Question

A bandpass filter has a center, or resonant, frequeng of \(80 \mathrm{krad} / \mathrm{s}\) and a quality factor of 8 . Find the bard width, the upper cutoff frequency, and the lowercult off frequency. Express all answers in kilohertz.

Short answer

Bandwidth is approximately 1.592 kHz, upper cutoff is 13.525 kHz, lower cutoff is 11.937 kHz.

Step-by-step solution

Understand the Definitions

The center frequency \( \omega_0 \) of a bandpass filter is given as \( 80\ \text{krad/s} \), and the quality factor \( Q \) is 8. We need to convert the frequency from radians per second to hertz by dividing by \( 2\pi \).

Calculate Bandwidth

The bandwidth \( BW \) of a bandpass filter is given by the formula \( BW = \frac{\omega_0}{Q} \). Given \( \omega_0 = 80\ \text{krad/s} = 80000\ \text{rad/s} \), substitute the values: \( BW = \frac{80000}{8} = 10000\ \text{rad/s} \).

Convert Bandwidth to Hertz

Convert the bandwidth to hertz by dividing by \( 2\pi \): \( BW_{Hz} = \frac{10000}{2\pi} \approx 1591.55\ \text{Hz} \). To convert Hz to kHz, divide by 1000: \( BW_{kHz} \approx 1.592\ \text{kHz} \).

Calculate Upper and Lower Cutoff Frequencies

The upper and lower cutoff frequencies are symmetrical around \( \omega_0 \). Use \( \omega_u = \omega_0 + \frac{BW}{2} \) and \( \omega_l = \omega_0 - \frac{BW}{2} \). Substituting the values: \( \omega_u = 80000 + \frac{10000}{2} = 85000\ \text{rad/s} \) and \( \omega_l = 80000 - \frac{10000}{2} = 75000\ \text{rad/s} \).

Convert Upper and Lower Cutoff Frequencies to Hertz

Convert the upper and lower cutoff frequencies to hertz by dividing by \( 2\pi \): \( f_u = \frac{85000}{2\pi} \approx 13524.89\ \text{Hz} \) and \( f_l = \frac{75000}{2\pi} \approx 11936.62\ \text{Hz} \). Convert these to kHz by dividing by 1000: \( f_u \approx 13.525\ \text{kHz} \) and \( f_l \approx 11.937\ \text{kHz} \).

Key concepts

Quality Factor

The quality factor, often denoted by the letter \( Q \), is a measure of how "selective" a bandpass filter is with respect to its center frequency. It indicates the sharpness of the peak of the frequency response curve. A higher \( Q \) means a narrower bandwidth, allowing the filter to be more selective.
This is particularly useful in applications such as radio tuning or signal processing where you need to isolate a narrow range of frequencies.

• In mathematical terms, \( Q \) is defined as the ratio of the resonant frequency \( \omega_0 \) to the bandwidth \( BW \).
• The formula is \( Q = \frac{\omega_0}{BW} \).
• For a given center frequency, a higher \( Q \) results in a smaller \( BW \), meaning the filter allows only a smaller range of frequencies around \( \omega_0 \) to pass through.

Understanding the quality factor helps in designing filters that can effectively differentiate between closely spaced frequency signals.

Cutoff Frequency

The cutoff frequencies of a bandpass filter are crucial points that define where the filter transitions from passing a signal to attenuating it. These frequencies are typically defined as the points where the output signal is reduced to half its power, corresponding to a drop of approximately 3 dB.
For a bandpass filter, there are two cutoff frequencies: the lower cutoff frequency \( low\omega_l \) and the upper cutoff frequency \( \omega_u \). Both are symmetrically placed around the center frequency \( \omega_0 \).

• The lower cutoff frequency \( \omega_l \) is calculated using \( \omega_l = \omega_0 - \frac{BW}{2} \).
• The upper cutoff frequency \( \omega_u \) is calculated using \( \omega_u = \omega_0 + \frac{BW}{2} \).
• The difference between the upper and lower cutoff frequencies gives the filter's bandwidth.

In practice, knowing the cutoff frequencies helps in understanding the filter's range of operation, ensuring that only the desired frequencies are emphasized while unwanted frequencies are suppressed.

Resonant Frequency

The resonant frequency, often denoted as \( \omega_0 \), is the center frequency of a bandpass filter. It's the frequency at which the filter allows signals to pass through with maximum amplitude, typically leading to the least attenuation.
This frequency is crucial because it determines the midpoint of the frequencies that the filter will pass.

• Mathematically, the resonant frequency is neither the upper nor the lower cutoff; rather, it is the frequency at the center of the filter's passband.
• A change in the resonant frequency will shift the passband of the filter, thereby altering which signals can pass through unimpaired.
• In electronic design, this frequency is often set based on the application needs, such as in musical instruments, radio stations, or communication devices.

Understanding the resonant frequency is key to designing filters that achieve the desired frequency response, ensuring optimal performance of electronic systems.