Ch. 31: Electrical Oscillations, LC Circuits, Alternating Current
Physics 2102
Jonathan Dowling
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• Electric charge Electric force on other electric charges Electric field, and electric potential • Moving electric charges : current • Electronic circuit components: batteries, resistors, capacitors • Electric currents Magnetic field Magnetic force on moving charges • Time-varying magnetic field Electric Field • More circuit components: inductors. • Electromagnetic waves light waves • Geometrical Optics (light rays). • Physical optics (light waves) OOssccillaillattoorrss inin PPhhyyssicicss Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system. All oscillators operate along the same principle: they are systems that can store energy in more than one way and exchange it back and forth between the different storage possibilities. For instance, in pendulums (and swings) one exchanges energy between kinetic and potential form.
In this course we have studied that coils and capacitors are devices that can store electromagnetic energy. In one case it is stored in a magnetic field, in the other in an electric field. AA mmeecchhaannicicaall oossccillaillattoorr
1 2 1 2 Etot = Ekin + E pot E = mv + k x tot 2 2 dx dEtot 1 & dv # 1 & dx # v = 0 = m$2v ! + k$2x ! = dt 2 % dt " 2 % dt " dt
dv 2 Newton’s law m k x 0 d x + = m + k x = 0 F=ma! dt dt 2 k Solution : x(t) = x cos(" t +! ) ! = 0 0 m
x0 : amplitude " : frequency
!0 : phase AAnn eeleleccttrroommaaggnneetticic oossccillaillattoorr Capacitor initially charged. Initially, current is zero, energy is all stored in the capacitor.
A current gets going, energy gets split between the capacitor and the inductor.
Capacitor discharges completely, yet current keeps going. Energy is all in the inductor.
The magnetic field on the coil starts to collapse, which will start to recharge the capacitor.
Finally, we reach the same state we started with (with opposite polarity) and the cycle restarts. EEleleccttrricic OOssccillaillattoorrss:: tthhee mmaatthh 1 1 q 2 E = E + E E = Li 2 + tot mag elec tot 2 2 C
dEtot 1 & di # 1 & dq # dq = 0 = L$2i ! + $2q ! i = dt 2 % dt " 2C % dt " dt & di # 1 0 = L$ ! + (q) (the loop rule!) % dt " C 2 d q q Compare with: d 2 x 0 = L + m + k x = 0 dt 2 C dt 2 Analogy between electrical and mechanical oscillations: x(t) = x cos(" t +! ) q = q cos(" t +! ) q ! x 1/C ! k 0 0 0 0 k 1 i ! v L ! M ! = ! = m LC EEleleccttrricic OOssccillaillattoorrss:: tthhee mmaatthh q q cos( t ) 1.5 = 0 " +!0
1 dq i = = #" q0 sin(" t +!0 ) 0.5 dt Charge 0 Current Time -0.5 1 2 1 2 2 2 Emag = Li = L" q0 sin (" t +!0 ) -1 2 2 2 -1.5 1 q 1 2 E = = q cos2 (" t +! ) ele 2 C 2C 0 0
1.2
1 And remembering that,
0.8 2 2 1 Energy in capacitor 0.6 cos x + sin x = 1, and ! = Energy in coil LC 0.4
0.2 1 2 0 Etot = Emag + Eele = q0 Time 2C The energy is constant and equal to what we started with.
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FM radio stations: frequency The inductor and capacitor is in MHz. in my car radio are usually 1 # = set at L = 1 mH & C = 3.18 LC pF. Which is my favorite 1 FM station? = rad / s 1!10"6 !3.18!10"12 = 5.61!108 rad / s (a) QLSU 91.1 # (b) WRKF 89.3 f = 2" (c) Eagle 98.1 WDGL = 8.93!107 Hz = 89.3MHz ExampleExample 22 • In an LC circuit, L = 40 mH; C = 4 F 1 1 µ " = = rad / s • At t = 0, the current is a LC 16x10!8 maximum; • ω = 2500 rad/s • When will the capacitor be • T = period of one complete cycle fully charged for the first = 2π/ω = 2.5 ms time? • Capacitor will be charged after 1/4 cycle i.e at t = 0.6 ms
1.5
1
0.5 Charge 0 Current Time -0.5
-1
-1.5 EExxaammpplele 33
• In the circuit shown, the switch is in position “a” for a long time. 1 mH E=10 V 1 µF It is then thrown to position “b.” a • Calculate the amplitude of the b resulting oscillating current.
i = #" q0 sin(" t +!0 ) • Switch in position “a”: charge on cap = (1 µF)(10 V) = 10 µC • Switch in position “b”: maximum charge on C = q0 = 10 µC • So, amplitude of oscillating current = 1 !q0 = (10µC) = 0.316 A (1mH )(1µF) EExxaammpplele 44 In an LC circuit, the maximum current is 1.0 A. If L = 1mH, C = 10 µF what is the maximum charge on the capacitor during a cycle of oscillation?
q = q0 cos(" t +!0 ) dq i = = #" q sin(" t +! ) dt 0 0
Maximum current is i0=ωq0 → q0=i0/ω
Angular frequency ω=1/√LC=(1mH 10 µF)-1/2 = (10-8)-1/2 = 104 rad/s
4 -4 Maximum charge is q0=i0/ω = 1A/10 rad/s = 10 C DDaammppeedd LLCC OOssccillaillattoorr
Ideal LC circuit without resistance: C oscillations go on for ever; = ω L (LC)-1/2 R Real circuit has resistance, dissipates Rt energy: oscillations die out, or are 2 ! Q L “damped” 1.0 U = e max 2C Math is complicated! Important points: 0.8 – Frequency of oscillator shifts away from = (LC)-1/2 0.6 ω E U – Peak CHARGE decays with time 0.4 constant = 2L/R 0.2 – For small damping, peak ENERGY decays with time constant = L/R 0.0 0 4 8 12 16 20 time (s) SSuummmmaarryy • Capacitor and inductor combination produces an electrical oscillator, natural frequency of oscillator is ω=1/√LC • Total energy in circuit is conserved: switches between capacitor (electric field) and inductor (magnetic field). • If a resistor is included in the circuit, the total energy decays (is dissipated by R).