PHY2049: Chapter 311Chapter 31: RLC CircuitsPHY2049: Chapter 312TopicsÎLC Oscillations Conservation of energyÎDamped oscillations in RLC circuits Energy lossÎAC current RMS quantitiesÎForced oscillations Resistance, reactance, impedance Phase shift Resonant frequency PowerÎTransformers Impedance matchingPHY2049: Chapter 313LC OscillationsÎWork out equation for LC circuit (loop rule)ÎRewrite using i = dq/dt ω (angular frequency) has dimensions of 1/tÎIdentical to equation of mass on springLC0qdiLCdt−− =2222200dq q dqLqCdt dtω+=⇒ + =2222200dx dxmkx xdt dtω+=⇒ + =1LCω=kmω=PHY2049: Chapter 314LC Oscillations (2)ÎSolution is same as mass on spring ⇒ oscillations qmaxis the maximum charge on capacitor θ is an unknown phase (depends on initial conditions)ÎCalculate current: i = dq/dtÎThus both charge and current oscillate Angular frequency ω, frequency f = ω/2π Period: T = 2π/ω Current and charge differ in phase by 90°()maxcosqq tωθ=+()()max maxsin siniq t i tωωθ ωθ=− + =− +kmω=PHY2049: Chapter 315Plot Charge and Current vs t()maxcosqq tω=()maxsinii tω=−ωtPHY2049: Chapter 316Energy Oscillations in LC CircuitsÎTotal energy in circuit is conserved. Let’s see why0di qLdt C+=0di q dqLidt C dt+=Multiply by i = dq/dtEquation of LC circuit()()221022Ld diqdt C dt+=Use22dx dxxdt dt=2211220dqLidt C⎛⎞+=⎜⎟⎜⎟⎝⎠UL+ UC= const221122constqLiC+=PHY2049: Chapter 317Oscillation of EnergiesÎEnergies can be written as (using ω2= 1/LC)ÎConservation of energy: ÎEnergy oscillates between capacitor and inductor Endless oscillation between electrical and magnetic energy Just like oscillation between potential energy and kinetic energy for mass on spring()222maxcos22CqqUtCCωθ== +() ()22222 2max11max22sin sin2LqULiLq t tCωωθ ωθ== += +2maxconst2CLqUUC+= =PHY2049: Chapter 318Plot Energies vs t()CUt()LUtSumPHY2049: Chapter 319LC Circuit ExampleÎParameters C = 20μF L = 200 mH Capacitor initially charged to 40V, no current initiallyÎCalculate ω, f and T ω = 500 rad/s f = ω/2π = 79.6 Hz T = 1/f = 0.0126 secÎCalculate qmaxand imaxqmax= CV = 800 μC = 8 × 10-4C imax= ωqmax= 500 × 8 × 10-4 = 0.4 AÎCalculate maximum energies UC= q2max/2C = 0.016J UL= Li2max/2 = 0.016J()()51/ 1/ 2 10 0.2 500LCω−==× =PHY2049: Chapter 3110LC Circuit Example (2)ÎCharge and currentÎEnergiesÎVoltagesÎNote how voltages sum to zero, as they must!()0.0008cos 500qt=()0.4sin 500dqitdt==−()()220.016cos 500 0.016sin 500CLUtUt==()/ 40cos 500CVqC t==()()max/ cos 500 40cos 500LV Ldi dt L i t tω==− =−PHY2049: Chapter 3111QuizÎBelow are shown 3 LC circuits. Which one takes the least time to fully discharge the capacitors during the oscillations? (1) A (2) B (3) CA B CCCCCC1/LCω=C has smallest capacitance, therefore highestfrequency, therefore shortest periodPHY2049: Chapter 3112RLC CircuitÎThe loop rule tells usÎUse i = dq/dt, divide by LÎSolution slightly more complicated than LC caseÎThis is a damped oscillator (similar to mechanical case) Amplitude of oscillations falls exponentially0di qLRidt C++=220dq Rdq qLdt LCdt++=() ()2/2maxcos 1/ / 2tR Lqq e t LC RLωθω−′′=+=−PHY2049: Chapter 3113Charge and Current vs t in RLC Circuit()qt()it/2tR Le−PHY2049: Chapter 3114RLC Circuit ExampleÎCircuit parameters L = 12mL, C = 1.6μF, R = 1.5ΩÎCalculate ω, ω’, f and T ω = 7220 rad/s ω’ = 7220 rad/s f = ω/2π = 1150 Hz T = 1/f = 0.00087 secÎTime for qmaxto fall to ½ its initial value t = (2L/R) * ln2 = 0.0111s = 11.1 ms # periods = 0.0111/.00087 ≈ 13()()61/ 0.012 1.6 10 7220ω−=×=()227220 1.5/0.024ωω′=− /21/2tR Le−=PHY2049: Chapter 3115RLC Circuit (Energy)0di qLRidt C++=Basic RLC equationMultiply by i = dq/dt20di q dqLiRidt C dt++ =2221122dqLi i Rdt C⎛⎞+=−⎜⎟⎜⎟⎝⎠Collect terms(similar to LC circuit)()2LCdUU iRdt+=−Total energy in circuitdecreases at rate of i2R(dissipation of energy)/tottR LUe−∼PHY2049: Chapter 3116Energy in RLC Circuit()CUt()LUtSum/tR
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