Design an RC high-pass filter that rejects 60.0

Design an RC high-pass filter that rejects 60.0 - Hz line noise from a circuit used in a detector. Your criteria are reduction of the amplitude of the line noise by a factor of $1000 .$ and total impedance at high frequencies of $2.00 \mathrm{k} \Omega$. a) What components will you use? b) What is the frequency range of the signals that will be passed with at least $90.0 \%$ of their amplitude?

Design an RC high-pass filter that rejects 60.0 - Hz line noise from a circuit used in a detector. Your criteria are reduction of the amplitude of the line noise by a factor of $1000 .$ and total impedance at high frequencies of $2.00 \mathrm{k} \Omega$.
a) What components will you use?
b) What is the frequency range of the signals that will be passed with at least $90.0 \%$ of their amplitude?

Key Concepts

RC High-Pass Filter

An RC high?pass filter is a circuit comprised of a resistor and a capacitor arranged so that signals above a specific frequency pass with minimal attenuation while lower-frequency signals, including unwanted noise, are effectively blocked. Its performance relies on the frequency?dependent reactance of the capacitor, which diminishes as the frequency increases.

Cutoff Frequency

The cutoff frequency, defined by the equation fc = 1/(2?RC), is the threshold where the output voltage is about 70.7% (or -3 dB) of the input voltage. This frequency determines the boundary between the passband (frequencies passed with higher amplitude) and the stopband (frequencies significantly attenuated), guiding the filter’s effectiveness.

Transfer Function

The transfer function of an RC high-pass filter, typically expressed as H(?) = (j?RC)/(1 + j?RC), mathematically models how the input signal is transformed by the filter across various frequencies. Its magnitude response explains how signal amplitudes are modified, thereby providing insight into the degree of attenuation at a particular frequency.

Attenuation Factor

Attenuation is the measure of how much a filter reduces the amplitude of an input signal at specific frequencies. Specifying an attenuation factor, such as reducing a problematic noise frequency by a factor of 1000, is critical in filter design to ensure that undesired noise levels are effectively minimized.

High Frequency Impedance

In an RC high-pass filter, as the frequency increases, the capacitor’s impedance tends toward zero, making the resistor the dominant component in determining the circuit’s overall impedance. Prescribing a particular high-frequency impedance, like 2.00 k?, is important for maintaining the integrity of the signals passing through the filter and ensuring compatibility with connected circuitry.

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