March 11, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 30March 11, 2005 Physics for Scientists&Engineers 2 2ReviewReview! If we have a single loop RLC circuit, the charge in the circuit as afunction of time is given by! Where! The energy stored in the capacitor as a function of time is given byq = qmaxe!Rt2 Lcos"t +#( )!=!02"R2L#$%&'(2!0=1LCUE=qmax22Ce!RtLcos2"t +#( )March 11, 2005 Physics for Scientists&Engineers 2 3iR=VRR= IRsin!tResistanceXC=1!CiC=VCXCsin(!t + 90°)CapacitiveReactanceiL=VLXLsin!t " 90°( )XL=!LInductiveReactanceReview (2)Review (2)! Time-varying emf! Time-varying emf VR with resistor! Time-varying emf VC with capacitor! Time-varying emf VL with inductorVemf= Vmaxsin!tMarch 11, 2005 Physics for Scientists&Engineers 2 4Review (3)Review (3)March 11, 2005 Physics for Scientists&Engineers 2 5Series RLC CircuitSeries RLC Circuit! Consider a single loop circuit that hasa resistor, a capacitor, an inductor, anda source of time-varying emf! We can describe the time-varying currentsin these circuit elements using a phasor I! The projection of I on the vertical axisrepresents the current flowing in thecircuit as a function of time• The angle of the phasor is given by !t - "! We can also describe the voltage in terms of a phasor V! The time-varying currents and voltages in the circuit canhave different phasesMarch 11, 2005 Physics for Scientists&Engineers 2 6Series RLC Circuit (2)Series RLC Circuit (2)! We can describe the current flowing in the circuitand the voltage across the various components• Resistor• The voltage vR and current iR are in phase with each other andthe voltage phasor vR is in phase with the current phasor I• Capacitor• The current iC leads the voltage vC by 90° so that the voltagephasor vC will have an angle 90° less than I and vR• Inductor• The current iL lags behind the voltage vL by so that voltagephasor vL will have an angle 90° greater than I and vRMarch 11, 2005 Physics for Scientists&Engineers 2 7Series RLC Circuit (3)Series RLC Circuit (3)! The voltage phasors for an RLC circuit are shown below! The instantaneous voltages across each of the componentsare represented by the projections of the respectivephasors on the vertical axisMarch 11, 2005 Physics for Scientists&Engineers 2 8Series RLC Circuit (4)Series RLC Circuit (4)! Kirchhof’s loop rules tells that the voltage drops across all the devicesat any given time in the circuit must sum to zero, which gives us! The voltage can be thought of as the projection of the vertical axis ofthe phasor Vmax representing the time-varying emf in the circuit asshown below! In this figure we have replaced the sum of the two phasors VL and VCwith the phasor VL - VCV ! vR! vC! vL= 0 " V = vR+ vC+ vLMarch 11, 2005 Physics for Scientists&Engineers 2 9Series RLC Circuit (5)Series RLC Circuit (5)! The sum of the two phasors VL - VC and VR must equalVmax so! Now we can put in our expression for the voltage across thecomponents in terms of the current and resistance or reactance! We can then solve for the current in the circuit! The denominator in the equation is called the impedance! The impedance of a circuit depends on the frequency of the time-varying emfVmax2= VR2+ VL! VC( )2Vmax2= IR( )2+ IXL! IXC( )2I =VmaxR2+ XL! XC( )2Z = R2+ XL! XC( )2March 11, 2005 Physics for Scientists&Engineers 2 10Series RLC Circuit (6)Series RLC Circuit (6)! The current flowing in an alternating current circuitdepends on the difference between the inductive reactanceand the capacitive reactance! We can express the difference between the inductivereactance and the capacitive reactance in terms of thephase constant "! This phase constant is defined as the phase differencebetween voltage phasors VR and VL - VC!= tan"1VL"VCVR#$%&'(= tan"1XL" XCR#$%&'(March 11, 2005 Physics for Scientists&Engineers 2 11Series RLC Circuit (7)Series RLC Circuit (7)! Thus we have three conditions for an alternating currentcircuit• For XL > XC, " is positive, and the current in the circuit will lagbehind the voltage in the circuit• This circuit will be similar to a circuit with only an inductor, except thatthe phase constant is not necessarily 90°• For XL < XC, " is negative, and the current in the circuit will lead thevoltage in the circuit• This circuit will be similar to a circuit with only a capacitor, except thatthe phase constant is not necessarily -90°• For XL = XC, " is zero, and the current in the circuit will be in phasewith the voltage in the circuit• This circuit is similar to a circuit with only a resistance• When " = 0 we say that the circuit is in resonanceMarch 11, 2005 Physics for Scientists&Engineers 2 12Series RLC Circuit (8)Series RLC Circuit (8)XL > XCXL < XCXL = XC! For XL = XC and " = 0 we get the maximum current in the circuit andwe can define a resonant frequency!L "1!C= 0 # !0=1LCMarch 11, 2005 Physics for Scientists&Engineers 2 13Real-lifeReal-life RLC Circuit (2)RLC Circuit (2)! Let’s study a real-life circuit•R = 10 !•L = 8.2 mH•C = 100 µF•Vmax = 7.5 V! We measure the current in the circuit as afunction of the frequency of the time-varying emf! We see the correct resonant frequency(peak at !/!0 = 1)•L and C must be accurate! However, our formula for the current(green line) using R = 10 ! does not agreewith the measurements• We must use R = 15.4 !• The inductor has a resistance even atresonanceMarch 11, 2005 Physics for Scientists&Engineers 2 14Resonant Behavior of RLC CircuitResonant Behavior of RLC Circuit! The resonant behavior of an RLC circuit resembles the responseof a damped oscillator! Here we show the calculated maximum current as a function ofthe ratio of the angular frequency of the time varying emfdivided by the resonant angular frequency, for a circuit withVmax = 7.5 V, L = 8.2 mH, C = 100 µF, and three resistances! One can see that as theresistance is lowered, themaximum current at theresonant angular frequencyincreases and there is a morepronounced resonant peakMarch 11, 2005 Physics for Scientists&Engineers 2 15Energy and Power in RLC CircuitsEnergy and Power in RLC Circuits! When an RLC circuit is in operation, some of the energy in the circuit isstored in the electric field of the capacitor, some of the energy isstored in the magnetic field of the inductor, and some energy isdissipated in the form of heat in the resistor! The energy stored in the capacitor
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