Exam #1

Problem #1

Consider the system with the transfer function 30/(s2 + 8s+7)

(5 pts) Is the system underdamped, overdamped, or critically damped?

We need to find z. The value of z tells us whether the system is overdamped or critically damped or underdamped.

Rearrange the polynomial into a more general form: t2s2 + 2tzs + 1:

 

So, t2 = 1/7, meaning t = 0.37796.

2tz = 8/7, so:

This is greater than 1, so the system is overdamped.


 

(20 pts) If x(t) = e-6t, find an expression for y(t). Show all work.

First invert x(t):

So:

Do partial fraction expansion:

Use Heaviside expansion to find the constants. Multiply by s+7, let s = -7:

Multiply by s+6, let s = -6:

Multiply by s+1, let s = -1:

So,

Invert:

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