Frequency response function (polar diagram)

In summary, the conversation discusses the process of plotting frequency response functions on polar diagrams and the difficulty in understanding the steps involved. The method involves squaring and multiplying certain terms to eliminate ωT and obtain a semi-circle plot on the Argand diagram. The speaker expresses their struggle in understanding this process and the lecturer's claim of it being a straightforward elimination.
  • #1
Sci-Fry
8
0
This isn't really a homework question but I felt this is the most relevant section for this. I therefore apologize for not following the standard post template.

I was going through my electrical engineering notes on frequency response functions. It was explaining how to plot frequency response functions on polar diagrams and had the following two steps, which I didn't quite understand. It got to this:

[tex]G(j\omega) = x + jy = \frac{1}{R}\frac{1-j\omega T}{1+(\omega T)^2}[/tex]

It is easy to write:

[tex]x=\frac{1}{R}\frac{1}{1+(\omega T)^2} ; y=\frac{1}{R}\frac{-\omega T}{1+(\omega T)^2}[/tex]

That's fine, but I didn't understand how they jumped to the next step:

From which it is straightforward to eliminate \omegaT to give

[tex](x-\frac{1}{2R})^2 + y^2 = (\frac{1}{4R})^2[/tex]

The point of writing it in this form is that the frequency response can now be plotted as a semi-circle on the Argand diagram. However, I don't understand how they did this step so easily. Is there something I'm not spotting? What is the method from getting from the previous step to this one?

Many thanks,
Sci-Fry
 
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  • #2
Hi Sci-Fry! :smile:

(have an omega: ω :wink:)

square y, and multiply by R2(1 + (ωT)2) …

that gives you (ωT)2/(1 + (ωT)2), which is 1 - … ? :smile:
 
  • #3
Hi tiny-tim, thanks for the response (and the omega :P)!

Hmmm... I'm probably being incredibly daft, but I see what you're doing...

[note: couldn't get some of the latex to work so I'm using mimtex off another server]

{1+%28%20\omega%20T%29^2}%20=%201%20-%20\frac{1}{1%20+%20%28%20\omega%20T%29^2}%20=%201%20-%20xR.gif


Is that what you were getting at? I'm still not sure what exactly to do after that. Playing around with the numbers gets me to the wrong answer:

mimetex.gif


mimetex.gif


That gets rid of the omega, but it's not what I'm looking for. I know I'm being stupid, just wish I knew where...
 
Last edited:
  • #4
hmm … now i look at it, my "1 -" is missing the point :redface:

square (x - 1/2R), and multiply by R2(1 + (ωT)2) :smile:
 
  • #5
Oh that makes sense now, got it. Thanks.

Still difficult to spot though. I don't get how the lecturer can call that a straightforward elimination :P
 

Related to Frequency response function (polar diagram)

What is a frequency response function (polar diagram)?

A frequency response function (FRF) is a graphical representation of the relationship between input and output signals of a system, as a function of frequency. It is typically displayed as a polar diagram, with the magnitude and phase of the response plotted against frequency.

What is the purpose of a frequency response function (polar diagram)?

The purpose of a frequency response function is to provide a comprehensive understanding of a system's behavior over a range of frequencies. It allows scientists to analyze the system's frequency characteristics and identify any resonances or other abnormalities that may affect its performance.

How is a frequency response function (polar diagram) measured?

A frequency response function is typically measured by applying a known input signal to the system and measuring the resulting output signal. This process is repeated at various frequencies to obtain a complete FRF. The data is then plotted on a polar diagram, with the magnitude and phase values calculated using mathematical techniques such as Fourier analysis.

What is the difference between magnitude and phase in a frequency response function (polar diagram)?

The magnitude of a frequency response function represents the ratio of output amplitude to input amplitude, measured in decibels (dB). It indicates the system's gain or attenuation at a specific frequency. The phase, on the other hand, represents the time delay between the input and output signals, measured in degrees. It shows the system's response time or latency at a specific frequency.

What are some applications of frequency response function (polar diagram) analysis?

Frequency response function analysis is widely used in various fields, including engineering, physics, and biomedicine. It is commonly used to study the behavior of mechanical, electrical, and acoustic systems, as well as to design and optimize control systems. It is also utilized in medical imaging and vibration analysis, among other applications.

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