Chapter 31: RLC Circuits
Chapter 31: RLC CircuitsPHY2049: Chapter 311
TopicsÎLCOscillations ConservationÎDamped EnergyÎACof energyoscillations in RLC circuitslosscurrent RMSÎForcedquantitiesoscillations Resistance,reactance, impedance Phaseshift Resonant frequency PowerÎTransformers ImpedancematchingPHY2049: Chapter 312
LC OscillationsÎWorkout equation for LC circuit (loop rule)qdi L 0CdtÎRewriteCusing i dq/dtd 2qqd 2q2 ωL 2 0 q 02Cdtdt ωLω 1LC(angular frequency) has dimensions of 1/tÎIdenticalmd 2xdt2to equation of mass on spring kx 0 d 2xdt2 ω x 02PHY2049: Chapter 31kω m3
LC Oscillations (2)is same as mass on spring oscillationskq qmax cos (ω t θ )ω mÎSolution qmaxis the maximum charge on capacitor θ is an unknown phase (depends on initial conditions)ÎCalculatecurrent: i dq/dti ω qmax sin (ω t θ ) imax sin (ω t θ )ÎThusboth charge and current oscillatefrequency ω, frequency f ω/2π Period: T 2π/ω Current and charge differ in phase by 90 AngularPHY2049: Chapter 314
Plot Charge and Current vs tq qmax cos (ωt )i imax sin (ωt )ωtPHY2049: Chapter 315
Energy Oscillations in LC CircuitsÎTotalenergy in circuit is conserved. Let’s see whydi qL 0Equation of LC circuitdt Cdiq dqL i 0dt C dt( )Multiply by i dq/dt( )Ld 21 d 2i q 02 dt2C dtd 1 2 1 q2 2 Li 2 0dt C dx 2dxUse 2xdtdt2q1 Li 2 1 const22 CUL UC constPHY2049: Chapter 316
Oscillation of EnergiesÎEnergiescan be written as (using ω2 1/LC)2q 2 qmaxUC cos 2 (ω t θ )2C2C2q2U L 12 Li 2 12 Lω 2 qmaxsin 2 (ω t θ ) max sin 2 (ω t θ )2C2qmaxÎConservation of energy: U C U L const2CÎEnergyoscillates between capacitor and inductor Endlessoscillation between electrical and magnetic energy Just like oscillation between potential energy and kinetic energyfor mass on springPHY2049: Chapter 317
UC (t )Plot Energies vs tU L (t )PHY2049: Chapter 31Sum8
LC Circuit ExampleÎParameters C 20μF L 200 mH Capacitor initially charged to 40V, no current initiallyÎCalculate ωω, f and T 500 rad/s f ω/2π 79.6 Hz T 1/f 0.0126 secÎCalculateω 1/ LC 1/( 2 10 5 ) ( 0.2 ) 500qmax and imax CV 800 μC 8 10-4 C ωqmax 500 8 10-4 0.4 A qmax imaxÎCalculate UCmaximum energies q2max/2C 0.016J UL Li2max/2 0.016JPHY2049: Chapter 319
LC Circuit Example (2)ÎChargeand currentq 0.0008cos ( 500t )dqi 0.4sin ( 500t )dtÎEnergiesU C 0.016cos 2 ( 500t )U L 0.016sin 2 ( 500t )ÎVoltagesVC q / C 40cos ( 500t )VL Ldi / dt Lω imax cos ( 500t ) 40cos ( 500t )ÎNotehow voltages sum to zero, as they must!PHY2049: Chapter 3110
QuizÎBeloware shown 3 LC circuits. Which one takes the leasttime to fully discharge the capacitors during theoscillations? (1)A (2) B (3) CCAω 1/ LCCCCBCCC has smallest capacitance, therefore highestfrequency, therefore shortest periodPHY2049: Chapter 3111
RLC CircuitÎTheloop rule tells usdiqL Ri 0dtCÎUsei dq/dt, divide by Ld 2qR dq q 02L dt LCdtÎSolutionq qmax eÎThisslightly more complicated than LC case tR / 2 Lcos (ω ′t θ ) ω ′ 1/ LC ( R / 2 L )2is a damped oscillator (similar to mechanical case) Amplitudeof oscillations falls exponentiallyPHY2049: Chapter 3112
Charge and Current vs t in RLC Circuitq (t )i (t )ePHY2049: Chapter 31 tR / 2 L13
