# Sketch the waveform to represent the transient response

## Homework Statement

Sketch, on a set of common axes, waveforms to represent the transient
response of circuits having transfer functions with the following parameters:

a) ζ = 0.5, ω = 1×10^3 rad s^-1
b) ζ = 0.2, ω = 2×10^3 rad s^-1
c) ζ = 2, ω = 1×10^3 rad s^-1

## Homework Equations

wd=wo√(1-ζ^2)
Time to peak overshoot = π/wd
overshoot = e-(ζπ)/(√(1-ζ^2))

## The Attempt at a Solution

So with this in mind for
a)
wd=(1*103)√(1-0.5^2) = 866.025
Time to peak overshoot = π/wd = 0.00363seconds
overshoot = e-(ζπ)/(√(1-ζ^2)) = 0.16303

b)
wd=(2*103)√(1-0.2^2) = 866.025
Time to peak overshoot = π/wd = 0.00363seconds
overshoot = e-(ζπ)/(√(1-ζ^2)) = 0.16303

c) will have no overshoot as ζ = 2>1 (which means no oscillation).

How do i actually draw these as graphs now? is there more info i need?

do i also need to use ?

rude man
Homework Helper
Gold Member
What's the input? Step? Delta? Ramp? ....
Given this you can solve your ODE but there are many places to look up under-damped, critically damped, and over-damped 2nd order systems.

This is what i'm unsure of, if it mentioned one of the above i could work through it via my notes... I've only encountered step change input for second order systems. Thus i understand which ones are under-damped (a/b) and which are over-damped c if this is the case. But how do i draw this? only thing i can see from the graph is y(t)=1 but how do i find x(t) & the time period...

rude man
Homework Helper
Gold Member
This is what i'm unsure of, if it mentioned one of the above i could work through it via my notes... I've only encountered step change input for second order systems. Thus i understand which ones are under-damped (a/b) and which are over-damped c if this is the case.
Assume step input then.
But how do i draw this? only thing i can see from the graph is y(t)=1
Say what? The output is a constant dc voltage? And what graph? You haven't provided one for us to look at.
You have the information you need to draw the output graphs. I suggest looking up the various possible graphs depending on ζ and ω for a step input. Widely available on the web.

Assume step input then.Say what? The output is a constant dc voltage? And what graph? You haven't provided one for us to look at.
Apologies I think i've gotten myself confused... I think im looking too deep into it. I'll be back when i have some form of graph! Thanks for your help @rude man

rude man
Homework Helper
Gold Member
Apologies I think i've gotten myself confused... I think im looking too deep into it. I'll be back when i have some form of graph! Thanks for your help @rude man
Good idea. Google "second order systems responses". The curves will all be there.

@rude man so after searching through second order system responses I have a graph that looks like this. How does it look?

#### Attachments

• TMA 3 Q4 Waveform Sketch.pdf
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rude man
Homework Helper
Gold Member
That's the idea.
Make sure yu understand which waveform corresponds to your values of ωn and ζ.

You should also understand how your ODE in your post 1 solves to those waveforms. That equation requires one or two initial conditions otherwise you get nothing from it.