Submitted by

Waqas Wali (54)Umer Habib Khan (47)

Talha Khan (37)Zia Ullah (23)

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

Contents Euler’s Method

Heun’s Method

Runge Kutta’s 4th Order Method

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.

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.

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

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

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.

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.

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

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

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.

How to Solve ODEs The methods are

1. Euler’s method.

2. Heun’s method.

3. Runge kutta’s 4th order method.

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

Euler’s Method Formula for Euler’s method is

))(,( 11 iiiiii xxyxfyy

Heun’s Method. Heun’s method, introduces a new idea to solve the

I.V.P.

))),(,(),((2

)( 11 iiiiiiii ythfytfytfh

tyy

Runge Kutta’s 4th Order