System Modeling Coursework Class 3: Mathematical Modeling of Electrical Systems P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email:
[email protected],
[email protected] Blog: www.godsfavouritechild.wordpress.com Web address: http://www.esnips.com/web/SystemModelingClassNotes
WARNING! • 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. • For best results, it is always suggested you read the source material. July – December 2008
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Contents • • • • • •
Basic elements of an electrical system Equations for the basic elements Numerical and its solutions Summary Unsolved Problems References
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Basic elements of the electrical system • The basic components that are used in electrical systems are resistance, inductance and capacitance. • These systems are analyzed by the application of Kirchhoff’s voltage and current laws.
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Governing equations for the basic elements VR (t) = R i(t) VL (t) = L VC (t) =
di(t) dt
1 idt + V (0) ∫ C
In Laplace domain, the equations are
VR ( s ) = I R ( s ) R VL ( s ) = LsI L ( s ) I c ( s ) = CsVc ( s )
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Mathematical functions of electrical quantities
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Problem No.1 • 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 Vi (t) =Ri(t) +L
di(t) 1 + ∫ i (t )dt dt C
1 V0 (t) = ∫ i (t )dt C Vi (s) =RI(s) +LsI(s) +
1 I(s) Cs
1 I ( s) Cs 1⎤ ⎡ Vi (s) = ⎢ R +Ls + ⎥ I(s) Cs ⎦ ⎣ V0 (s) 1 = 1⎤ 1 Vi (s) ⎡ R +Ls + ⎢⎣ Cs ⎥⎦ Cs V0 (s) =
V0 (s) 1 = Vi (s) ⎡⎣ LCs 2 + RCs + 1⎤⎦ July – December 2008
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Problem No.2 • 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 • Always identify the basic elements in the given circuit • Write equations for the basic elements using governing laws for the circuit • Relate them in terms of specified input and output • If necessary, do the computation in laplace domain
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Review Questions • 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 • 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… • THY will be done
Thanks for listening… July – December 2008
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