- Information
- AI Chat
Transfer Fuctions Notes
Process Control Systems 3B
Durban University of Technology
Preview text
TRANSFER FUCTIONS
NOTES
[Acceleration of mass M =D2(x + y)] The mass-spring damper system shown in Fig. 2 represents the suspension for an automobile in which B is the shock absorber, K is the spring, and M is the mass of the car.
Construct the grounded-chair representation, and then determine the equation relating f and x and that relating y and x.
Figure 2.
Solution
Grounded-chair representation (Fig. 2)
Note: Remember the steps to be followed in constructing the grounded chair
representation ( RAVEN page 29).
Figure 2.
Equation relatingfand x The series combination of spring K and B is in parallel with M. Application of the laws for parallel and series elements gives:
x
therefore
MD 2 (K + BD)
x (1) MD 2 + BD + K
Equation relating y and x
f = MD 2 y (2) The desired relationship between y and x is obtained by eliminating f between equations (1) and (2).
MD 2 (K + BD) i. MD 2 y x MD 2 + BD+K
Example 13
By means of block-diagram reduction techniques, find the transfer ftnction of the system,
, for the configuration illustrated in Fig. 2.
Figure 2.
Solution
(1)
(2)
(3)
(4)
Substituting equation (3) into (4), gives (5)
Substituting equation (2) into (5), gives C(s) =
(6)
Substituting equation (1) into (6) C(s) =
-G3(s)
Thus
[1 +A(s) +B(s) + ID(s)] C(s) =
therefore
where
-
- -
Substituting equation (1) into (6) C(s) =
- G3(s)
Thus
therefore ID(s)]
where A(s) =
D(s) = G3(s) H3(s)•, and
From the above, it is clear, by inserting a sign reverser, causes a negative feedback.
Example 15
Determine the transfer function, E(s)/F(s) (also called the error ratio), of the system shown in Fig. 2.
Solution
(1)
- G3(s)
R(s) = GI - 1-
Substituting equation (4) into (3), gives
Substituting equation (4) into (2)., gives
(2)
(3)
(4)
(5)
R(s) = GI - H2 (s)G3 (s) K(s)
Substituting equation (6) into (5), gives
(6)
K(s) = - H2 (s)G3 (s) K(s)]- (s) K(s) = (s)G3 (s) K(s)- (s) K(s) thus
[1 + therefore G 2 (s)Gl (7) 1 + G 2 (s)H2 (s)G3 (s) + (s)G3 (s)
Substituting equation (7) into (4), gives G3 (s)G2 (s)Gl (8) 1 + (s)H2 (s)G3 (s) + (s)G3 (s)
=
Figure 2)
Refer to paragraph 2.11 for the standard form of a single-loop feedback system. Replace the cascaded elements in loop 2, with one element: Fig. 2(b).
Figure 2(b) Eliminate loop 2: Fig. 2(c).
Replace the cascaded elements in loop 3, with one element: Fig. 2(d).
Eliminate loop 3: Fig. 2@).
Transfer Fuctions Notes
Course: Process Control Systems 3B
University: Durban University of Technology
- Discover more from:
