Chapter 2 Modeling in the Frequency Domain

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Chapter 2 Modeling in the Frequency Domain

• 1. Laplace Transform Review
• 2. The Transfer Function
• 3. Linearization

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• The Laplace transform is defined as
• The inverse Laplace transform, which allows us to
  find
• given , is

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1. Laplace Transform Review
Table 2.1 Laplace transform table

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Table 2.2 Laplace transform theorems
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Inverse Laplace Transform

• Partial-Fraction Expansion
• To find the inverse Laplace transform of a
  complicated function, we can convert the function
  to a sum of simpler terms for which we can know
  the Laplace transform of each term. The result is
  called a partial-fraction expansion.
• Case 1. Root of the denominator are real and
  distinct.
• Case 2. Root of the denominator are real and
  repeated.
• Case 3. Root of the denominator are complex or
  imaginary.

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Case 1 Root of the denominator are real and
distinct. where constants

• Case 2 Root of the denominator are real and
  repeated.
• where constants

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Case 3 Root of the denominator are complex or
imaginary.
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2. The Transfer Function

• The transfer function of a linear system is
  defined as the ratio
• of the Laplace transform of the output variable
  to the Laplace
• transform of the input variable, with all initial
  conditions
• assumed to be zero.
• Consider a general nth-order, linear,
  time-invariant differential
• equation

Figure 2.2 Block diagram of a transfer function
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Table 2.3Voltage-current, voltage-charge, and
impedance relationships for capacitors,
resistors, and inductors (passive linear
components)
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Electric Networks
In electric circuits, transfer functions can be
obtained using Kirchhoffs voltage law and
summing voltages around loops (Mesh Analysis)
and/or Kirchhoffs current law and summing
Currents flowing from nodes (Nodal Analysis)
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Find the transfer function relating the
capacitor, , to the input voltage, .
Figure 2.3 RLC network
Figure 2.5 Laplace-transformed network
Figure 2.4 Block diagram of series RLC
electrical network
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Figure 2.6a. Two-loop electrical networkb.
transformed two-loop electrical networkc.
block diagram
Given the network of Figure 2.6 (a), find
the transfer function,
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Figure 2.10a. Operational amplifierb.
schematic for an inverting operational amplifier
c. inverting operational amplifier configured
for transfer function realization. Typically, the
amplifier gain, A, is omitted.
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Operational Amplifiers

• An operational amplifier has the following
  characteristics
• Differential inputs,
• High input impedance, (ideal)
• Low output impedance, (ideal)
• High constant gain amplification, (ideal)

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Figure 2.12 General noninverting operational
amplifier circuit
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Figure 2.13 Noninverting operational amplifier
circuit for
Find the transfer function
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Table 2.4Force-velocity, force-displacement, and
impedance translational relationships for
springs, viscous dampers, and mass
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Figure 2.15 a. Mass, spring, and damper system
b. block diagram
Figure 2.16 a. Free-body diagram of mass,
spring, and damper systemb. transformed
free-body diagram
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Figure 2.17 a. Two-degrees-of-freedom
translationalmechanical systemb. block diagram
Find the transfer function
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Figure 2.18a. Forces on M1 due only to motion of
M1b. forces on M1 due only to motion of M2c.
all forces on M1
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Figure 2.19a. Forces on M2 due only to motion of
M2b. forces on M2 due only to motion of M1c.
all forces on M2
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Table 2.5Torque-angular velocity, torque-angular
displacement, and impedance rotational
relationships for springs, viscous dampers, and
inertia
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Figure 2.22 a. Physical system b. schematic
c. block diagram
Find the transfer function
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Figure 2.23a. Torques on J1 due only to the
motion of J1b. torques on J1 due only to the
motion of J2c. final free-body diagram for J1
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Figure 2.24a. Torques on J2 due only to the
motion of J2b. torques on J2 due only to the
motion of J1c. final free-body diagram for J2
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Figure 2.27A gear system
Figure 2.28Transfer functions for a. angular
displacement in lossless gears and b. torque in
lossless gears
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Figure 2.29a. Rotational system driven by
gearsb. equivalent system at the output after
reflection of input torquec. equivalent
system at the input after reflection of impedances
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Figure 2.30a. Rotational mechanical system with
gearsb. system after reflection of torques and
impedances to the output shaftc. block diagram
Find the transfer function
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Figure 2.31Gear train
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Figure 2.32a. System using a gear trainb.
equivalent system at the inputc. block diagram
Find the transfer function
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Figure 2.35 DC motora. schematic12b. block
diagram
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Figure 2.36Typical equivalentmechanical loading
on a motor
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Figure 2.37DC motor driving a rotational
mechanical load
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Figure 2.45a. Linear systemb. nonlinear system
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Figure 2.46Some physical nonlinearities
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The linearization can obtained using the Taylor
series expansion, which expresses the value of a
function in terms of the value of that function
at a particular point (operation point or
equilibrium point), and derivatives evaluated at
that point.The Taylor series is represented
byFor small excursions of from ,
we can neglect higher-order terms.
Linearization
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Figure 2.47Linearization about a point A
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Figure 2.48Linearization of 5 cos x about x
?/2
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Example Linearize the following
dynamic equation for small excursions about