7#()IY)JaJaPPPPQLQLQL QXFQ(QQ&QxPRd RS*TkP SR>SSYmeSSSSSSPhysics 310 TRANSIENT RESPONSE AND ELECTRICAL RESONANCE (1) Feynman, Lectures on Physics, Vol. I Chap. 23-25. (2) Purcell, Electricity and Magnetism, Ch. 8 (3) Page & Adams, Principles of Electricity, p. 333-361; 502-509. See Appendix of these notes for brief analysis of resonant and coupled circuits. ****************************************************************************** These guidelines supplement the detailed instructions on the following pages. I. a) and b) "Qualitative" descriptions are all that's needed, but be sure to describe the waveforms in terms of the natural physical quantities: i for L, q for C, v for R. Sketch the responses (a total of 3 x 2 x 2 = 12 sketches). c) There is no need to measure R if the decade boxes are used. C should be measured in the P360 lab for ALL values used on the variable capacitor box. Measurement of Rs (source impedance) is required. II. Be careful not to confuse Q' with Q. Q' is the resonant quality; Q is the stored capacitive charge. One photo required per circuit. Care should be taken when wiring VR and VC simultaneously - what do you do about 'scope grounds? One photo required here. The critically and overdamped circuits require sketches, qualitative descriptions, and comparisons to theory. III. Use the same 2 circuits as in II. The phase comparison is to be done as a function of frequency as well, using either Lissajou figures or subtraction techniques (n.b. one value must be obtained both ways to show that you understand the technique). Do not confuse f = n with w = 2pn. IV. A page or two to show you understand these derivations is all that's needed. V - VI. Component matching is important. Even then there will be unresolvable matchings which will cause imperfect results. Vary R and L (within reason) to maximize the beats in V1(t) and V2(t) while minimizing them in V1 V2. I. Transient Response: LR and RC Circuits Important: In connecting circuits and oscilloscope avoid multiple grounds. It's easy to short out a component. To avoid, connect circuits so that one end of the component you want to observe in the oscilloscope is connected to the signal generator ground. Using the function generator to give a square wave, study the transient response of an LR and RC circuit. a) Describe qualitatively the voltage across the resistor and capacitor for the RC circuit. Consider the three cases: T>> RC; T ~dBA6()_ RC; T < DBA10()_ RC T = 1/2 period of square wave b) Same for the LR circuit for T>> L/R; T ~DBA6()_ L/R; T<< L/R c) Take a Polaroid picture of the voltage across the resistor and capacitor. Plot the voltage vs. time on a semilog graph and find the time constant. Compare your measurement with theoretical prediction. Estimate error in your measurement. Measure all the components on a General Radio Bridge, or other device. Note that the function generator has a finite internal impedance (about 50 W) which influences the measurement. Devise a way to determine this internal impedance. II. Transient Response LCR (See Appendix) As Feynman points out, there are several different possible behaviors for a harmonic oscillator (or, in this case, an LRC circuit). The behavior depends on the relative values of the resonant frequency wo and the damping constant g. The words associated with these behaviors are Under-damped. wo > g/2 Critically damped. wo = g/2 Over-damped. wo < g/2 One can also express these conditions in terms of the figure of merit or quality factor of the system, Q'. (Commonly the quality factor is called Q, but Q is used for charge in these notes).  