Determining Response of Overdamped System

[Ivan Antunovic]

Homework Statement

When we have a damped system,we have 3 types of responses of the system,what really determines how our system will behave?

Homework Equations

$$\alpha = R/2L$$
$$\Omega = 1/LC$$

The Attempt at a Solution

Overdamped system solution is $$x(t)=A1*exp(\alpha+\sqrt(\alpha^2-\omega^2) + A2* exp-(\alpha+\sqrt(\alpha^2-\omega^2)$$,this green exponential decay is when one of the constants is zero I guess,because then we have exp(-t),what about other curves?

 

[donpacino]

I suggest you look at the system in the frequency domain.

 

[Mark44]

Homework Statement

When we have a damped system,we have 3 types of responses of the system,what really determines how our system will behave?

Homework Equations

$$\alpha = R/2L$$
$$\Omega = 1/LC$$

The Attempt at a Solution

Overdamped system solution is $$x(t)=A1*exp(\alpha+\sqrt(\alpha^2-\omega^2) + A2* exp-(\alpha+\sqrt(\alpha^2-\omega^2)$$,

Your solution should be a function of t, which doesn't appear on the right side above.

this green exponential decay is when one of the constants is zero I guess,because then we have exp(-t),what about other curves?

The system is modeled by a 2nd order differential equation. The roots of the characteristic equation for the differential equation determine whether the system is overdamped, underdamped, or critically damped.

 

[Ivan Antunovic]

Homework Statement

When we have a damped system,we have 3 types of responses of the system,what really determines how our system will behave?

Homework Equations

[tex][/B]\alpha = R/L[/tex]
[tex]\Omega = 1/LC[/tex]

The Attempt at a Solution

Overdamped system solution is [tex] x(t)=A1*exp(\alpha+sqrt(\alpha^2-\omega^2) + A2* exp-(\alpha+sqrt(\alpha^2-\omega^2)[/tex],this green exponential decay is when one of the constants is zero I guess,because then we have exp(-t),what about other curves?[/B]

Your solution should be a function of t, which doesn't appear on the right side above.The system is modeled by a 2nd order differential equation. The roots of the characteristic equation for the differential equation determine whether the system is overdamped, underdamped, or critically damped.

Yes,I know that roots determine system behaviour,what I wonder is about DAMPED system,we have 3 variations of the response and I forgout to multiply by t

 

[Ivan Antunovic]

I am really sad that none actually understood my question,what I figured out,those 3 graphs are determined by initial conditions, for the blue one if the capacitor has initial voltage as on the picture it will produce a current that has the same direction as the initial current of the inductor,that's why we have a little 'spike' over value of 1.(they are being added up)

The red graph ,is when we have initial voltage on the capacitor producing a current in ic in the opposite direction of the initial current in the inductor ,and therefore we have a negative 'spike' ,because in the scenario of the blue graph currents add up,and here one current will have bigger value than the other one.

What about the green graph,I am not quite sure about it?

 

[NascentOxygen]

I am really sad that none actually understood my question,what I figured out,those 3 graphs are determined by initial conditions, for the blue one if the capacitor has initial voltage as on the picture it will produce a current that has the same direction as the initial current of the inductor,that's why we have a little 'spike' over value of 1.(they are being added up)

The red graph ,is when we have initial voltage on the capacitor producing a current in ic in the opposite direction of the initial current in the inductor ,and therefore we have a negative 'spike' ,because in the scenario of the blue graph currents add up,and here one current will have bigger value than the other one.

What about the green graph,I am not quite sure about it?

The middle graph is what would result when the capacitor has an initial voltage but there is no initial current in the inductor.

I think you might have the direction of initial I_L around the wrong way for red and blue, but anyway your idea sounds spot on.

I'm not sure that the values of RC and L that give rise to the red curve would, without change, also be able to produce the blue and green curves, but a simulation would easily confirm this is so just by setting different values for I_L(0).

 

[Ivan Antunovic]

The middle graph is what would result when the capacitor has an initial voltage but there is no initial current in the inductor.

I think you might have the direction of initial I_L around the wrong way for red and blue, but anyway your idea sounds spot on.

I'm not sure that the values of RC and L that give rise to the red curve would, without change, also be able to produce the blue and green curves, but a simulation would easily confirm this is so just by setting different values for I_L(0).

Do you have a suggestion for some good simulation program?Maybe proteus demo?

