AC CircuitsLecture OutlinePhasorsPhasors for L,C,RSeries LCR AC CircuitPhasors: LCRSlide 7Lecture 20, ACT 3Conceptual QuestionPhasors:LCRPhasors:TipsSlide 12Lagging & LeadingSlide 14Lecture 21, ACT 1Slide 16ResonanceSlide 18Power in LCR CircuitSlide 20Slide 21Power in RLCThe Q factorSlide 24Conceptual Question 2Lecture 21, ACT 2Slide 27Power TransmissionPowerPoint PresentationTransformersIdeal Transformers (no load)Ideal TransformersTransformers with a LoadLecture 21, ACT 3Slide 35Slide 36Slide 37Slide 38Physics 1304: Lecture 18, Pg 1AC CircuitsAC Circuits imRimL imCmN2N1(primary)(secondary)ironV2V1x ..,0.0r1nr10 1 200.51f( )xg( )xxim002omR0R=RoR=2RoPhysics 1304: Lecture 18, Pg 2Lecture OutlineLecture OutlineDriven Series LCR Circuit:•General solution•Resonance condition»Resonant frequency»“Sharpness of resonance” = Q•Power considerations»Power factor depends on impedanceTransformers•Voltage changes•Faraday’s Law in action gives induced primary current.•Power considerationsText Reference: Chapter 33.4-6Physics 1304: Lecture 18, Pg 3PhasorsPhasorsA phasor is a vector whose magnitude is the maximum value of a quantity (eg V or I) and which rotates counterclockwise in a 2-d plane with angular velocity . Recall uniform circular motion:The projections of r (on the vertical y axis) execute sinusoidal oscillation. iLtLmcosi C tC m cosiRtRmsinx r t cosy r t sinV Ri tR R m sin•R: V in phase with iVQCtC m sin•C: V lags i by 90V LdidttLLm sin•L: V leads i by 90xyyPhysics 1304: Lecture 18, Pg 4Phasors for Phasors for L,C,RL,C,RitVRitVLitVCSuppose:tiimsinV Ri tR m sinVCi tC m1cosV Li tL m costiVR0VC0ii0VLPhysics 1304: Lecture 18, Pg 5Series LCRSeries LCRAC CircuitAC Circuit•Back to the original problem: the loop equation gives:Here all unknowns, (im,) , must be found from the loop eqn; the initial conditions have been taken care of by taking the emf to be: m sint. •To solve this problem graphically, first write down expressions for the voltages across R,C, and L and then plot the appropriate phasor diagram.LCRLd QdtQCRdQdttm22 sin•Assume a solution of the form:i i tm sin( ) Phasors: LCRPhasors: LCR•Assume:From these equations, we can draw the phasor diagram to the right. imRimL imCmLCR•Given: mtsini i tm sin( ) Qitm cos( )didti tm cos( )V Ri Ri tR m sin( ) V LdidtLi tL m cos( )VQC Ci tC m 1 cos( )This picture corresponds to a snapshot at t=0. The projections of these phasors along the vertical axis are the actual values of the voltages at the given time.Physics 1304: Lecture 18, Pg 7Phasors: LCRPhasors: LCR•The phasor diagram has been relabeled in terms of the reactances defined from: imRmimXCimXLLCRXCC1X LL The unknowns (im,) can now be solved for graphically since the vector sum of the voltages VL + VC + VR must sum to the driving emf.Physics 1304: Lecture 18, Pg 8Lecture 20, ACT 3Lecture 20, ACT 3A driven RLC circuit is connected as shown. For what frequencies of the voltage source is the current through the resistor largest?(a) small(b) large(c) 1LCLCRConceptual Conceptual QuestionQuestionA driven RLC circuit is connected as shown. For what frequencies of the voltage source is the current through the resistor largest?(a) small(b) large(c)LCR 1LC• This is NOT a series RLC circuit. We cannot blindly apply our techniques for solving the circuit. We must think a little bit.• However, we can use the frequency dependence of the impedances (reactances) to answer this question.• The reactance of an inductor = XL = L. • The reactance of a capacitor = XC = 1/(C).• Therefore, • in the low frequency limit, XL 0 and XC . • Therefore, as 0, the current will flow mostly through the inductor; the current through the capacitor approaches 0.• in the high frequency limit, XL and XC 0. • Therefore, as , the current will flow mostly through the capacitor, approaching a maximum imax = /R.Physics 1304: Lecture 18, Pg 10Phasors:LCRPhasors:LCRimRmim(XL-XC)imRmimXCimXLXCC1X LLZ R X XL C 22tan X XRL Cm m L Ci R X X2 2 22 iR X XZmmL Cm 22Phasors:TipsPhasors:Tips• This phasor diagram was drawn as a snapshot of time t=0 with the voltages being given as the projections along the y-axis.imRmimXCimXLyximRimXLimXCm“Full Phasor Diagram”From this diagram, we can also create a triangle which allows us to calculate the impedance Z:X XL CZR“ Impedance Triangle”• Sometimes, in working problems, it is easier to draw the diagram at a time when the current is along the x-axis (when i=0).Physics 1304: Lecture 18, Pg 12Phasors:LCRPhasors:LCRWe have found the general solution for the driven LCR circuit:X LLXCC1Z R X XL C 22RXLXCtan X XRL CiZmm i Zmthe loop eqnXL - XCi i tm sin( ) imRimXLimXCmLagging & LeadingLagging & LeadingThe phase between the current and the driving emf depends on the relative magnitudes of the inductive and capacitive reactances.RXLXCtan X XRL CiZmmX LLXCC1XL > XC > 0 current LAGS applied voltageRXLXCXL < XC < 0 current LEADSapplied voltageXL = XC = 0 current IN PHASE applied voltageRXLXCPhysics 1304: Lecture 18, Pg 14Conceptual Conceptual QuestionQuestionThe series LCR circuit shown is driven by a generator with voltage = msint. The time dependence of the current i which flows in the circuit is shown in the plot.How should be changed to bring the current and driving voltage into phase?(a) increase (b) decrease (c) impossible• Which of the following phasors represents the current i at t=0?1B1ALCR0x ..,0r1nr12 4 61011.011.01f( )x6.81.53 xioim-im0it(a)(b)(c)iiiPhysics 1304: Lecture 18, Pg 15Lecture 21, ACT
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