming that phasor voltage Van has phase angle zero, • Find phasor currents Ia and Iab and (note – Iab is inside • the motor delta windings) • Find the three phase motor Q and S • How much capacitive kVAr (three-phase) should be connected in • parallel with the motor to improve the net power factor to 0.95? • Assuming no change in supply voltage, what will be the new • after the kVArs are added?

42. Now Work a Delta-Wye Conversion Example

43. 3. Transformers

44. Φ jXs Rs Ideal Transformer jXm Rm 7200:240V Turns ratio 7200:240 (30 : 1) (but approx. same amount of copper in each winding) 7200V 240V Single-Phase Transformer

45. Isc + Vsc - Short circuit test: Short circuit the 240V-side, and raise the 7200V-side voltage to a few percent of 7200, until rated current flows. There is almost no core flux so the magnetizing terms are negligible. Calculate Φ Turns ratio 7200:240 (but approx. same amount of copper in each winding) Short Circuit Test jXs Rs Ideal Transformer jXm Rm 7200:240V 7200V 240V

46. Ioc + Voc - Open circuit test: Open circuit the 7200V-side, and apply 240V to the 240V-side. The winding currents are small, so the series terms are negligible. Calculate Φ Turns ratio 7200:240 (but approx. same amount of copper in each winding) Open Circuit Test jXs Rs Ideal Transformer jXm Rm 7200:240V 7200V 240V

47. jXs Rs Ideal Transformer jXm Rm 7200:240V 7200V 240V Ideal Transformer 7200:240V 7200V 240V Single-Phase TransformerImpedance Reflection by the Square of the Turns Ratio

48. Φ jXs Rs Ideal Transformer jXm Rm 7200:240V Turns ratio 7200:240 (30 : 1) (but approx. same amount of copper in each winding) 7200V 240V See Grady 2007, pp. 76-77 Now Work a Single-Phase Transformer Example Open circuit and short circuit tests are performed on a single - phase, 7200:240V, 25kVA, 60Hz distribution transformer. The results are: · Short circuit test (short circuit the low - voltage side, energize the high - voltage side so that rated current flows , an d measure P and Q ). Measure d P = 400W, Q = 200VAr . s c s c s c s c · Open circuit test (open circuit the high - voltage side, apply rated voltage to the low - voltage side , and measure P and Q ). Measure d P = 100W, Q = 250VAr . oc oc oc oc Determine the four impedance val ues (in ohms) for the transformer model shown.

49. Wye-Equivalent One-Line Model A N jXs Rs Ideal Transformer jXm Rm N1 : N2 • Values for one of the transformer windings, on side 1 • Can reflect to side 2 using either individual transformer turns ratio N1:N2, or three-phase bank line-to-line turns ratio (which are identical ratios) A three-phase transformer can be three separate single-phase transformers, or one large transformer with three sets of windings N1:N2 N1:N2 N1:N2 Y - Y

50. Wye-Equivalent One-Line Model A N Ideal Transformer • Converting side 1 impedances from delta to equivalent wye • Can reflect to side 2 using either individual transformer turns ratio N1:N2, or three-phase • bank line-to-line turns ratio (which are identical ratios) For Delta-Delta Connection Model, Convert the Transformer to Equivalent Wye-Wye N1:N2 N1:N2 N1:N2 Δ - Δ