The document discusses mathematical modeling of electrical systems and circuits using transfer functions. It explains that electrical networks can be modeled by representing the components as impedances and applying Kirchhoff's laws. Transfer functions are obtained by taking the Laplace transform of the network equations. Simple examples are worked out showing how to write the mesh or nodal equations, solve for the output variable, and obtain the transfer function relating an output such as voltage to the input.

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