RC Low-Pass Filter Frequency Analysis

Vocabulary flashcards covering the frequency response, transfer function, cut-off values, and high-frequency behavior of an RC low-pass filter based on the lecture notes.

Transfer Function H(j\text{\textomega}) of an RC Low-Pass Filter

H(j\text{\textomega}) = \frac{1}{1 + j\text{\textomega} RC}

Module |H(j\text{\textomega})|

|H(j\text{\textomega})| = \frac{1}{\sqrt{1 + (\text{\textomega} RC)^2}}

Argument \text{ extphi}(\text{ extomega})

\text{\textphi}(\text{\textomega}) = -\arctan(\text{\textomega} RC)

Time Constant $RC$

For $R = 2\,\text{k}\Omega$ and $C = 10\,\text{nF}$, $RC = 2000 \times 10^{-8} = 2 \times 10^{-5}\,\text{s}$

Cut-off Pulsation \text{ extomega}c

\text{\textomega}_c = \frac{1}{RC} = \frac{1}{2 \times 10^{-5}} = 50\,000\,\text{rad/s}

Cut-off Frequency fc

f_c = \frac{\text{\textomega}_c}{2\pi} \approx \frac{50\,000}{2\pi} \approx 7\,960\,\text{Hz} \approx 8\,\text{kHz}

Gain in dB at Cut-off Frequency

At \text{\textomega} = \text{\textomega}_c, $|H| = \frac{1}{\sqrt{2}}$, resulting in $\text{Gain(dB)} = 20 \log\left(\frac{1}{\sqrt{2}}\right) \approx -3\,\text{dB}$

High-Frequency Behavior (\text{ extomega} >> \text{ extomega}c)

For \text{\textomega} \gg \text{\textomega}_c, |H| \approx \frac{1}{\text{\textomega} RC}. The gain decreases with a slope of $-20\,\text{dB/decade}$, the phase tends toward $-90^\circ$, passing low frequencies and attenuating high frequencies.