AC POWER CALCULATION Instantaneous, average and reactive power Apparent Power and Power Factor

1. AC POWER CALCULATION Instantaneous, average and reactive power Apparent Power and Power Factor Complex Power SEE 1023 Circuit Theory Dr. Nik Rumzi Nik Idris

2. i(t) Passive, linear network Instantaneous, Average and Reactive Power + v(t)  Instantaneous power absorbed by the network is, p =v(t).i(t) Let v(t) = Vm cos (t + v) and i(t) = Imcos(t + i) Which can be written as v(t) = Vm cos (t + v  i) and i(t) = Imcos(t)

3. v i v(t) = Vm cos (t + v  i) and i(t) = Imcos(t) p = Vm cos(t + v – i ) . Im cos(t) Example when v  i = 45o 45o positive p = power transferred from source to network Instantaneous Power (p) negative p = power transferred from network to source

4. p = = AVERAGE POWER (watt) = REACTIVE POWER (var) v(t) = Vm cos (t + v  i) and i(t) = Imcos(t) p = Vm cos(t + v – i ) . Im cos(t) Using trigonometry functions, it can be shown that: Which can be written as p = P + Pcos(2t)  Qsin(2t)

5. p =

6. p = Example for v-i = 45o

7. p = p = P + P cos(2t)  Q sin(2t) P = average power Q = reactive power

8. p = P + P cos(2t)  Q sin(2t) P = AVERAGE POWER • Useful power – also known as ACTIVE POWER • Converted to other useful form of energy – heat, light, sound, etc • Power charged by TNB Q = REACTIVE POWER • Power that is being transferred back and forth between load and source • Associated with L or C – energy storage element – no losses • Is not charged by TNB • Inductive load: Q positive, Capacitive load: Q negative

9. Voltage and current are in phase, p = P = average power = p = p = Power for a resistor Q = reactive power = 0

10. Voltage leads current by 90o, p = p = v i p = Q = reactive power = Power for an inductor P = average power = 0

11. Voltage lags current by 90o, p = p = v i p = Q = reactive power = Power for a capacitor P = average power = 0

12. = Apparent Power and Power Factor Consider v(t) = Vm cos (t + v) and i(t) = Imcos(t + i) We have seen, Is known as the APPARENT POWER VA

13. Apparent Power and Power Factor We can now write, is known as the POWER FACTOR The term For inductive load, (v  i) is positive  current lags voltage  lagging pf For capacitive load, (v  i) is negative  current leads voltage  leading pf

14. Apparent Power and Power Factor

15. Irms= 5- 40o + VL  Vrms= 25010o +  Load Source Apparent Power and Power Factor (lagging) Power factor of the load = cos (10-(-40)) = cos (50o) = 0.6428 Apparent power, S = 1250 VA Active power absorbed by the load is 250(5) cos (50o)= 1250(0.6428) = 803.5 watt Reactive power absorbed by load is 250(5) sin (50o)= 1250(0.6428) = 957.56 var

16. Complex Power Defined as: (VA)  Where, and and If we let (VA)

17. Complex Power (VA) Where,

18. Irms= 5- 40o + VL  Vrms= 25010o +  Load Source S 957.56 var 50o 803.5 watt Complex Power The complex power contains all information about the load We have seen before: Apparent power, S = 1250 VA Active power, P = 803.5 watt Reactive power, Q = 957.56 var With complex power, S = 25010o (5-40o) VA S = 1250 50o VA S = (803.5 + j957.56) VA |S| = S = Apparent power = 1250 VA

19. We know that Complex Power Other useful forms of complex power P Q

20. We know that  For a pure resistive element, For a pure reactive element, Complex Power Other useful forms of complex power

21. Conservation of AC Power Complex, real, and reactive powers of the sources equal the respective sums of the complex, real and reactive powers of the individual loads

22. But Conservation of AC Power Complex, real, and reactive powers of the sources equal the respective sums of the complex, real and reactive powers of the individual loads Ss = Ps +jQs = (P1 + P2 + P3) + j (Q1 + Q2 + Q3)

23. I ZTh + V  AC linear circuit VTh +  ZL Maximum Average Power Transfer Max power transfer in DC circuit can be applied to AC circuit analysis What is the value of ZL so that maximum average power is transferred to it?

24. I ZTh + V  VTh +  ZL Maximum Average Power Transfer What is the value of ZL so that maximum average power is transferred to it?

25. I ZTh + V  VTh +  ZL Maximum Average Power Transfer What is the value of ZL so that maximum average power is transferred to it? ZTh= RTh + jXTh ZL= RL + jXL and P max when

26. I ZTh + V  VTh +  ZL and P max when Maximum Average Power Transfer What is the value of ZL so that maximum average power is transferred to it? XL = XTh , RL= RTh