Chapter 4 Transfer Function and Block Diagram Operations

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Chapter 4 Transfer Function and Block Diagram
Operations

• 4.1 Linear Time-Invariant Systems
• 4.2 Transfer Function and Dynamic Systems
• 4.3 Transfer Function and System Response
• 4.4 Block Diagram Operations for Complex
  Systems

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4.1 Linear Time-Invariant Systems (1)

• LTI Systems

Differential Equation Formulation
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4.1 Linear Time-Invariant Systems (2)

• Solution Decomposition
• y(t)y(I.C., system)y(system, input)
• y(I.C., system)yh(t)
• I.C.-dependent solution
• Homogeneous solution
• Natural response
• Zero-input response
• y(system, input)yp(t)
• Forcing term dependent solution
• Particular solution
• Forced response
• Zero-state response

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4.1 Linear Time-Invariant Systems (3)

• Solution Modes
• Characteristic equation
• Eigen value
• Solution modes

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4.1 Linear Time-Invariant Systems (4)
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4.1 Linear Time-Invariant Systems (5)

• Output Response
• (1) y(t)yh(t)yp(t)
• yh(t) Linear combination of solution
  modes
• yp(t) Same pattern and characteristics
  as the forcing function
• The RH side of LTI model affects only
  the coefficients of solution modes.
• The LH side of LTI model dominates the
  solution modes of the transient
• response.
• (2) y(t)ys(t)yt(t)
• yt(t) Transient solution
• ys(t) Steady state solution
• Transient solution is contributed by
  initial condition and forcing function.

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4.2 Transfer Function and Dynamic Systems (1)

• Input is transfered through system G to
  output.
• Definition
• Key points Linear, Time-Invariant, Zero
  initial condition

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4.2 Transfer Function and Dynamic Systems (2)

• Pierre-Simon Laplace (1749 1827)

• Monumental work Traite de mécanique céleste

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4.2 Transfer Function and Dynamic Systems (3)

• Laplace Transform
• Definition
• Time function
• Existence Condition
• Inverse Laplace Transform
• Signals that are physically realizable
  (causal) always has a Laplace transform.

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4.2 Transfer Function and Dynamic Systems (4)

• Important Properties
• 
  t Domain
  s Domain
• Linearity
• Time shift
• Scaling
• Final value theorem
• Initial value theorem
• Convolution
• Differentiation
• Integration

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4.2 Transfer Function and Dynamic Systems (5)

• Signal
• Unit impulse
• Unit step
• Ramp
• Exponential decay
• Sine wave
• Cosine wave

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4.2 Transfer Function and Dynamic Systems (6)

• Fundamental Transfer Function of Mechanical
  System
• Elements Function
  Block Diagram T.F. Example
• Static element
• (Proportional element)
• Integral element
• Differential element
• Transportation lag

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4.2 Transfer Function and Dynamic Systems (7)

• States and Constitutive Law of Physical Systems
  

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4.2 Transfer Function and Dynamic Systems (8)

• Analog Physical Systems

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4.2 Transfer Function and Dynamic Systems (9)

• Inverse Laplace Transform and Partial Fraction
  Expansion

• Roots of D(s)0
• (1) Real and distinct roots
• From Laplace transform pairs

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4.2 Transfer Function and Dynamic Systems (10)

• (2) Real repeated roots
• From Laplace transform pairs

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4.2 Transfer Function and Dynamic Systems (11)

• (3) Complex conjugate pairs with real distinct
  roots
• From Laplace transform properties and
  pairs

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4.2 Transfer Function and Dynamic Systems (12)

• Dynamic System Equation and Transfer Function
  
• Differential Equation and Transfer Function
• Differential Equation
  Transfer Function
• Problems associated with differentiation of
  noncontinuous functions, ex. step function,
  impulse function.

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4.2 Transfer Function and Dynamic Systems (13)

• Integral Equation and Transfer Function
• The transfer function of a system is the
  Laplace transform of its
• impulse response

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4.3 Transfer Function and System Response (1)

• Transfer Function G(s)
• Rational T.F.
• Irrational T.F.
• Proper T.F.

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4.3 Transfer Function and System Response (2)

• Response by T.F.
• Partial fraction expansion is employed to
  find y(t).

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4.3 Transfer Function and System Response (3)

• Ex

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4.3 Transfer Function and System Response (4)

• Poles, Zeros, and Pole-zero Diagram
• For an irreducible proper rational
  transfer function G(s),
• a number (real or complex) is said to
  be
• Pole-zero diagram
• Representation of poles and zeros
  distribution by using x and o,
• respectively in complex plane along with
  static gain.
• Ex
• Ex
• Characteristic Equation
• i.e.
• characteristic roots The roots of
  characteristic equation
• i.e. The poles of G(s).

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4.3 Transfer Function and System Response (5)

• Impulse Response of Poles Distribution

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4.3 Transfer Function and System Response (6)

• Effects of Poles and Zeros
• A pole of the input function generates the
  form of the forced response.
• A pole of the transfer function generates
  the form of the natural response.
• The zeros and poles of transfer function
  generate the amplitude for both the forced and
  natural responses.
• The growth, decay, oscillation, and their
  modulations determined by the impulse response of
  the poles distribution.

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4.3 Transfer Function and System Response (7)

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4.4 Block Diagram Operations for Complex
Systems (1)

• Fundamental Operations
• Signal operation
• Summer
  Y(s)X1(s)X2(s)
• Comparator Y(s)X1(s)-X2(s)
• Take-off point Y(s)X1(s)
• Component combinations
• Serial
• Parallel
• Feedback

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4.4 Block Diagram Operations for Complex
Systems (2)

• Moving junction / sequence
• Ahead of a block
• Past a block
• Exchange sequence

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4.4 Block Diagram Operations for Complex
Systems (3)

• Negative Feedback System

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4.4 Block Diagram Operations for Complex
Systems (4)

• Loading Effect
• Cascade

Realization
Isolated Amp by 741OP
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4.4 Block Diagram Operations for Complex
Systems (5)

• History of Operational Amplifier

OP was first built with vacuum tubes. Originally
designed by C. A. Lovell of Bell Lab. and was
used to control the movement of artillery during
World War ?.
1965 Fairchild develops the first OpAmp
(operational amplifier) generally used throughout
the industry--a milestone in the linear
integrated circuit field.
1968 Fairchild introduces an OpAmp (operational
amplifier) that is one of the earliest linear
integrated circuits to include temperature
compensation and MOS capacitors.
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4.4 Block Diagram Operations for Complex
Systems (6)

• Operational Amplifier

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4.4 Block Diagram Operations for Complex
Systems (7)

• Network 1
• Network 2

Loading effect
Loading effect
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4.4 Block Diagram Operations for Complex
Systems (8)
Note For MIMO System

Output Vector
Transfer Matrix
Input Vector
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4.4 Block Diagram Operations for Complex
Systems (9)

• Ex Armature control DC servomotor
• Static characteristics (Ideal)

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4.4 Block Diagram Operations for Complex
Systems (10)
Dynamic characteristics

• I/O Block Diagram Reduction

Total Response
Command Response
Disturbance Response
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4.4 Block Diagram Operations for Complex
Systems (11)
Model Reduction

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4.4 Block Diagram Operations for Complex
Systems (12)

• Static gain is dominated by feedback gain
  Kb1/Km in system dynamics.
• Key points Linear time-invariant motor
• No load
• No delay
• No damping
• No inertia
• No resistance
• No inductance