Sketch waveform to represent the transient response

[creative10]

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

The Attempt at a Solution

Ive been looking through all my work books but I can't seem to find the equations to convert the information provided into a graph, I am just looking for someone to point me in the right direction thanks in advance.

 

[gneill]

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

The Attempt at a Solution

Ive been looking through all my work books but I can't seem to find the equations to convert the information provided into a graph, I am just looking for someone to point me in the right direction thanks in advance.

Hi Creative10, Welcome to Physics Forums.

ζ is known as the damping factor, so it makes sense to investigate it. Look up in your text or online: RLC circuit damping. You should be able to come across examples with graphs of the response for the various cases.

 

[creative10]

Hi gneill

Thanks for the advice, I've been reading through a number of websites on damping and I think the equation I need is the following below, am I on the right tracks? just need to work out how to obtain the value of K

x^..+2ζωex^.+ω^2x=Kω^2y

 

[gneill]

Hi gneill

Thanks for the advice, I've been reading through a number of websites on damping and I think the equation I need is the following below, am I on the right tracks? just need to work out how to obtain the value of K

x^..+2ζωex^.+ω^2x=Kω^2y

I'm not positive about that equation. It's incomplete as shown (missing exponents).

My instinct would be to write the transfer function for an RLC circuit in Laplace form and take the inverse Laplace to find the equation for the time domain. Or, solve the differential equation for the RLC circuit by other means. Then plug in the given parameters to get the particular curves.

But really, the question only asks you to sketch the curves, not to solve the equations and plot them. You should be able to find enough information/examples to do that without solving any equations.

 

[creative10]

Your right it doesn't ask me to solve the equations and I have a waveform example here that's quite similar so I could use that, thanks again.

 

[Gremlin]

I'm not positive about that equation. It's incomplete as shown (missing exponents).

My instinct would be to write the transfer function for an RLC circuit in Laplace form and take the inverse Laplace to find the equation for the time domain. Or, solve the differential equation for the RLC circuit by other means. Then plug in the given parameters to get the particular curves.

But really, the question only asks you to sketch the curves, not to solve the equations and plot them. You should be able to find enough information/examples to do that without solving any equations.

What would the two axis of the graphs be? When you look at damping graphs on the internet the axis are often different?

 

[gneill]

What would the two axis of the graphs be? When you look at damping graphs on the internet the axis are often different?

The response of an electronic system to a stimulus is most often portrayed as a plot of the output versus time, the output having the same units as that of the input stimulus. This is not always the case for example the desired "output" might be a particular current that occurs as a result of a given voltage input, but that's less common than, say, a ratio of voltage output to voltage input.

When looking at system responses in general the particular units of what is considered stimulus and what is considered the output don't matter. It's the mathematical relationship between them that is important.

 

[Gremlin]

I have no idea where to begin with this.

The question gives us:

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

The only thing i can find that relates ζ & ω is ζ = α/ω

a) α = 1000
b) α = 800
c) α = 4000

Past that i have no idea. The only graph i can see in my course notes is attached.

 

[gneill]

Wikipedia's entry on the RLC Circuit has a reasonable overview of damping on transient response and the equations and constants involved. Your course notes should also have some coverage of the differential equation solutions for the three cases: under damped, critically damped, and over damped. Take a look there and at the graph they give showing responses for various amounts of damping.

 

[Gremlin]

I can see the time to peak overshoot which = π/ω_d

where ω_d = ω₀√1-ζ²

 

[Electest]

Hi Gneill

I've been looking at this and was trying the following

ωd = ωo*SQRT(1 − ζ^2)

Where wo= 1x10^3 and ζ=2

But you can't SQRT a negative number

1x10^3*SQRT(1-2^2)

1000*SQRT(-3)

Any pointers

Thanks

 

[Electest]

Unless it's complex??

 

[gneill]

Hi Gneill

I've been looking at this and was trying the following

ωd = ωo*SQRT(1 − ζ^2)

Where wo= 1x10^3 and ζ=2

But you can't SQRT a negative number

1x10^3*SQRT(1-2^2)

1000*SQRT(-3)

Any pointers

Thanks

Consider what is implied by ζ being greater than unity. What form does the resulting response curve take?

 

[Gremlin]

Look at pages 48 & 49 of 4-1 - i think the answers are there - although I'm still trying to pick the bones out of it myself.

 

[Gremlin]

Consider what is implied by ζ being greater than unity. What form does the resulting response curve take?

It means it's over damped and there is no overshoot.

 

[Gremlin]

Wikipedia's entry on the RLC Circuit has a reasonable overview of damping on transient response and the equations and constants involved. Your course notes should also have some coverage of the differential equation solutions for the three cases: under damped, critically damped, and over damped. Take a look there and at the graph they give showing responses for various amounts of damping.

So for a & b the graph will show no overshoot - but for c it will as ζ >1.

