DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
TUTORIAL 3
(ANSWER SCHEME)
1. Write the differential equation for the system below.
R(s)
C(s)
s 5 2s 4 4s 3 s 2 3
6
5
4
3
2
s 7 s 3s 2 s s 3
The transfer function,
Cross-multiplying,
Taking the inverse Laplace transform and assuming zero initial conditions,
2. Write the differential equation that is mathematically equivalent to the block diagram below.
Assume that r(t) = 3t3.
R(s)
s5
C(s)
s 4 2 s 3 5s 2 s 1
3s 4 2 s 3 4s 2 5s 2
The transfer function,
Cross-multiplying,
Taking the inverse Laplace transform and assuming zero initial conditions,
DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
Substituting r(t) = 3t3 yields,
3. A system is described by the following differential equation:
d 2x
dt 2
With the initial conditions x (0)
2
1 , x (0)
dx
dt
3x 1
1 . Show a block diagram of the system, giving its
transfer function and all pertinent inputs and outputs.
d 2x
dt 2
2
dx
dt
3x
r (t )
Finding the Laplace transform for each element of the differential equation,
Substituting the transformed expressions into the differential equation yields,
DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
The block diagram is then,
+
+
4. Find the transfer function, G(s) = V 0(s)/V i(s), for each network below.
a) Writing the equation for nodes,
Solve for,
b) Theveninzing,
Using voltage divider rule,
DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
5. Find the transfer functions, G(s) = V L(s)/V(s), for each of the networks below.
a)
Writing the equation for nodes,
Rearranging the second equation yields,
Substituting I1(s) into the first equation,
or
DNT 354 - Control Principle
Knowing that,
Therefore,
b)
Solving for I2(s),
Therefore,
Universiti Malaysia Perlis (UniMAP)
DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
6. Find the transfer function, G(s) = V o(s)/V i(s), for each of the operational amplifier circuits below.
a)
Solving for Z1(s) and Z2(s),
Therefore,
DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
b) Solving for Z1(s) and Z2(s),
Therefore,
7. Find the transfer function, G(s) = X1(s)/F(s), for the translation mechanical system below.
Writing the equations of motion, where x 2(t) is the displacement of the right member of spring,
Adding the equations,
From which,
DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
8. Find the transfer function, G(s) = X2(s)/F(s), for the translation mechanical system below.
Writing the equations of motion,
Solving for X2(s),
From which,
9. Find the transfer function, G(s) = X2(s)/F(s), for the translation mechanical system below. (Hint: Place
a zero mass at x 2(t)).
DNT 354 - Control Principle
Universiti Malaysia Perlis (UniMAP)
Let X1(s) be the displacement of the left member of the spring and X 3(s) be the displacement of the
mass.
Writing the equations of motion,
Solving for X2(s),
Thus,
10. Find the transfer function, G(s) = X 1(s)/F(s) as figure below.
Solving for M2,
DNT 354 - Control Principle
Solving for M1,
Solving for transfer function X 1(s)/F(s),
Universiti Malaysia Perlis (UniMAP)