System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: pr.venkat@manipal.edu, prv_i@yahoo.com

Blog: www.godsfavouritechild.wordpress.comWeb address: http://www.esnips.com/web/SystemModelingClassNotes

Class 3: Mathematical Modeling of Electrical Systems

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WARNING!

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I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation.

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For best results, it is always suggested you read the source material.

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Contents

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Basic elements of an electrical system•

Equations for the basic elements

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Numerical and its solutions•

Summary

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Unsolved Problems•

References

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Basic elements of the electrical system

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The basic components that are used in electrical systems are resistance, inductance and capacitance.

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These systems are analyzed by the application of Kirchhoff’s voltage and current laws.

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Governing equations for the basic elements

R

L

C

V (t) = R i(t)di(t)V (t) = Ldt

1V (t) = (0)C

idt V+∫

In Laplace domain, the equations are

( ) ( )

( ) ( )

( ) ( )

R R

L L

c c

V s I s R

V s LsI s

I s CsV s

=

=

=

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Mathematical functions of electrical quantities

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Problem No.1

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Consider the electrical system with Resistor, Inductor and Capacitor in Series shown in figure. Find the Transfer function of the system

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Solution to Problem No.1

i

0

di(t) 1V (t) =Ri(t) +L ( )dt

1V (t) = ( )

i t dtC

i t dtC

+ ∫

∫i

0

i

0

i

1V (s) =RI(s) +LsI(s) I(s)

1V (s) = ( )

1V (s) = R +Ls I(s)

V (s) 11 1V (s) R +Ls

Cs

I sCs

Cs

Cs Cs

+

⎡ ⎤+⎢ ⎥⎣ ⎦

=⎡ ⎤+⎢ ⎥⎣ ⎦

02

i

V (s) 1V (s) LCs RCs 1

=⎡ ⎤+ +⎣ ⎦

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Problem No.2

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Find the transfer function of the following circuit in Figure.

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Solution to Problem No.2

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Solution to Problem No.2

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Summary

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Always identify the basic elements in the given circuit

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Write equations for the basic elements using governing laws for the circuit

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Relate them in terms of specified input and output•

If necessary, do the computation in laplace

domain

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Review Questions

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Given the electrical network as in figure, solve for the transfer function I2

(s)/V(s) of the system.

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Review Questions

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Find the transfer function G(s) = V0

(s)/Vi

(s) for the networks in figure (a) and (b).

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References

• MIT OCW material: Lecturer: Penmen Gohari

amongst others…

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And, before we break…

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THY will be done

Thanks for listening…