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MSU PHY 184 - PHY184-Lecture29n

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March 10, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 29March 10, 2005 Physics for Scientists&Engineers 2 2ReviewReview! Consider a circuit consisting of aninductor L and a capacitor C! The charge on the capacitor as afunction of time is given by! The current in the inductor as afunction of time is given by! where ! is the phase and "0 is the angular frequencyq = qmaxcos!0t +"( )!0=1LC=1LCi = !imaxsin("0t +#)March 10, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2)! The energy stored in the electricfield of the capacitor C as a functionof time is! The energy stored in the magnetic field of the inductor L as a functionof time is! The total energy stored in the circuit is given byUE=qmax22Ccos2!0t +"( )UB=L2imax2sin2(!0t +")U = UE+ UB=qmax22CMarch 10, 2005 Physics for Scientists&Engineers 2 4RLC CircuitRLC Circuit! Now let’s consider a single loop circuitthat has a capacitor C and aninductance L with an added resistance R! We observed that the energy of a circuitwith a capacitor and and an inductorremains constant and that the energy translated fromelectric to magnetic and back gain with no losses! If there is a resistance in the circuit, the current flow inthe circuit will produce ohmic losses to heat! Thus the energy of the circuit will decrease because ofthese lossesMarch 10, 2005 Physics for Scientists&Engineers 2 5RLC Circuit (2)RLC Circuit (2)! The rate of energy loss is given by! We can rewrite the change in energy of the circuit as afunction time as! Remembering that i = dq/dt and di/dt = d2q/dt2 we canwritedUdt= !i2RdUdt=ddtUE+ UB( )=qCdqdt+ Lididt= !i2RqCdqdt+ Lididt+ i2R =qCdqdt+ Ldqdtd2qdt2+dqdt!"#$%&2R = 0March 10, 2005 Physics for Scientists&Engineers 2 6RLC Circuit (3)RLC Circuit (3)! We can then write the differential equation! The solution of this differential equation is! whereLd2qdt2+dqdtR +qC= 0q = qmaxe!Rt2 Lcos"t +#( )!=!02"R2L#$%&'(2!0=1LCMarch 10, 2005 Physics for Scientists&Engineers 2 7RLC Circuit (4)RLC Circuit (4)! Now consider a single loop circuit that contains a capacitor, an inductorand a resistor! If we charge the capacitor then hook it up to the circuit, we willobserve a charge in the circuit that varies sinusoidally with time andwhile at the same time decreasing in amplitude! This behavior with time is illustrated belowMarch 10, 2005 Physics for Scientists&Engineers 2 8RLC Circuit (5)RLC Circuit (5)! The charge varies sinusoidally with but the amplitude isdamped out with time! After some time, no charge remains in the circuit! We can study the energy in the circuit as a function oftime by calculating the energy stored in the electric fieldof the capacitor! We can see that the energy stored in the capacitordecreases exponentially and oscillates in timeUE=12q2C=12qmaxe!Rt2 Lcos"t +#( )$%&'()2C=qmax22Ce!RtLcos2"t +#( )March 10, 2005 Physics for Scientists&Engineers 2 9Alternating CurrentAlternating Current! Now we consider a single loop circuitcontaining a capacitor, an inductor,a resistor, and a source of emf! This source of emf is capable producinga time varying voltage as opposed thesources of emf we have studied in previous chapters! We will assume that this source of emf provides asinusoidal voltage as a function of time given by! where " is that angular frequency of the emf and Vmax isthe amplitude or maximum value of the emfVemf= Vmaxsin!tMarch 10, 2005 Physics for Scientists&Engineers 2 10Alternating Current (2)Alternating Current (2)! The current induced in the circuit will also vary sinusoidally withtime! This time-varying current is called alternating current! However, this current may not always remain in phase with thetime-varying emf! We can express the induced current as! where the angular frequency of the time-varying current is thesame as the driving emf but the phase ! is not zero! Note that traditionally the phase enters here with a negative sign! Thus the voltage and the current in the circuit are notnecessarily in phasei = I sin!t "#( )March 10, 2005 Physics for Scientists&Engineers 2 11Circuit with ResistorCircuit with Resistor! To begin our analysis of RLC circuits, let’s startwith a circuit containing only a resistor and asource of time-varying emf as shown to the right! Applying Kirchhof’s loop rule to this circuit we get! where vR is the voltage drop across the resistor! Substituting into our expression for the emf as a function oftime we get! Remembering Ohm’s Law, V = iR, we getVemf! vR= 0 " Vemf= vRvR= VRsin!tiR=vRR= IRsin!tMarch 10, 2005 Physics for Scientists&Engineers 2 12! Thus we can relate the current amplitude and the voltage amplitude by! We can represent the time varying current by a phasor IR and the time-varying voltage by a phasor VR as shown below! The current flowing through the resistor and the voltage across theresistor are in phase, which means that the phase difference betweenthe current and the voltage is zeroCircuit with Resistor (2)Circuit with Resistor (2)VR= IRRMarch 10, 2005 Physics for Scientists&Engineers 2 13Circuit with CapacitorCircuit with Capacitor! Now let’s address a circuit that contains a capacitorand a time varying emf as shown to the right! The voltage across the capacitor is given byKirchhof’s loop rule! Remembering that q = CV for a capacitor we can write! We would like to know the current as a function of time ratherthan the charge so we can writevC= VCsin!tq = CvC= CVCsin!tiC=dqdt=d CVCsin!t( )dt=!CVCcos!tMarch 10, 2005 Physics for Scientists&Engineers 2 14Circuit with Capacitor (2)Circuit with Capacitor (2)! We can rewrite the last equation by defining a quantity that issimilar to resistance and is called the capacitive reactance! Which allows us to write! We can now express the current in the circuit as! We can see that the current and the time varying emf are out ofphase by 90°XC=1!CiC=VCXCcos!tiC= ICcos!t = ICsin!t + 90°( )March 10, 2005 Physics for Scientists&Engineers 2 15Circuit with Capacitor (3)Circuit with Capacitor (3)! We can represent the time varying current by a phasor ICand the time-varying voltage by a phasor VC as shown below! The current flowing this circuit with only a capacitor issimilar to the expression for the current flowing in acircuit with only a resistor except that the current is outof phase with the emf by 90°March 10, 2005 Physics for Scientists&Engineers 2 16Circuit with Capacitor (4)Circuit with Capacitor (4)! We can also


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MSU PHY 184 - PHY184-Lecture29n

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