Topic: Modelling of physical systems: Electrical System

1. Unit:1
Systems and
their
representations
Topic: Modelling of physical systems:
Electrical System

2. Electrical System

3. Transfer Function of a Electrical System
Using KVL:
Using, i(t) =
𝑑𝑞(𝑡)
𝑑𝑡
Using, q t = C𝑉𝑐(𝑡)
Taking, Laplace Transform
Final Transfer Function
Find Transfer Function of the system:
𝑽 𝒄(𝒔)
𝑽(𝒔)

4. Example 1
Find Transfer Function of the system:
𝑸 (𝒔)
𝑽(𝒔)
Here, input is voltage and output is the
charge (q(t)) displacement
Using KVL:
Using, i(t) =
𝑑𝑞(𝑡)
𝑑𝑡
Taking, Laplace Transform
2 1
( ) ( ) ( ) ( )Ls Q s RsQ s Q s V s
C
  
2
( ) 1
1( ) ( )
Q s
V s Ls Rs
C

 

5. Comparing with Mechanical System
2
( ) 1
1( ) ( )
Q s
V s Ls Rs
C

 
Comparing Electrical System with Mechanical System
We can say:
B
Mechanical Electrical
Force (F) Voltage (V)
Displacement (x) Charge (q)
Mass (M) Inductance (L)
Damper (B) Resistance (R)
Spring (K) Reciprocal of capacitance (1/C)
Velocity (Ve) Current (I)

6. Example 2
Find Transfer Function of the system:
𝑰 𝟐 (𝒔)
𝑽(𝒔)
Taking, Laplace Transform
For Loop 1:
For Loop 2:
Solving the equation

7. Mechanical Gear

8. Mechanical Gear
We are assuming the system to be lossless thus no energy is wasted in the form of heat or anything
as the gears turn, the distance travelled along
each gear’s circumference
is the same. Thus
1 1 2 2r r  1 2 2
2 1 1
r N
r N


 
As ratio number of teeth ∝ ratio of radius

9. Mechanical Gear
T1 can be reflected to the output by multiplying by N2/N1. The
result is shown in below figure ,from which we write the equation of motion as

10. Mechanical Gear
equivalent system at the input after reflection of impedances
Thus, Rotational mechanical impedances can be reflected through gear trains by multiplying the
mechanical impedance by the ratio

11. Example 1:
Find the transfer function, θ2(s)/T1(s)
Step1: Step2:
Step3: