Physics 116A NotesFall 2004David E. PellettDraft v.0.9• Notes Copyright 2004 David E. Pellett unless stated otherwise.• References:– Text for course:Fundamentals of Electrical Engineering, second edition, by LeonardS. Bobrow, published by Oxford University Press (1996)– Others as noted1Th´evenin; Circuits with L, R and C – ODEs• Th´evenin and Norton equivalent (continued)– Examples– Maximum power transfer• Inductor, capacitor basics• Circuit differential equations(time domain circuit analysis)• RC, LR, LRC circuit natural response(once over lightly; more in 116B)2Thevenin Equivalent3Norton Equivalent4Th´evenin e and r Determination• To find Th´evenin equivalent e and r1. Remove external load from the two black box output nodes2. The voltage across the two output nodes Voc= e3. Set independent sources in the black box to zero, leaving dependentsources as-is4. Determine resistance between two output nodes to find r(a) If result is a resistor network, combine to find r(b) In general, may have to connect an external independent source Vacross the output nodes, calculate I, find r = V/I• For Norton– Short output nodes and find Iss= i– Find r as for Th´evenin– Could also just find Th´evenin and convert to Norton with i = e/r– Etc. (variations possible on these themes)5Maximum Power TransferFor what value of RLwill PL, the power transferred to RL, be maximum?(Note PL= 0 for RL= 0 or RL→ ∞, nonzero for finite RL)Find V (voltage divider) and t hen PLV =eRLRL+ rPL= V2/RL=e2RL(RL+ r)2To maximize, take derivative of PLwith respect to RLand set equal to 0,solve for RL. Prove to yourself that max. is when RL= r.6Time Domain Circuit Analysis with L, R, C• Topics are covered in Ch. 3 of text• Understand IV relations and energy stored for inductors and capacitors• Remainder of Ch. 3 (3.3 – 3.6) once over lightly– KVL, KCL still apply at any instant– Get ODEs for i(t), v(t) for circuits involving L, R, C– Natural (transient) response∗ Example: RC circuit, find vC(t)for initial condition: vC= v0at t = 0(result is decaying exponential, as proven in 9 series:vC= v0e−t/τwhere τ ≡ RC)∗ Example: Series LRC circuit, find vC(t)for initial conditions: vC= v0, i = 0 at t = 0(over-, under-, and critically-damped solutions: see text for details)– Driven response with pulse input deferred until Physics 116B• Next: Steady-state (driven) response to sinusoidal input voltageand AC Circuit analysis (major topic this quarter)7Inductor Basicsself-inductance(ΦB∝ i)8Inductors (continued)vL= Ldidt• Inductor is a short circuit for DC• Opening switch in series with inductor carrying current will cause spark– Do you see why?• Integral relation for current in inductor:i(t) =1LZt−∞vL(t0) dt0• The current through an inductor can’t change instantaneously– unless we have a delta function voltage spike• Energy stored in inductorwL(t) =12Li2– Do you see how and where energy is stored?9Capacitor BasicsNote no net charge is collecting on capacitor10Capacitors (continued)i = CdvCdt• Capacitor is open circuit for DC (blocks DC)• Integral relation for capacitor voltage:vC(t) =1CZt−∞i(t0) dt0• The voltage across a capacitor can’t change instantaneously– unless we have a delta function current spike• Energy stored in capacitorwC(t) =12CvC2– Do you see how and where energy is stored?11RC Natural Response12RC Natural Response PlotNote that the graph has a title telling what it is, the axes are labeled andunits are given. . . a word to the wise for the graphs in your lab logbook13Series LRC Natural Response14Series LRC Natural ResponseThis substitution leads (after some calculus and algebra) to a quadratic eq’nfor s with rootss1= −α −pα2− ωn2, s2= −α +pα2− ωn2.• The resulting ODE solution,y(t) = A1es1t+ A2es2tis called critically damped if α = ωn, overdamped if α > ωnand under-damped if α < ωn. A1and A2are constants of integration.– Critically damped: roots are real and equal, f(t) = (A1t + A2)e−αt– Overdamped: roots are real and unequal, f(t) = A1es1t+ A2es2t– Underdamped: roots are complex conjugates,f(t) = e−αt(A1e−jωdt+ A2ejωdt) ≡ Be−αtcos(ωdt − φ)where ωd≡pωn2− α2and B and φ are constants of integration.– Note that for electronics, one gets used to the notation j ≡√−1• See Sec. 3.5 of text for details.15Series LRC Natural Response: