Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether and treat the full circuit. et s recap! circuits: In a previus lab we fund that fr a charging capacitr, the vltage lp law gives, q I 0. Here the ac supply has vltage ε. In terms f the charge we may write, dq q. (. Then It. The diagram belw shws the capacitr charging up. t We may slve this t find, q t) 1 e cnstant is t e. The time Yu find t1/ frm lking at the either the vltage gain acrss the capacitr r vltage drp ver the resistr. When the vltage is half the max r min value measure t1/ frm the time scale, using the relevant time/div. scaling n the scpe. Yu can use t1/ t find the time cnstant τc. A very similar situatin arises with circuits. circuits: When yu have a battery (r signal generatr) with vltage, ɛ, and cnnect a resistance and inductance t it, the vltage lp equatin gives us, di t I 0, which can be slved fr current I(t) as, It 1 e, here the time cnstant is. (B = ε) 11-1
ircuits, Damped Frced Harmnic Mtin Physics 6 ab Yu find t1/ frm lking at the either the vltage drp acrss the inductr r vltage build up n the resistr. When the vltage is half the max r min value measure t1/ frm the time scale, using the relevant time/div. scaling n the scpe. S finally we cme t circuits circuit: This is a perfect example f a damped driven harmnic scillatr which yu will find thrughut yur studies in physics in mechanics as well as ptics and electrnics. Yu can Ggle it and find dem s nline. heck ut fr example http://en.wikipedia.rg/wiki/_circuit. Instead f a direct ac supply we will be using a mutual inductance frm a primary cil. The cil shwn here is the secndary. We will cnnect the primary t a square wave signal generatr. We ve dne this befre in the last experiment ab 10. k at the vltage drps arund the circuit, and using dq I, we get, d q dq q. I di q. Divide this by This secnd rder differential equatin can be slved by standard methds, t cmplementary functin etc, t find, q( t) qe cst. Frm which we find q( t) X X the vltage acrss the capacitr as ( t) where tan and where 1 is the natural r resnance frequency f the scillatr. We define the impedance as Z X X supply max vltage Z I. Then we can write the. Yu can find the analysis nline and in Halliday and esnick Physics l II. The impedance Z is a cmplex quantity, Z i yu can find the magnitude and phase in the usual way (n the next page). X X 11 -
ircuits, Damped Frced Harmnic Mtin Physics 6 ab Z X X And i Z Z e where tan 1 X X Nte that resistance is the real part f impedance and the cmplex part is the reactance. When energy flws arund a circuit, the resistance will cause energy lss as heat dissipatin, the reactance stres the energy in either the electric field (fr the capacitr) r the magnetic field (fr the inductr). This energy can be released when the capacitr discharges r the slenid (inductr) pwers dwn. S with resistance yu lse energy, with reactance yu can get the energy back. X X Maybe yu dn t want t invlve secnd rder differential equatins. OK let s slve the prblem anther way. et s lk at the vltage drp acrss the resistr: I Isin t Taking I t ) I t ( sin. Nte that the vltage acrss the resistr is in phase with the current. et s lk nly at the vltage drp acrss the capacitr: q 1 I I cs t I sint Here yu can see that the current reaches a maximum ¼ cycle (r π/) ahead f. urrent leads vltage by π/ in a capacitr. We define the capacitive reactance as 1 X then we get, IX sint. et s lk at the vltage drp acrss the inductr: di I sint Here we can see that the vltage acrss the inductr is π/ ut f phase with the current and it lags behind the current by π/. Take the inductive reactance t be, then the vltage acrss the inductr is, X X I sint. ltage leads current by π/ in an inductr. 11-3
ircuits, Damped Frced Harmnic Mtin Physics 6 ab As a memry aid, remember EI the IE man. In EI, the E is the vltage, is the inductr, and I is the current. S, fr an inductr,, the vltage, E, leads the current, I, since E cmes befre I in EI. In IE, the I is the current, is the capacitr, and E is the vltage. S, fr a capacitr,, the current, I, leads the vltage, E, since I cmes befre E in IE. Ok let s tie everything tgether nw the slutin withut slving a differential equatin fllws: ltage drp arund the circuit is: 0 Use sin t, then we get; sin sin t I sin t I X sin t I X sin t t I sin t X X cs t This can be slved via trig methds t give sin t I X X sin t Where the impedance is given by Z X X vltage f the supply, hence simply put the max I Z yu can cancel the sine terms n bth sides! Als f where f is the frequency in Hz r per secnd. The frequency f = 1/T where T is the perid f the scillatin. The impedance is a minimum when X X. Then the natural frequency is given by 1. 11-4
