Chapter 6 Applications of the Laplace Transform



Chapter 6 Applications of the Laplace Transform

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

Review of Resistive Network

1) Elements

[pic]

2) Superposition

[pic]

3) KVL and KCL

[pic]

4) Equivalent Circuits

[pic]

5) Nodal Analysis and Mesh Analysis

[pic]

Mesh analysis

[pic] Solve for I1 and I2.

Characteristics of Dynamic Network

Dynamic Elements ( Ohm’s Law: ineffective

1) Inductor

[pic]

2) Capacitor

[pic]

3) Example (Problem 5.9):

[pic]

Why so simple? Algebraic operation!

Dynamic Relationships (not Ohm’s Law) Complicate the analysis

Using Laplace Transform

[pic]

Define ‘Generalized Resistors’ (Impedances)

[pic] [pic]

[pic] As simple as resistive network!

Solution proposed for dynamic network:

All the dynamic elements ( Laplace Trans. Models.

( [pic] As Resistive Network

Key: Laplace transform models of (dynamic) elements.

Laplace transform models of circuit elements.

1) Capacitor

[pic]

Important: We can handle these two ‘resistive network elements’!

2) Inductor

[pic]

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

4)Sources

[pic]

5) Mutual Inductance (Transformers)

[pic]

(make sure both i1 and i2 either away

or toward the polarity marks to make

the mutual inductance M positive.)

Circuit (not transformer) form:

[pic]

Benefits of transform

(

Let’s write the equations from this circuit form:

The Same

( Laplace transform model: Obtain it by using inductance model

[pic]

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

[pic]

Assumption: [pic]

*Review of Resistive Network

1) short-circuit current through the load: [pic]

2) Equivalent Impedance or Resistance [pic]or [pic]:

A: Remove all sources

B: Replace [pic] by an external source

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

D: Voltage / Current ( [pic]

*Solution

1) Find [pic]

[pic]

[pic]

2) Find [pic]

[pic]

condition: 1 ohm = 3/s

or I(s) = 0

=>I(s) = 0 =>Zs = (

3)

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

[pic]

Question: What are i0 and v0?

What is [pic]?

Solution

1) Laplace Transformed Circuit

[pic]

2) KVL Equations

[pic]

Important: Signs of the sources!

3) Simplified (Standard form)

[pic]

[pic]

4. Transfer Functions

1. Definition of a Transfer Function

1) Definition

[pic]

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

[pic]

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)

[pic]

2) ( input

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

In this case: X(s) = L [( (t)] = 1

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.

3) 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( j2(f ) or H( j( ): frequency response function of the system

(Replace s in H(s) by j2(f or j()

|H( j2(f )| or |H( j( )|: amplitude response function

(H(j2(f) or (H( j( ): Phase response function

2. Properties of Transfer Function for Linear, Lumped stable systems

1) Rational Function of s

Lumped, fixed, linear system =>

[pic]

Corresponding differential equations:

[pic]

(2) [pic] 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: [pic]

(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)

[pic] where degree N’(s) X(s) = 1/s

[pic] ([pic] not bounded!)

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

i.e., [pic]

Why?

[pic]

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

[pic]

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

3. Components of System Response

[pic]

Because x(t) is input, we can assume

[pic]

Laplace transform of the differentional equation

[pic]

D(s): System parameters

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

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

[pic]

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

[pic]

From another point of view:

Transient Response: Approaches zero as t(∞

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!

[pic]

(2) zero-state response? No prolbem!

[pic]

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

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

Example 6-7

Input [pic]

Output [pic]

Initial capacitor voltage: [pic]

RC = 1 second

Solution

1) Find total response

[pic]

2) Find zero-input response and zero-state response

Zero-input response: [pic]

Zero-state response:

[pic]

3) Find transient and forced response

[pic]

Which terms go to zero as t((?

[pic]

What are the other terms:

[pic]

4. Asymptotic and Marginal Stability

System: (1) Asymptotically stable if [pic] as t(( (no input) for all

possible initial conditions, y(0), y’(0), … y(n-1)(0)

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

(2) Marginally stable

[pic] all t>0 and all initial conditions

(3) Unstable

[pic]grows without bound for at least some values

of the initial condition.

