Jan 1320

# Jan 1320 - EE 102 JAN 13 JAN.20 IP OP Relation u v pairs...

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Unformatted text preview: EE 102 JAN 13 . JAN .20 IP OP Relation u . , v . pairs Our start Example dv dt k v t k u t can be generalized to d k f . dt k 1 f . for k Generalises to d n v t dt n k n 1 a k d k v t dt k k m b k d k u t dt k n th order differential equation. The coefficient of the highest degree is nonzero and normalized to be unity. On the right side the integer m is aribitrary no constraint placed Our next stage example is n 2 d 2 v t dt 2 a 1 d v t dt a v t b 1 d u t dt b u t We can mention 2 canonical examples Printed by Mathematica for Students ELECTRIC CIRCUIT EXAMPLE : L d 2 Q t dt 2 R dQ t dt G Q t u t v t G Q t Output volts u t Input Volts d 2 dt 2 Q t R L d dt Q t G L Q t u t L d 2 dt 2 v t R L d dt v t G L v t G L u t R resistance L Inductance; G capacitance 1 C Input Volts STRUCTURE MECHANICS Example m d 2 dt 2 v t D d dt v t K v t u t m mass D damping K Elastic Constant Input Force You are NOT required to know this at this time This is our IP OP RELATION 2 Jan 1320.nb Printed by Mathematica for Students HOW DO SOLVE FOR v . given u . ?? TECHNIQUE LAPLACE TRANSFORMS NOT ONLY FOR SOLVING THE EQUATION BUT TO STUDY SYSTEM STABILITY We need to begin with the notion of A Right Half Plane in the COMPLEX PLANE z x i y x, y RIGHT HALF PLANE Consists of all points z such that x Re z Σ We prefer to use the letter s instead of z : Right HalfPlane s Re s Σ Specifying sigma specifies the half plane. Laplace Transform : of a function W t of time t defined on the HalfLine : t W s e st W t dt Re s Σ a Jan 1320.nb 3 Printed by Mathematica for Students Limit T T e st W t dt Σ a is called the "Abcissa of Convergence" The domain of definition is always a Right Half Plane It is required that ' W . is Laplace Transformable ' e Σ t W t dt for any Σ Σ a Here are some functions which are NOT Laplace Transformable : W t e t 2 t Problem at upper limit W t t 3 2 t Problem at the lower limit zero Fortunately such functions occur rarely for us Temporarily we may use F s for W s A function defined on the Time Domain right halfline...
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Jan 1320 - EE 102 JAN 13 JAN.20 IP OP Relation u v pairs...

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