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lecture3 - Lec 3 System Modeling Transfer Function Model...

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1 Lec 3. System Modeling Transfer Function Model Model of Mechanical Systems Model of Electrical Systems Model of Electromechanical Systems Reading: 3.1-3.3, 3.6-3.8 LTI Systems Given by Differential Equations LTI systems modeling practical systems are often given by ODE Take Laplace transform (assuming zero initial condition) Transfer function :
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2 (Rational) Transfer Functions Roots of B ( s ) are called the zeros of H ( s ) ( z 1 ,…,z m ) Roots of A ( s ) are called the poles of H ( s ) ( p 1 ,…,p n ) System is called an n -th order system Poles and zeros occur in complex conjugate pairs, e.g. if 2+ j 3 is a pole (or zero), so is 2- j 3. Pole-Zero plot: Standard Forms of (Rational) Transfer Function Ratio of polynomials: Factored (or product) form: Sum form (assume poles are distinct): p 1 ,…, p n : poles, r 1 ,…, r n : residues
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3 A Geometric Interpretation of Residues Pole zero plot: Distance to zeros Distance to poles (except itself) Remark: if a pole is very close to a zero, its residue will be small.
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