Chapter 6 Applications of the Laplace Transform

EE 422G Notes: Chapter 6 Instructor: Zhang

Chapter 6 Applications of the Laplace Transform

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

1. Review of Resistive Network

1) Elements

2) Superposition

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3) KVL and KCL

4) Equivalent Circuits

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5) Nodal Analysis and Mesh Analysis

Mesh analysis Solve for I1 and I2.

2. Characteristics of Dynamic Network

Dynamic Elements Ohm’s Law: ineffective

1) Inductor

2) Capacitor

3) Example (Problem 5.9):

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Why so simple? Algebraic operation!Dynamic Relationships (not Ohm’s 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 Laplace Trans. Models.

As

Resistive Network Key: Laplace transform models of (dynamic) elements.

3. Laplace transform models of circuit elements.

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1) Capacitor

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

2) Inductor

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

4)Sources

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5) 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:

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Benefits of transform

Let’s write the equations from this circuit form:

The Same

Laplace transform model: Obtain it by using inductance model

Just ‘sources’ and ‘generalized resistors’ (impedances)!

4. 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

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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 *Solution 1) Find

2) Find

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condition: 1 ohm = 3/s or I(s) = 0 =>I(s) = 0 =>Zs =

3)

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

Question: What are i0 and v0? What is ?

Solution1) Laplace Transformed Circuit

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Vtest(s)

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

Why this direction?

a

ZL


2) KVL Equations

Important: Signs of the sources!

3) Simplified (Standard form)

6-4 Transfer Functions

1. Definition of a Transfer Function

(1) Definition System analysis: How the system processes the input to form the output, or

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Why this direction?


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)

(2) input

The resultant output y(t) to (t) input: unit impulse responseIn 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( j2f ) or H( j ): frequency response function of the system

(Replace s in H(s) by j2f or j) |H( j2f )| or |H( j )|: amplitude response function H(j2f) or H( j ): Phase response function

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

(1) 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.,

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Why?

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

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

3. Components of System Response

Because x(t) is input, we can assume Laplace transform of the differentional equation

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D(s)

C(s)

N(s)


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 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!

(2) zero-state response? No prolbem!

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 Output Initial capacitor voltage:

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RC = 1 second

Solution(1) Find total response

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

Zero-input response:

Zero-state response:

(3) Find transient and forced response Which terms go to zero as t?

What are the other terms:

4. Asymptotic and Marginal StabilitySystem: (1) Asymptotically stable if 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

all t>0 and all initial conditions

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(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

(1) Necessary condition All (when is used) any => system unstable!

Why? Denote to esnure stability

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

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be at least a Re(pj) <= 0 (i.e., ) => system unstable

(2) 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

(1) Basic Method

Formation of Routh ArrayNumber of sign changes in the first column of the array=> number of poles in the r. h. p.

Example 6-8

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

Example 6-9

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

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Case 2: whole row is zero (must occur at odd power row) construct an auxiliary polynomial and the perform differentiation

Example: best way.

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Example 6-11

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S

Replace 0


(3) Application: Can not be replaced by MatLab Range of some system parameters.

Example:

Stable system

to ensure system stable!

6-6 Frequency Response and Bode Plot

Transfer Function

Frequency Response

Amplitude Response:

Real positive number: function of

Phase Response:

Interest of this section

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Why?


In particular, obtain

What are these?

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:

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(2) s

Can we plot it?

: Can we plot for them?

Phase s: :

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(3)

step 1: Coordinate systemsstep 2: corner frequency step 3: Label 0.1c, c , 10c

step 4: left of c :

step 5: right of c :

Why?

If

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Point 1

Point 2

Line


If

Example: 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

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Example : (Ts + 1)±N

(4) (Complex --- Conjugate poles)

Step 3 : Before : Right of : point 1: ( , ) point 2: ( , )

Example:

Actual and (show Fig 6-20)What’s resonant frequency: reach maximum: Under what condition we have a resonant frequency:

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Line


: see fig 6-21

What about : ?

2. Bode plots: More than one factors

Can we sum two plots into one?Can we sum two plots into one? Yes!

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3. MatLab

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

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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

4. Summer

5. Single-loop system

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Let’s find Closed-loop transfer function Equation (1) Equation (2)

6. More Rules and Summary: Table 6-1

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

Example 6-15: Armature- Controlled dc servomotor Input : Ea (armature voltage) Output : (angular shift)

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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.

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Chapter 6 Applications of the Laplace Transform

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