Filters and Bode plots
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| filter/ˈfɪltə/ | bộ lọc |
| Bode plot | Biểu đồ Bode |
| corner frequency | tần số góc |
| decibel | Đê-xi-ben |
Filters and Bode: the design is straight lines
- A filter passes a band and attenuates the rest; the Bode plot draws $20\log_{10}|H|$ against log frequency, where each pole costs −20 dB/decade above its corner frequency $ω_{c} = 1/(RC)$ and reads −3 dB, −45° at the corner. The decibel arithmetic is $20\log_{10} 10 = 20$ per decade.
- The asymptotes are the design tool and the exact curve the check; a stopband floor bounds the corner from above, a passband ceiling from below — and the two can conflict.
$R = 1.6$ kΩ, $C = 0.1$ µF: $ω_{c} = 6250$ rad/s. Low-pass at $10ω_{c}$: asymptote −20 dB, exact −20.04 dB. Swap the output to the resistor and the high-pass is the exact mirror (machine-verified): $-20.04$ dB at $+84.3°$ for $0.1ω_{c}$, $-3.01$ dB at $+45°$ at the corner, $-0.04$ dB at $+5.7°$ for $10ω_{c}$ — phase now leading, stop-band below.
{exercise:uni-ee4x-filter}
A first-order RC low-pass at 10 times its corner frequency reads —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
One pole rolls off 20 dB per decade past the corner; at the corner itself −3 dB; −40 dB is two decades (or two poles).
A high-pass RC at its corner frequency has gain and phase of
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The high-pass is the low-pass mirror: −3.01 dB at the corner but the phase LEADS +45 degrees; at 0.1 wc it reads −20.04 dB at +84.3 degrees.
One pole above its corner costs how much gain per decade?
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
20log10(10) = 20 dB per decade per pole; at 10wc the RC low-pass reads −20.04 dB exact against the −20 asymptote.