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

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2 Instantaneous, Average and Reactive Power + v(t) i(t) Passive, linear network Instantaneous power absorbed by the network is, p =v(t).i(t) Let v(t) = V m cos ( t + v ) and i(t) = I m cos( t + i ) Which can be written as v(t) = V m cos ( t + v i ) and i(t) = I m cos( t)

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3 p = V m cos( t + v – i ). I m cos( t) v i Instantaneous Power (p) Example when v i = 45 o positive p positive p = power transferred from source to network negative p negative p = power transferred from network to source 45 o

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4 v(t) = V m cos ( t + v i ) and i(t) = I m cos( t) p = V m cos( t + v – i ). I m cos( t) p = p = P + Pcos(2 t) Qsin(2 t) Using trigonometry functions, it can be shown that: = AVERAGE POWER (watt) = REACTIVE POWER (var) Which can be written as

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5 p =

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6 Example for v - i = 45 o

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

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8 P = AVERAGE POWER Q = REACTIVE POWER p = P + P cos(2 t) Q sin(2 t) Useful power – also known as ACTIVE POWER Converted to other useful form of energy – heat, light, sound, etc Power charged by TNB 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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9 Power for a resistor p = Voltage and current are in phase, p = P = average power = Q = reactive power = 0

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

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

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12 Apparent Power and Power Factor Consider v(t) = V m cos ( t + v ) and i(t) = I m cos( t + i ) We have seen, = APPARENT POWER Is known as the APPARENT POWER VA

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

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14 Apparent Power and Power Factor

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

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

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17 Complex Power (VA) Where,

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

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19 Complex Power Other useful forms of complex power We know thatPQ

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20 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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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

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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 S s = P s +jQ s = (P 1 + P 2 + P 3 ) + j (Q 1 + Q 2 + Q 3 ) But

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23 Maximum Average Power Transfer Max power transfer in DC circuit can be applied to AC circuit analysis ZLZL + V I Z Th V Th + AC linear circuit average What is the value of Z L so that maximum average power is transferred to it?

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24 Maximum Average Power Transfer ZLZL + V I Z Th V Th + average What is the value of Z L so that maximum average power is transferred to it?

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25 Maximum Average Power Transfer ZLZL + V I Z Th V Th + average What is the value of Z L so that maximum average power is transferred to it? Z Th = R Th + jX Th Z L = R L + jX L P max when and

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26 Maximum Average Power Transfer ZLZL + V I Z Th V Th + average What is the value of Z L so that maximum average power is transferred to it? P max when and X L = X Th, R L = R Th

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