A high-pass RC filter with a crossover frequency of 1000 Hz uses a 100 Ω resistor. What is the value of the capacitor?

Verified step by step guidance

1. Understand the problem: A high-pass RC filter is a circuit that allows signals with frequencies higher than the crossover frequency to pass through while attenuating lower frequencies. The crossover frequency (f_c) is given as 1000 Hz, and the resistor (R) has a value of 100 Ω. We need to find the capacitance (C).
2. Recall the formula for the crossover frequency of an RC circuit: $f$_c=12πRC. Rearrange this formula to solve for the capacitance (C): $C=\frac{1}{2\pi Rf}$_c.
3. Substitute the given values into the formula: $C=\frac{1}{2\pi \times 100\times 1000}$. Ensure that the units are consistent (resistance in ohms, frequency in hertz).
4. Simplify the denominator: Calculate the product of $2\pi \times 100\times 1000$. This will give the value of the denominator in the capacitance formula.
5. Finally, compute the capacitance (C) by dividing 1 by the result of the denominator. The unit of capacitance will be in farads (F). If needed, convert the result to microfarads (μF) for practical use by multiplying by $10^6$.

Key Concepts

High-Pass RC Filter

A high-pass RC filter is an electronic circuit that allows signals with a frequency higher than a certain cutoff frequency to pass through while attenuating signals with lower frequencies. It consists of a resistor (R) and a capacitor (C) arranged in a specific configuration. The cutoff frequency is determined by the values of R and C, and it is the frequency at which the output signal is reduced to 70.7% of the input signal.

Crossover Frequency

The crossover frequency, also known as the cutoff frequency, is the frequency at which the output voltage of the filter is equal to 70.7% of the input voltage. For a high-pass filter, frequencies above this point are allowed to pass, while those below are attenuated. It is calculated using the formula f_c = 1/(2πRC), where f_c is the crossover frequency, R is the resistance, and C is the capacitance.

Capacitance Calculation

To find the value of the capacitor in an RC circuit, you can rearrange the formula for the crossover frequency. Given the crossover frequency (f_c) and the resistance (R), the capacitance (C) can be calculated using C = 1/(2πf_cR). This relationship shows how the capacitor's value directly influences the filter's behavior and its ability to pass or block certain frequencies.