Lecture 24 Outline: Circuit Analysis, Inverse, LTI Systems Announcements: Reading: “6: Laplace Transform” pp. 27-63, 67-71.5 HW 8 posted, due Wed. Free 1-day extension OceanOne Robot Tour will be after class May 27 (11:30-12:20) Lunch provided afterwards. Can arrange separate tour for those w/conflicts Laplace Analysis of Circuits: 1st order LPF Bode Plots Inverse of Rational Laplace Transforms Causality and Stability in LTI Systems LTI Systems described by differential equations Example: 2nd order Lowpass Systems

Review of Last Lecture Laplace Transforms and Properties LTI analysis: Convolution Property Equivalence of Systems Systems in Series and Parallel One-Sided Signals have One-Sided ROCs Two-sided signals have ROCs that are strips Magnitude/Phase of Fourier Transform easy to obtain from Laplace Rational Form: ROCROCxROCh

Example: First-order LPF “s-domain circuit analysis of KVL equations” Circuit branch relations: Resistor: v(t)=Ri(t)V(s)=RI(s) Capacitor: i(t)=Cv(t)I(s)=sCV(s) Inductor: v(t)=Li(t)I(s)=sCV(s) Use Laplace to solve KVL circuit eqns using algebra instead of DEs RC=t ROC h(t) causal, so the ROC is right-sided, and H(jw) exists  ROC defined implicitly

Bode Plot for 1st Order LPF Bode Plots: show magnitude/phase on dB scale Plots 10log10|H(jw)|2=20log10|H(jw)| vs. w (dB) Can plot exactly or via straight-line approximation Poles lead to 20dB per decade decrease in Bode plot, Zeros lead to 20 dB per decade increase Frequency w in rad/s is plotted on a log scale for w0 only 20 log10|H(jw)| (dB) Bode Exact H(jw) (rad)

Inversion of Rational Laplace Transforms , Extract the Strictly Proper Part of X(s) If M<N, is strictly proper, proceed to next step If MN, perform long division to get , where Invert D(s) to get time signal: Follows from and The second term is strictly proper Perform a partial fraction expansion: Invert partial fraction expansion term-by-term For right-sided signals: Obtain coefficients via residue method Examples given in Reader and Section

Causality and Stability in LTI Systems Causal LTI systems: impulse response h(t)H(s) LTI system is causal if h(t)=0, t<0, so is h(t) right-sided For H(s) rational, a causal system has its ROC to the right of the right-most pole Step response is h(t)u(t) H(s)/s Stable LTI System LTI system is bounded-input bounded-output (BIBO) stable if all bounded inputs result in bounded outputs A system is stable iff its impulse response is absolutely integrable; true if H(jw) exists and H(s) proper rational A causal system with H(s) rational is stable if & only if all poles of H(s) lie in the left-half of the s-plane Equivalently, all poles have Re(s)<0 ROC defined implicitly for causal stable LTI systems

LTI Systems Described by Differential Equations (DEs) Finite-order constant-coefficient linear DE system Poles are roots of A(s), zeros are roots of B(s) If system is causal: for x(t)=d(t), initial conditions are zero: y(0-)=y(1)(0-)=…=y(N-1)(0-) ROC of H(s) is right-half plane to the right of the right-most pole If initial conditions not zero, must specific the ROC of H(s) Can solve DEs with non-zero initial conditions using the unilateral Laplace transform: Not covered in this class as we focus on causal stable systems Extra credit reading: “Laplace” pp. 63.5-67, example pp. 71-72.5

Second Order Lowpass System + - x(t) y(t) natural frequency wn and damping coefficient z Poles in left-half of s-plane h(t) causal H(jw) exists We first factor H(s): Three regimes: Underdamped: 0<z<1, gs distinct, complex conjugates Critically Dampled: z=1, gs equal and real Overdamped: 1<z<, gs distinct and real

Frequency Response: Underdamped: 0<z<1, gs distinct, complex conjugates Critically Dampled: z=1, gs equal and real Overdamped: 1<z<, gs distinct and real

Main Points Laplace allows circuit analysis using simple algebra Invert rational Laplace Xfms with partial fraction expansion A causal system with H(s) rational is stable if & only if all poles of H(s) lie in the left-half of the s-plane (all poles have Re(s)<0) ROC defined implicitly for causal stable LTI systems Systems described by differential equations easily characterized using Laplace analysis If system is causal: for x(t)=d(t), initial conditions are zero ROC of H(s) is right-half plane to the right of the right-most pole Second order systems characterized by 3 regimes: underdampled, critically damped, overdamped