Analyze the transient response for two under-damped LCR circuits, one with "high" quality (Q' 10) and one with "low" quality (Q' 5). Determine the frequency of oscillation, the decay constant, and any other quantities you think are important. Compare your results with theoretical predictions. Put VR into the horizontal terminals and VC into the vertical terminals of the oscilloscope. Take a picture of the pattern and explain it in detail. Can you write down an equation which describes the pattern? Study the behavior of a critically damped and an over-damped circuit and see how your results compare with theoretical predictions. III. Steady State Response of LCR Circuits Measure the amplitude of the charge oscillation as a function of the driving frequency w. Perform the measurement for two different values of Q'. Plot the amplitude and the square of the amplitude vs w. Determine wo and Q' from your graphs and compare with theoretical predictions as well as with your results from part II. Also measure the phase between the charge oscillation and driving voltage for these two circuits. Compare your results with theory. IV. Coupled Oscillations (see Appendix) Show that the equations of motion for the two identical inductively coupled oscillators pictured below are LC f(d2V1,dt2) + RC f(dV1,dt) + V1 + MC f(d2V2,dt2) = 0 LC f(d2V2,dt2) + RC f(dV2,dt) + V2 + MC f(d2V1,dt2) = 0 V1 is the voltage across capacitor in circuit 1 and V2 is the voltage across the capacitor in circuit 2. M is the mutual inductance. The solution to these equations shows that this circuit will have two resonant frequencies, w+ and w-, given by w+ = f(wo,r(1 + f(M,L))) w- = f(wo ,r(1 - f(M,L))) wo = F(1,r (LC)) Note: when M = 0, w+ = w- = wo (V1 + V2) and (V1 - V2) are the normal modes of the system. (V1 + V2) will oscillate at frequency w+ and (V1 - V2) at frequency w-. If one looks at the transient response of the circuit, V1 or V2 separately, beating of frequency wB = (w+ - w-) will be observed. This means that the two circuits are interchanging energy at the frequency wB. You should prove these statements or look up their proof in a reference. The normal modes can be easily shown by adding the two equations to get V1 + V2 and subtracting the two to get V1 - V2. V. Coupled Oscillators, Transient Response Be sure that your circuit elements are matched as well as possible. Examine transient response of the circuit. Photograph V1 vs. t and measure the beat frequency. Photograph V1(t) + V2(t) and V1(t) - V2(t), the normal modes, and determine w+ and w-. SUGGESTED CIRCUIT FOR MEASURING V1 + V2 AND V1 - V2 Alternately, you can use one of the isolation amplifiers and the second scope input for measuring the voltage across C2. Use the scope adding and subtracting features for getting V1 + V2 and V1 - V2. VI. Coupled Oscillators, Steady State Response Measure V1 or V2 as a function of frequency, driving the circuit with a sine wave from the function generator. Determine w+ and w- from these data. Also measure the amplitude for V1 + V2 and V1 - V2 and determine w+ and w- from these data. Appendix on Resonant Circuits Here we discuss, briefly, the transient and steady state solutions for LCR circuits. Detailed derivation of the results can be found in Feynman Vol. 1 Chaps. 23 and 24; Purcell Electricity and Magnetism Chap. 8. [N.B. In this appendix, Q is charge and Q' is the resonant quality] RSG VR = RI = R f(dQ,dt) VL = L F(dI,dT) = L f(d2Q,dt2) Vc = Q/C Equation of motion: f(d2Q,dt2) + g f(dQ,dt) + wo2Q = V/L g = f(R,L) , wo2 = f(1,LC) , w2 = wo2 - g2/4 Transient Response V = 0 Homogeneous solution: Q = e-gt/2 bbc[(Aeiwt + Be-iwt) (a) Under-damped: wo2 > g2/4 Q = Ae-gt/2 cos (wt + f) charge undergoes damped oscillations. (b) Critical Damping: wo2 = g2/4 Q = (A+Bt) e -gt/2 Prove this!! (c) Over-damped: wo2 < g2/4 Q = e-gt/2 bbc[(Ae+r(g2/4 - wo2) t + Be-r(g2/4 - wo2) t) For large g, Q decays with time constant RC. (This comes from the first term). Quality Factor of circuit = Q' (not to be confused with charge)!! Various definitions: Energy stored in circuit at any time. E = Q2/2C + LI2/2 = f(Q2,2C) + f(L,2) bbc((f(dQ,dt))2 Consider under-damped case: When f(dQ,dt) = 0 all the energy is stored in the capacitor. When Q = 0 all the energy is stored in the inductor. EC = 1/2 f(Q2,C) = f(A2 e-gt,2C) T = f(1,g) = f(L,R) = time for stored energy to decrease to 1/e of its value. Q' (quality factor) = 2p times number of oscillations during the time T. Q' = woT = f(woL,R) Equivalent definitions Q' = wo f(energy stored, average power dissipated) Q' = f(wo,Dwo) where Dwo is the bandwidth of the steady-state resonance curve. Steady State Solution V = Vo cos wt Q = f(Vo