Coupled Systems Based Models-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

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Lecture Slides

on

Modeling and Simulation

Lecture: Coupled Systems based Models

Models : electrical systems and Resonance

C

R

L

V

I

Simple Mathematical Model for RLC Circuit

Current in circuit = I

Voltage Source = V

Voltage drop across resistance = RI

Voltage drop across capacitor = Q/C

Voltage drop across inductor = L dI/dt

Kirchoff’s law:

All voltage drops = Applied Voltage

RI CQ V

dt

dI

L   

Initial conditions:

Current (t = 0) =I(0) = 0

Charge (t = 0) = Q(0) = 0.

Mathematical modeling:

For the left loop, the voltage drops across each element of the circuit yields:

Voltage from the source = voltage drop across L + Voltage drop across R

Models for Coupled Electrical circuits

12  1  dI 1 (^) / dt  4 ( I 1  I 2 )

or 4 1 4 2 12 1   I  I  dt

dI

Where I1 is the current in first loop and I2 is the current in the second loop.

For the next loop again the voltage balance gives:

 0  6 I (^) 2  4 ( I 2  I 1 ) 4 I 2 dt

When we differentiate it we get the following balance equation:

2

2 1 0 10 4 4 I dt

dI

dt

dI    0. 4 0. 4 2 0 2 1   I  dt

dI

dt

dI

This model is a linear first order non-Homogeneous, ODE Based model

Initial conditions are : I1(0) = 0.0; I2(0) =0.

1   I  I  dt

dI

2 1   I  dt

dI

dt

dI

C = 0.25 farad

R 1 =4 ohms

L = 1 henry

V = 12 Volts

R 2 =4 ohms

Switch

t = 0

I 1 I 2 I (^1) I 2

The current in Tank 2

increases from zero

exponentially and reaches to

a peak value then it

eventually decays to zero.

The current in loop-1 grows

to a peak values and then

exponentially to a saturation

value of 3.

Model in MATLAB/SIMULINK

Example 1: Models for electrical coupled system

1   I  I  dt

dI 2

2 1

1. 4 0. 4 I dt

dI

dt

dI  

Initial conditions are :

I 1 (0) = 0.0; I 2 (0) =0.

0 2 4 6 8 10

0

1

2

3

4

5

current

time

current in loop-1: I 1 current in loop-2: I 2

Example 2: Another type of Coupled Electrical system

This has two coupled loops. The Kirchoff’s law for voltage balance will give the model equations:

C = 0.25 farad

R 1 =4 ohms

L = 1 henry

V = 12 Volts

L 2 =2 henry

Switch

t = 0

I 1 I 2 I (^1) I 2

This model is a linear, second order non-Homogeneous, ODE Based model

Initial conditions are :

I1(0) = 0.0; I2(0) =0.

1   I  I  dt

dI

C = 0.25 farad

R 1 = 0. ohms

L = 1 henry

V = 12 Volts L 2 = henry

Switch

t = 0

I 1 I 2 I (^1) I 2

2

2 1 2

2

2

0 0. 2 0. 2 2 I dt

dI

dt

dI

dt

d I    

Example 2: Another type of Coupled Electrical system

Model in MATLAB/SIMULINK

Example 2: Models for electrical coupled system

Initial conditions are :

I 1 (0) = 0.0; I 2 (0) =0.

1   I  I  dt

dI 2

2 1 2

2

2

1. 2 0. 2 2 I dt

dI

dt

dI

dt

d I   

I1(t)

I2(t)

I2(t)

-0.4I1 + 0.4I2+

0.2dI1/dt - 0.2dI2/dt +2.I

example 2 : coupled electrical model

dI1/dt

I To Workspace

I To Workspace

Scope

Scope

1 s Integrator

1 s Integrator

1 s Integrator

-0.

Gain

Gain

Gain

Gain

-0. Gain

12 Constant

The current in loop-1 increases and reaches a saturation value of 30. However, the current in loop-2 first increases to a peak value and then oscillates with decreasing amplitude,

Example 2: Models for electrical coupled system

Initial conditions are :

I 1 (0) = 0.0; I 2 (0) =0.

Model in MATLAB/SIMULINK

1   I  I  dt

dI

2

2 1 2

2

2

1. 2 0. 2 2 I dt

dI

dt

dI

dt

d I   

0 5 10 15 20

0

5

10

15

20

25

30

current

time

current in loop - 1

0 5 10 15 20

-1.

-1.

-0.

current

time

current in loop-

C = 0.25 farad

R 1 = 0.4 ohms

L = 1 henry

Sinusoidal

source

L 2 =2 henry

Switch

t = 0

I 1 I 2 I (^1) I 2

This has two coupled loops. One loop has a voltage source which is AC

type with V= Asin(wt). Let A = 12 and w can be changed from 0.1 to 2.5.

We want to study what will happen when we change this w value.

Example 3: AC based electrical coupled system

So, first develop mathematical model.

The Kirchoff’s law for voltage balance will give the model equations:

This model is a linear, second order, non-Homogeneous, ODE Based model

Initial conditions are :

I 1 (0) = 0.0; I 2 (0) =0.

Example 3: AC based Coupled Electrical system

C = 0.25 farad

R 1 = 0. ohms

L = 1 henry

Sinusoidal

source L 2 = henry

Switch

t = 0

I 1 I 2 I (^1) I 2

This has two coupled loops.

The Kirchoff’s law for

voltage balance gives the

model equations:

I I A  wt  dt

dI (^1)  0. 4 1  0. 4 2  sin

2

2 1 2

2

2 0 0. 2 0. 2 2 I dt

dI

dt

dI

dt

d I    

Driving force = Asin(wt); A = 12, w = 0.

Example 3: AC based electrical coupled system

Initial conditions are :

I 1 (0) = 0.0; I 2 (0) =0.

Effect of angular frequency parameter (w) on currents

I I A wt dt

dI

1. 4 1 0. 4 2 sin 1    2 2 1 2

2

2

1. 2 0. 2 2 I dt

dI

dt

dI

dt

d I   

Current in loop – 1 Current in loop^ –^2

RESONANCE

Driving force = Asin(wt); A = 12, w = 0.

Example 3: AC based electrical coupled system

Initial conditions are :

I 1 (0) = 0.0; I 2 (0) =0.

Effect of angular frequency parameter (w) on currents

I I A wt dt

dI

1. 4 1 0. 4 2 sin 1    2

2 1 2

2

2

1. 2 0. 2 2 I dt

dI

dt

dI

dt

d I   

Current in loop – 1 Current in loop^ –^2

RESONANCE

Driving force = Asin(wt); A = 12, w = 0.

Example 3: AC based electrical coupled system

Initial conditions are :

I 1 (0) = 0.0; I 2 (0) =0.

Effect of angular frequency parameter (w) on currents

I I A wt dt

dI

1. 4 1 0. 4 2 sin 1    2

2 1 2

2

2

1. 2 0. 2 2 I dt

dI

dt

dI

dt

d I   

Current in loop – 1 Current in loop^ –^2

RESONANCE