Which of the following is true about the cut-off frequency ωB of a first-order low-pass filter? Suppose that the DC gain is 1.

a.When the input is a sinusoid operating at ωB, the output is phase shifted from the input phase by −45∘.
b.The Bode magnitude plot rolls off by −40dB/decade after the corner frequency, at ωB.
c.When the input is a sinusoid operating at ωB, the output amplitude is scaled by a factor of 0.707≈−3dB.
d.Inputs operating at frequencies far below ωB are weakly attenuated (i.e. the amplitudes of outputs are approximately equal to the amplitudes of inputs.)
e.Inputs operating at frequencies far above ωB have phase shifts of close to −90∘.

c,d

The correct options are:

d. Inputs operating at frequencies far below ωB are weakly attenuated (i.e. the amplitudes of outputs are approximately equal to the amplitudes of inputs.)

e. Inputs operating at frequencies far above ωB have phase shifts of close to −90∘.

Explanation:

a. This statement is false. In a first-order low-pass filter, when the input is a sinusoid operating at the cut-off frequency ωB, the output phase shift is actually -90°, not -45°.

b. This statement is false. The Bode magnitude plot of a first-order low-pass filter rolls off by -20dB/decade after the corner frequency, at ωB.

c. This statement is false. When the input frequency is at the cut-off frequency ωB, the output amplitude is scaled by a factor of 0.707, which is approximately -3dB. This is a reduction of 3 decibels, not -3dB.

d. This statement is true. Inputs operating at frequencies far below ωB (i.e., in the passband) are weakly attenuated in a first-order low-pass filter. The amplitudes of the outputs are approximately equal to the amplitudes of the inputs.

e. This statement is true. Inputs operating at frequencies far above ωB (i.e., in the stopband) have phase shifts close to -90° in a first-order low-pass filter.

To determine which of the options is true about the cut-off frequency ωB of a first-order low-pass filter, we can analyze the characteristics of such a filter.

A first-order low-pass filter has a transfer function of the form H(s) = K / (s + ωB), where s is the complex frequency variable and K is the DC gain.

a. When the input is a sinusoid operating at ωB, the output is phase-shifted from the input phase by -45∘:
To determine the phase shift, we can substitute s = jωB (j is the imaginary unit) into the transfer function. The magnitude of j is always 1, and when multiplied by ωB, it results in a phase shift of 90∘. However, the negative sign in the transfer function contributes an additional -45∘ phase shift. Therefore, option a is true.

b. The Bode magnitude plot rolls off by -40dB/decade after the corner frequency, at ωB:
To determine the slope of the magnitude plot, we can take the logarithm of the transfer function magnitude and differentiate it with respect to logarithmic frequency. In the case of a first-order low-pass filter, the slope is -20dB/decade after the corner frequency. Since -40dB/decade is steeper, option b is not true.

c. When the input is a sinusoid operating at ωB, the output amplitude is scaled by a factor of 0.707 ≈ -3dB:
The transfer function magnitude at the corner frequency is 1/√2 ≈ 0.707. This corresponds to a loss of -3dB (decibels) in amplitude. Therefore, option c is true.

d. Inputs operating at frequencies far below ωB are weakly attenuated (i.e., the amplitudes of outputs are approximately equal to the amplitudes of inputs):
In a low-pass filter, frequencies below the corner frequency are passed through with minimal attenuation. Therefore, option d is true.

e. Inputs operating at frequencies far above ωB have a phase shift close to -90∘:
At frequencies much higher than the corner frequency, the phase shift tends to approach -90∘. Therefore, option e is true.

To summarize, the correct options are a, c, d, and e.

c, d are obviously true

b. Bode rolls off at 20db/decade on simple RC filterss
think about a, e.