application of od es
TRANSCRIPT
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Submitted by
Waqas Wali (54)Umer Habib Khan (47)
Talha Khan (37)Zia Ullah (23)
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Contents
Problem statement
How to analyze PN junction
Behavior of Resistor
Behavior of Capacitor
Behavior of PN junction
RC Model
Ordinary Differential Equation
How to solve ODE
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Contents Euler’s Method
Heun’s Method
Runge Kutta’s 4th Order Method
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Problem statement We have to analyze the behavior of semi-conductor
devices used in electronics devices i.e. in computer.
The most common semi-conductor device is transistor and in turn the PN junction.
So our problem is to analyze the PN junction.
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How to analyze PN junction PN junction can be analyzed by constructing there
lumped circuit having with very simple circuit components comprising of resistors and capacitors.
The lumped circuit can be named as “RC-model”.
Before going through the RC-model we will first throw some light on resistor and capacitor separately.
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Behavior of resistor A resistor is an electrical element that introduce
hindrance to the flow of current.
From Ohm’s law, the equation of voltage across the resistor is.
IRV
IRV-V 01
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Behavior of capacitor A capacitor is a circuit element that is use to store
electrical charge.
The equation of voltage across the capacitor is
Idt
dVC
Idt
VVdC
)( 01
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Behavior of PN junction PN junction also resist the electrical current from
flowing and hence provide resistance.
It also stores charge across its junction as the capacitors do hence behaving like capacitor.
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Behavior of PN junction The depletion layer of a PN junction is the interface of the
junction, and (as the name implies) is depleted of charge carriers.
The bias voltage across the junction determines the
capacitance of the depletion layer.
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RC-model Considering the capacitive and resistive nature of a pn
junction we can represent it by an RC circuit.
dtdVRCVV
Idt
dVC
and
IRVV
ccin
c
cin
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Ordinary Differential Equation A differential equation is an algebraic equation that
contains some derivatives
e.g.
Derivative indicates a change in a dependent variable with respect to an independent variable.
xydx
dy35
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How to Solve ODEs There are different methods to solve ordinary
differential equations.
Each method have its own advantage depending upon the computation time and accuracy.
These methods trade off between computation time and accuracy.
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How to Solve ODEs The methods are
1. Euler’s method.
2. Heun’s method.
3. Runge kutta’s 4th order method.
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Euler’s Method.
Euler’s method is a numerical technique to solve ordinary differential equations of the form
Only first order ordinary differential equations can be solved by using Euler’s method.
0)0(),,( yyyxfdx
dy
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Euler’s Method Formula for Euler’s method is
))(,( 11 iiiiii xxyxfyy
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Heun’s Method. Heun’s method, introduces a new idea to solve the
I.V.P.
))),(,(),((2
)( 11 iiiiiiii ythfytfytfh
tyy
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Runge Kutta’s 4th Order