Can we plot these on a graph using the same axis as in my attachment in #8? The only thing that concerns me is the ωo 2 x 10³ in B rather than ωo 1 x 10³ in the other 2 transfer functions.

 

[gneill]

It means it's over damped and there is no overshoot.

And more importantly, no oscillations ("ringing") superimposed on the output signal. What does that imply about any frequency associated with the output signal?

As loosely related mathematical analogy, what do imaginary roots of a quadratic equation imply about x-intercepts of plotted function?

 

[Gremlin]

And more importantly, no oscillations ("ringing") superimposed on the output signal. What does that imply about any frequency associated with the output signal?

As loosely related mathematical analogy, what do imaginary roots of a quadratic equation imply about x-intercepts of plotted function?

I'm not sure in all honesty - that it's the same as the input?

 

[gneill]

So for a & b the graph will show no overshoot - but for c it will as ζ >1.

Can we plot these on a graph using the same axis as in my attachment in #8? The only thing that concerns me is the ωo 2 x 10³ in B rather than ωo 1 x 10³ in the other 2 transfer functions.

You might consider that the plot time axes have been "normalized" so that all presented waveforms will have a unit time constant on the axis. Or you might consider scaling the plot with the "unlike" frequency (a higher frequency will "compress" the plot in the time dimension accordingly). Just be sure to include a description of what you've done and any assumptions you've made.

 

[Gremlin]

I've got this question to a point where I'm happy with it. Thanks for your time.

 

[gneill]

I'm not sure in all honesty - that it's the same as the input?

The input is a step function. There's no one frequency associated with it. The output, on the other hand, may show "ringing".

 

[Electest]

Hi Gneill/Gremlin

I've had a good read of my learning material and done some googling, so trying to mate sense of it

I was calculating Td , Tr , Tp , Ts and Mp, to sketch my graph, which was fine for ζ less than 1.

I realized that things wouldn't be straight forward as β = cos^−1 (ζ )

I have seen plenty of sketches showing values of zeta up to 5, but how do I go about plotting these.

Thanks

 

[gneill]

Hi Gneill/Gremlin

I've had a good read of my learning material and done some googling, so trying to mate sense of it

I was calculating Td , Tr , Tp , Ts and Mp, to sketch my graph, which was fine for ζ less than 1.

I realized that things wouldn't be straight forward as β = cos^−1 (ζ )

I have seen plenty of sketches showing values of zeta up to 5, but how do I go about plotting these.

Thanks

Investigate the solutions to the differential equation. The form of the solution for the various damping cases is different. The wikipedia article I mentioned previously (for the RLC circuit) shows the mathematical forms.

Also note that the question asks for sketches, not plots. You may be overworking the problem.

 

[Electest]

I know that they only say sketch, but they give you values of zeta and wo, which can be used to calculate:

1. Delay time, Td2. Rise time, Tr3. Peak time, Tp4. Maximum overshoot, Mp5. Settling time, Ts

Which is fine for the first 2 lines, which are under damped (zeta=<1), but have throw in the third where zeta=>1, so you have to go a step further in your study.

Confused [emoji20]

 

[donpacino]

I know that they only say sketch, but they give you values of zeta and wo, which can be used to calculate:

1. Delay time, Td2. Rise time, Tr3. Peak time, Tp4. Maximum overshoot, Mp5. Settling time, Ts

Which is fine for the first 2 lines, which are under damped (zeta=<1), but have throw in the third where zeta=>1, so you have to go a step further in your study.

Confused [emoji20]

ask yourself if a system is overdamped, is there an overshoot?

 

[Electest]

No, and I can see that clearly on many graphs on the subject.

 

[donpacino]

No, and I can see that clearly on many graphs on the subject.

so an overdamped system will slowely rise to the end value, kind of like a first order response. You'll see that it will never actually reach 100%, and will not have any overshoot

 

[Electest]

Yes, I also understand that too. I guess the formulas I'm using are only valid for values up to zeta=1. I think that, what I'm currently studying only scratches the surface of this subject.
Why I've questioned this and because they don't just give you a value of Zeta, but also include a value of an I damped natural frequency wo. Why give both values to sketch a graph after all?

Are there any relationships between different values of Zeta, as you can clearly see the waveforms differ in frequency (looking at values of Zeta from 0.1, 0.2, 0.3 etc)

Thanks

 

[donpacino]

Yes, I also understand that too. I guess the formulas I'm using are only valid for values up to zeta=1. I think that, what I'm currently studying only scratches the surface of this subject.
Why I've questioned this and because they don't just give you a value of Zeta, but also include a value of an I damped natural frequency wo. Why give both values to sketch a graph after all?

Are there any relationships between different values of Zeta, as you can clearly see the waveforms differ in frequency (looking at values of Zeta from 0.1, 0.2, 0.3 etc)

Thanks

you need zeta and the natural frequency to find the resonant frequency

 

[Electest]

Sorry a few typos above. Don't know how to edit.
Undamped natural frequency (wo)