RLC Circuit ExampleÎCircuit Lparameters 12mL, C 1.6μF, R 1.5ΩÎCalculate ωω, ω’, f and T 7220 rad/s ω’ 7220 rad/s f ω/2π 1150 Hz T 1/f 0.00087 secÎTimeω 1/( 0.012 ) (1.6 10 6 ) 7220ω ′ 72202 (1.5/ 0.024 )2ωfor qmax to fall to ½ its initial value e tR / 2 L 1/ 2 t (2L/R) * ln2 0.0111s 11.1 ms # periods 0.0111/.00087 13PHY2049: Chapter 3114
RLC Circuit (Energy)diqL Ri 0dtCBasic RLC equationdiq dq2L i Ri 0dtC dtMultiply by i dq/dtd 1 2 1 q2 2Li iR 2 2dt C Collect terms(similar to LC circuit)d(U L U C ) i 2 RdtTotal energy in circuitdecreases at rate of i2R(dissipation of energy)U tot e tR / LPHY2049: Chapter 3115
Energy in RLC CircuitUC (t )U L (t )Sume tR / LPHY2049: Chapter 3116
C 20μF L 200 mH Capacitor initially charged to 40V, no current initially ÎCalculate ω, f and T ω 500 rad/s f ω/2π 79.6 Hz T 1/f 0.0126 sec ÎCalculate q max and i max q max CV 800 μC 8 10-4 C i max ωq max 500 8 10-4 0.4 A ÎCalculate maximum energies U C q2 max/2C 0.016J U L Li 2 max/2 0.016J
Lab. Supervisor: 1 Experiment No.14 Object To perform be familiar with The Parallel RLC Resonance Circuit and their laws. Theory the analysis of a parallel RLC circuits can be a little more mathematically difficult than for series RLC circuits so in this tutorial about parallel RLC circuits only pure components are assumed in this tutorial to .
Received December 2015. Volume 8, Number 4, 2015 TEACHING RLC PARALLEL CIRCUITS IN HIGH-SCHOOL PHYSICS CLASS Alpár Simon Abstract: This paper will try to give an alternative treatment of the subject "parallel RLC circuits" and "resonance in parallel RLC circuits" from the Physics curricula for the XIth grade from Romanian high-schools,
Parallel RLC Circuit The RLC circuit shown on Figure 6 is called the parallel RLC circuit. It is driven by the DC current source Is whose time evolution is shown on Figure 7. Is R L C iL(t) v -iR(t) iC(t) Figure 6 t Is 0 Figure 7 Our goal is to determine the current iL(t) and
driven RLC circuit coupled in re ection. (b) A capactively driven RLC circuit coupled in re ection. (c) An inductively driven RLC circuit coupled to a feedline. (d) A capacitively driven RLC circuit coupled to a feedline. (e) A capacitively driven RLC tank circuit coupled in transmission.
Figure 7.1: A series RLC circuit The series RLC circuit has three possible output voltages. These are the three voltages across the three components of the circuit. We can use an impedance analysis to determine the gain of the circuit. The voltage divider analysis, given by Equation 6.1 of the previous lab, is also applicable to RLC circuits.
Lab Report 2 RLC Circuits Author: Muhammad Obaidullah 1030313 Mirza Mohsin 1005689 Ali Raza 1012542 Bilal Arshad 1011929 Supervisor: Dr. Montasir Qasymeh Section 1 October 12, 2012. Abstract In this lab we were educated in series and parallel RLC circuit analysis and achieving reso-nance frequency in a series RLC circuit. 1 Introduction When we .
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
Contemporary Electric Circuits, 2nd ed., Prentice-Hall, 2008 Class Notes Ch. 9 Page 1 Strangeway, Petersen, Gassert, and Lokken CHAPTER 9 Series–Parallel Analysis of AC Circuits Chapter Outline 9.1 AC Series Circuits 9.2 AC Parallel Circuits 9.3 AC Series–Parallel Circuits 9.4 Analysis of Multiple-Source AC Circuits Using Superposition 9.1 AC SERIES CIRCUITS
