Lecture 2 7 Instantaneous Power in AC Circuit

Lecture 2 -7 Instantaneous Power in AC Circuit

Fig. (3) The capacitive (reactive power) waveform.

From the previous analysis it is clear that there are two types of power in AC circuits: 1. A power transfers from the source to the load during all the time of the cycle. This power due to the existence of the resistive part and is called the real power. Hence, it is an oscillatory unidirectional part of the AC power. 2. The power that transfer alternatively from source to load and from load to source at each quarter cycle, hence it oscillate at double frequency of the applied voltage. This is called the imaginary power. Hence, it is a bidirectional part of the AC power.

Ex. 1: For the circuit shown below determine the instantaneous apparent (total) power, assume that the sinusoidal waveform current of frequency= (50 Hz) passing through the circuit and of an rms value= 2 A, by summation of the instantaneous power on each element. The following waveforms show each one of these powers as a function of time

Inst. Power in Resistance 45 40 35 Power (W) 30 25 20 P res =P real 15 10 5 0 0 2 4 6 8 10 Time (ms) 12 14 16 18 20

Inst. Power in Res. and Inductance 50 40 30 Power (W) 20 10 P res =P real Pind 0 0 5 10 -20 -30 Time (s) 15 20

Inst. Power in R, L, and C 80 60 Power (W) 40 20 P res =P real 0 0 2 4 6 8 10 14 16 18 20 Pind Pcap -20 -40 -60 -80 12 Time (s)

Determination of Inst. Reactive Power 80 60 Power (W) 40 20 P res =P real 0 0 2 4 6 8 10 -20 12 14 16 18 20 Pind Pcap Preac=Pimag -40 -60 -80 Time (s)

All Power Components 80 60 Power (W) 40 20 P res =P real Pind 0 0 2 4 6 8 10 12 14 16 18 20 Pcap Preac=Pimag -20 P appparent -40 -60 -80 Time (s)

The Main Three Types of Power 50 Instantaneous Power (W) 40 30 20 P real 10 P reactive P apparent 0 0 2 4 6 8 10 -20 -30 Time (ms) 12 14 16 18 20

The Power Triangle CCW 50 40 30 20 P real 10 P reactive P apparent 0 -10 -20 -30 CCW 0 5 10 15 20

t 0 0. 0005 0. 0015 0. 0025 0. 0035 0. 0045 0. 0055 0. 0065 0. 0075 0. 0085 0. 0095 0. 0105 0. 0115 0. 0125 0. 0135 0. 0145 0. 0155 0. 0165 0. 0175 0. 0185 0. 0195 0. 02 t (ms) 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 4. 5 5 5. 5 6 6. 5 7 7. 5 8 8. 5 9 9. 5 10 10. 5 11 11. 5 12 12. 5 13 13. 5 14 14. 5 15 15. 5 16 16. 5 17 17. 5 18 18. 5 19 19. 5 20 P res =P real 0 0. 978868 3. 819654 8. 244282 13. 81964 19. 99997 26. 18031 31. 75567 36. 18031 39. 02112 40 39. 02115 36. 18038 31. 75576 26. 18041 20. 00008 13. 81974 8. 244368 3. 819716 0. 978901 2. 82 E-10 0. 978835 3. 819591 8. 244196 13. 81954 19. 99987 26. 18021 31. 75559 36. 18025 39. 02108 40 39. 02118 36. 18044 31. 75585 26. 18051 20. 00019 13. 81984 8. 244454 3. 819779 0. 978934 1. 13 E-09 Pind 0 6. 1803311. 755716. 180319. 02112019. 021116. 180411. 75576. 18039 -5. 3 E-05 6. 180284 11. 75565 16. 1803 19. 02111 20 19. 02116 16. 18039 11. 75578 6. 180436 0. 000106 6. 1802311. 755616. 180319. 02112019. 021216. 180411. 75586. 180490. 000166. 180183 11. 75557 16. 18024 19. 02107 20 19. 02119 16. 18046 11. 75587 6. 180537 0. 000212 Pcap 0 18. 541 35. 26709 48. 54099 57. 06337 60 57. 06342 48. 54109 35. 26722 18. 54116 0. 000159 18. 540935. 26748. 540957. 06336057. 063548. 541235. 267318. 54130. 0003218. 5407 35. 26683 48. 5408 57. 06327 60 57. 06352 48. 54127 35. 26748 18. 54146 0. 000478 18. 540635. 266748. 540757. 06326057. 063648. 541435. 267618. 54160. 00064 - Preac=Pimag 0 6. 180335 11. 7557 16. 18033 19. 02112 20 19. 02114 16. 18036 11. 75574 6. 180385 5. 31 E-05 6. 1802811. 755716. 180319. 02112019. 021216. 180411. 75586. 180440. 000116. 180234 11. 75561 16. 18027 19. 02109 20 19. 02117 16. 18042 11. 75583 6. 180486 0. 000159 6. 1801811. 755616. 180219. 02112019. 021216. 180511. 75596. 180540. 00021 - P appparent 0 7. 159203 15. 57535 24. 42461 32. 84076 39. 99997 45. 20145 47. 93604 47. 93605 45. 2015 40. 00005 32. 84086 24. 42472 15. 57546 7. 159303 7. 96 E-05 5. 201427. 936075. 201530. 000117. 159069 15. 5752 24. 42446 32. 84063 39. 99987 45. 20138 47. 93601 47. 93608 45. 20157 40. 00016 32. 841 24. 42487 15. 57561 7. 159437 0. 000186 5. 201357. 936095. 20160. 00021 -