 

[donpacino]

LT SPICE
http://www.linear.com/designtools/software/

 

[donpacino]

I am really sad that none actually understood my question,what I figured out,those 3 graphs are determined by initial conditions, for the blue one if the capacitor has initial voltage as on the picture it will produce a current that has the same direction as the initial current of the inductor,that's why we have a little 'spike' over value of 1.(they are being added up)

The red graph ,is when we have initial voltage on the capacitor producing a current in ic in the opposite direction of the initial current in the inductor ,and therefore we have a negative 'spike' ,because in the scenario of the blue graph currents add up,and here one current will have bigger value than the other one.

What about the green graph,I am not quite sure about it?

write the equations for the circuit out. In your initial post you gave us no indication of what this system was...

 

[Ivan Antunovic]

write the equations for the circuit out. In your initial post you gave us no indication of what this system was...

I wrote that it's a overdamped system ,and gave you the equation of the system,I really don't see how equations going to help you out with this one?

 

[donpacino]

If you either
A. disregaurd our advice
or
B. don't give us the information we need to give you advice

don't be upset when you are not happy with the results.

Looking at the governing equations (preferably in the frequency domain) will tell you how the component values and the initial conditions will effect the transient conditions.

where did you get the equation from post #1?
is there any other information you are withholding from us?

 

[donpacino]

what is your backround in electronics?

IL(0-) cannot be anything but 0 (assuming the switch closes at t=0)

if it opens at t=0 that is a different story

 

[amsi]

Check your equation again.

 

[Ivan Antunovic]

Well man,I've made a mistake,I forgot to add a voltage/current source in the parallel,my background is very good at electronics,but this doesn't have to do anything with electronics,it's circuit theory.

This equation is from lecture notes , MIT.

Okay this picture is from a book 'Linear and non linear electrical circuits',it shows the equation
http://postimage.org/][/PLAIN]
image hosting sites

And writing down the equation will give me only general solution to the equation,it doesn't say what determines those 3 graphs,it's obvious that initial conditions determine those 3 graphs.And I wasn't upset,I just had a feeling that none understood me.

 

[donpacino]

Well man,I've made a mistake,I forgot to add a voltage/current source in the parallel,my background is very good at electronics,but this doesn't have to do anything with electronics,it's circuit theory.

This equation is from lecture notes , MIT.

Okay this picture is from a book 'Linear and non linear electrical circuits',it shows the equation
[/PLAIN]
image hosting sites

And writing down the equation will give me only general solution to the equation,it doesn't say what determines those 3 graphs,it's obvious that initial conditions determine those 3 graphs.And I wasn't upset,I just had a feeling that none understood me.

ok. if you think there is a large difference between electronics and circuit theory, then what is your background in circuit theory.

developing a mathematical model of the system is very important. from the mathematical model you can see what parameters will effect the outputs in which way. If you have a mathematical model in the frequency domain, you can clearly see the poles and zeros. from there evaluating the poles and zeros will show which parameters will effect the output.

what is the actual question you are trying to answer? all of it.
Are you given a circuit? or are you only given graphs?

If you are given graphs, you can make no assumptions about the circuit, and you need to answer the question mathmatically. where the switch is, and where the sources are will effect the output drastically, if there is even a circuit.

 

[donpacino]

look up dominant zeros.
it may help you, if you either derive a mathematical expression for the circuit, or don't have a circuit and only need a mathematical answer.

 

[Ivan Antunovic]

The professor asked me 'what determines those 3 graphs' that was the question.

 

[donpacino]

The professor asked me 'what determines those 3 graphs' that was the question.

ok... if that was the entire question, and there is no included or implied circuit, then i would recommend looking up dominant zeros. Or the answer is simply the initial condition

 

[Ivan Antunovic]

ok... if that was the entire question, and there is no included or implied circuit, then i would recommend looking up dominant zeros. Or the answer is simply the initial condition

Sorry I am not native english speaker, what do you mean looking up dominant zeros?

 

[donpacino]

Sorry I am not native English speaker, what do you mean looking up dominant zeros?

have you done any work with transfer functions in the frequency domain?
do you know what poles or zeros are?

if you have not worked with the frequency domain this explanation will not be sufficient.

a dominant zero is when a system has a zero that is smaller or close to the smallest pole of the system. http://wolfweb.unr.edu/~fadali/ee370/TimeResp.pdf

look at the above link. page 4 will kind of explain what poles and zeros are.
you can see on page 43-46 the effects of the zeros. you should then be able to explain the effects of the graphs you showed above.

