Decibels and the Bode Asymptote
Replace the curve with two lines
Sweep the frequency and watch the gap between the real curve (gold) and the two asymptotes (dashed). Far above or below cutoff they nearly touch, but at one place the gap is largest. Find that corner, the cutoff frequency.
Decibels turn multiplying into adding
A decibel is twenty times the log of the gain, dB = 20·log10|H|. Because it is a log, multiplying gains becomes adding decibels. Cascade two stages in series and the total in decibels is just the sum of each stage’s decibels. A gain of 1 (pass) is 0 dB, a gain of 1/√2 (half power) is −3 dB, a gain of 1/10 is −20 dB, and 1/100 is −40 dB.
On log axes the curve becomes straight
Place frequency on a log axis and the two ends of a first-order response become straight. Far below cutoff the gain is nearly 1, a horizontal 0 dB line; far above, it falls a steady −20 dB for every tenfold rise in frequency, a line of constant slope. These two lines are the asymptotes, and where they meet is the cutoff frequency. The real curve simply rounds that corner smoothly.
At the corner it misses by exactly 3 dB
The asymptotes say the response bends at 0 dB at the corner, but the real curve is already down at −3 dB there. So the largest error of the straight-line approximation is exactly at the cutoff frequency, and its value is exactly 3 dB. One octave (a factor of two) away from the corner the error shrinks to 1 dB, and farther out it is nearly zero. So drawing the two asymptotes and just dipping the corner by 3 dB makes a hand-drawn Bode plot almost match the real one.
Back to the first screen
As you swept frequency, the gap between the real curve and the asymptotes was nearly zero far out but opened widest, to 3 dB, at the cutoff frequency. It means that instead of drawing one curve, two straight lines with the corner dipped by 3 dB were enough. Because decibels and the log axis straightened the curve into lines, a frequency response that looked complicated could be sized up quickly by hand.