1. Lecture 24 Outline: Circuit Analysis, Inverse, LTI Systems
Announcements:
Reading: “6: Laplace Transform” pp ,
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

2. 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

3. 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

4. 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)

5. 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

6. 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

7. 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 , example pp

8. 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

9. 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

10. 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