Chapter 10 Engr8 Circuit Analysis Dr Curtis Nelson Chapter 10 Objectives Understand the following AC power concepts: Instantaneous power; Average power; Root Mean Squared (RMS) value; Reactive power; Coplex power; Power factor; Power factor correction. Understand how axiu real power is calculated and delivered to a load in an AC circuit. Engr8 - Chapter 10, Nilsson 10e 1
Instantaneous Power Instantaneous power is the power easured at any given instant in tie. In DC circuits, power is easured in watts as: v P vi i R R In AC circuits, voltage and current are tie-varying, so instantaneous power (also easured in watts) is tievarying. Power is still easured as: v P vi i R R Instantaneous AC Power p (t) v(t) i(t) In an AC circuit, voltage and current are expressed in general for as: v t) V cos( w t + q ) ( v i t) I cos( w t + q ) ( i Instantaneous power is then: p( t) VI cos( w t + qv)cos( wt + qi) Engr8 - Chapter 10, Nilsson 10e
Instantaneous Power - continued p( t) VI cos( w t + qv)cos( wt + qi) Using the trig identity cos( x )cos( y) 0.5cos( x - y) + 0.5cos( x + y) it can be shown that instantaneous power is: VI p t) ( v i v i {cos( q -q ) + cos(wt + q + q )} Note that the instantaneous power contains a constant ter as well as a coponent that varies with tie at twice the input frequency. Instantaneous Power Plot VI p t) ( v i v i {cos( q -q ) + cos(wt + q + q )} Engr8 - Chapter 10, Nilsson 10e 3
Instantaneous Power - continued VI p t) ( v i v i {cos( q -q ) + cos(wt + q + q )} The equation above can be further anipulated as follows: cos( a+ b) cosacos b - sina sinb cos( wt + q - q ) cos( q - q )cos( wt) -sin( q - q )sin( wt) v i v i v i pt ( ) V I cos( q ) cos( )cos( ) v - q + V I q v - q wt - V I sin( qv - q )sin( wt ) i i i Note that the instantaneous power still contains a constant ter as well as coponents that vary with tie at twice the input frequency. Instantaneous Power Plot pt ( ) V I cos( q ) cos( )cos( ) v - q + V I q v - q wt - V I sin( qv - q )sin( wt ) i i i Engr8 - Chapter 10, Nilsson 10e 4
Instantaneous vs. Average Power While instantaneous power is an interesting atheatical developent for AC circuits, it is not all that useful for deterining how uch power a circuit generates or absorbs since it varies with tie. Average Power is a better indicator of power in AC circuits. Average power is soeties called real power, because it describes the power in a circuit that is transfored fro electric to nonelectric energy, such as heat. Average Power The average power over an arbitrary interval fro t 1 to t is: where p(t) is the instantaneous power. When the power is periodic with period T, the average power can be calculated over any one period as: Engr8 - Chapter 10, Nilsson 10e 5
Average Power - continued pt ( ) V I cos( q ) cos( )cos( ) v - q + V I q v - q wt - V I sin( qv - q )sin( wt ) i i i To calculate average power, note that the last two ters in the equation above will integrate to zero since the average value of sin and cosine signals over one period is zero. Therefore, average power in AC circuits is: P AVG VI cos( q -q ) v i Exaple - Power Calculations Calculate the instantaneous and average power if: v(t) 80cos(10t + 0º)V i(t) 15cos(10t + 60º)A P inst (t) 459.6 + 600cos(0t + 80º)W P avg 459.6W Engr8 - Chapter 10, Nilsson 10e 6
Average Power for Circuit Eleents P AVG VI cos( q -q ) v i Since the voltage and current are in phase across a resistor, the average power absorbed by the resistor R is: P R, AVG 1 V R The average power absorbed by a purely reactive eleent is zero, since the current and voltage are 90 degrees out of phase: P C P, AVG L, AVG 0 Exaple - Average Power Calculate the average power absorbed by the resistor and inductor and find the average power supplied by the voltage source. P R 9.6W absorbed P L 0W P Source 9.6W delivered Engr8 - Chapter 10, Nilsson 10e 7
