You need to track your units:okay ,,
Assumed the intersection points with w-axis X and K
i.e.
X with the left skirt is +26db
26 = 20 log10(10Hz/X Hz)
1.3= log10(10Hz/X Hz) =====> 10Hz/X Hz = 10^1.3
X= 10 /10^1.3 = 0.5 ??? Is it Right??
Good! You are asking whether or not the answer makes sense instead of just blindly accepting whatever your calculator spits out.K with the right skirt is -26db
-26 = 20log10(K Hz/20,000 Hz)
-1.3= log10(K Hz/20,000 Hz) =====> K Hz/20,000 Hz = 10^-1.3
K = 1002.37 HZ It's is wrong because on the fig. 10^5 < K < 10^6
You need to track your units:
26dB = 20dB log10(10Hz/X)
1.3 = log10(10Hz/X)
10Hz/X = 10^1.3
X = 10Hz /10^1.3 = 0.5Hz
See how the units work out? Most mistakes you make will screw up the units, so tracking the units will let you catch most (not all) mistakes.
Good! You are asking whether or not the answer makes sense instead of just blindly accepting whatever your calculator spits out.
It might help to drop back and look at the fundamental relationship involved
y(f) = m*log10(f/fo) + b
Don't worry about the fo in the equation. You can take it to be any frequency you want, such as 1Hz. It doesn't matter because it will cancel out eventually. But it needs to be there in the equation as given because the logarithm function is transcendental and can't have an argument that has units.
You know:
y(20kHz) = 26dB
m = -20dB/decade
You want to find x such that:
y(x) = 0dB
So, from the known data point, we have
y(f) = m*log10(f/fo) + b
y(20kHz) = -20dB/decade*log10(20kHz/fo) + b = 26dB
b = 26dB + 20dB/decade*log10(20kHz/fo)
Substituting this into the unknown data point:
y(x) = 0dB = m*log10(x/fo) + b
0dB = -20dB/decade*log10(x/fo) + 26dB + 20dB/decade*log10(20kHz/fo)
20dB/decade*log10(20kHz/fo) - 20dB/decade*log10(x/fo) = -26dB
20dB/decade*[log10(20kHz/fo) - log10(x/fo)] = -26dB
[log10(20kHz/fo) - log10(x/fo)] = -26dB / (20dB/decade) = -1.3decades
log10[(20kHz/fo)/(x/fo)] = -1.3decades
log10(20kHz/x) = -1.3decades
20kHz/x = 10^(-1.3decades) = 0.501 modify 10^(-1.3decades) = 0.05
x = 20kHz/ 0.0501 ~= 400kHz
Note that the unit of "decade" is very analogous to "radian". It is dimensionless but conveys useful information to help us keep track of the meaning of certain numbers.
by Duane Benson
by Duane Benson
by Jake Hertz