hapter 31: RL ircuits PHY049: hapter 31 1
L Oscillations onservation of energy Topics Damped oscillations in RL circuits Energy loss A current RMS quantities Forced oscillations Resistance, reactance, impedance Phase shift Resonant frequency Power Transformers Impedance matching PHY049: hapter 31
L Oscillations Work out equation for L circuit (loop rule) q di L 0 = L Rewrite using i = dq/ d q q d q L + = 0 + ω q= 0 ω (angular frequency) has dimensions of 1/t Identical to equation of mass on spring d x d x m + kx= 0 + ω x= 0 ω = ω = 1 L k m PHY049: hapter 31 3
L Oscillations () Solution is same as mass on spring oscillations q= q t+ 1 q max is the maximum charge on capacitor θ is an unknown phase (depends on initial conditions) harge oscillation Angular frequency ω, frequency f = ω/π Period: max cos ( ω θ ) ω = L T = 1/ f = π / ω = π L Example: L =.0 H, = 0.5 μf ( ) 6 6 3 ω = 1/.0 0.5 10 = 1/ 10 = 10 rad/s f T = ω/ π = 1000 / 6.83 = 159Hz = 1/ f = 0.0063 sec PHY049: hapter 31 4
alculate current: i = dq/ L Oscillations (3) Thus both charge and current oscillate Same angular frequency ω, frequency f = ω/π Period: max q= q t+ max cos ( ω θ ) ( ) sin( ) i = ωq sin ωt+ θ i ωt+ θ T = π / ω = π L max urrent and charge differ in phase by 90 max More on that later! ( ω θ) ( ω θ π ) i = i sin t+ = i cos t+ + / max PHY049: hapter 31 5
Plot harge and urrent vs t q= q ωt max cos ( ) i = i ωt max sin ( ) ωt PHY049: hapter 31 6
Energy Oscillations in L ircuits Total energy in circuit is conserved. Let s see why di q L + = 0 Equation of L circuit L di i q dq + = 0 Multiply by i = dq/ Ld ( ) 1 d ( i + q ) = 0 Use 1 dx = x dx d 1 Li 1 q + = U L U 0 1 Li 1 q + = const U L + U = const PHY049: hapter 31 7
Oscillation of Energies Energies can be written as (using ω = 1/L) U max cos q q = = + onservation of energy: ( ωt θ ) 1 1 q max sin max ( ) sin ( ) UL = Li = Lω q ωt+ θ = ωt+ θ max q + UL = = const Energy oscillates between capacitor and inductor U Endless oscillation between electrical and magnetic energy Just like oscillation between potential energy and kinetic energy for mass on spring PHY049: hapter 31 8
U t ( ) ( ) Plot Energies vs t U t Sum L PHY049: hapter 31 9
Parameters = 0μF L = 00 mh L ircuit Example apacitor initially charged to 40V, no current initially alculate ω, f and T ω = 500 rad/s f = ω/π = 79.6 Hz T = 1/f = 0.016 sec alculate q max and i max q max = V = 800 μ = 8 10-4 i max = ωq max = 500 8 10-4 = 0.4 A alculate maximum energies U = q max/ = 0.016J U L = Li max/ = 0.016J ( 5 )( ) ω = 1/ L = 1/ 10 0. = 500 PHY049: hapter 31 10
L ircuit Example () harge and current dq q 0.0008cos( 500t) i = = 0.4sin 500t Energies U tot = q max/ = 0.016J Voltages = ( ) ( ) ( ) L U = 0.016cos 500t U = 0.016sin 500t ( ) V = q/ = 40cos 500t max ( ) ( ) V = Ldi / = Lωi cos 500t = 40cos 500t L So voltages sum to zero, as they must! PHY049: hapter 31 11
Quiz Below are shown 3 L circuits. Which one takes the least time to fully discharge the capacitors during the oscillations? (1) A () B (3) (4) All are same A B ω = 1/ L tot has smallest capacitance, therefore highest frequency, therefore shortest period PHY049: hapter 31 1
The loop rule tells us di q L + Ri+ = 0 Use i = dq/, divide by L d q R dq q + + = L L 0 RL ircuit Solution slightly more complicated than L case tr /L ( ω θ) ω ( ) q= qmaxe cos t+ = 1/ L R/ L This is a damped oscillator (similar to mechanical case) Amplitude of oscillations falls exponentially PHY049: hapter 31 13
harge and urrent vs t in RL ircuit qt ( ) it ( ) / e tr L PHY049: hapter 31 14
ircuit parameters RL ircuit Example L = 1mH, = 1.6μF, R = 1.5Ω alculate ω, ω, f and T 6 ω = 70 rad/s ω = 1/ 0.01 1.6 10 = 70 ω = 70 rad/s f = ω/π = 1150 Hz T = 1/f = 0.00087 sec ( )( ) Time for q max to fall to ½ its initial value ( ) t = (L/R) * ln = 0.0111s = 11.1 ms # periods = 0.0111/.00087 13 So every 13 periods the charge amplitude falls by a factor of ω = 70 1.5/ 0.04 = 719.7 ω tr /L e = 1/ PHY049: hapter 31 15
L di RL ircuit (Energy) q + Ri+ = 0 Basic RL equation di q dq L i+ Ri + = 0 Multiply by i = dq/ d 1 1 q Li + = d ( U L + U ) = i R / Utot e tr L i R ollect terms (similar to L circuit) Total energy in circuit decreases at rate of i R (dissipation of energy) PHY049: hapter 31 16
Energy in RL ircuit U ( ) t U L ( ) Sum t / tr L e Energy time constant = L/R PHY049: hapter 31 17
A ircuits with RL omponents Enormous impact of A circuits Power delivery Radio transmitters and receivers Tuners Filters Transformers Basic components R L Driving emf Now we will study the basic principles PHY049: hapter 31 18
A ircuits and Forced Oscillations RL + driving EMF with angular frequency ω d ε = ε sin ω m d t di q L Ri msin + + = ε ω General solution for current is sum of two terms Transient : Falls exponentially & disappears Steady state : onstant amplitude Ignore tr / L i e cosω t PHY049: hapter 31 19