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**Chapter 11 AC Power Analysis**

Chapter Objectives: Know the difference between instantaneous power and average power Learn the AC version of maximum power transfer theorem Learn about the concepts of effective or Rms value Learn about the complex power, apparent power and power factor Understand the principle of conservation of AC power Learn about power factor correction Huseyin Bilgekul Eeng224 Circuit Theory II Department of Electrical and Electronic Engineering Eastern Mediterranean University EEE 224

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**An Electical Power Distribution Center**

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**Apparent Power and Power Factor**

The Average Power depends on the Rms value of voltage and current and the phase angle between them. The Apparent Power is the product of the Rms value of voltage and current. It is measured in Volt amperes (VA). The Power Factor (pf) is the cosine of the phase difference between voltage and current. It is also the cosine of the angle of load impedance. The power factor may also be regarded as the ratio of the real power dissipated to the apparent power of the load.

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**Apparent Power and Power Factor**

Not all the apparent power is consumed if the circuit is partly reactive. Purely resistive load (R) θv– θi = 0, Pf = 1 P/S = 1, all power are consumed Purely reactive load (L or C) θv– θi = ±90o, pf = 0 P = 0, no real power consumption Resistive and reactive load (R and L/C) θv– θi > 0 θv– θi < 0 Lagging - inductive load Leading - capacitive load P/S < 1, Part of the apparent power is consumed

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**Power equipment are rated using their appparent power in KVA.**

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**Apparent Power and Power Factor**

Both have same P Apparent Powers and pf’s are different Generator of the second load is overloaded

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**Apparent Power and Power Factor**

Overloading of the generator of the second load is avoided by applying power factor correction.

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Complex Power The COMPLEX Power S contains all the information pertaining to the power absorbed by a given load.

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Complex Power The REAL Power is the only useful power delivered to the load. The REACTIVE Power represents the energy exchange between the source and reactive part of the load. It is being transferred back and forth between the load and the source The unit of Q is volt-ampere reactive (VAR)

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**Resistive Circuit and Real Power**

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**Inductive Circuit and Reactive Power**

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**Inductive Circuit and Reactive Power**

If the average power is zero, and the energy supplied is returned within one cycle, why is a reactive power of any significance? At every instant of time along the power curve that the curve is above the axis (positive), energy must be supplied to the inductor, even though it will be returned during the negative portion of the cycle. This power requirement during the positive portion of the cycle requires that the generating plant provide this energy during that interval, even though this power is not dissipated but simply “borrowed.” The increased power demand during these intervals is a cost factor that must that must be passed on to the industrial consumer. Most larger users of electrical energy pay for the apparent power demand rather than the watts dissipated since the volt-amperes used are sensitive to the reactive power requirement. The closer the power factor of an industrial consumer is to 1, the more efficient is the plant’s operation since it is limiting its use of “borrowed” power.

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**Capacitive Circuit and Reactive Power**

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Complex Power The COMPLEX Power contains all the information pertaining to the power absorbed by a given load. Real Power is the actual power dissipated by the load. Reactive Power is a measure of the energy exchange between source and reactive part of the load.

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Power Triangle The COMPLEX Power is represented by the POWER TRIANGLE similar to IMPEDANCE TRIANGLE. Power triangle has four items: P, Q, S and θ. a) Power Triangle b) Impedance Triangle Power Triangle

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Power Triangle Finding the total COMPLEX Power of the three loads.

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Power Triangle

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**Real and Reactive Power Formulation**

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**Real and Reactive Power Formulation**

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**Real and Reactive Power Formulation**

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**Real and Reactive Power Formulation**

P is the REAL AVERAGE POWER Q is the maximum value of the circulating power flowing back and forward

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**Real and Reactive Powers**

REAL POWER CIRCULATING POWER

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**Real and Reactive Powers**

Vrms =100 V Irms =1 A Apparent power = Vrms Irms =100 VA From p(t) curve, check that power flows from the supply into the load for the entire duration of the cycle! Also, the average power delivered to the load is 100 W. No Reactive power.

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**Real and Reactive Powers**

Power Flowing Back Vrms =100 V Irms =1 A Apparent power = Vrms Irms =100 VA From p(t) curve, power flows from the supply into the load for only a part of the cycle! For a portion of the cycle, power actually flows back to the source from the load! Also, the average power delivered to the load is 50 W! So, the useful power is less than in Case 1! There is reactive power in the circuit.

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Practice Problem 11.13: The 60 resistor absorbs 240 Watt of average power. Calculate V and the complex power of each branch. What is the total complex power?

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Practice Problem 11.13: The 60 resistor absorbs 240 Watt of average power. Calculate V and the complex power of each branch. What is the total complex power?

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**Practice Problem 11. 14: Two loads are connected in parallel**

Practice Problem 11.14: Two loads are connected in parallel. Load 1 has 2 kW, pf=0.75 leading and Load 2 has 4 kW, pf=0.95 lagging. Calculate the pf of two loads and the complex power supplied by the source. LOAD 1 2 kW Pf=0.75 Leading LOAD 2 4 kW Pf=0.95 Lagging

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**Conservation of AC Power**

The complex, real and reactive power of the sources equal the respective sum of the complex, real and reactive power of the individual loads. b) Loads in Series a) Loads in Parallel For parallel connection: Same results can be obtained for a series connection.

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**Complex power is Conserved**

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