Question

Figure 2 shows the electrical system (band-stop or notch filter). These circuits are used to "filter out" or attenuate input signals within a particular frequency band. For example, notch filters are used on aerospace vehicles to remove the mechanical vibrations that are transmitted to the onboard sensors such as gyroscopes and accelerometers. Derive the transfer function G(s) of the band-stop or notch filter where voltage eo is the desired output and source voltage ein() is the input. L Output voltage C eo Figure 2

Answer 1

Applying KVL in the loop eim (t) - iR + Ldi/dt + 1/c integral i dt (1) For deriving function all initial condition should be assumed to be i.e. i(0) = 0: Initial capacitor v_c (0) = 0 voltage Applying Laplace transform an both side eq (1) eim (s) = R I(s) + SL I(s) 6 1/LS I(s) I(s) = eim (s)/R + SL + 1/LS output voltage: e(t) = eim (t) - iR (3) Applying Laplace transform on both sides we get e_0 (s) = eim (s) - RI (s) \r\n substituting I(s) from eq (2), we get e_0 (s) = eim (s) - R (e_in (s)/R + Sl + 1/CS) e_0 (s) = (SL + 1/CS/R + SL + 1/CS) e_0 (s)/e i(s) = s^2 LC + 1/S^2 LC + SCR + 1 G(s) = e_0 (s)/e i(s) = s^2 + 1/LC/s^2 + R/L S + 1/LC Dividing by LC for both numerator & denominator