Phase margin in laplacian equations

[goldsmith]

Hello My friends!
I want to know that how are the laplacian equations and bood Diagram of this filter.( and relation between them.)

Is it possible guide me please?
Best Regards
Goldsmith

 

[jimito13]

It is quite easy to do it by yourself!Assuming an ideal opamp (Infinite Gain,Infinite Rin,Zero Rout) and based on uo=A(u+ - u-) and the Kirchoff's law around the opamp you can extract the desired transfer function.
If you assume a more complex model for the opamp the procedure is identical but maybe computations are more time consuming.
Having the transfer function you can sketch the magnitude and phase response or simulate your designed circuit and compare the results from theory.

 

[goldsmith]

Dear jimito13
Thanks for your reply. i can write the transfer function in s plane as simple as. but i can not find the phase margin from that.
and i can not draw the bood diagram . is it possible that you guide me , please?
Best Regards
Goldsmith

 

[jimito13]

i can write the transfer function in s plane as simple as. but i can not find the phase margin from that.

You can't find the Phase Margin via theory and equations or via simulation?
Remember PM definition : It is the distance of the phase for the frequency in which the small-signal gain crosses zero dB from the -180deg.
Now,let's apply the above definition :
1.)Take the magnitude of your transfer function (substitute s=jω) and make it equal to 1(thus 0dB).Solve for ω and as a result you get the frequency (ω0) in which the small-signal gain crosses zero dB.
2.)Take the phase response of your system and substitute the variable ω with ω0.Let's name ω1 the result of your calculations.
3.)PM=ω1-(-180deg)=ω1+180deg

As for simulation it's much easier,just remember the definition of PM and apply the above steps at the magnitude and phase response that will get after a small-signal analysis.

 

[goldsmith]

Dear jimito13
Thanks for your valuable and helpful reply. but is it possible that you do it for this circuit. i confused really. and i think if i see its solution from you as an example , i will understand it perfectly.
Again thank you for your attention and guide.
Sincerely
Goldsmith

 

[RobG]

An important point that seems to being missed is that phase margin is measured with the feedback loop broken. That is, it is an open loop characteristic. The circuit is shown in closed loop. You need to break the loop. Any good textbook on circuit design or control theory should have an explanation.

If you want to derive the equation you don't want to use infinite gain - the gain has to drop off as frequency increases. An opamp gain of w0/s is a good approximation. w0 is the point where the opamp gain crosses zero dB.

 

[LvW]

Yes, RobG is right.
Everything that Jimito tries to explain is related to the LOOP GAIN only.
That means: Open the feedback loop, inject at this point a test signal and measure the voltage ratio at the opening.
However, very often this creates problems: DC operating point is altered or even lost and loading conditions change at that node.
Therefore, some special methods have to be applied (e.g. as proposed by Middlebrook).

 

[jimito13]

Thanks guys,i totally forgot to mention that all my previous posts were concerning the loop gain of the circuit.Thanks again for the supplemental information you provided and sorry from the original poster for this omission.

 

[goldsmith]

Dear Friends!
Hi
Thank you for all of your replies and attentions.
But i confused . i can find the place of pole and zero of that circuit and i can find its H(s) . but i can not find its phase margin and bood diagram and i'm really confused and i want to understand it . Guide me , please .
Thank you
Goldsmith

 

[LvW]

Goldsmith, the desired Bode diagram (BODE !!!) consists of a graphical representation of the loop gain magnitude (in dB) and the loop gain phase versus frequency.
Then, follow the instructions as outlined in Jimito's posting #4.