(4) Asymptotically stable (internally stable)

=>must be BIBO stable. (external stability)

[pic]

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):

[pic]

Other method to determine (just) stability without factorization?

Routh Array

1) Necessary condition

All [pic] (when [pic] is used)

⇨ any [pic] => system unstable!

Why?

Denote [pic] to esnure stability

[pic]

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

be at least a Re(pj) system unstable

2) Routh Array

Question: All [pic] 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

1) Basic Method

[pic]

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

[pic][pic]

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

verification:

[pic]

Example 6-9

[pic][pic]

[pic] [pic]

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

2) Modifications for zero entries in the array

Case 1: First element of a row is zero

⇨ replace 0 by ε (a small positive number)

Example 6-10

[pic][pic]

[pic]

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

[pic][pic]

3) Application: Can not be replaced by MatLab

Range of some system parameters.

Example: [pic][pic]

[pic] [pic]

Stable system

[pic] to ensure system stable!

6. Frequency Response and Bode Plot

Transfer Function [pic]

Frequency Response

[pic]

Amplitude Response: [pic]

Real positive number: function of [pic]

Phase Response: [pic]

Interest of this section

In particular, obtain

[pic]

What are these?

[pic]

[pic]

Important Question: What is a Bode Plot?

How to obtain them without much computations?

Asymptotes only!

1. Bode plots of factors

1) Constant factor k:

[pic]

(2) s

[pic]

[pic]

Can we plot it?

[pic]: Can we plot [pic] for them?

Phase s:[pic]

[pic]: [pic]

[pic]

(3) [pic]

[pic]

step 1: Coordinate systems

step 2: corner frequency

[pic]

step 3: Label 0.1(c, (c , 10(c

step 4: left of (c : [pic]

step 5: right of (c : [pic]

Why?

[pic]

If

[pic]

If

[pic]

Example: [pic]

What is T : T = 0.2

What is (c : (c = 1/T = 5

Example : 0.2s + 1

Example : (0.2s + 1)2, (0.2s + 1)-2

Example : (Ts + 1)±N

(4) [pic] (Complex --- Conjugate poles)

Step 3 : Before [pic] : [pic]

Right of [pic] :

point 1: ([pic] , [pic])

point 2: ([pic] , [pic])

Example: [pic]

Actual [pic] and ( (show Fig 6-20)

What’s resonant frequency: reach maximum: [pic]

Under what condition we have a resonant frequency:

[pic]

[pic] : see fig 6-21

What about : [pic]?

2. Bode plots: More than one factors

[pic]

Can we sum two [pic] plots into one?

Can we sum two [pic] plots into one?

Yes!

3. MatLab

[pic]

Show result in fig 6-24

6.7 Block Diagrams

1. What is a block diagram?

Concepts: Block, block transfer function,

Interconnection, signal flow, direction

Summer

System input, system output

Simplification, system transfer function

2. 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.

3. Cascade connection

[pic][pic]

4. Summer

[pic]

5. Single-loop system

[pic]

[pic]

[pic]

[pic][pic]

Let’s find [pic][pic] Closed-loop transfer function

Equation (1) [pic][pic]

Equation (2) [pic][pic]

[pic][pic]

6. More Rules and Summary: Table 6-1

[pic]

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

[pic]

[pic]

[pic]

Example 6-15: Armature- Controlled dc servomotor

Input : Ea (armature voltage)

Output : [pic] (angular shift)

Can we obtain [pic]?

Example 6-16 Design of control system

[pic]

Design of K such that closed loop system stable.

[pic]

Routh Array: [pic][pic]

[pic] [pic]

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

-----------------------

D(s)

[pic]

[pic]

C(s)

ZL

Why this direction?

Why this direction?

a

(Will I(s) be zero? We don’t know yet!)

Vtest(s)

[pic]

[pic]

N(s)

[pic]

[pic]

S

[pic]

[pic]

[pic]

Replace 0

Why?

[pic]

Line

Point 1

Point 2

[pic]

[pic]

[pic]

Line

[pic]

[pic]

[pic]

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