Effective or RMS Value Soeties using average AC power values can be confusing. For instance, the average DC power absorbed by a resistor is P V M I M while the average AC power is P V M I M /. By introducing a new quantity called effective value, the forulas for the average power absorbed by a resistor can be ade the sae for dc, sinusoidal, or any general periodic wavefor. The effective value of a periodic voltage is the DC voltage that delivers the sae average power to a resistor as the periodic AC voltage. Effective or RMS Value For any periodic function x(t), the effective or root ean squared (rs) value, is given by: X rs 1 T ò T 0 x dt Engr8 - Chapter 10, Nilsson 10e 8
Effective or RMS Value For exaple, the effective value of I I cos t is 1 T Irs I cos tdt T ò w 0 I T 1 Irs (1 cos t)dt T ò + w 0 Average power can then be written in two ways: P AVG P AVG V VI rs I rs cos( q -q ) v I cos( q -q ) v i i Effective Values of Coon Wavefors Engr8 - Chapter 10, Nilsson 10e 9
Textbook Proble 10.14 Nilsson 10 th a) Find the rs value of the periodic voltage shown in the figure below. b) If the voltage is applied across a 40Ω resistor, find the average power dissipated by the resistor. V rs 8.8V RMS P 0W Coplex Power The average power in capacitors and inductors 0 because voltage and current have a 90º phase difference. However, there is still voltage across, and current through, inductors and capacitors so they are part of the overall power picture. We now investigate the part that inductors and capacitors play, leading us to define a new quantity called Coplex Power. Engr8 - Chapter 10, Nilsson 10e 10
Coplex Power - Triangle Coplex power contains both a real part (average power) and an iaginary part (reactive power) that together are represented in a power triangle. Coplex Power - Units Engr8 - Chapter 10, Nilsson 10e 11
Coplex Power The Equations P AVG V RMS I RMS cos(θ V - θ i ) P REACTIVE V RMS I RMS sin(θ V - θ i ) P APPARENT V RMS I RMS watts volt-aps-reactive volt-aps Coplex Power The Angle θ V - θ i is known as the Power Factor Angle. cos(θ V - θ i ) is known as the Power Factor (PF) For a purely resistive load, PF1; For a purely reactive load, PF0; For any practical circuit, 0 PF 1. Engr8 - Chapter 10, Nilsson 10e 1
Apparent Power & Power Factor Power factor is also defined as: average power P PF apparent power V I eff avg eff Coplex Power and Ipedance Triangles S Q IZI X q P q R Power triangle Ipedance triangle P pf cosq S R Z Engr8 - Chapter 10, Nilsson 10e 13
Exaple: Power Factor Calculate the power factor of the circuit below as seen by the source. What is the average power supplied by the source? pf 0.936 (lagging) P AVG 118W Leading and Lagging Power Factor pf is lagging if the current lags the voltage (θ V - θ I is positive like in an inductive load); pf is leading if the current leads the voltage (θ V - θ I is negative like in a capacitive load). Engr8 - Chapter 10, Nilsson 10e 14
Exaple Proble Find the average power delivered to each of the two loads, the apparent power supplied by the source, and the power factor of the cobined loads. P avg 88 W P avg1 144 W P app S 70 VA PF 0.6 (lagging) Textbook Proble 10.17 Nilsson 10 th For i g 50 + j0 A (rs), find the average and reactive power for the current source and state whether it is absorbing or delivering this power. P AVG -50.0 W (delivering) P REAC 5.0 VAR (absorbing) Engr8 - Chapter 10, Nilsson 10e 15
Textbook Proble 10.18 Nilsson 10 th Find the average power, the reactive power, and the apparent power absorbed by the load if v g 150cos50t V. P AVG 180 W P REAC 90 VAR P APP 180 + j90 VA More on Coplex Power S P + jq V rs I rs " V - " I V rs " V I rs -" I V rs I* rs Engr8 - Chapter 10, Nilsson 10e 16
