Transfer Function and Mathematical Modeling
Transfer Function
Poles And Zeros of a Transfer Function
Properties of Transfer Function
Advantages and Disadvantages of T.F.
Translation motion
Rotational motion
Translation-Rotation counterparts
Analogy system
Force-Voltage analogy
Force-Current Analogy
Advantages
Example
2. Transfer Function
Poles And Zeros of a Transfer Function
Properties of Transfer Function
Advantages and Disadvantages of T.F.
3. Transfer function gives us the relationship between the
input and the output and hence describes the system.
In control systems, the input is represented as r(t)
(instead of x(t)) and the output is represented as c(t)
(instead of y(t)).
4. Once the concept of Transfer function is understood,
Let us move a little further.
Let the Transfer function of a system be represented
as G (s).
i.e. G (s) =
Or more generally G(s) =
Where C(s) :- Laplace transform of output ;
R(s) :- Laplace transform of input.
Both C(s) and R(s) are polynomials in s.
i.e. G(s) =
Vo (s)
Vi (s)
C(s)
R(s)
5. =
Poles :- poles of a transfer function are the values of
the Laplace transform variable, s, that cause the
transfer function to become infinite.
When s=a1, s=a2, s=a3…s=an, the transfer function
G(s) becomes infinite.
Hence a1,a2,a3…an are the poles of the transfer
function.
If we equate the denominator of the transfer function
to zero, we obtain the poles of the system.
K(s-b1) (s-b2) (s-b3)…(s-bm)
(s-a1) (s-a2) (s-a3)…(s-an)
6. Zeros :- zeros of a transfer function are the values of
the Laplace transform variable, s, that cause the
transfer function to become zero.
When s=b1, s=b2, s=b3…s=bm, the transfer function
G(s) becomes equal to zero.
Hence b1,b2,b3…bm are the zeros of the transfer
function.
If we equate the numerator of the transfer function to
zero, we obtain the zeros of the system.
7. 1. The transfer function of a system is the Laplace
transform of its impulse response for zero initial
conditions.
2. The transfer function can be determined from
system input-output pair by taking ratio Laplace of
output to Laplace of input.
3. the transfer function is independent of the inputs to
the system.
4. The system poles/zeros can be found out from
transfer function.
5. The transfer function is defined only for linear time
invariant systems. It is not defined for non-linear
systems.
8. Advantages :-
1. It is a mathematical model that gives the gain of the
given block/system.
2. Integral and differential equations are converted to
simple algebraic equations.
3. Once the transfer function is known, any output for
any given input, can be known.
4. System differential equation can be obtained by
replacement of variable ‘s’ by ‘d/dt’
5. The value of transfer function is dependent on the
parameters of the system and independent of the
input applied.
9. Disadvantages :-
1. Transfer function is valid only for Linear Time Invariant
systems.
2. It does not take into account the initial conditions. Initial
conditions loose their significance.
3. It does not give any idea about how the present output
is progressing.
11. As stated earlier translation
motion refer to a type of motion in which a body or an object
moves along a linear axis rather than a rotation axis.
Translation motion involves
moving left or right , forward or back , up and down.
The following three basic element viz.
1). Mass
2). Spring
3). Damper
12. • A model of the mass element assumes that the
mass in concentrated at the body.
• The Displacement of the mass always take place in
the direction of the applied force.
14. If a mobile phone on the table needs to be pushed
from one place two other, we needs to apply force.
The force that we apply will have to overcome this
friction.
Friction exists between a moving body and a fixed
support or also between moving surface.
while friction opposes motion, it is not always
unesirable
16. • In case of a spring , we require force to deform the
spring.
• Here the force is proportional to the displacement.
• Net displacement on application of force f(t) at and X1
and X2
F(t) = K [ X1(t) - X2(t)]
17. In such system, force get
replaced by Torque(T), displacement by angular
displacement ( ), velocity by angular velocity( )
and acceleration by angular acceleration( ).
• The following three element viz.
1). Inertia J
2). Damper
3). Spring
18. In rotation motion, we have a concept of inertia.
T =
2). Damper
• As stated earlier, it’s behavior is similar to that in
translation motion.
T(t) = B .
19. • Like the damper, the spring is also similar to the one
studied in translation motion.
T = K (t)
20. Sr. No. Translation Motion Rotational Motion
1 Mass (M) Inertia (J)
2 Damper (B) Damper (B)
3 Spring (K) Spring (K)
4 Force (F) Force (T)
5 Displacement (X) Angular Displacement
6 Velocity = v Angular Velocity =
21. There are two main electrical analogous system :
1). Force-voltage analogy
2). Force-current analogy
Now that we have Discussed mechanical system as well
electrical system, it is worth nothing that exists a analogy-
similarity in their equations.
According to Newton’s law, the applied force will be used
up to cause displacement in the spring, acceleration to
the mass against the friction force.
+ + K x (t)
22. we get,
F(t) = M S2
X(s) + s B(s) X(s) + K x(s)
• This equation is called equilibrium equation of the
mechanical system.
23. • Here force is analogous to voltage.
v(t) = Ri +L + . dt
put, i=
v(t) = R + L + q
24. Translation Electrical Rotational
Force F Voltage - V Torque - T
Mass M Inductance - L Inertia J
Dumper B Resistance - R Damper -B
Spring K Elasticity – D = Spring K
Displacement X Charge -q Displacement
Velocity -V Current - i = Velocity - ω
ω
25. • Standard equation for a Translational system is
+ + K x (t)
The following analogies
F V
M L
B R
K
X Q
26. Standard equation for a Rotational system is
T = J + B . + K
The following analogies
• T V
J L
B R
K
Q
27. Here force is analogous to current.
I = + ∫ V. dt C
put, v =
I = . + + C
i(s)=1/R.S. .S
28. Translation Electrical Rotational
Force F Voltage - i Torque - T
Mass M Capacitance - C Inertia J
Spring K Resistance of inductance
-
Spring K
Damper B Conductance = Damper - B
Displacement X Flux linkege - Displacement -
Velocity Voltage V = Velocity
29. Standard equation for a Translation system is
+ + K x (t)
The following analogies
• F I
• M C
B
K
X
30. Standard equation for a Rotational system is
T = J + B . + K
The following analogies
• T I
J C
B
• K
Q
31. • Equation of the system can be converted into
another.
• Trial design in one system involving changinge of the
values M , B , K may be costlier than changing in R ,
L , C .
