15. LCR Oscillators. S. G. Rajeev. March 17, 2009

Transcription

1 15 LCR Osciators S G Rajeev March 17, 2009 We studied how direct current fows in a circuit that has a resitance and a capacitor As we turn the circuit on, the current decreases to zero rapidy We wi study a new kind of circuit where the current osciates: it increases to a positive vaue then decreases to a negative vaue and so on, back and forth in time These osciators are the basis of many devices that we use every day The tuner in your TV or radio is an exampe, as are eectronic watches In most modern devices, the osciations are reguated using additiona components ike transistors or crystas We wi study ony the most basic exampe with inductance, capacitance and a resistance 1 Inductance We saw that a coi carrying an aternating current can induce an eectromotive force on a nearby coi: ir can drive a current on it This emf on the second coi E 2 is proportiona to the derivative of the current on the first coi: if the current is constant there is no induction Thus E 2 = M di 1 This constant M is caed the mutua inductance of the two cois Using Faraday s aw we can derive a formua for it M = µa N 1N 2 where A is the cross-sectiona area the two cois share, the ength of the cois and N 1, N 2 the number of turns on them Aso, µ is the magnetic permeabiity of the materia inside the cois Of course, the second coi wi induce an emf on the first as we E 1 = M di 2 Any time that magnetic fux across the area encosed by a wire changes, there wi be an induced emf on it Thus a varying current can induce an emf even on the wire carrying it: this is caed sef-inductance By Lens z aw, the sign of this emf wi be such that it opposes the change of the current So 1

2 E = L di The constant L, caed sef-inductance (shortened to inductance) has a formua as we, derived from Faraday s aw L = µa N 2 Again Nis the number of turns in the coi, its ength and A the area To have a arge inductance, the coi can be fied with a materia with a arge magnetic permeabiity ( ike iron) The unit of inductance is Henry Ceary a Henry is the same as Vots(1/Amperes)secs 11 An Exampe A 381 m-ong coi containing 225 oops is wound on an iron core (average µ = 1850µ 0 ) aong with a second coi of 115 oops The oops of each coi have a radius of 100 cm The mutua inductance is M = µa N 1N 2 = π π(001) = H 381 The sef inductance of the first coi is L 1 = µa N 2 1 whie that of the second coi is = π 10 7 π(001) = H L 2 == µa N 2 2 = π 10 7 π(001) = H 2 An LC Circuit Inductance represents the inertia of an eectric current The more the inductance, the more it wi oppose changes in the current An inductor is happiest (has no induced emf) when it has a constant current fowing through it A Capacitor on the other hand, is in a hurry to get rid of the eectric charge stored in it: it is happiest (has zero emf) when it carries no charge By putting an inductance and a capacitance together we can trick them into osciating back and forth between their states of maximum and minimum emf In any rea circuit, the wires carrying the current wi have some resitance But as a first step we ignore this and consider an idea circuit with ony L and C Aso, as a first step we assume there is no externa emf appied: no battery or AC power source The emf of the inductance and the capacitance must add up to zero 2

3 L di + Q C = 0 But reca that current is the rate of discharge of the charge on the capacitor Q : Combining these we get di = 1 LC Q dq = I This pair of differentia equations can be soved in terms of famiiar functions What is a pair of functions so that the derivative of one is a constant times the other? Reca that d sin θ dθ = cos θ So for a constant ω, d cos θ dθ = sin θ d sin ωt = ω cos ωt d cos ωt = ω sin ωt This ooks a itte more ike what we want If we put Q(t) = Q 0 cos ωt for some constant Q 0 ( you can check that it is the vaue of the charge at time t = 0), we wi have Continuing with this = ωq 0 sin ωt di = ω2 Q 0 cos ωt 3

