Transcript Chapter 6 The Laplace Transform and the Transfer Function

Chapter 6 The Laplace
Transform and the Transfer
Function Representation
6.1 Laplace Transform of a Signal
• Example 6.1 Laplace Transform of Exponential
Function
– x(t) = e-bt u(t)
– X(s) = 1/(s+b)
– Region of convergence Re s > -b
• Example 6.2 Fourier Transform from Laplace
Transform
– For x(t) above, X(ω) =1/(jω + b), where s=jω in the
Laplace transform—assuming all is well with the
region of convergence.
• Example 6.3 Laplace Transforms Using Symbolic
Manipulation (page 284)
6.2 Properties of the Laplace
Transform
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Ex. 6.4 Linearity
Ex. 6.5 Laplace Transform of a Pulse
Ex. 6.6 Time Scaling
Ex. 6.7 Unit-Ramp Function
Ex. 6.8 Multiplication of an Exponential by t
Ex. 6.9 Multiplication by an Exponential
Ex. 6.10 Laplace Transform of a Cosine
Ex. 6.11 Multiplication of an Exponential by Cosine and Sine
Ex. 6.12 Multiplication by Sine
Ex. 6.13 Differentiation
Ex. 6.14 Integration
Ex. 6.15 Convolution
Note: Initial Value Theorem and Final Value Theorem are for
Laplace transform only.
6.3 Computation of the Inverse
Laplace Transform
• The inverse is difficult to calculate directly.
• An algebraic technique is discussed in this
section.
– The transform X(s) = B(s)/A(s), where B(s) and A(s)
are polynomials, is said to be a rational function of s
since it is the ratio of polynomials.
– Then we can solve for the zeros of both numerator
(zeros) and denominator (poles)
– If the poles are distinct then
• X(s) = c1/(s-p1) + c2/(s-p2) +…+ cN/(s-pN)
• And x(t) = c1exp(p1t) +…+cNexp((pNt)
6.3 Examples
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Ex. 6.17 Distinct Pole Case
Ex. 6.18 Complex Pole Case
Ex. 6.19 Completing the Square
Ex. 6.20 Equating Coefficients
Ex. 6.21 Repeated Poles
Ex. 6.22 Powers of Quadratic Terms
Ex. 6.23 Order of B(s) is greater than order of A(s): M =
3; N = 2.
Ex. 6.24 General Form of a Signal
Ex. 6.25 Limiting Value
Ex. 6.26 Use of Matlab
Ex. 6.27 Transform containing an Exponential
6.4 Transform of the Input/Output
Differential Equation
• 6.4.1 First Order Case
– System Equation: dy(t)/dt + ay(t) = bx(t)
– Take the Laplace Transform:
• sY(s) – y(0-) + a Y(s) = bX(s)
– Then we have:Y(s)(s + a) = bX(s) + Y(0-)
– And so:Y(s) = {bX(s)/(s+a)} + {Y(0-)/(s+a)}
– If the initial condition is 0, then
• Y(s) = {b/(s+a)} X(s)
• and so H(s) = b/(s+a)
6.4.2 Second Order Case
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System Equation:
d2y(t)/dt +a1dy(t)/dt +a0y(t) = b1(dx(t)/dt) +b0x(t)
Laplace Transform:
s2Y(s) – y(0-)s – y’(0-) + a1[sY(s) – y(0-)] + a0Y(s) =
b1sX(s) + b0X(s)
Y(s) ={y(0-)s – y’(0-) + a1 y(0-) }/{s2 + a1s +a0}+
[b1s+b0]/ {s2 + a1s +a0} X(s)
If initial conditions are 0, then
Y(s) = [b1s+b0]/ {s2 + a1s +a0} X(s)
And so the transfer function is H(s) = [b1s+b0]/ {s2
+ a1s +a0}
6.5 Transform of the Input/Output
Convolution Integral
• Y(s) = H(s) X(s)
• Poles and zeros of the system function can be
plotted in the s-plane (see Example 6.33, figure
6.5)
6.6 Direction Construction of the
Transfer Function
• Interconnections of Integrators, adders, subtracters, and
scalar multipliers.
– See Figure 6.15.
• Transfer Functions of Block Diagrams
– Parallel Interconnection: Y(s) = [H1(s) +H2(s)] X(s)
– Series Connection: Y(s) = [H2(s) H1](s) X(s)
• Feedback Connection (Figure 6.20)
– H(s) = H1(s)/[1-H1(s)H2(s)]