 

[NascentOxygen]

Take a closer look. The 3 step responses pictured in this graph are clearly not illustrating a variation in poles and zeros or damping.

 

[donpacino]

Take a closer look. The 3 step responses pictured in this graph are clearly not produced by a variation in poles and zeros or damping.

i respectfully disagree. the red and blue lines are produced by dominant zeros.

edit: the red and blue lines CAN BE produced by dominant zeros.

 

[NascentOxygen]

There a more than 2 plots. They all show a system that starts at a common level, and falls back to a lower level.

The peak in the blue plot can be explained only by a change in initial conditions.

 

[donpacino]

There a more than 2 plots. They all show a system that starts at a common level, and falls back to a common level.

a systems poles and zeros do not effect the start level or end level

 

[NascentOxygen]

Poles and zeros do not cause a ZSR to initially veer off in the wrong direction, which is what the blue curve does.

 

[donpacino]

Poles and zeros do not cause a ZSR to initially veer off in the wrong direction, which is what the blue curve does.

(-s+5)/(s^2+4*s+13)

look at the step response of the tf I posted
matlab code below

sys=tf([-1 5],[1 4 13])
step(sys)

 

[donpacino]

step response above

 

[donpacino]

A while back I had a very good reference (it was lecture notes) about the effects of zeros and how to estimate them. I will attempt to find it.

 

[NascentOxygen]

(-s+5)/(s^2+4*s+13)

Keep your eye on the subject line. This thread is about an over-damped system.

Thanks for the MATLAB code.

 

[donpacino]

Keep your eye on the subject line. This thread is about an over-damped system.

Thanks for the MATLAB code.

I can switch to an overdamped system. it will not effect the initial shoot in the other direction

(-s+5)/(s^2+8*s+13)

 

[donpacino]

 

[NascentOxygen]

OP's system needs precisely 50% swing in the wrong direction, then no overshoot upon settling. Experimenting with a few values, it looks like a zero around 1.8 will achieve that.

BUT is there any parameter you can now vary to get the other two faster settling plots to fit OP's graph? I think not.

I do think you are trying to solve a problem different from the OP and his professor: he doesn't know about zeros, he proposes a realisation having no zeros, and I think it can be demonstrated using the simple RLC series circuit with different initial inductor current.

 

[donpacino]

OP's system needs precisely 50% swing in the wrong direction, then no overshoot upon settling. Experimenting with a few values, it looks like a zero around 1.8 will achieve that.

BUT is there a parameter you can vary to get the other two faster settling plots to fit OP's graph? I think not.

I do think you are trying to solve a problem different from the OP and his professor: he doesn't know about zeros, and he proposes a realisation having no zeros, and I think it can be demonstrated using the simple RLC series circuit with different initial inductor current.

I can agree with that statement. The problem is there was no circuit in the initial problem. I made an assumption that the answer had to be mathematical in nature.

just as a note (even though this is spelling out a potential answer which is frowned upon)
if you use the following MATLAB code, you will get answers that more or less mimic the results above. (exaggerated results )

% sys1: no zeros
% sys2: dominant zero in right half plane
% sys3: dominant zero in left half plane

sys1 = tf( [ 1 ] , [1 8 13] )
sys2 = tf( [-1 1] , [1 8 13] )
sys2 = tf( [ 1 1] , [1 8 13] )

step(sys1, sys2, sys3)

 

[NascentOxygen]

It's remarkable how closely you've been able to match the original plots. I don't think you stand any chance of being able to implement this using a passive circuit---a single zero with a changeable sign would seem to rule this out.

On the other hand, were this asked in a control subject, your answer would probably be the one that the instructor wanted. So well done.

 

[Ivan Antunovic]

Hello donpacino,I have worked in the ferquency domain for the transfer functions for DC motors, asynchrouns motors,RC filters etc,but poles and zeroes are not analysed so much,it focuses more on the control. I think this is covered at the next semester in a subject called 'Automatic regulation'.

And sorry for the late answer I am really busy with the finals.

My professor doesn't want from me to think as mathematician , he wants me to think with my own head and think as a engineer and a physicist,he wants from me to describe him a problem and give him solution without equations actually by logical thinking.