Question

Match the gain of the filter with the frequencies in the low pass filter

Frequency Gain of the filter

1. f < fH i. VO/Vin ≅ AF/√2

2. f=fH         ii. VO/Vin ≤ AF

3. f>fH         iii. VO/Vin ≅ AF

(a) 1-i,2-ii,3-iii

(b) 1-ii,2-iii,3-i

(c) 1-iii,2-ii,3-i

(d) 1-iii,2-i,3-ii

I got this question during an online interview.

This intriguing question comes from First Order Low Pass Butterworth Filter in chapter Active Filters of Linear Integrated Circuits

Answer 1

In a Low Pass Filter, the frequency response and the corresponding gain at different frequencies behave as follows:

1. For frequencies less than the cutoff frequency fHf_HfH​ (i.e., f<fHf < f_Hf<fH​):

   • The filter passes the signal with minimal attenuation, and the gain is approximately AFA_FAF​ (the maximum gain).
   • This corresponds to VO/Vin≈AFVO/Vin \approx A_FVO/Vin≈AF​, so the gain is essentially flat at low frequencies.

2. At the cutoff frequency fHf_HfH​ (i.e., f=fHf = f_Hf=fH​):

   • The gain drops to 1/21/\sqrt{2}1/2​ of the maximum gain AFA_FAF​, corresponding to the point where the filter's response is reduced by 3 dB.
   • This corresponds to VO/Vin≈AF/2VO/Vin \approx A_F / \sqrt{2}VO/Vin≈AF​/2​.

3. For frequencies greater than the cutoff frequency fHf_HfH​ (i.e., f>fHf > f_Hf>fH​):

   • The gain continues to decrease as the frequency increases and ultimately approaches zero at very high frequencies.
   • This corresponds to VO/Vin≤AFVO/Vin \leq A_FVO/Vin≤AF​ (since the gain decreases with increasing frequency).

Now, let's match these points with the provided options:

Frequencies and Gain Correspondence:

• f < f_H: Gain is VO/Vin≈AFVO/Vin \approx A_FVO/Vin≈AF​, which corresponds to iii.
• f = f_H: Gain is VO/Vin≈AF/2VO/Vin \approx A_F/\sqrt{2}VO/Vin≈AF​/2​, which corresponds to i.
• f > f_H: Gain decreases, so VO/Vin≤AFVO/Vin \leq A_FVO/Vin≤AF​, which corresponds to ii.

Correct Matching:

• 1. f<fHf < f_Hf<fH​ → iii: VO/Vin≈AFVO/Vin \approx A_FVO/Vin≈AF​
• 2. f=fHf = f_Hf=fH​ → i: VO/Vin≈AF/2VO/Vin \approx A_F/\sqrt{2}VO/Vin≈AF​/2​
• 3. f>fHf > f_Hf>fH​ → ii: VO/Vin≤AFVO/Vin \leq A_FVO/Vin≤AF​

Thus, the correct answer is:

(d) 1-iii, 2-i, 3-ii