Control System

1. Mathematical Modeling of Electrical
Systems
Control Systems – MCE
3406
Ashraf AlShalalfeh,
M.Sc.
ashalalfeh@hct.ac.a

2. Electric Circuits Modeling
procedure:
• Apply the transfer function to the mathematical
modeling of electric circuits including passive
networks
• Configure Electric networks components we work
with, It consists of three passive linear
components: resistors, capacitors, and inductors.“
• Combine electrical components into circuits,
decide on the input and output, and find the
transfer function. Our guiding principles are
Kirchhoff s laws.

3.  The transfer function approaches to the
mathematical modeling of electrical circuits
 Circuits via mesh analysis
 Circuits via nodal analysis
Electrical Network Transfer Function:

4.  The transfer function to the mathematical modeling of electric
circuits.
Electrical Network Transfer Function
TABLE 2.3 Voltage-current, Voltage-charge, and impedance relationships for capacitors, registers, and inductors

5.  Simple Circuits via Mesh Analysis
 For the capacitor,
 For the resistor,
 For the inductor,
 The transfer function, impedance:
 Step 1. Laplace transform,
 Step 2. The component values => their impedance values
-> It's similar to the case of dc circuits.
Electrical Network Transfer Function
(2.67)
1
( ) ( )
V s I s
Cs

( ) ( )
V s RI s

( ) ( )
V s LsI s

( )
( )
( )
V s
Z s
I s

( ), ( ), ( ) ( ), ( ), ( )
c c
v t i t v t V s I s V s

(2.68)
(2.69)
(2.70)
(2.71)

6.  Ex 2.7: Find the transfer function relating the capacitor
voltage, Vc(s), to the input voltage, V(s), in Figure 2.3:
 Sol) Using figure 2.5 and writing a mesh equation using the
impedances as we would use resistor values in a purely resistive
circuit
Solving for I(s)/V(s).
Electrical Network Transfer Function
 
2
1
1
( )
I s
V s Ls R
Cs

 
   
1
Ls R I s V s
Cs
 
  
 
 
(2.73)
(2.74)
TABLE 2.3 RLC network

7.  Voltage across the capacitor, Vc(s) , is the product of the current and
the impedance of the capacitor.
Electrical Network Transfer Function
   
1
c
V s I s
Cs
 (2.75)

8.  To solve Complex circuits via mesh analysis steps
 Step1. Replace passive element values with their impedances.
 Step2. Replace all sources and time variables with their Laplace
transform.
 Step3. Assume a transform current and a current direction in each
mesh.
 Step4. Write Kirchhoff's voltage law around each mesh.
 Step5. Solve the simultaneous equations for the output.
 Step6. Form the transfer function
Electrical Network Transfer
Function

9.  Ex 2.10 Transfer function – multiple loops
 Given the network of Figure 2.6(a), find the transfer function,
I2(s)/V(s).
Laplace Transform Review
TABLE 2.6 a. Two-loop electrical network ;
b. transformed two-loop electrical network;
c. block diagram;

10.  Solution:
Around Mesh1, where I1(s) flows,
Around Mesh2, where I2(s) flows,
combining term
Laplace Transform Review
     
1 1 1 2
{ } ( )
R I s Ls I s I s V s
  
       
2 1 2 2 2
1
{ } 0
Ls I s I s R I s I s
Cs
   
     
1 1 2 ( )
R Ls I s LsI s V s
  
   
1 2 2
1
0
LsI s Ls R I s
Cs
 
    
 
 
(2.78)
(2.79)
(2.80a)
(2.80b)

11.  Use Cramer’s rule
 Forming the transfer function
Laplace Transform Review
 
   
 
1
2
0
R Ls V s
LsV s
Ls
I s


 
 
 
1
2
1
R Ls Ls
Ls Ls R
Cs
 
   
  
 
 
 
 
     
2
2
2
1 2 1 2 1
I s Ls LCs
G s
V s R R LCs R R C L s R
  
    
(2.81)
(2.82)

12. 8/27/2022