ircuits, Damped Frced Harmnic Mtin Physics 6 ab The energy in the circuit slshes back and frth between the capacitr and the inductr the scillatins are damped ut by the resistance in the circuit. The capacitr charges when the cil pwers dwn, then the capacitr discharges and the cil pwers up and s n. There is a natural frequency t this scillatin yu shuld try t determine fr yur circuit. With n resistance it wuld be the natural frequency ω0 described abve. What Yu Need T D: Experiment 1 A) Take the secndary cil f 000 turns and hk it up t the multi-meter t determine its resistance. Write that dwn. B) Get the 0.05 μf capacitr (lwer vltage rating will be fine) and stick it int the circuit bard. ) Set the signal generatr t square wave pulse, abut 60 Hz (use the 100 Hz range fr this). D) Hk up the primary cil 400 turns t the signal generatr turn n the signal. Slide the plastic rd thrugh bth cils t hld them in place and have them tuching sme tape might be used t hld them tgether thrughut the experiments. E) nnect the secndary cil in series with the capacitr. Then attach the scpe H 1, t the +/ end f the capacitr. See Fig 1. Fr setup. F) nnect the psitive end f the primary cil t the EXT trigger surce f the scpe. We will be using the ext trigger fr this experiment. G) Turn n the scillscpe. Set vlts/div H 1 t 00m and time/div t 500 μs. hse menu, ext trigger surce fr H 1. 11-5
ircuits, Damped Frced Harmnic Mtin Physics 6 ab With everything switched n yu shuld be seeing a damped scillatry curve like the ne in the pht belw. If necessary press the run/stp buttn and use the hrizntal shift knb t get the full damped curve in view. Nte the red lead n the right bttm f the scpe is the Ext trigger. This is ging t the psitive end f the primary cil, which in turn is cnnected t the signal generatr. is the resistance f the secndary cil (inductr) which yu measured with the multi-meter ~ 50Ω. Yu shuld see nice damped scillatins as shwn abve. Using cursrs recrd ltage and time fr each peak f the signal. Plt the data in excel and fit an expnent t yur curve. ecall this is damped harmnics frm equatin t q( t) q e cs t (Nte: Since the signal is scillating nly cllect the magnitude f the peaks) 1) Find the time cnstant fr the decay τ? ) What is t1/? alculate fr the inductr. 3) Natural scillatin f the circuit ω? 4) alculate using ω mpare this value t yur calculatin frm part. 11-6
ircuits, Damped Frced Harmnic Mtin Physics 6 ab Experiment Nw hk up the resistance bx in series with the secndary cil as shwn belw. Yu need t see what happens when yu add in extra external resistance in series with the resistance frm the secndary cil. This shuld damp ut the scillatins faster and prduce ver damping. What fllws is a sequence f phts shwing hw the scillatins are gradually damped by adding larger resistrs t the circuit. Eventually yu get ver-damping, which is the case where yu hardly see any scillatin at all. The full resistance in the circuit is = 0 + where is the cil (inductr) resistance which yu can measure using the meter prvided befre yu hk up the cil int the circuit. is apprx 50 Ω fr all. Make sure t ntice hw the scillatins are damped ut by adding resistance! 11-7
ircuits, Damped Frced Harmnic Mtin Physics 6 ab 0 = 100 Ω frm resistance bx. 0 = 300 Ω 0 = 600 Ω 0 = 1000 Ω near critical damping. 11-8
ircuits, Damped Frced Harmnic Mtin Physics 6 ab Over-damping has n scillatin at all just expnential decay. k up under damped, critically damped and ver damped scillatin nline! POST AB EXEISE nsider the fllwing circuit cnsisting f a capacitr = 0.05 μf, and a cil f inductance = 0.4 mh and internal resistance =.0 Ω. The capacitr is charged t = 10 vlts and then the switch is clsed. The vltage acrss the capacitr as displayed n the scillscpe is shwn belw. Predict the expected values f: a) Free scillatin perid T and frequency ω0 = π f. b) Time cnstant fr the decay, τ. Use the half life methd yu have dne befre t 1. ln HINT: Nte that the vltage acrss the capacitr is expected t vary as; q( t) t t ( t) e cs(ft) IZe cs( t) Successive peaks ccur apprximately when the csine becmes unity and are rughly at intervals at perid T. A plt f the curve thrugh the successive peaks is an t apprximatin t the curve e. (A mre accurate result fr the perid wuld be gained by differentiating c and setting that equal t zer... find where c becmes zer find the freq r perid frm that.) 11-9