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ECX 5241DISTRIBUTED PARAMETER SYSTEMS



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velocity field in a fluid


Basically a Lump parameter system is described by a system of differential equations.

In this system space and time variables are absent or negligibly small.

Unlike Lump parameter systems, the Distributed parameter systems cannot be modeled purely by ordinary differential equations.

As a result Distributed parameter model is more general for this physical system.



Divergence theorem


Volume integral of divergence of a vector field is equals to total outward flux through the closed surface that bounds the volume

v .A dv = s A.dS



Significance of the theorem
Provision to evaluate closed surface integral by volume integral of divergence of the vector field or volume integral of the divergence of a vector field by closed surface is the significance of the theorem.


Helmholtz’s theorem

A vector field is determined to within an additive constant if both divergence and curls are specified in
every where.
Simply it says a vector field can completely specify in terms of its curl and divergence alone

DISTRIBUTED PARAMETER SYSTEMS  ECn 5241


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Objectives

Give an idea about DISTRIBUTED PARAMETER SYSTEMS and LUMP PARAMETER SYSTEMS with examples
Discuss about Del () operator and it associated functions
Discuss, analysis of divergence theorem, stroke’s theorem and helmholtz’s theorem with examples

DPS & LPS

DPS

Distributed parameter system is a system which the Parameters of system vary according to space and time, these system can be described mathematically by Partial Differential Equations.
Ex.; Transmission lines, Velocity of the fluid, Temperature of the fluid at different points, Fluid pressure... Etc.

LPS

The lump parameter system is a system witch reaches each element simultaneously. There is no propagation time, that mean lump parameters are always depend on the space.
Ex:- RLC circuit model, Hybrid parameters in electronic system.
However, if the frequency is so high that the length of the cable is of the same order of magnitude as the wavelength of the signal this obviously won't work, simply because phenomena like standing waves (and more generally impedance mismatch) become important.  Then we have to use distributed parameters to describe the cable and properties like the impedance is now a function of position Z(x), i.e. the parameters are distributed.
The line voltage V(x) and the current I(x) can be expressed in the frequency domain as