Lecture Slides
on
Modeling and Simulation
Lecture: Coupled Systems based Models
Models : electrical systems and Resonance
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Coupled Systems Based Models-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation
These lecture slides are delivered at The LNM Institute of Information Technology by Dr. Sham Thakur for subject of Mathematical Modeling and Simulation. Its main points are: Continuous, Pdfs, Random, Variables, Probability, Uniform, Distribution, Solution, Program, Generate
Typology: Slides
2011/2012
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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
- 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
- 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
- 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
- 4 1 0. 4 2 sin 1 2 2 1 2
2
2
- 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
- 4 1 0. 4 2 sin 1 2
2 1 2
2
2
- 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
- 4 1 0. 4 2 sin 1 2
2 1 2
2
2
- 2 0. 2 2 I dt
dI
dt
dI
dt
d I
Current in loop – 1 Current in loop^ –^2
RESONANCE