Given: A series-resonant filter with a resonant frequency of 100 kHz has a quali

Question

Given: A series-resonant filter with a resonant frequency of 100 kHz has a quality factor of 15. V_i is the input voltage and V_o is the output voltage. Required: a. Determine the capacitance value, C, in pF for the filter. b. Determine the resistance value, R, in k ohm for the filter. c. Determine the filter's gain in decibels, G_db, at a frequency of 200 kHz. d. Determine what type of filter this is. Solution: C = pF R = k ohm G_db = db and the filter is a High-pass, Low-pass, bandpass or Band-Reject filter.

Explanation / Answer

Answer:- a) We have the value of resonating frequency is 100 kHz, but mathematically for a series R-L-C resonating circuit the expression for resonating frequency is-

fr = 1 / [ 2 x 3.14 x (L x C)0.5 ]

=> 100 x 103 = 1 / [ 2 x 3.14 x (50 x 10-3 x C)0.5 ]

=>C = 50.66 x 10-12 F = 50.66 pF

b) We have the value of quality factor Q = 15 and mathematically-

Q = (L/C)0.5 / R

=> R = (L/C)0.5 / Q

=> R = 2.094 kOhm

c) Filter gain is the ratio of Vo to the Vi i.e Vo / Vi .

by KVL we can write Vi = I x ( R + XL + XC ) = I x [2094+ (j x 2 x 3.14 x 200 x 0.05) + (-j x 2 x 3.14 x 200 x 50.66 x 10-12) ]

Vi = I x (2094 + j62.83) ....................................(1)

where I is the current flowing in the circuit at 200 Hz frequency. Note XL = j x 2 x pie x frequency x inductance, and Xc = -j x 2 x pie x frequency x capacitance.

similarly for same time output voltage is the value of voltage drop accross capacitor i.e

Vo = I x XL = I x (-j x 2 x 3.14 x 200 x 50.66 x 10-12 ) = I x (-j x 6.36 x 10-8 ) ....................................(2)

we can get value of gain in linear form by dividing the magnitude of Vo by Vi.

G = | (-j6.36x10-8) | / | (2094 + j62.83) |

G = (6.36x10-8) / (2094.94)

G = 3.0358x10-11

in dB it is written as GdB = 20 log10(G) = -210.35 dB

d) The given filter is a Low Pass Filter since output is taken from capacitor.

The filter works as Band Pass Filter if output is taken accross Resistor.

The filter works as High Pass Filter if output is taken accross Inductor.

And the filter works as Band Reject Filter if output is taken from series connection end of Inductor and Capacitor.

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