For the band-pass filter in Fig. 22.37:
a. Determine the critical frequencies for the low- and high-pass filters.
b. Using only the critical frequencies, sketch the response characteristics.
c. Determine the actual value of $V_o$ at the high-pass critical frequency calculated in part (a), and compare it to the level that defines the upper frequency for the pass-band.

Question

For the band-pass filter in Fig. 22.37:
a. Determine the critical frequencies for the low- and high-pass filters.
b. Using only the critical frequencies, sketch the response characteristics.
c. Determine the actual value of [latex]V_o[/latex] at the high-pass critical frequency calculated in part (a), and compare it to the level that defines the upper frequency for the pass-band.

Accepted Answer

a. High-pass filter:

[latex]f_c=\frac{1}{2\pi R_1C_1}=\frac{1}{2\pi (1 kΩ)(1500 pF)}=106.1 kHz [/latex]

Low-pass filter:

[latex]f_c=\frac{1}{2\pi R_2C_2}=\frac{1}{2\pi (40 kΩ)(4 pF)}=994.72 kHz [/latex]

b. In the mid-region of the pass-band at about 500 kHz, an analysis of the network reveals that [latex]V_o ≅ 0.9V_i[/latex] , as shown in Fig. 22.38. The bandwidth is therefore defined at a level of [latex]0.707(0.9V_i ) = 0.636V_i[/latex] , as also shown in Fig. 22.38.

c. At f = 994.72 kHz,

[latex]X_{C_1}=\frac{1}{2\pi fC_1} ≅ 107 Ω\[0.5cm] and\hspace{30 pt} X_{C_2}=\frac{1}{2\pi fC_2}=R_2 = 40 kΩ [/latex]

resulting in the network in Fig. 22.39.

The parallel combination [latex]R_1\parallel (R_2-jX_{C_2})[/latex] is essentially 0.976 kΩ ∠0° because the [latex]R_2 - X_{C_2}[/latex] combination is so large compared to the parallel resistor [latex]R_1[/latex].

Then

[latex]V^{\prime}=\frac{0.976 kΩ ∠0°(V_i )}{0.976 kΩ - j 0.107 kΩ} ≅0.994V_i ∠6.26°\[0.5cm] with\[0.5cm] V_o=\frac{(40 kΩ ∠-90°)(0.994V_i ∠6.26°)}{40 kΩ - j 40 kΩ}\[0.5cm] V_o ≅ 0.703V_i ∠-39° \[0.5cm] so\; that\[0.5cm] V_o ≅ 0.703V_i \qquad at \qquad f = 994.72 kHz[/latex]

Since the bandwidth is defined at [latex]0.636V_i[/latex] the upper cutoff frequency will be higher than 994.72 kHz, as shown in Fig. 22.38.

Question 22.8: For the band-pass filter in Fig. 22.37: a. Determine the cri...

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