The Laplace Transform HW

Transfer Functions:

1. Find the transfer functions of the following systems: a) &y y x+ =4 3 b) && & &y y y x x+ + = −4 20 2 c) &&& && & && &y y y y x x x− + + = − +3 4 8 4 2

2. Find the transfer function of

+

-

R1 R2

C1 C2 x(t) y(t)

+

-

Give the result for C1=C2=100µf, R1 =R2 =2000Ω

3. Find the transfer function of the following circuit where R1=R2=1000Ω and C=100µf.

a)

4. For the system given below,

&& &y y y x+ + =8 116 116

a) Find the transfer function. b) Give the poles and zeros. c) Give the general form of the response y(t) to a step input (do not solve explicitly). d) Use MATLAB to plot the step response (put your name in the title of the plot).

5. Repeat Problem 4 for the system given below. In addition, compare the types of poles of this system to those in Problem 4 and use this to explain the resulting behavior seen in the step response plots.

&& &y y y x+ + =8 12 12

R2 + +

y(t) R1 C

- x(t)

-

6. Simplify the block diagram to find the transfer function

X(s) Y(s)

H1(s)

H2(s)

H4(s)

H3(s) +

+ + -

Give the transfer function H(s)=Y(s)/X(s) for H1(s)=2, H2(s)=10/s, H s s3

01

20 ( )

. =

+ , H s

s4 2

4 ( ) =

+ 7. Reduce the block diagram to one block.

8. Find the transfer function of the following circuit in terms of R1, R2, C, and L. Now, suppose that R1=R2=2000Ω, C = 100µf, L = 10mH. Determine the poles of the circuit.

H3(s)

H4(s)

H1(s)

H2(s)

X(s) Y(s)

- +

+

+

+ y(t) -

R1

C

+ x(t) -

R2

L