Chapter 7 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

Instantenous AC Power Instantenous Power p(t) is the power at any instant of time.

Instantenous AC Power Instantenous Power p(t) is the power at any instant of time. The instantaneous power is composed of two parts. A constant part. The part which is a function of time.

Trigonometri identity

Instantenous and Average Power The instantaneous power p(t) is composed of a constant part (DC) and a time dependent part having frequency 2ω. Instantenous Power p(t)

Instantenous and Average Power

Average Power The average power P is the average of the instantaneous power over one period .

Average Power A resistor has (θv-θi)=0º so the average power becomes: The average power P, is the average of the instantaneous power over one period . A resistor has (θv-θi)=0º so the average power becomes: P is not time dependent. When θv = θi , it is a purely resistive load case. When θv– θi = ±90o, it is a purely reactive load case. P = 0 means that the circuit absorbs no average power.

Instantenous and Average Power Example 1 Calculate the instantaneous power and average power absorbed by a passive linear network if:

Average Power Problem Practice Problem 11.4: Calculate the average power absorbed by each of the five elements in the circuit given.

Average Power Problem

Maximum Average Power Transfer Finding the maximum average power which can be transferred from a linear circuit to a Load connected. a) Circuit with a load b) Thevenin Equivalent circuit Represent the circuit to the left of the load by its Thevenin equiv. Load ZL represents any element that is absorbing the power generated by the circuit. Find the load ZL that will absorb the Maximum Average Power from the circuit to which it is connected.

Maximum Average Power Transfer Condition Write the expression for average power associated with ZL: P(ZL). ZTh = RTh + jXTh ZL = RL + jXL

Maximum Average Power Transfer Condition Therefore: ZL = RTh - XTh = ZTh will generate the maximum power transfer. Maximum power Pmax For Maximum average power transfer to a load impedance ZL we must choose ZL as the complex conjugate of the Thevenin impedance ZTh.

*Maximum Average Power Transfer Practice Problem 11.5: Calculate the load impedance for maximum power transfer and the maximum average power.

Maximum Average Power Transfer

Maximum Average Power for Resistive Load When the load is PURELY RESISTIVE, the condition for maximum power transfer is: Now the maximum power can not be obtained from the Pmax formula given before. Maximum power can be calculated by finding the power of RL when XL=0. ● RESISTIVE LOAD ●

Maximum Average Power for Resistive Load Practice Problem 11.6: Calculate the resistive load needed for maximum power transfer and the maximum average power.

Maximum Average Power for Resistive Load RL Notice the way that the maximum power is calculated using the Thevenin Equivalent circuit.

a) AC circuit Effective or RMS Value The EFFECTIVE Value or the Root Mean Square (RMS) value of a periodic current is the DC value that delivers the same average power to a resistor as the periodic current. a) AC circuit b) DC circuit

Effective or RMS Value of a Sinusoidal The Root Mean Square (RMS) value of a sinusoidal voltage or current is equal to the maximum value divided by square root of 2. The average power for resistive loads using the (RMS) value is:

Effective or RMS Value Practice Problem 11.7: Find the RMS value of the current waveform. Calculate the average power if the current is applied to a 9  resistor. 4t 8-4t

An Electical Power Distribution Center

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.

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

Power equipment are rated using their appparent power in KVA.

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

Apparent Power and Power Factor Overloading of the generator of the second load is avoided by applying power factor correction.

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

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)

Resistive Circuit and Real Power

Inductive Circuit and Reactive Power

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.

Capacitive Circuit and Reactive Power

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.

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

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

Power Triangle

Real and Reactive Power Formulation

Real and Reactive Power Formulation

Real and Reactive Power Formulation

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

Real and Reactive Powers REAL POWER CIRCULATING POWER

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.

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.

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?

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?

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

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.

Complex power is Conserved

Power Factor Correction The design of any power transmission system is very sensitive to the magnitude of the current in the lines as determined by the applied loads. Increased currents result in increased power losses (by a squared factor since P = I2R) in the transmission lines due to the resistance of the lines. Heavier currents also require larger conductors, increasing the amount of copper needed for the system, and they require increased generating capacities by the utility company. Since the line voltage of a transmission system is fixed, the apparent power is directly related to the current level. In turn, the smaller the net apparent power, the smaller the current drawn from the supply. Minimum current is therefore drawn from a supply when S = P and QT = 0. The process of introducing reactive elements to bring the power factor closer to unity is called power-factor correction. Since most loads are inductive, the process normally involves introducing elements with capacitive terminal characteristics having the sole purpose of improving the power factor.

Power Factor Correction Increasing the power factor without altering the voltage or current to the load is called Power Factor Correction Inductive Load with improved power factor correction Original Inductive Load Effect of capacitor on total current Power triangle of power factor correction

Power Factor Correction Increasing the power factor without altering the voltage or current to the load is called Power Factor Correction. Qc = Q1 – Q2 = P (tan θ1 - tan θ2) = ωCVrms2 Q1 = S1 sin θ1 = P tan θ1 Q2 = P tan θ2 P = S1 cos θ1

Power Factor Correction The process of increasing the power factor without altering the voltage or current to the original load is called power factor correction. Power factor correction is necessary for economic reasons. The capacitance value needed to change the pf angle from 1 to 2 . Similarly the inductance value needed to change the pf angle from 1 to 2 for a capacitive load.

Power Factor Correction

Power Factor Correction

Power Factor Correction

Practice Problem 11.15: Find the value of the capacitance needed to correct a load of 140 kVAR at 0.85 lagging pf to unity pf. The load is supplied by a 110 Volt (rms), 60 Hz line.

Applications: Power Measurement Wattmeter is the instrument for measuring the average power. Two coils are used, the high impedance Voltage coil and the low impedance Current coil. Wattmeter measures the average power given by: Wattmeter Wattmeter connected to the load

Problem 11-74