Network Analysis
Filters are frequency-selective circuits. They pass desired frequency components, suppress unwanted ones, and shape both amplitude and phase using the impedance behavior of resistors, capacitors, and inductors.
A filter is an electrical network that allows certain frequency components to pass while attenuating others. It does this because reactive elements do not behave the same way at every frequency.
Key idea: filters exploit the fact that circuit impedance changes with frequency.
Animated explanation
Output is almost equal to input at low frequency, then it rolls off after cutoff.
Passes low frequencies and attenuates high frequencies.
Passes high frequencies and suppresses low frequencies.
Passes only a band around a chosen center frequency.
Rejects a band while passing lower and higher frequencies.
A low-pass filter passes low frequencies and reduces high-frequency content. In a simple RC low-pass network, the output is taken across the capacitor.
Vo = Vi x ZC / (R + ZC)
Vo = Vi x 1 / (1 + j omega RC)
H(omega) = 1 / (1 + j omega RC)
Key points:
fc = 1 / (2 pi RC)
A high-pass filter does the opposite: it blocks low-frequency content and allows higher-frequency content to appear at the output.
Vo = Vi x R / (R + 1 / j omega C)
Vo = Vi x j omega RC / (1 + j omega RC)
H(omega) = j omega RC / (1 + j omega RC)
A band-pass filter transmits a band around a center frequency and suppresses frequencies below and above that range.
A band-stop filter removes a narrow or broad frequency band while passing signals below and above that rejected region.
The frequency where output magnitude falls to about 70.7 percent of the passband value.
BW = f2 - f1
Q = f0 / BW
fc = 1 / (2 pi RC) = 159 Hz approximately
For the RC low-pass case, frequencies well below 159 Hz pass with small attenuation, while frequencies above that point begin to fall off strongly.
Filters are frequency-selective circuits that shape signals by using the frequency-dependent behavior of reactive elements, making them essential in signal control, communication, and analog design.
