Circuits II EE221. Instructor: Kevin D. Donohue. Instantaneous, Average, RMS, and Apparent Power, and, Maximum Power Transfer, and Power Factors

Transcription

1 Crcuts II EE1 Unt 3 Instructor: Ken D. Donohue Instantaneous, Aerage, RMS, and Apparent Power, and, Maxmum Power pp ransfer, and Power Factors

2 Power Defntons/Unts: Work s n unts of newton-meters or joules same as energy. 1 joule s work done for 1 ampere passed through 1 ohm for 1 second, or work done by a force of 1 Newton appled oer 1 meter. Power s a measure of the rate at whch Power s a measure of the rate at whch work s done and s n unts ofjoules per seconds, 746 Watts n one horsepower.

3 Power Conerson Electrc motors and generators conert electrcal power to mechancal power and se ersa. Other deces exst that conert electrc Other deces exst that conert electrc power to lght, heat, sound,. and se ersa

4 Instantaneous and Aerage Power he nstantaneous power s the power absorbed by an element at an nstance of tme. In an electrc crcut ths s gen by: P( t ( t ( t where (t s the current through the element and (t s the oltage drop oer the element. Aerage Power oer some tme nteral s gen by: 1 P AV ( t ( t dt 0 If the and product s perodc, then P AV can be reduced to the ntegraton oer a sngle perod. For random/non-perodc d sgnals goes to nfnty. t

5 Instantaneous Power Instantaneous power s smply computed by the product of functons. For snusodal problems, trg denttes or phasors can be used to smplfy the products for easer aerage power formulae. Example: Wrte a Matlab scrpt that plots the nstantaneous power of a cosne oltage across an mpedance load. Wrte the scrpt so the oltage waeform and mpedance can be easly changed and nstantaneous power replotted.

6 Matlab Scrpts A scrpt s a seres of Matlab command lne nstructons typed n a text fle and stored wth an *.m extenson. hs s referred to as an mfle. o run these commands change your current drectory assocated wth the Matlab work space to the one contanng the mfle. hen type the base name of the fle n the Matlab command lne. Or on the edtor menu there s a green play arrow that wll also execute the program.

7 Matlab Scrpts For ths example comments are ncluded n the scrpt, and are dentfed by text followng a % character. he functons plot and cos are used n ths scrpt. ype help plot or help cos to get a full descrpton of these functons n the Matlab workspace.

8 Matlab Scrpts % hs scrpt wll plot the nstantaneous power absorbed % by an mpedance load wth a snusodal oltage oer t. % Set the parameters of the Voltage sgnal f = 1000; % Frequency of sgnal n Hz ph = 30; % Phase of sgnal n degrees A=; % ampltude of sgnal %Set mpedance alue zm = 8; % Impedance magntude zp = -40; % Impedance phase n degrees % Create a tme axs of perods oer whch to plot the power % Create a tme axs of perods oer whch to plot the power tp = 1/f; % Determne perod t = *tp*[0:5000]/5000; % Make a 5001 pont tme axs (row ector oer perods

9 Matlab Scrpts % Create a ector of ponts for oltage = A*cos(*p*f*t + p*ph/180; % Create a ector of ponts for the current (adjust magntude and phase % based on mpedance = (A/zm*cos(*p*f*t ( + p*(ph-zp/180; p ; % ake an element by element product to get power p =.*; % Plot t plot(t,p ttle(['instantaneous Power '] xlabel(['seconds'] ylabel(['watts']

10 Result 0.6 Instantaneous Power Watts Seconds x 10-3 What s the meanng of negate nstantaneous power?

11 Aerage Power n Perodc Sgnals Gen a snusodal oltage and current n a dece: Show: P ( t A cos( t t A cos( t AV ( 1 A A cos(

12 Aerage Power n Perodc Sgnals Gen a phasor representaton of a oltage and current n a dece: Vˆ Iˆ A A Show: P AV 1 * Re 1 VI ˆˆ A A cos(

13 Conseraton of Power In a gen crcut the aerage power absorb (denoted by poste alues equals the power delered (denoted by negate alues. For a crcut wth N elements the sum of all power s zero: 0 N 1 P Note: hese are all real or aerage power alues.

14 Passe Sgn Conenton It s assumed that poste charge enterng the poste termnal of an element mples power absorbed by the element. herefore, charge leang the poste termnal of an element mples power suppled or delered d by the element. If the words absorbed or suppled are not gen wth a power alue, power absorbed wll be assumed. (A negate sgn wll mply power suppled. I I V V

15 Example Aerage Power Fnd the aerage power each of the elements gen s (t = 3cos(1000tA 10 a s a 5Ω 10mH P 5Ω =34.61W P L =P C =0 P 5Ω =1.13W P CCVS =-5.54W P s =-50.19W 80µF 5Ω

16 Maxmum Power ransfer Gen a héenn crcut wth load Z L : Z th V th Z L Show that for a maxmum power transfer to the load: Zˆ ˆ * L Z th And the maxmum power transfer s: P max ˆ V th 8Re[ ˆ * Z th ]

17 Example Maxmum Power Fnd the mpedance of the load Z to result n the maxmum power transfer, and fnd the resultng power. Assume s (t = 110cos(377t V s 40µF 8Ω 1Ω 7.5mH Z Z= I th = A V th = V P z =3.98W

18 Root Mean Square (RMS Values he RMS alue of a perodc current or oltage s ts DC equalent alue for delerng aerage power to a resstor. P P 1 ( t dt R ( t dt R RI AV rms AV 1 1 I rms ( t dt ( t dt R 1 R ( t dt V R rms 1 V rms ( t dt

19 RMS Formula for Snusods he RMS alue for a snusod d of any frequency or phase s ts ampltude dded by square root of. 1 I rms A cos ( t dt I I I rms rms rms A A 4 A 4 exp( j( t exp( j( t exp( j( t exp(0 exp( j( t A 4 dt exp( j( t exp( j( t dt dt dt I rms A 4 0 A

20 Apparent Power and Power Factor Apparent power (S for snusodal waeforms s the ( t A product of the RMS oltage and cos( t current magntudes wthout ( t A cos( t regard to ther phase offsets. It s n unts of Volt-Amps (VA. Vˆ A he cosne of ther phase dfference s the power factor Iˆ A (pf. P AV 1 Re 1 * 1 VI ˆˆ A A cos( V I cos( S V rms I rms (apparent (pp power P S cos( (real power Q AV S sn( pf cos( Sˆ V rms I rms exp (reacte power (power factor rms j( (complex power rms

21 Leadng and Laggng he sgn of phase dfference between the oltage and current s related to the angle of the mpedance of the load. he terms leadng and laggng are used to descrbe ths property: 0 pf 0 0 pf leadng (Capacte Load laggng (Inducte Load (Only Real Load

22 Example An ndustral load consumes 10 kw at a power factor of.95 leadng. he oltage across the load s 0 Vrms (assume zero phase. Determne the current drawn by the load (ncludng phase. Result I Arms

23 Example Fnd complex power suppled by source (both real and reacte, and power factor at the source. V S 15 -j10 + 5kW.9 Laggng 6kVA.8 Leadng Vrms V Result V s pf.85 Leadng kvrms S s 111kW 68.6 kvar