4 In other words This is what we want if we set or di = ω2 Q ω 2 = 1 LC ω = 1 LC is Thus the charge and the current osciate periodicay in time The period T = 2π ω and the frequency is f = ω 2π = 1 2π LC At time t = 0 the charge is at a maximum vaue Q 0 The capacitor discharges If there was no inductance, the charge woud drop to zero very quicky But the inductance produces an emf that opposes the change in current But this causes the capacitor to overshoot and get itsef charged in the opposite poarity It discharges again and overshoots itsef so that it a charge of Q 0 again Then the whoe process repeats itsef It is ike having two poitica parties Party L aways opposes whatever change that Party Q is trying to make The country wi go back and forth between extremes 3 Mechanica Anaogy Imagine a mass attached to a spring When the spring is not extended, the force on the mass is zero: it is in equiibrium If the mass is dispaced by x from this equiibrium point, there is a force kx acting on it The sign of the force is in the direction that opposes the dispacement Thus ma = kx where ais the acceeration Reca that it is the rate of change of veocity, which is itsef the rate of change of position: v = dx 4

5 Thus we have a = dv v = dx dv = k m x This is just ike what we had above: the charge on the capacitor is ike the dispacement The current is ike the veocity ( more precisey the negative of the veocity) The inductance is ike the mass and the reciproca of capacitance is ike the spring constant k This makes sense because inductance is a form of inertia, according to Lenz s aw 4 Energy in an Osciator The energy in a capacitor is Q2 2C It is a form of potentia energy The energy in an inductor is due to the current fowing through it Thus it is a form of kinetic energy, due to the motion of eectrons It can aso be thought of as the energy of the magnetic fied inside the cois This energy is 1 2 LI2 The sum of these two is conserved in an idea LC osciator (with no resistance) 5 Adding Resistance 1 2 LI2 + Q2 2C = constant But in the rea word we are aways osing energy to heat: the wires that carry the current wi have resistance The osciations wi die out eventuay uness this energy is repenished from externa sources If we incude the votage drop across a resistor, In other words and sti L di + Q RI = 0 C L di + RI Q C = 0 5

6 For sma R,there shoud sti be an osciation, but each cyce, the maximum charge wi be a itte ess than the one before We can try a guess ike Q = Q 0 e γt cos ω t which woud capture this behavior By putting it into the differentia eqation we wi get γ = R 2L, and 1 ω = LC R2 4L 2 L If the resistance is arge (R > 2 C ) there are no osciations: the charge just decreases to zero exponentiay 6 Phase of Aternating Current An aternating current is described its maximum current, its frequency and its phase I(t) = I 0 sin[ωt + φ] Across the resistor the emf is proportiona to the current maximum when the current is a maximum But across an inductor, the emf is RI = RI 0 sin[ωt + φ] L di = LI 0ω cos[ωt + φ] = (Lω)I 0 sin[ωt + φ π 2 ] It reaches a Thus in addition to mutipying the current by Lω, we must aso shift the phase of the current to get the emf: when the current is a maximum, the emf of an inductance is actuay zero A capacitance aso shifts the phase but by the opposite amount The emf across a capacitor is Q C ; and Thus Q = 1 ω I 0 cos[ωt + φ] and 6

7 Q C = 1 ωc I 0 sin[ωt + φ π 2 ] A phasor (not the same as in Star Trek) is an eectrica engineer s trick for representing these facts convenienty A phasor is ike a vector: it has a magnitude and a direction A resistor is represented by a phasor of ength R pointing aong the horizonta axis An inductance is a phasor of ength Lω pointed aong the vertica axis and a capacitance a phasor or ength 1 Cω verticay downward from the origin Phasors added just ike vectors in the pane Thus an LCR cicruit wi have a phasor represented by the sum of the components ( R, ωl 1 ωc ) The magnitude of this phasor is caed impedance ( Z = R 2 + ωl 1 ) 2 ωc The direction is found by recaing that the resistance is the horizonta component of a phasor of ength Z cos θ = R Z The impedance reates the rms votage of an AC circuit to its rms current: there is something simiar to Ohm s aw 7 Driven Osciators V rms = ZI rms Because of resitance an osciator wi ose its current rapidy uness we give it some externa emf If this externa is itsef periodic we have a driven osciator It is simiar to a person giving a swing a push at periodic intervas so that the swing keeps osciating The frequency of this externa emf does not have to be the same as the natura frequency of the circuit If we appy an emf of anguar frequency ω, the current in an LR circuit is given by I rms = V rms Z = V rms R 2 + ( ) ωl 1 2 Thus the current is a maximum when the frequency of the driving emf is equa to the natura frequency of the circuit: ωl = 1 ωc ω = 1 LC This is the phenomenon of resonance ωc 7