Applications of the Laplace Transform

Part One: Analysis of Network (6-2, 6-3)

Review of Resistive Network

Elements

2) Superposition

3) KVL and KCL

4) Equivalent Circuits

5) Nodal Analysis and Mesh Analysis

Mesh analysis

Solve for I₁ and I₂.

Characteristics of Dynamic Network

Dynamic Elements Ohms Law: ineffective

1) Inductor

Capacitor

3) Example (Problem 5.9):

Why so simple? Algebraic operation!

Dynamic Relationships (not Ohms Law) Complicate the analysis

Using Laplace Transform

Define Generalized Resistors (Impedances)

As simple as resistive network!

Solution proposed for dynamic network:

All the dynamic elements Þ Laplace Trans. Models.

Þ As Resistive Network

Key: Laplace transform models of (dynamic) elements.

Laplace transform models of circuit elements.

Capacitor

Important: We can handle these two resistive network elements!

2) Inductor

3) Resistor V(s) = RI(s)

4)Sources

Mutual Inductance (Transformers)

(make sure both i₁ and i₂ either away

or toward the polarity marks to make

the mutual inductance M positive.)

Circuit (not transformer) form:

Benefits of transform

Lets write the equations from this circuit form:

The Same

Laplace transform model: Obtain it by using inductance model

Just sources and generalized resistors (impedances)!

Circuit Analysis: Examples

Key: Remember very little, capable of doing a lot

How: follow your intuition, resistive network

Little to remember: models for inductor, capacitor and mutual inductance.

Example 6-4: Find Norton Equivalent circuit

Assumption:

*Review of Resistive Network

1) short-circuit current through the load:

2) Equivalent Impedance or Resistance or :

A: Remove all sources

B: Replace by an external source

C: Calculate the current generated by the external source point a

D: Voltage / Current Þ

*Solution

1) Find

2) Find

Vtest(s)

	(Will I(s) be zero? We dont know yet!)

condition: 1 ohm = 3/s

or I(s) = 0

=>I(s) = 0 =>Zs ¥

Example 6-5: Loop Analysis (including initial condition)

Question: What are i₀ and v₀?

What is ?

Why this direction?

Solution

Laplace Transformed Circuit

Why this direction?

KVL Equations

Important: Signs of the sources!

Simplified (Standard form)

Transfer Functions

Definition of a Transfer Function

Definition

System analysis: How the system processes the input to form the output, or

Input : variable used and to be adjusted

to change or influence the output.

Can you give some examples for input and output?

Quantitative Description of how the system processes

the input to form the output: Transfer Function H(s)

d input

The resultant output y(t) to d (t) input: unit impulse response

In this case: X(s) = L [d (t)] =

Y(s) = Laplace Transform of the unit impulse response

=> H(s) = Y(s)/X(s) = Y(s)

Therefore: What is the transfer function of a system?

Answer : It is the Laplace transform of the unit impulse response

of the system.

Facts on Transfer Functions

* Independent of input, a property of the system structure and parameters.

* Obtained with zero initial conditions.

(Can we obtain the complete response of a system based on its transfer

function and the input?)

* Rational Function of s (Linear, lumped, fixed)

* H(s): Transfer function

H( j2pf ) or H( jw : frequency response function of the system

(Replace s in H(s) by j2pf or jw

|H( j2pf )| or |H( jw |: amplitude response function

ÐH(j2pf) or ÐH( jw Phase response function

Properties of Transfer Function for Linear, Lumped stable systems

Rational Function of s

Lumped, fixed, linear system =>

Corresponding differential equations:

(2) all real! Why? Results from real system components.

Roots of N(s), D(s): real or complex conjugate pairs.

Poles of the transfer function: roots of D(s)

Zeros of the transfer function: roots of N(s)

Example:

(3) H(s) = N(s)/D(s) of bounded-input bounded-output (BIBO) stable

system

* Degree of N(s) £ Degree of D(s)

Why? If degree N(s) > Degree D(s)

where degree N (s) <degree D(s)

Under a bounded-input x(t) = u(t) => X(s) = 1/s

( not bounded!)

* Poles: must lie in the left half of the s-plan (l. h. p)

i.e.,

Why?

(Can we also include k=1 into this form? Yes!)

* Any restriction on zeros? No (for BIBO stable system)

Components of System Response

Because x(t) is input, we can assume

Laplace transform of the differentional equation

D(s): System parameters

C(s): Determined by the initial conditions (initial states)

Initial-State Response (ISR) or Zero-Input Response (ZIR):

Zero-State Response (ZSR) (due to input)

From another point of view:

Transient Response: Approaches zero as tà

Forced Response: Steady-State response if the forced

response is a constant

How to find (1) zero-input response or initial-state response? No problem!

