Review If we have a single loop RLC circuit the charge in the circuit as a function of time is given by q qmax e Physics for Scientists Engineers 2 Rt 2L cos t Where R 02 2L Spring Semester 2005 Lecture 30 2 0 1 LC The energy stored in the capacitor as a function of time is given by UE March 11 2005 Physics for Scientists Engineers 2 1 March 11 2005 Review 2 Rt 2 qmax e L cos 2 t 2C Physics for Scientists Engineers 2 2 Review 3 Time varying emf Vemf Vmax sin t Time varying emf VR with resistor iR Resistance VR I R sin t R Time varying emf VC with capacitor XC 1 C iC VC sin t 90 XC Time varying emf VL with inductor XL L March 11 2005 iL VL sin t 90 XL Capacitive Reactance Inductive Reactance Physics for Scientists Engineers 2 3 March 11 2005 Physics for Scientists Engineers 2 4 Series RLC Circuit Series RLC Circuit 2 We can describe the current flowing in the circuit and the voltage across the various components Consider a single loop circuit that has a resistor a capacitor an inductor and a source of time varying emf Resistor We can describe the time varying currents in these circuit elements using a phasor I The voltage vR and current iR are in phase with each other and the voltage phasor vR is in phase with the current phasor I The projection of I on the vertical axis represents the current flowing in the circuit as a function of time Capacitor The current iC leads the voltage vC by 90 so that the voltage phasor vC will have an angle 90 less than I and vR The angle of the phasor is given by t Inductor We can also describe the voltage in terms of a phasor V The current iL lags behind the voltage vL by so that voltage phasor vL will have an angle 90 greater than I and vR The time varying currents and voltages in the circuit can have different phases March 11 2005 Physics for Scientists Engineers 2 5 March 11 2005 Series RLC Circuit 3 Physics for Scientists Engineers 2 6 Series RLC Circuit 4 Kirchhof s loop rules tells that the voltage drops across all the devices at any given time in the circuit must sum to zero which gives us The voltage phasors for an RLC circuit are shown below V vR vC vL 0 V vR vC vL The voltage can be thought of as the projection of the vertical axis of the phasor Vmax representing the time varying emf in the circuit as shown below The instantaneous voltages across each of the components are represented by the projections of the respective phasors on the vertical axis March 11 2005 Physics for Scientists Engineers 2 In this figure we have replaced the sum of the two phasors VL and VC with the phasor VL VC 7 March 11 2005 Physics for Scientists Engineers 2 8 Series RLC Circuit 5 Series RLC Circuit 6 The sum of the two phasors VL VC and VR must equalVmax so 2 max V V VL VC 2 R The current flowing in an alternating current circuit depends on the difference between the inductive reactance and the capacitive reactance 2 Now we can put in our expression for the voltage across the components in terms of the current and resistance or reactance 2 Vmax IR IX L IXC 2 We can express the difference between the inductive reactance and the capacitive reactance in terms of the phase constant 2 We can then solve for the current in the circuit I Vmax R 2 X L XC This phase constant is defined as the phase difference between voltage phasors VR and VL VC 2 The denominator in the equation is called the impedance Z R 2 X L XC V V X XC tan 1 L C tan 1 L R VR 2 The impedance of a circuit depends on the frequency of the timevarying emf March 11 2005 Physics for Scientists Engineers 2 9 March 11 2005 Physics for Scientists Engineers 2 Series RLC Circuit 7 10 Series RLC Circuit 8 Thus we have three conditions for an alternating current circuit For XL XC is positive and the current in the circuit will lag behind the voltage in the circuit XL XC This circuit will be similar to a circuit with only an inductor except that the phase constant is not necessarily 90 XL XC XL XC For XL XC is negative and the current in the circuit will lead the voltage in the circuit This circuit will be similar to a circuit with only a capacitor except that the phase constant is not necessarily 90 For XL XC is zero and the current in the circuit will be in phase with the voltage in the circuit This circuit is similar to a circuit with only a resistance L When 0 we say that the circuit is in resonance March 11 2005 Physics for Scientists Engineers 2 For XL XC and 0 we get the maximum current in the circuit and we can define a resonant frequency 11 March 11 2005 1 0 C 0 1 LC Physics for Scientists Engineers 2 12 Real life RLC Circuit 2 Resonant Behavior of RLC Circuit The resonant behavior of an RLC circuit resembles the response of a damped oscillator Let s study a real life circuit R 10 L 8 2 mH C 100 F Vmax 7 5 V We measure the current in the circuit as a function of the frequency of the timevarying emf We see the correct resonant frequency peak at 0 1 However our formula for the current green line using R 10 does not agree with the measurements Here we show the calculated maximum current as a function of the ratio of the angular frequency of the time varying emf divided by the resonant angular frequency for a circuit with Vmax 7 5 V L 8 2 mH C 100 F and three resistances One can see that as the resistance is lowered the maximum current at the resonant angular frequency increases and there is a more pronounced resonant peak L and C must be accurate We must use R 15 4 The inductor has a resistance even at resonance March 11 2005 Physics for Scientists Engineers 2 13 March 11 2005 Energy and Power in RLC Circuits Physics for Scientists Engineers 2 Energy and Power 2 When an RLC circuit is in operation some of the energy in the circuit is stored in the electric field of the capacitor some of the energy is stored in the magnetic field of the inductor and some energy is dissipated in the form of heat in the resistor We define the root mean square rms current to be I rms 2 So we can write the average power as P I rms R rms voltage Therefore the energy transferred from the source of emf to the circuit is transferred to the resistor rms time varying emf …
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