cos (wt + f),Lr((wo2 - w2)2 + (gw)2)) tan f = f(- gw,(wo2 - w2)) The charge oscillation has a maximum at w @ wo. How is "width" of curve related to Quality factor of circuit? The phase of the charge oscillation relative to the driving voltage, f, is -p/2 at w = wo; approaches zero as w 0; and approaches -p as w . Coupled Oscillations  V1 = capacitor voltage in the primary circuit. V2 = capacitor voltage in the secondary circuit. Equations (1) and (2) are Kirchhoff's Law for the two loops. Transient Response V = 0. Equations of motion are: LC f(d2V1,dt2) + RC f(dV1,dt) + V1 + MC f(d2V2,dt2) = 0 (1) LC f(d2V2,dt2) + RC f(dV2,dt) + V2 + MC f(d2V1,dt2) = 0 (2) Define V+ = V1 + V2 V- = V1 - V2 add and subtract (1) and (2) (L + M) C f(d2V+,dt2) + RC f(dV+,dt) + V+ = 0 (L - M) C f(d2V-,dt2) + RC f(dV-,dt) + V- = 0 Note: In coordinates in which V+ and V- are the normal modes, the equations are uncoupled. Solution, as before, given by V+ = A e -g+t cos (w+ t + d+) V- = Be -g-t cos (w- t - d-) ws(2,+) = f(1,(L + M) C) ws(2,-) = f(1,(L - M) C) g+ = f(R,2(L + M)) g- = f(R,2(L - M)) V1 = 1/2 (V+ + V-) V2 = 1/2 (V+ - V-) V1 = Ae-g+t cos (w+t + d+) + Be-g-t cos (w-t + d-) which can be transformed to V1 = A' cos [f(1,2) (w+ + w-)t + D+] cos [f(1,2) (w+ -w-)t + D-] + B' sin [f(1,2) (w+ + w-)t + D+] sin [f(1,2) (w+- w-) t + D-] The solution displays beats of frequency wB = f(w+ - w-,2) superimposed on an oscillation frequency o(-,w) = f(w+ + w-,2) Also D+ = f(d+ + d-,2) D- = f(d+ + d-,2) -- uAl<.HHHHd WORDHH<.HHHHd WORDHH<.HHHHd WORDHH<.HHHHd WORDHH<eH.HHHHd WORDHHcos ) ) Steady state: V = Vo cos wt Write down solution. Show that the plots of V+ and V- vs w peak at w+and w-, respectively, and that the plots of V1 and V2 vs w have two peaks, approximately at w+ and w-. Q = e-gt/2 blc[ (Ae+r(g2/4 - wo2)) t + brc](Be-r(g2/4 - wo2) t) iples of Electricity, p. 333-361; 502-509. See Appendix of these notes for 8-0mn;<BC #*+4589STXY`   } ~ BCDGH_`bdevwx{|?@t @)  @) @)P QtuKO`acdgiwx{~#DEF    TFGIJKNOPUVZ[cdhisyz:;<@ABGHIBCGHijnow TU\]fgno @ @     SKLLMNSTU  +,GHLMNVXyz~     @J*@ @ N!-./079:=CORSTUZ[\efglqrvy !"#$%&)*+,-       O-0489:<?@ABCDEFGRS   ( ) 6 7 H I Q [ \ !! ! !!!!!!!"!#!&!!!!!!!!!!!")"*","-"."/"3"9";"<">"u"""""ݸ         H""""""""""""""""""""""""""""""####################$$$$ $ $:$;$$$$$$$$$%% % %%%%%%%-%7%8%9%:%=  @*@    P%=%>%O%P%Z%[%h%i%j%k%n%o%}%%%%%%%%%%%%%%%%%%%%%%&&&&&&&#&$&.&/&V&W&]&^&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&''''     R'''$'7'8':'N'O'P'f'g'i'r't'w'x'}'~'''''''''''''''''''''''''''''''(((((( (((( (!("($(%(&(+(,(-(I(L(P(Q(R(U(V(W(\(](^(h(k(o(p(q(s(t(u({   V({(|(}(((((((((((((((((()))))))) ) )))) )!)")#)%)&)')()))-).)/)5)6)7):);)<)=)>)A)B)C)H)I*|********************  @ @  N 8o2f$u[ pôyh_VMC !0h !0h!0h0h!0|h!0|h!0|h !0h !0|h!0ph !0h8 !0h ! 0 `! 0! 0! 0 p G 3Kh?AB;Yi1Rs˹}th_VM !80h!0h !n0h !-0h !%0h !0h !0h!0h !0h!0h !0h!0h!0h0h !0h !0h !0h !0hg$O`t=_yG  , ` |䐐xne[O !0h !0h!0 !08 !08h !0h !0h!0h! 0! 0!0!0!0h!0h !%0h !0h !0h0h | !!m!!!""3"t""""#`#$$ $$$%-%}%%%&6&&ĥuk\\S!0h!0h D !0h !0h!0h !e0h !0h!0h !0h !'0h0h !#0h !0h !0h !0h !0h&&'4'b'''(/(((() ))@)A)E)F)G)H)I*****+Q+S+T+U+V+X+Y+Z+[˭˭˭uggYggg !0h !0h 0h !0h! !0h !0h !0h !0h !0h!0hl!0h8<H !0h"  (I@)I\ !u$(I,,*4:P %&4&&'0'((@(I!!!!!!! tF-"%='({* p |&* !"#$!" HH(FG(HH********+++++++#+$+F+G+H+M+N+O+P+Q+R+V+W+X+Y+Z+[+a+b+h+i+n+o+p+q+s+u+x+y+|+}+~+++++++++++++++++++++++++++++++++    N+[+b+++)&@)I\ !$(S)&,,*4: @ Y&&F&&'B'((&(a(o())))&!!!!!!!!!!!!!!!! tF-"%='({*+& p |&+[+ !"#$'!" HH(FG(HH(d@=/R@H -:LaserWriter ChicagoNew YorkGenevaMonacoVenice San FranciscoN Helvetica NarrowTimes HelveticaCourierSymbol! Avant Garde"New Century SchlbkSantiagoI New Century Schlbk ItalicB New Century Schlbk BoldBI New Century Schlbk BoldItcmmi8 MT ExtraXXX(K%euJO DJODJOD "#$)+.456789:;<?@CDFIKNTXY&&&-&0&J&K&a&b&&' ''''#'$','/'3'4'6'7'8'9'_'a'o'p'r's'~''''''(((((*(+(3(4(Q(S(`(a(l(m(n(((((((((((())))))))$)%)&+[+h+i+n+p+q+s+u+x+y+|+}+~++++++++++++++++++++++G&&'''8'9'O'P*|'''(( *((((!("*($(%(&*(O(](^*(`*(m(](q:(r(()))))))!)"***************+++#+$+F+G+M+N+R+)A)G+Ue+Transient Response and Electrical ResonanceP310 LMDepartment of PhysicsDepartment of Physics