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[SOLVED] Determining the y value in dB from log x-axis

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Dear friends,

I am sorry to post a very basic question,

I have a problem on determining the y value in db for a given x value in Bode plot.

as an example to explain, if I have unity gain Butterworth 4th order LPF response with -80 dB/decade, suppose fc = 5 MHz,
it is straight forward that one decade from 5 MHZ which is 50 MHz will give me gain = -80 dB.

I know from the simulation that gain at f = 10 MHz is about -24 dB, but I don't know how mathematically calculating it or to calculate other different values t o find gain from frequency and vice virsa



Thank you
 

Absolute value of transfer function.
For example 1st order RC filter has transmittance:
\[ H(s)=\frac{1}{1+\frac{s}{\omega_c}} \]
After \[s \rightarrow i \omega \] , taking absolute value and converting to decibels:
\[20 log [|H(i\omega)|] = 20 log \left (\frac{1}{\sqrt{1+(\frac{\omega}{\omega_c})^2}} \right ) \]

You can also estimate value knowing that single pole results with roll-off of -6dB per octave or estimates pass-band as simple N order pole:
\[ |H(i\omega)| \approx \frac{1}{\sqrt{1+\left ( \frac{f}{f_c} \right )^{2 N} }} \xrightarrow{f \gg f_c} \left ( \frac{f_c}{f} \right )^N \]
 
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If you are interested in the gain when the filter characteristic is rolling off and close to the asymptote than if filter order is N, dc gain is 1 and corner frequency is fc, the gain (i.e. transfer) at some frequency f is (approximately if frequency values close to fc):

(fc/f)^N=gain

So, for fc=5MHz, f=10MHz and N=4

(5/10)^4=1/16=62.5e-3 or -24dB
 
Thank you Dominik and Suta,

Equation for Dominik can only applied for the first order system as it represent the transfer function of first order LPF

For Suta it can work with any order, I have tested and almost gives accurate result

Thank you very much guys, problem is solved with your help
 

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