VI. Analyzing Dynamic Systems in the Frequency-Domain

Motivation for Analyzing Dynamic Systems in the Frequency-Domain

dynamic system models will now be analyzed in teh frequency-domain using the lapalce transform

What does the Laplace Transform allow? - Motivation

i. different equations to be conviently solved via algebra

ii. complicated systems to be decomposed into simpler functional blocks

iii. information system dynamic characterizations w/o the need to solve the differential equations

Unilateral LT of a Signal f(t)

the complex fcn: F(s) = integral from 0- to infinity (f(t) * e^(-st) dt where s is the complex frequency (s = sigma + jw)

LT converts a time-domain fcn to a frequency-domain fcn

bc the LT is an infinite integral, the LT DNE for all signals

What signals have LTs?

ones that grow no faster than an exponential

signals routinely used in dynamic systems analysis

Laplace Transform Tables

LT Properties

4 of them: linearity, time-differentiation, time interaction, and time-shifting

Linearity - LT Property

if f1(t) ←> F1(s) + f2(t) ←> F2(s), then af1(t) + bf2(t) ←> aF1(s) + bF2(s) where a/b are any constants

Time-differentiation - LT Property

if f(t) ←> F(s), then f.(t) ←> sF(s) - f(0-)

f..(t) ←> s²F(s) - sf(0-) - f.(0-)

Time Integration - LT Property

if f(t) ←> F(s0, then the running integral of 0- to infinity (f(tau)dtau) ←> (1/s)F(s)

differentiating/integrating in time corresponds to multiplying/dividing by s in frequency (LT converts calculus to algebra)

Time Shifting - LT Property

if f(t) ←> F(s), then f(t - to) ←> F(s) e^(-sto) to > 0

LT of g(t) = u9t) - u(t-T) Example

LT of y.(t) + ay(t) = 0 for some y(0) Example

1st-order, linear, time-invariant, autonomous equation

LT of y..(t) + (a+b)y.(t) + aby(t) = Ax(t) Example

Inverse LT Formula

f(t) = integral from c-jinfinity to c+jinfinity (F(s) e^(st) d(t)) → contour integral

Contour Integral is what?

more complicated to solve than the LCCDE in the time-domain

Alternative Method to Solving the Inverse LT

via partial fraction expansion (PFE) → applicable when Y(s) is a rational fcn (ratio of polynomials)

Partial Fraction Expansion (PFE) Steps

1. brea Y(s) up into simplier components

2. find inverse LT of each component using tabl elook-up

3. sum the inverse LTs to get y(t)

this works bc the LT is linear

Impulse Response

time-domain approach that was used to determine the impulse response of an LCCDE that only works when the right-hand side is bx(t) and doesn’t for more complicated right-hand sides (e.g., x(t) + x.(t))

LT may be applied to easily determine this of any LCCDE

Show that impulse(t) ←> 1 - Impulse Response Examples

Find the impulse response for y..(t) + (a+b)y.(t) + aby(t) = Ax(t)

System Impedance

the feature of the system that produces the opposition that the (generalized) flow encounters and the (generalized) effort that is needed to produce the flow

ex. systemic arterial circulation

Other BioE System Impedance Examples

electrical impedance when dealing w/ recording instruments

electrical impedance of biological tissues when dealing with electrical stimulation devices (pacemakers and neural stimulators)

mechano-impedance of the lung ventilatory system (pushing air into the lungs)

mechanical impedance of movement round joints and tissues

Impedance of a Network (Z(s))

the ratio of the LT of the efort at its entry point (E(s)) to the LT of the flow at this point (F(s))

What does Z(s) (impedance of a network) do?

allows an alternate analysis of networks w/o using node/loop laws

provides a single expression for all network elements as a kind of simplification of multiple elements and their arrangement

Impedance of Generalized Elements

resistance - Z(s) = R

PE storage/capacitance element - Z(s) = 1/sc

KE storage/inductance element - Z(s) = sL

Impendance of a Network of Elements

combine impednaces of individual components

Combining Series Impedances

loop law so Z(s) = Z₁(s) + Z₂(s) + …

just add them together

Combining Paralllel Impedances

node law so Z(s) = (Z₁(s) * Z₂(s)) / (Z₁(s) + Z₂(s))

Combining Impedances Example

Impedance to Differentiation

the diff. eqn. relating e to f at the entry port can be easily determine

start with definition of Z(s) (= E(s) / F(s)), cross-multiply, then determine LCCDE by inspection based on LT properties

Impedance to Differentiation Examples

Transfer Function - Motivation

a transfer fcn is an s-domain fcn by which the LT of an input (X(s)) is multiplied by it to produce the LT of the output (Y(s)): Y9s) = H(s) * X(s) → H(s) = Y(s) / X(s) (transfer fcn)

system impedance (Z(s)) is a transfer fcn specifically relating the LT of the input flow (F(s)) to the LT of the output effort (E(s)) at the entry port of the system: E(s) = Z(s) * F(s)

transfer fcn is general and related any pair of input and output vars. of aystem

What are transfer functions useful for?

synthesis - combining subsystems to determine overall system behaviot

analysis - determing behavior of a asystem w/o having to solve the differential equations

Transfer System Definition

Comments about the Transfer Function (H(s))

can be determined from an LCCDE and vice versa by inspection (w/ sⁱ ←> di/dtⁱ)

if x(t) = impulse (t), then X(s) = 1 so therefore from earlier, the inverse LT of H(s) is the impulse response

completely characterizes the LCCDE in the sense that it can be used to determine the output to varioys inputs when the system is at zero ICs (rest) → therefore system block diagrams are often labeled with H(s)