Coplex Power - Suary Real (Average) power: Reactive power: Apparent power: Coplex power: P Re(S) Scos( q - q ) Watts Q I(S) Ssin( q - q ) VAR S + S VrsIrs P Q VA S P jq V I * + RMS RMS VA Q 0 for resistive loads (unity power factor); Q < 0 for capacitive loads (leading power factor); Q > 0 for inductive loads (lagging power factor). v v i i Exaple - Apparent Power and Power Factor A series connected RC load draws a current of when the applied voltage is i ( t) 4cos(100pt + 10 ) A v ( t) 10cos(100pt - 0 ) V Find the apparent power and the power factor of the load and deterine the eleent values that for the load. S 40 V pf 0.866 (leading) C 1.µF R 6.0Ω Engr8 - Chapter 10, Nilsson 10e 17
Power Factor Correction The process of increasing the power factor without altering the voltage or current to the original load is known as power factor correction. Most loads are inductive. The power factor of the load can be iproved (to ake it closer to unity) by installing a capacitor in parallel with an inductive load. a) Original inductive load b) Inductive load with iproved power factor Power Factor Correction Capacitors Engr8 - Chapter 10, Nilsson 10e 18
Power Factor Correction Capacitors Power Factor Correction by Power Triangle The triangle below shows that the reactive power (Q 1 ) in a large inductive load can be reduced by an aount (Q C ) by adding a capacitor in parallel to the load. Q 1 Ptan θ 1 Q Ptan θ Q C Q 1 Q P(tan θ 1 tan θ ) Engr8 - Chapter 10, Nilsson 10e 19
Power Factor Correction by Power Triangle But Q C is also the aount of reactive power the capacitor ust provide: Q C V X rs C wcv rs The value of required parallel capacitance is then: Q C wv C rs P(tan q - tan q ) 1 wv rs Exaple - PF Correction #1 When connected to a 10 V RMS, 60Hz power line a load absorbs 4kW at a lagging power factor of 0.8. Find the value of capacitance necessary to raise the pf to 0.95 C 310.4µF Engr8 - Chapter 10, Nilsson 10e 0
Exaple - PF Correction # Find the value of parallel capacitance needed to correct a load of 140kVAR at 0.85 lagging pf to unity pf. Assue that the load is supplied by a 110V RMS, 60Hz line. C 30,691µF Textbook Proble 10.33 Nilsson 10 th A group of sall appliances on a 60 Hz syste requires 0 kva at 0.85 pf lagging when operated at 15 V rs. The ipedance of the feeder supplying the appliances is 0.01+ j0.08 Ω. Find: a) The rs value of the voltage at the source end of the feeder. b) The average power loss in the feeder. V S 133.48 V RMS P REAC 56 W Engr8 - Chapter 10, Nilsson 10e 1
Power Generation Hydro-Power Generation Solar Power Nuclear Power Generation Fossil Fuel Generation Wave Power Generation Gravity Light Project Maxiu Power Transfer An independent voltage source in series with an ipedance Z th delivers a axiu average power to that load ipedance Z L when it is the conjugate of Z th. Z L Z th * Engr8 - Chapter 10, Nilsson 10e
Maxiu Average Power Transfer In rectangular for, the Thevenin ipedance and load ipedances are: Z Z Th L R R L Th + jx + jx For axiu average power transfer, the load ipedance Z L ust be equal to the coplex conjugate of the Thevenin ipedance Z Th (i.e. the iaginary parts of the ipedance cancel each other out): Z R + jx R - jx Z L L L Z Z L Th * Th L Th Th * Th Maxiu Average Power Transfer When Z L Z * TH, the axiu power delivered to the load is: P ax V 4R TH, RMS Th If the load is purely resistive, then the value of R L that axiizes P L is the agnitude of the Thevenin ipedance. Note that the 4 factor in the denoinator above becoes 8 if the voltage is expressed as an average or peak value. Engr8 - Chapter 10, Nilsson 10e 3
Exaple: Maxiu Average Power Deterine the load ipedance Z L that axiizes the average power drawn fro the circuit below. Calculate the axiu average power. Z L * Z Th.9333 - j4. 4667W Pax. 3675W Chapter 10 Suary Understand the following AC power concepts: Instantaneous power; Average power; Root Mean Squared (RMS) value; Reactive power; Coplex power; Power factor; Power factor correction. Understand how axiu real power is calculated and delivered to a load in an AC circuit. Engr8 - Chapter 10, Nilsson 10e 4