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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

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**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)

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**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

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**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(2t) Qsin(2t) = AVERAGE POWER (watt) = REACTIVE POWER (var)

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p =

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p = Example for v-i = 45o

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**p = P + P cos(2t) Q sin(2t)**

P = average power Q = reactive power

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**p = P + P cos(2t) Q sin(2t)**

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

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

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

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

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**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

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**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

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**Apparent Power and Power Factor**

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**Apparent Power and Power Factor**

Irms = 5- 40o Vrms = 25010o 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

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

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Complex Power (VA) Where,

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Complex Power The complex power contains all information about the load Irms = 5- 40o Vrms = 25010o Load + Source VL We have seen before: Apparent power, S = VA Active power, P = watt Reactive power, Q = var With complex power, S = 25010o (5-40o) VA S = 1250 50o VA S var S = ( j957.56) VA |S| = S = Apparent power 50o = VA 803.5 watt

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

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**Complex Power Other useful forms of complex power We know that **

For a pure resistive element, For a pure reactive element,

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**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

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**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

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**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?

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**Maximum Average Power Transfer**

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

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**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

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**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

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