1. Submitted by
Waqas Wali (54)
Umer Habib Khan (47)
Talha Khan (37)
Zia Ullah (23)

2.  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

3. Contents
 Euler’s Method
 Heun’s Method
 Runge Kutta’s 4th Order Method

4. 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.

5. 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.

6. 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.
V1 - V0 IR
V IR

8. Behavior of capacitor
 A capacitor is a circuit element that is use to store
electrical charge.
 The equation of voltage across the capacitor is
d (V 1 V 0 )
C I
dt
dV
C I
dt

9. 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.

10. 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.

11. RC-model
 Considering the capacitive and resistive nature of a pn
junction we can represent it by an RC circuit.
V in Vc IR
and
C dV c I
dt
V in Vc RC dV c
dt

12. Ordinary Differential Equation
 A differential equation is an algebraic equation that
contains some derivatives
e.g. dy
5 y 3x
dx
 Derivative indicates a change in a dependent variable
with respect to an independent variable.

13. 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.

14. How to Solve ODEs
 The methods are
1. Euler’s method.
2. Heun’s method.
3. Runge kutta’s 4th order method.

15. Euler’s Method.
 Euler’s method is a numerical technique to solve
ordinary differential equations of the form
dy
f ( x , y ), y ( 0 ) y0
dx
 Only first order ordinary differential equations can be
solved by using Euler’s method.

16. Euler’s Method
 Formula for Euler’s method is
yi 1 yi f ( x i , y i )( x i 1 xi )

17. Heun’s Method.
 Heun’s method, introduces a new idea to solve the
I.V.P.
h
yi 1 y (ti ) ( f (ti, y i ) f (ti , yi
1 hf ( t i , y i )))
2

18. Runge Kutta’s 4th Order