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 1

§ 4.1 Linear Time-Invariant Systems (1) LTI Systems: Differential Equation Formulation 2

§ 4.1 Linear Time-Invariant Systems (2) Solution Decomposition: y(t)=y(I.C., system)+y(system, input) y(I.C., system)=y h (t) I.C.-dependent solution Homogeneous solution Natural response Zero-input response y(system, input)=y p (t) Forcing term dependent solution Particular solution Forced response Zero-state response 3

§ 4.1 Linear Time-Invariant Systems (3) Solution Modes: Characteristic equation Eigen value Solution modes 4

§ 4.1 Linear Time-Invariant Systems (4) 5

§ 4.1 Linear Time-Invariant Systems (5) Output Response: (1) y(t)=y h (t)+y p (t) y h (t): Linear combination of solution modes y p (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)=y s (t)+y t (t) y t (t): Transient solution y s (t): Steady state solution Transient solution is contributed by initial condition and forcing function. 6

§ 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 7

§ 4.2 Transfer Function and Dynamic Systems (2) Pierre-Simon Laplace (1749 ~ 1827) Monumental work “ Traite de mécanique céleste ” 8

§ 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. 9

§ 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 10

§ 4.2 Transfer Function and Dynamic Systems (5) Signal: Unit impulse Unit step Ramp Exponential decay Sine wave Cosine wave 11

§ 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 12

§ 4.2 Transfer Function and Dynamic Systems (7) States and Constitutive Law of Physical Systems: 13

§ 4.2 Transfer Function and Dynamic Systems (8) Analog Physical Systems: 14

§ 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 15

§ 4.2 Transfer Function and Dynamic Systems (10) (2) Real repeated roots From Laplace transform pairs 16

§ 4.2 Transfer Function and Dynamic Systems (11) (3) Complex conjugate pairs with real distinct roots From Laplace transform properties and pairs 17

§ 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. 18

§ 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 19

§ 4.3 Transfer Function and System Response (1) Transfer Function G(s): Rational T.F. Irrational T.F. Proper T.F. 20

§ 4.3 Transfer Function and System Response (2) Response by T.F.: Partial fraction expansion is employed to find y(t). 21

§ 4.3 Transfer Function and System Response (3) Ex: 22

§ 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). 23

§ 4.3 Transfer Function and System Response (5) Impulse Response of Poles Distribution 24

§ 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. 25

§ 4.3 Transfer Function and System Response (7) 26

§ 4.4 Block Diagram Operations for Complex Systems (1) Fundamental Operations: Signal operation Summer Y(s)=X 1 (s)+X 2 (s) Comparator Y(s)=X 1 (s)-X 2 (s) Take-off point Y(s)=X 1 (s) Component combinations Serial Parallel Feedback 27

§ 4.4 Block Diagram Operations for Complex Systems (2) Moving junction / sequence Ahead of a block Past a block Exchange sequence 28

§ 4.4 Block Diagram Operations for Complex Systems (3) Negative Feedback System: 29

§ 4.4 Block Diagram Operations for Complex Systems (4) Loading Effect: Cascade Realization Isolated Amp by 741OP 30

§ 4.4 Block Diagram Operations for Complex Systems (5) History of Operational Amplifier: 1965 Fairchild develops the first OpAmp (operational amplifier) generally used throughout the industry--a milestone in the linear integrated circuit field. 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 Ⅱ Fairchild introduces an OpAmp (operational amplifier) that is one of the earliest linear integrated circuits to include temperature compensation and MOS capacitors. 31

§ 4.4 Block Diagram Operations for Complex Systems (6) Operational Amplifier: 32

§ 4.4 Block Diagram Operations for Complex Systems (7) Network 1: Network 2: Loading effect 33

§ 4.4 Block Diagram Operations for Complex Systems (8) Note: For MIMO System Output Vector Transfer Matrix Input Vector 34

Ex: Armature control DC servomotor Static characteristics (Ideal) § 4.4 Block Diagram Operations for Complex Systems (9) 35

§ 4.4 Block Diagram Operations for Complex Systems (10) I/O Block Diagram Reduction Dynamic characteristics Total Response Command Response Disturbance Response 36

§ 4.4 Block Diagram Operations for Complex Systems (11) Model Reduction 37

§ 4.4 Block Diagram Operations for Complex Systems (12) Static gain is dominated by feedback gain K b =1/K m in system dynamics. Key points: Linear time-invariant motor No load No delay No damping No inertia No resistance No inductance 38