 

[goldsmith]

Dear LvW
Hi
Thanks for your attention and reply.
I think the H(s) is : (R5)/(R1R5sc+R1)
And i think it has not zero . and it has a stable pole at left of s plane .Are they right?
what should i do? should i convert it to the db gain? is its formula 20log(H(s)) for this example?
Best Regards
Goldsmith

 

[LvW]

Dear LvW
Hi
Thanks for your attention and reply.
I think the H(s) is : (R5)/(R1R5sc+R1)
And i think it has not zero . and it has a stable pole at left of s plane .Are they right?
what should i do? should i convert it to the db gain? is its formula 20log(H(s)) for this example?
Best Regards
Goldsmith

At first, write H(s) in "normal form" by dividing numerator and denominator by R5.
Then you can follow the rules for constructing the BODE diagram: Approximate H(s) in dB by two straight lines:
A horizontal line for very low frequencies and another line with a slope of -20dB/dec (frequency axis in log scale!) for very high frequencies.
Both lines intersect at the pole frequency 1/R1C.
(Further details regarding the Bode diagram can be found elsewhere)

 

[RobG]

I don't know if the equation is correct (sign is wrong for negative feedback), but do NOT convert it to dB. Just find where the magnitude is equal to 1. Then calculate the phase angle at that point.

---------- Post added at 08:56 ---------- Previous post was at 08:43 ----------

ok... revisiting this, you apparently aren't looking for phase margin - you are just finding the closed loop response. So ignore what I just said. The equation looks correct (except that it should be negative) and LvW explains correctly how to get the approximate bode plot.

 

[goldsmith]

Dear RobG
Hi
Thank you for your attention.
you're right . i forget to write the minus at side of equation.
Thank you
Goldsmith

---------- Post added at 11:54 ---------- Previous post was at 11:50 ----------

Dear LvW
What is your mean by normal form? if i divide the equation by R5 , the equation will be : -(1)/((R1sc)+(R1/R2)). is it your mean?
and how can i find , that when the magnitude is equal to 1 ?

---------- Post added at 11:58 ---------- Previous post was at 11:54 ----------

i confused ! Is it possible that you say your mean more clear than this , please?
Best Regards
Goldsmith

 

[varunkant2k]

I think the H(s) is : (R5)/(R1R5sc+R1)
Goldsmith

Yes, It must be correct. And phase shift for this formula is [-tan-1(R5C.W0)] and phase margin will be
-180 - [-tan-1(R5C.W0)].
System is stable with single pole.

 

[goldsmith]

i confused ! Is it possible that you say your mean more clear than this , please?
Best Regards
Goldsmith

---------- Post added at 12:03 ---------- Previous post was at 11:59 ----------

Dear varunkant2k
Hi
Thanks for your helpful reply. is it possible that you explain that where this formula([-tan-1(R5C.W0)]) is from? and about bode diagram drawing i really confused !
Best Regards
Goldsmith

 

[RobG]

Dear RobG
Dear LvW
What is your mean by normal form? if i divide the equation by R5 , the equation will be : -(1)/((R1sc)+(R1/R2)). is it your mean?
and how can i find , that when the magnitude is equal to 1 ?
Best Regards
Goldsmith

Oops, both LvW and I messed up. You should divide by R1, not R5. Then your equation will be -(R5/R1)/(1+s*R5*C1).

 

[varunkant2k]

RobG is correct, I guess now you can find the PM.

 

[goldsmith]

Dear RobG
Thank you for your attention and helpful post .
What will be next step for bode drawing? and why we say that this form is normal form?
Thanks
Goldsmith

 

[LvW]

Goldsmith, Hi again.

May I give you one advice?
It seems to me that you really are a beginner in electronics and system theory (transfer function, normal form, loop gain, Bode diagram, phase margin, oscillation condition...).
And you are very busy here in the forum.
Therefore, my recommendation is (in your own interest): Make an economical use of your time by reading a suitable textbook rather than asking many many questions here in the forum. It does not help to much to get answers to a variety of questions - without understanding the backgrounds.

This thread is a good example:
In your first posting you ask for a "bood" diagram (I only could guess that you were referring to BODE) and the transfer function of a filter.
In the second posting you ask for the phase margin (do you know it's meaning?).
The next questions concern poles and zeros and how to convert the transfer functions to the dB domain.
And your last problem was how to find the frequency with |H(s)|=1. Do you know the meaning of the complex frequency "s" ?

Believe me, it's better first to use a suitable book and to study some basics.
Regards