July 8, 200410- Sinusoidal Steady-State Power CalculationsInstantaneous PowerPower for Purely Resistive CircuitsPower for Purely Inductive CircuitsPower for Purely Capacitive Circuit
The Power factorJuly 8, 2004 10- Sinusoidal Steady-State Power Calculations • All electrical energy is supplied in the form of sinusoidal voltages and currents. • Our interest is the average power delivered to or supplied from a pair of terminals as a result of sinusoidal voltages and currents Instantaneous Power v - + Back Box i Instantaneous Power is the power at any instant of time is vip= where )cos(vmtVvθω+= i )cos(imtIθω+= Let’s shift both voltage and current by iθ cos( )mvvV tiωθθ=+− i )cos( tImω= Therefore, instantaneous power becomes ttIVvipivmmωθθωcos)cos(−+== (1.1) To simplify this, we can use the following trigonometric identity )cos(21)cos(21cosβαβαβα++−=cos )2cos(2)cos(2ivmmivmmtIVIVpθθωθθ−++−= (1.2) We know that βαβαβαsinsincoscos)cos(−=+ Therefore, instantaneous power tIVtIVIVpivmmivmmivmmωθθωθθθθ2sin)sin(22cos)cos(2)cos(2−−−+−= (1.3) ECE309-05--1In this equation, • the first term is constant, • the 2nd and 3rd show that instantaneous power has twice frequency of the voltage or current. • It has negative for some portion of cycle. That means, energy stored in the inductors o capacitors is now being extracted. Average and Reactive Power. We can divide the instantaneous power (last equation) in three terms. tQtPPpωω2sin2cos−+= (1.4) where P is called the average power or real power )cos(2ivmmIVPθθ−= [W] (1.5) Q is called the reactive power )sin(2ivmmIVQθθ−=[VAR] (1.6) The average power be represented in the following form too ∫+=TttpdtTP001 T is period of the sinusoidal function. ECE309-05--2Power for Purely Resistive Circuits If the circuit between the terminal is purely resistive, there is no phase different between voltage and current, which means ivθθ= , so that instantaneous power will be (1 cos 2 )2mmVIptω=+ This is called as the instantaneous real power. • It has some average value • Frequency is 2ω • Can never be negative, that means power can not be extracted from purely resistive circuit Power for Purely Inductive Circuits If the circuit between the terminal is purely inductive, there is phase different between voltage and current, The current lags the voltage by 90° which means ( ), so that instantaneous power will be D90−=viθθD90+=−ivθθ sin 22mmVIptω=− This is called as the instantaneous reactive power. • The average will be zero. • Frequency is 2ω • When p is positive, energy is being stored in the magnetic fields associate with the inductive elements, • When p is negative, energy is being extracted from the magnetic fields. ECE309-05--3Power for Purely Capacitive Circuit If the circuit between the terminal is purely capacitive, there is phase different between voltage and current, The current leads the voltage by 90° which means ( ), so that instantaneous power will be D90+=viθθD90−=−ivθθ sin 22mmVIptω= This is called as the instantaneous reactive power. • The average will be zero. • Frequency is 2ω • When p is positive, energy is being stored in the capacitive elements • When p is negative, energy is being extracted from the capacitive elements. ECE309-05--4The Power factor The angle ivθθ− is the power factor angle. The power factor is )cos(ivpfθθ−= (1.7) The reactive factor is )sin(ivrfθθ−= Depends on the phase, the current lags the voltage by 90° , we call lagging power factor )cos(ivpfθθ−= and the current laeads the voltage by 90° , we call leading power factor )cos(vipfθθ−= ECE309-05--5Example: )15cos(100D+= tvωV. )15sin(4D−= tiωA Let’s convert current i(t) in cos function 4cos( 15 90 ) 4cos( 105 )it tωω=−−=−DD D the average power or real power will be )cos(2ivmmIVPθθ−= WP 100))105(15cos(2)4)(100(−=−−= (Network inside the box is delivering average power to the terminal) The reactive power )sin(2ivmmIVQθθ−= VARQ 21.173))105(15sin(2)4)(100(=−−= (Inside the box is absorbing magnetizing vars at its terminal.) The instantaneous power will be 100 100cos 2 173.21sin 2pttωω=− − − ECE309-05--6The RMS Value and Power Calculation The RMS value of a periodic sinusoidal function is defined as a square root of the mean value of the square sinusoidal function. Also, the RMS value can be called as effective value of the Ex: The RMS value of () cos( )mvt V tωθ=+ is (1.8)00221cos ( )2tTmrms mtVtdtTωθ+=+∫VV (1.9) = Let’s look at the average power delivered to the resistor +-R Vmcos(ωt+θ) 002211cos ( )tTmtPVtRTωθ+=+∫dt 222rms mVVPRR== If we know the current on the resistor, we can find the average power delivered to the resistor as 222mrmsIPIR R== The previous pages we give the average power as )cos(2ivmmIVPθθ−= (1.10) This can be written in terms of RMS value cos( )22mmviVIPθθ=− cos( )rms rms v iPVIθθ=− (1.11) ECE309-05--7The reactive power )sin(2ivmmIVQθθ−= sin( )rms rms v iQVIθθ=− Example: , , 110rmsVV= 50R =Ω Find Vm and the Average power delivered to the resistor 2mrmsVV = Æ 2 110 2 155.56mrmsV===VV 2211024250rmsVPWR== =