Circuits IIEE221Unit 3Unit 3Instructor: Kevin D. DonohueInstantaneous, Average, RMS, and Apparent Power, and, Maximum Power ppTransfer, and Power FactorsPower Definitions/Units:Power Definitions/Units:Work is in units of newton-metersor joulesWork is in units of newtonmetersor joulessame as energy.1 j l i k d f 1 d 1 joule is work done for 1 ampere passed through 1 ohm for 1 second, or work done by a force of 1 Newton applied over 1 yppmeter. Power is a measure of the rate at which Power is a measure of the rate at which work is done and is in units ofjoules per seconds,746 Watts in one horsepower.Power ConversionPower ConversionElectric motors and enerators convert Electric motors and generators convert electrical power to mechanical power and vise versavise versa.Other devices exist that convert electric Other devices exist that convert electric power to light, heat, sound, …. and vise versaInstantaneous and Average PowerInstantaneous and Average Power The instantaneous power is the power absorbed by an element at an instance of time. In an electric circuit this is given by:)()()( titvtPwhere i(t) is the current through the element and v(t) is the voltage drop over the element. Average Power over some time interval Tis given by:TAVdttitvTP )()(1If the iand vproduct is periodic, then PAV can be reduced to the integration over a single period. For d/idi i l T t ifiitT0random/non-periodic signals Tgoes to infinity.Instantaneous PowerInstantaneous PowerInstantaneous power is simply computed by the Instantaneous power is simply computed by the product of 2 functions. For sinusoidal problems, trig identities or phasors can be used to simplify the products for easier average power formulae the products for easier average power formulae.  Example: Write a Matlab script that plots the pppinstantaneous power of a cosine voltage across an impedance load. Write the script so the voltage waveform and impedance can be easily changed pygand instantaneous power replotted.Matlab ScriptsMatlab ScriptsA script is a series of Matlabcommand line A script is a series of Matlabcommand line instructions typed in a text file and stored with an *.m extension. This is referred to as an mfile as an mfile. To run these commands change your To run these commands change your current directory associated with the Matlab work space to the one containing the mfile Then type the base name of the the mfile. Then type the base name of the file in the Matlab command line. Or on the editor menu there is a green “play” arrow that will also execute the program that will also execute the program.Matlab ScriptsMatlab ScriptsFor this example comments are included in For this example comments are included in the script, and are identified by text following a “%” character following a % character. The functions plotand cosare used in this The functions plotand cosare used in this script. Type help plot or help cos to get a full description of these functions in the fpffMatlab workspace.Matlab ScriptsMatlab Scripts% This script will plot the instantaneous power absorbedpp p% by an impedance load with a sinusoidal voltage over it.% Set the parameters of the Voltage signalf = 1000; % Frequency of signal in Hz;qygph = 30; % Phase of signal in degreesA=2; % amplitude of signal%Set impedance value%S pzm = 8; % Impedance magnitudezp = -40; % Impedance phase in degrees% Create a time axis of 2 periods over which to plot the power% Create a time axis of 2 periods over which to plot the powertp = 1/f; % Determine periodt = 2*tp*[0:5000]/5000; % Make a 5001 point time axis (row vector) over 2 periodsMatlab ScriptsMatlab Scripts% Create a vector of points for voltage pgv = A*cos(2*pi*f*t + pi*ph/180);% Create a vector of points for the current (adjust magnitude and phase% based on impedancei = (A/zm)*cos(2*pi*f*t + pi*(ph-zp)/180);()(pp(pp));% Take an element by element product to get powerp = v.*i;% Plot itplot(t,p) p(,p)title(['Instantaneous Power '] ) xlabel(['Seconds']) ylabel(['Watts'])ResultResult0.50.6Instantaneous Power0.30.40.50.10.2Watts0020406081121416182-0.1000.20.40.60.811.21.41.61.82x 10-3SecondsWhat is the meaning of negative instantaneous power?What is the meaning of negative instantaneous power?Average Power in Periodic SignalsAverage Power in Periodic SignalsGiven a sinusoidal voltage and current in a device:Given a sinusoidal voltage and current in a device:)cos()(vvtAtvShow:)cos()(iitAti)cos(21ivivAVAAPAverage Power in Periodic SignalsAverage Power in Periodic SignalsGiven a phasor representation of a voltage and Given a phasor representation of a voltage and current in a device:AVˆiivvAIAVˆShow:)(1ˆˆR1*AAIVP)cos(2Re2ivivAVAAIVPConservation of PowerConservation of PowerIn a given circuit the average power absorb (denoted In a given circuit the average power absorb (denoted by positive values) equals the power delivered (denoted by negative values).For a circuit with Nelements the sum of all power is zero:NNiiP10Note: These are all real or average power valuesor average power values.Passive Sign ConventionPassive Sign ConventionIt is assumed that positive charge entering the positive terminalof entering the positive terminalof an element implies power absorbed by the element.Th f h l h IVTherefore, charge leaving the positive terminal of an element implies power supplied or dl d b h lIdelivered by the element.If the words absorbedor suppliedare not givenwith a power value, Vgwpw u,power absorbed will be assumed. (A negative sign will imply power supplied).suppl ed).Example Average PowerExample Average PowerFind the average power each of the elements given i(t) = 3cos(1000t)Aelements given is(t) = 3cos(1000t)Aia10ia ia25Ω10mHis80µF5ΩP=34 61WP25Ω=34.61WPL=PC=0P5Ω=21.13WP5 54WPCCVS=-5.54WPis=-50.19WMaximum Power TransferGiven a Thévenin circuit with load ZL:ZhZthZLVthShow that for a maximum power transfer to the load:Show that for a maximum power transfer to the load:*ˆˆthLZZ And the maximum power transfer is: ˆˆ*2maxthVP]ˆRe[8*maxthZExample Maximum PowerExample Maximum PowerFind the impedance of the load Z to result in the maximum power transfer and find the resulting maximum power transfer, and find the resulting power. Assume vs(t) =