Chapter 31: RLC Circuits

PHY2049: Chapter 31 1 Topics

ÎLC Oscillations ‹ Conservation of energy ÎDamped oscillations in RLC circuits ‹ Energy loss ÎAC current ‹ RMS quantities ÎForced oscillations ‹ Resistance, reactance, impedance ‹ Phase shift ‹ Resonant frequency ‹ Power ÎTransformers ‹ Impedance matching

PHY2049: Chapter 31 2 LC Oscillations

ÎWork out equation for LC circuit (loop rule) qdi C −−L =0 L Cdt

ÎRewrite using i = dq/dt dq22 q dq 1 Lq+=⇒00 +ω 2 = ω = dt22C dt LC

‹ ω (angular frequency) has dimensions of 1/t

ÎIdentical to equation of mass on spring dx22 dx k mkx+=⇒00 +ω 2 x = ω = dt22 dt m

PHY2049: Chapter 31 3 LC Oscillations (2)

ÎSolution is same as mass on spring ⇒ oscillations k qq=+max cos()ω tθ ω = m

‹ qmax is the maximum charge on capacitor ‹ θ is an unknown phase (depends on initial conditions)

ÎCalculate current: i = dq/dt

iq=−ω maxsin(ωθ t + ωθ) =− i max sin( t + )

ÎThus both charge and current oscillate ‹ Angular frequency ω, frequency f = ω/2π ‹ Period: T = 2π/ω ‹ Current and charge differ in phase by 90°

PHY2049: Chapter 31 4 Plot Charge and Current vs t qq= max cos(ω t)

ii= − max sin(ω t)

ωt

PHY2049: Chapter 31 5 Energy Oscillations in LC Circuits

ÎTotal energy in circuit is conserved. Let’s see why di q L +=0 Equation of LC circuit dt C

di q dq Li+=0 Multiply by i = dq/dt dt C dt

2 Ld221 d dx dx (iq)+=( ) 0 Use = 2x 22dt C dt dt dt

2 2 dq⎛⎞ 112 q 11Li2 +=0 Li +=const ⎜⎟22 22C dt⎝⎠ C UL + UC = const PHY2049: Chapter 31 6 Oscillation of Energies

ÎEnergies can be written as (using ω2 = 1/LC) q2 q2 Ut==max cos2 ()ω +θ C 22CC q2 ULiLq==112222ω sin()ωθ t += ωθmax sin 2() t + L 22max 2C q2 ÎConservation of energy: UU+=max =const CL2C

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

PHY2049: Chapter 31 7 Plot Energies vs t Sum UtC ( ) UtL ( )

PHY2049: Chapter 31 8 LC Circuit Example

ÎParameters ‹ C = 20μF ‹ L = 200 mH ‹ Capacitor initially charged to 40V, no current initially

ÎCalculate ω, f and T −5 ‹ ω = 500 rad/s ω ==×1/LC 1/() 2 10() 0.2 = 500 ‹ f = ω/2π = 79.6 Hz ‹ T = 1/f = 0.0126 sec

ÎCalculate qmax and imax -4 ‹ qmax = CV = 800 μC = 8 × 10 C -4 ‹ imax = ωqmax = 500 × 8 × 10 = 0.4 A ÎCalculate maximum energies 2 2 ‹ UC = q max/2C = 0.016J UL = Li max/2 = 0.016J

PHY2049: Chapter 31 9 LC Circuit Example (2)

ÎCharge and current dq qt= 0.0008cos( 500 ) it==−0.4sin() 500 dt ÎEnergies 22 UtUtCL==0.016cos( 500) 0.016sin( 500 ) ÎVoltages

VqCC ==/ 40cos( 500 t)

VL ==− Ldi/ dt Lω imax cos( 500 t) =− 40cos( 500 t) ÎNote how voltages sum to zero, as they must!

PHY2049: Chapter 31 10 Quiz

ÎBelow are shown 3 LC circuits. Which one takes the least time to fully discharge the capacitors during the oscillations? ‹ (1) A ‹ (2) B ‹ (3) C C C C C C

A B C

C has smallest capacitance, therefore highest ω =1/ LC frequency, therefore shortest period

PHY2049: Chapter 31 11 RLC Circuit

ÎThe loop rule tells us di q LRi++=0 dt C

ÎUse i = dq/dt, divide by L dq2 Rdq q ++=0 dt2 Ldt LC

ÎSolution slightly more complicated than LC case

−tR/2 L 2 qq=+=−max ecos()ωθω′′ t 1/ LCRL / 2 ()

ÎThis is a damped oscillator (similar to mechanical case) ‹ Amplitude of oscillations falls exponentially

PHY2049: Chapter 31 12 Charge and Current vs t in RLC Circuit

qt( ) it( ) e−tR/2 L

PHY2049: Chapter 31 13 RLC Circuit Example

ÎCircuit parameters ‹ L = 12mL, C = 1.6μF, R = 1.5Ω

ÎCalculate ω, ω’, f and T ω =×=1/ 0.012 1.6 10−6 7220 ‹ ω = 7220 rad/s ()( ) 2 ‹ ω’ = 7220 rad/s ω′ =−72202 () 1.5/0.024 ω ‹ f = ω/2π = 1150 Hz ‹ T = 1/f = 0.00087 sec −tR/2 L ÎTime for qmax to fall to ½ its initial value e =1/2 ‹ t = (2L/R) * ln2 = 0.0111s = 11.1 ms ‹ # periods = 0.0111/.00087 ≈ 13

PHY2049: Chapter 31 14 RLC Circuit (Energy)

di q LRi++=0 Basic RLC equation dt C

di q dq LiRi++2 =0 Multiply by i = dq/dt dt C dt

dq⎛⎞2 Collect terms 11Li22+=− i R ⎜⎟22 (similar to LC circuit) dt⎝⎠ C Total energy in circuit d 2 ()UU+=− iR decreases at rate of i2R dt LC (dissipation of energy) −tR/ L Uetot ∼ PHY2049: Chapter 31 15 Energy in RLC Circuit

UtC ( ) UtL ( ) Sum

e−tR/ L

PHY2049: Chapter 31 16