 # AC POWER CALCULATION Instantaneous, average and reactive power

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

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)

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

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)

p =

p = Example for v-i = 45o

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

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

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

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

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

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

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

Apparent Power and Power Factor

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

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

Complex Power (VA) Where,

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

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

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

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

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

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?

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

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

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