Physics 3 Jonathan Dowling Lecture 35: FRI 7 APR Electrical Oscillations, LC Circuits, Alternating Current I Nikolai Tesla
What are we going to learn? A road map Electric charge è Electric force on other electric charges è Electric field, and electric potential Moving electric charges : current Electronic circuit components: batteries, resistors, capacitors Electric currents è Magnetic field è Magnetic force on moving charges Time-varying magnetic field è Electric Field More circuit components: inductors. Electromagnetic waves è light waves Geometrical Optics (light rays. Physical optics (light waves
Energy Density in E and B Fields u E = ε E u B = B µ
Oscillators in Physics Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system. All oscillators 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. We have studied that inductors and capacitors are devices that can store electromagnetic energy. In the inductor it is stored in a B field, in the capacitor in an E field.
PHYS: A Mechanical Oscillator E = K +U = ENERGY E = mv + k x de dt = = dv m v dt + dx k x dt v = x (t a = v (t = x (t m dv dt + k x = d x m + k x = dt : x( t = x cos( ω t + Solution φ Newton s law F=ma! ω = k m x : ω : φ : amplitude frequency phase
PHYS3 An Electromagnetic LC Oscillator Capacitor initially charged. Initially, current is zero, energy is all stored in the E-field of the capacitor. Energy Conservation: U tot = U B +U E A current gets going, energy gets split between the capacitor and the inductor. U B = L i U E = q C Capacitor discharges completely, yet current keeps going. Energy is all in the B-field of the inductor all fluxed up. The magnetic field on the coil starts to deflux, which will start to recharge the capacitor. U tot = L i + q C Finally, we reach the same state we started with (with opposite polarity and the cycle restarts.
Electric Oscillators: the Math U tot = U B + U E U tot = L i + q C du tot dt = = di L i dt + C dq q dt Energy Cons. V L + V C = = L di dt + ( C q Or loop rule! Both give Diffy-Q: Solution to Diffy-Q: d q L + dt = q C i = q (t i (t = q (t q = q cos(ω t + ϕ i = q (t = q ω sin(ω t + ϕ ω LC LC Frequency In Radians/Sec i (t = q (t = ω q cos(ω t + ϕ
T a LC T b L(C T c L(C / T b > T a > T c ω LC eq T = π ω LC eq C a eq C b eq c C eq = C = C = C /
Electric Oscillators: the Math q = q cos(ω t + ϕ i(t = q (t = q ω sin(ω t + ϕ i (t = q (t = ω q cos(ω t + ϕ Energy as Function of Time Voltage as Function of Time U B = L [ i ] = L [ q ω sin(ω t + ϕ ] V L = L i (t = Lω q cos(ω t + ϕ U E = [ q] C = [ C q cos(ω t + ϕ ] V C = C q(t [ ] = C q cos(ω t + ϕ [ ]
LC Circuit: At t= /3 Of Energy U total is on Capacitor C and Two Thirds On Inductor L. Find Everything! (Phase φ =? U B (t = L [ q ω sin(ω t + ϕ ] U E ( t = [ C q cos(ω t + ϕ ] U B ( = L [ q ω sin(ϕ ] = U total / 3 U E ( = [ C q cos(ϕ ] = U total / 3 U B ( U E ( = L q ω sin(ϕ C q cos(ϕ = U total / 3 U total / 3 LC q ω sin(ϕ = q cos(ϕ ω = / q = VC LC tan(ϕ = ϕ = arctan ( / = 35.3 q = q cos(ω t + ϕ i(t = q ω sin(ω t + ϕ i (t = ω q cos(ω t + ϕ V L (t = q C cos(ω t + ϕ V C (t = q C cos(ω t + ϕ
Analogy Between Electrical And Mechanical Oscillations d q q =L + dt C ω= LC d x m +k x = dt k ω= m q = q cos(ω t + ϕ x(t = x cos(ω t + φ i = q (t = qω sin(ω t + ϕ v = x (t = ω x sin(ω t + ϕ i (t = q (t = ω q cos(ω t + ϕ a = x (t = ω x cos(ω t + ϕ q x i v /C k L m Charqe q -> Position x Current i=q -> Velocity v=x Dt-Current i =q -> Acceleration a=v =x
LC Circuit: Conservation of Energy.5.5 Charge Current Time -.5 q = q cos( ω t + φ dq i = = ω q sin( ω t + φ dt U B = Li = Lω q sin (ω t + ϕ - -.5..8.6.4. Time Energy in capacitor Energy in coil U E = cos q C = C q cos (ω t + ϕ And remembering that, x + sin x =, and ω = U tot = U B + U E = C q LC The energy is constant and equal to what we started with.
LC Circuit: Phase Relations.5.5 Time -.5 - -.5 Charge Current q = q cos( ω t + φ dq i = = ω q sin( ω t + φ dt Take ϕ = as origin of time. q cos(ω t i sin(ω t Trigamarole: sin(ω t π / = cos(ω t The current runs 9 out of phase with respect to the charge.
.5.5 ω LC Time -.5 - T = π ω Charge Current -.5 t = T t = T / 4 t = T / t = 3T / 4 t = T
.5.5 Time -.5 Charge Current - -.5 (a T / t = (b T V c = q / C t = T / 4 t = T (c T / t = T / (d T / 4 t = 3T / 4
Example : Tuning a Radio Receiver The inductor and capacitor in my car radio have one program at L = mh & C = 3.8 pf. Which is the FM station? (b WRKF 89.3 What is wavelength of radio wave? FM radio stations: frequency is in MHz. ω = LC = rad/s 6 3.8 = 5.6 8 rad/s How about for WJBO 5 AM? f = ω π = 8.93 7 Hz = 89.3 MHz
In an LC circuit, L = 4 mh; C = 4 µf At t =, the current is a maximum; When will the capacitor be fully charged for the first time?.5.5 Time -.5 - -.5 t = t = T / t = T / 4 t = 3T / 4 Example Charge Current t = T ω = LC = rad/s 8 6x ω = 5 rad/s T = period of one complete cycle T = π/ω =.5 ms Capacitor will be charged after T=/4 cycle i.e at t = T/4 =.6 ms
Example 3 In the circuit shown, the switch is in position a for a long time. It is then thrown to position b. Calculate the amplitude ωq of the resulting oscillating current. mh b mf a E= V i = ω q sin( ω t + φ Switch in position a : q=cv = ( mf( V = mc Switch in position b : maximum charge on C = q = mc So, amplitude of oscillating current = ωq = (mh(µf (µc =.36 A
Example 4 In an LC circuit, the maximum current is. A. If L = mh, C = mf what is the maximum charge q on the capacitor during a cycle of oscillation? i q = q cos( ω t + φ dq = = ω q sin( ω t + φ dt Maximum current is i =ωq Maximum charge: q =i /ω Angular frequency w=/ LC=(mH mf / = ( -8 / = 4 rad/s Maximum charge is q =i /ω = A/ 4 rad/s = 4 C