Plot frquency resonse from transfer function in complex frequency domain

[Steve Collins]

I am going through a past paper for an upcoming exam and I want to check that I am approaching this question correctly.

H(s) = 1/(s² + 8485.28s + 36x10^-6)

Calculate and plot frequency response for 0 rad/s, 500 rad/s, 1000 rad/s and 10000 rad/s.

I am under the impression that the 's' term can be replaced with jω which gives:

H(s) = 1/(jω² + 8485.28jω + 36x10^-6)

where ω is the frequency.

I then replace jω with the the frequencies given above.

Is this correct?

 

[rude man]

You replace ω, not jω. Then you get a complex number for which you have to compute its magnitude.

 

[Steve Collins]

so for 500 rad/s,

H(s) = 1/(jω² + 8485.28jω + 36x10^-6)

= 1/(j500² + 8485.28 x j500 + 36x10^-6)

= 1/(j250000 + j4242640 + 36x10^-6)

= 1/(36x10^-6 + j4492640)

So,

r= 1/√[(36x10^-6)² + 4492640²]

= 1/4492640

I've obviously done something wrong here as the imaginary number is so small that it is having no effect on the outcome of the magnitude.

 

[rude man]

If your original H(s) was correct thenn what you did is also correct (but H(s) should be H(jw) of course.

Looks like your transfer function has a high gain at zero frequency (dc) of nearly 30,000 but starts to "roll off" almost immediately (around 5e-9 rad/s if I computed correctly!) at a 20dB/decade rate. At ~8500 rad/s it picks up another 20 dB/decade. (Don't feel bad if you don't dig all this lingo, I'm mostly talking to myself here ...).



This looks very much like the gain of a typical operational amplifier BTW.

 

[Steve Collins]

I'm not a million miles away from what you're saying. I'll continue looking at this tomorrow when I'll attempt the plot and probably have some more questions.

cheers for your help

steve

 

[rude man]

OK, and happy to explain anything you're curious about.
Cheers hence also!

 

[Steve Collins]

I've converted my answers to dBs and got

0 rad/s... 88.874dB
500 rad/s... -133.05dB
1000 rads/s... -139.541dB
10000 rad/s... -165.337dB

Does the frequency along the x-axis have to be logarithmic? If so is there a set standard for spacing?

 

[rude man]

Does the frequency along the x-axis have to be logarithmic? If so is there a set standard for spacing?

Yes, the standard way is a log-log plot, i.e. dB on the y-axis and log₁₀(frequency) on the x axis. The resulting plot consists of straight sections called asymptotes.

The reason is that that way the asymptotes are straight lines. Take H(s) = 1/(Ts+1); the gain plot is a straight line until you get to radian frequency 1/T, then the gain is another straight line but angling downward with slope = 20dB/10:1 change in frequency. The actual gain follows this approximately, with the max. gain error at ω = 1/T. At that point the actual gain is 3dB less than the dc gain. At dc it's 100% accurate and very closely so at high frequencies like ω = 100/T. You should be exposed to how the asymptotes approximate the actual gain in the course of your studies. It gets trickier if you have complex-conjugate poles in your H(s).

Not sure what you mean by "spacing". The x-axis has frequency in either Hz or rad/s on log-log paper. Every major division is 10:1 change, e.g. 10Hz, 100Hz, 1000Hz, etc. On the y-axis every major division is another 20 dB.

 

[Steve Collins]

Not sure what you mean by "spacing". The x-axis has frequency in either Hz or rad/s on log-log paper. Every major division is 10:1 change, e.g. 10Hz, 100Hz, 1000Hz, etc. On the y-axis every major division is another 20 dB.

I was wondering how to draw a graph with log scale, but looking at the front sheet of the past paper I am working through I see that log paper was provided, obviously!

 

[rude man]

I was wondering how to draw a graph with log scale, but looking at the front sheet of the past paper I am working through I see that log paper was provided, obviously!

OK, I see what you meant. No one has ever asked me to draw my own log paper either!

But if someone does give you linear-linear, just mark the x-axis in powers of 10: 10⁰Hz, 10¹Hz, 10²Hz etc. Half-way between 10 and 100 for example would be √(10*100) = 31.6 Hz etc.

And BTW I think I told you wrong: when you mark off dB on the y axis, that scale is linear in dB. It's the dB themselves that make it effectively logarithmic. So your graph paper should be linear on the y-axis and logarithmic on the x.