Physics 116A Notes Fall 2004 David E. Pellett Draft v.0.9 Notes Copyright 2004 David E. Pellett unless stated otherwise. References: Text for course: Fundamentals of Electrical Engineering, second edition, by Leonard S. Bobrow, published by Oxford University Press (1996) Others as noted 1
Thévenin; Circuits with L, R and C ODEs Thévenin 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) 2
Thevenin Equivalent 3
Norton Equivalent 4
Thévenin e and r Determination To find Thévenin equivalent e and r 1. Remove external load from the two black box output nodes 2. The voltage across the two output nodes V oc = e 3. Set independent sources in the black box to zero, leaving dependent sources as-is 4. 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 V across the output nodes, calculate I, find r = V/I For Norton Short output nodes and find I ss = i Find r as for Thévenin Could also just find Thévenin and convert to Norton with i = e/r Etc. (variations possible on these themes) 5
Maximum Power Transfer For what value of R L will P L, the power transferred to R L, be maximum? (Note P L = 0 for R L = 0 or R L, nonzero for finite R L ) Find V (voltage divider) and then P L V = er L R L + r P L = V 2 /R L = e2 R L (R L + r) 2 To maximize, take derivative of P L with respect to R L and set equal to 0, solve for R L. Prove to yourself that max. is when R L = r. 6
Time 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 v C (t) for initial condition: v C = v 0 at t = 0 (result is decaying exponential, as proven in 9 series: v C = v 0 e t/τ where τ RC) Example: Series LRC circuit, find v C (t) for initial conditions: v C = v 0, 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 voltage and AC Circuit analysis (major topic this quarter) 7
Inductor Basics self-inductance (Φ B i) 8
Inductor is a short circuit for DC Inductors (continued) v L = L di dt Opening switch in series with inductor carrying current will cause spark Do you see why? Integral relation for current in inductor: i(t) = 1 L t v L (t ) dt The current through an inductor can t change instantaneously unless we have a delta function voltage spike Energy stored in inductor w L (t) = 1 2 Li2 Do you see how and where energy is stored? 9
Capacitor Basics Note no net charge is collecting on capacitor 10
Capacitors (continued) i = C dv C dt Capacitor is open circuit for DC (blocks DC) Integral relation for capacitor voltage: v C (t) = 1 C t i(t ) dt The voltage across a capacitor can t change instantaneously unless we have a delta function current spike Energy stored in capacitor w C (t) = 1 2 Cv C 2 Do you see how and where energy is stored? 11
RC Natural Response 12
RC Natural Response Plot Note that the graph has a title telling what it is, the axes are labeled and units are given... a word to the wise for the graphs in your lab logbook 13
Series LRC Natural Response 14
Series LRC Natural Response This substitution leads (after some calculus and algebra) to a quadratic eq n for s with roots s 1 = α α 2 ω 2 n, s 2 = α + α 2 ω 2 n. The resulting ODE solution, y(t) = A 1 e s 1t + A 2 e s 2t is called critically damped if α = ω n, overdamped if α > ω n and underdamped if α < ω n. A 1 and A 2 are constants of integration. Critically damped: roots are real and equal, f(t) = (A 1 t + A 2 )e αt Overdamped: roots are real and unequal, f(t) = A 1 e s 1t + A 2 e s 2t Underdamped: roots are complex conjugates, f(t) = e αt (A 1 e jω dt + A 2 e jω dt ) Be αt cos(ω d t φ) where ω d ω n 2 α 2 and 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. 15
Series LRC Natural Response: Plots 16