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. Instantaneous, Average and Reactive Power
i(t)
Passive, linear network +
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(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 v i
positive p = power transferred from source to network
Instantaneous
Power (p)
negative p = power transferred from network to source

4. 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:
p =
Which can be written as
p = P + Pcos(2t)  Qsin(2t)
= AVERAGE POWER (watt)
= REACTIVE POWER (var)

5. p =

6. p =
Example for
v-i = 45o

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

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

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

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

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

17. Complex Power
(VA)
Where,

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

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

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

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. 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)
But

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

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

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

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