Physics 116A Notes Fall 2004

Transcription

1 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

2 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

3 Thevenin Equivalent 3

4 Norton Equivalent 4

5 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

6 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

7 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 ( ) 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

8 Inductor Basics self-inductance (Φ B i) 8

9 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

10 Capacitor Basics Note no net charge is collecting on capacitor 10

11 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

12 RC Natural Response 12

13 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

14 Series LRC Natural Response 14

15 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

16 Series LRC Natural Response: Plots 16