Question

For the band-pass filter shown in Figure \(30.25,\) how can the width of the frequency response be increased? a) increase \(R_{1}\) b) decrease \(C_{1}\) c) increase \(R_{2}\) d) increase \(C_{2}\) e) do any of the above

Short answer

Answer: Decrease \(C_{1}\).

Step-by-step solution

Option a) Increase \(R_{1}\)

Increasing \(R_1\) will increase the time constant of the low-pass filter section \(\tau_1 = R_1C\). In the case of a series resonant circuit, \(\tau = RC\). Thus, increasing \(\tau\) will decrease the width of the frequency response. So, this option is incorrect.

Option b) Decrease \(C_{1}\)

Decreasing \(C_1\) will decrease the time constant of the low-pass filter section \(\tau_1 = R_1C_1\). In case of a series resonant circuit, \(\tau = RC\). Thus, decreasing \(\tau\) will increase the width of the frequency response. So, this option is correct.

Option c) Increase \(R_{2}\)

Increasing \(R_2\) will increase the time constant of the high-pass filter section \(\tau_2 = R_2C_2\). In case of a series resonant circuit, \(\tau = RC\). Thus, increasing \(\tau\) will decrease the width of the frequency response. So, this option is incorrect.

Option d) Increase \(C_{2}\)

Increasing \(C_2\) will increase the time constant of the high-pass filter section \(\tau_2 = R_2C_2\). In case of a series resonant circuit, \(\tau = RC\). Thus, increasing \(\tau\) will decrease the width of the frequency response. So, this option is incorrect.

Option e) Do any of the above

As we have seen previously, option b) is the correct answer. Option e) implies that all the other options are viable to increase the width of the frequency response, which is not accurate. Therefore, this option is incorrect. To summarize, to increase the width of the frequency response of the band-pass filter, we should decrease \(C_{1}\). The correct answer is option b).

Key concepts

Electrical Resonance

Electrical resonance occurs when a system's inductive reactance and capacitive reactance are equal in magnitude, but opposite in phase, resulting in a net reactive impedance of zero at a particular frequency. This frequency is known as the resonant frequency. In filters, like the band-pass filter mentioned in the exercise, resonance dictates the central frequency where the passband is centered.

In context with the exercise provided, electrical resonance plays a crucial role in determining the operating frequency range of the band-pass filter. Implementing changes in the circuit components can shift the resonance point, affecting the filter's ability to select a narrower or wider band of frequencies.

Time Constant

The time constant, typically denoted as \( \tau \), is a measure of the time it takes for a system to respond to changes. In electrical circuits, the time constant is the product of resistance (R) and capacitance (C), expressed as \( \tau = RC \). It reflects how quickly a capacitor charges to approximately 63% of its maximum charge or discharges to about 37% of its initial charge.

Regarding the band-pass filter from the exercise, by adjusting the time constant you influence the filter's frequency response. Decreasing the capacitance \( C_1 \) for the low-pass portion, as shown in the solution, lowers its time constant, which in turn increases the width of the frequency response, allowing a broader range of frequencies to pass through.

High-Pass Filter

A high-pass filter is designed to allow frequencies above a certain cutoff frequency to pass through while attenuating frequencies below this threshold. The behavior of a high-pass filter is characterized by its time constant \( \tau = R_2C_2 \). This time constant determines how quickly the filter responds to changes in frequency.

In the step-by-step solution to the exercise, increasing the resistance \( R_2 \) is identified as a method that would actually reduce the width of the frequency response, not increase it. This is because a larger time constant results in a slower response of the high-pass section of the band-pass filter, leading to a narrower frequency range being passed.

Low-Pass Filter

A low-pass filter allows frequencies below a certain cutoff frequency to pass through while reducing the amplitude of frequencies above this threshold. Similar to the high-pass filter, its behavior is also defined by a time constant \( \tau = R_1C_1 \) for the low-pass section of the band-pass filter.

In terms of the exercise, reducing the capacitance \( C_1 \) decreases the time constant, which is the correct solution to increase the width of the frequency response. This change allows the low-pass part of the band-pass filter to allow a wider range of frequencies to pass, thereby expanding the bandwidth of the filter.