(2) zero-state response? No prolbem!

How to find (1) transient response? All terms which go to 0 as tà¥

(2) forced response? All terms other than transient terms.

Example 6-7

Input

Output

Initial capacitor voltage:

RC = second

Solution

Find total response 

Find zero-input response and zero-state response

Zero-input response:

Zero-state response:

Find transient and forced response

Which terms go to zero as tà¥

What are the other terms:

Asymptotic and Marginal Stability

System: (1) Asymptotically stable if as tॠ(no input) for all

possible initial conditions, y(0), y(0), y^(n-1)(0)

è Internal stability, has nothing to do with external input/output

(2) Marginally stable

all t>0 and all initial conditions

(3) Unstable

grows without bound for at least some values

of the initial condition.

(4) Asymptotically stable (internally stable)

=>must be BIBO stable. (external stability)

6-5 Routh Array

1. Introduction

System H(s) = N(s)/D(s) asymptotically stable all poles in l.h.p (not

include jw axis.

How to determine the stability?

Factorize D(s):

Other method to determine (just) stability without factorization?

Routh Array

Necessary condition

All (when is used)

ð     any => system unstable!

Why?

Denote to esnure stability

When all Re(p_j) > 0 , all coefficients must be greater than zero. If some coefficient is not greater than zero, there must

be at least a Re(p_j) <= 0 (i.e., )

=> system unstable

Routh Array

Question: All implies system stable?

Not necessary

Judge the stability: Use Routh Array (necessary and sufficient)

2. Routh Array Criterion

Find how many poles in the right half of the s-plane

Basic Method

Formation of Routh Array

Number of sign changes in the first column of the array

=> number of poles in the r. h. p.

Example 6-8

sign: Changed once =>one pole in the r.h.p

verification:

Example 6-9

Sign: changed twice => two poles in r.h.p.

Modifications for zero entries in the array

Case 1: First element of a row is zero

ð     replace 0 by ε (a small positive number)

Example 6-10

Case 2: whole row is zero (must occur at odd power row)

construct an auxiliary polynomial and the perform differentiation

Example: best way.

Example 6-11

S

Application: Can not be replaced by MatLab

Range of some system parameters.

Example:

Stable system

to ensure system stable!

Frequency Response and Bode Plot

Transfer Function

Frequency Response

	Why?

Amplitude Response:

Real positive number: function of

Phase Response:

Interest of this section

In particular, obtain

What are these?

Important Question: What is a Bode Plot?

How to obtain them without much computations?

Asymptotes only!

Bode plots of factors

Constant factor k:

(2) s

Can we plot it?

: Can we plot for them?

Phase s:

:

(3)

step 1: Coordinate systems

step 2: corner frequency

step 3: Label 0.1w_c w_c w_c

Line

Point 1

step 4: left of w_c :

Point 2

step 5: right of w_c :

Why?

If

If

Example:

What is T : T = 0.2

What is w_c w_c = 1/T = 5

Example : 0.2s + 1

Example : (0.2s + 1)², (0.2s + 1)^-2

Example : (Ts + 1)^N

(4) (Complex --- Conjugate poles)

Step 3 : Before :

Right of :

Line

point 1: ( , )

point 2: ( , )

Example:

Actual and z (show Fig 6-20)

Whats resonant frequency: reach maximum:

Under what condition we have a resonant frequency:

: see fig 6-21

What about : ?

Bode plots: More than one factors

Can we sum two plots into one?

Can we sum two plots into one?

Yes!

MatLab

Show result in fig 6-24

6.7 Block Diagrams

What is a block diagram?

Concepts: Block, block transfer function,

Interconnection, signal flow, direction

Summer

System input, system output

Simplification, system transfer function

Block

Assumption: Y(s) is determined

by input (X(s)) and block transfer

function (G(s)). Not affected by

the load.

Should be vary careful in

analysis of practical systems about the accuracy of this assumption.

Cascade connection

Summer

5. Single-loop system

Lets find Closed-loop transfer function

Equation (1)

Equation (2)

6. More Rules and Summary: Table 6-1

Example 6-14: Find Y(s)/X(s)

Example 6-15: Armature- Controlled dc servomotor

Input : Ea (armature voltage)

Output : (angular shift)

Can we obtain ?

Example 6-16 Design of control system

Design of K such that closed loop system stable.

Routh Array:

System stable if k>0. If certain performance is required in addition to the stability, k must be further designed.