Question 6.FOM.2: AC Line Interference Filter One application of narrow-band f...
AC Line Interference Filter
One application of narrow-band filters is in rejecting interference due to AC line power. Any undesired 60-\mathrm{Hz} signal originating in the AC line power can cause serious interference in sensitive instruments. In medical instruments such as the electrocardiograph, 60-Hz notch filters are often provided to reduce the effect of this interference { }^{2} on cardiac measurements. Figure 6.23 depicts a circuit in which the effect of 60-\mathrm{Hz} noise is represented by way of a 60-Hz sinusoidal generator connected in series with a signal source \left(\mathbf{V}_{S}\right), representing the desired signal. In this example we design a 60-Hz narrow-band (or notch) filter to remove the unwanted 60-Hz noise.
{ }^{2} See Example 13.3 and Section 15.2 for further information on electrocardiograms and line noise, respectively.
Known Quantities– R_{S}=50 \Omega.
Find– Appropriate values of L and C for the notch filter.
Assumptions– None.
Analysis– To determine the appropriate capacitor and inductor values, we write the expression for the notch filter impedance:
Z_{\|}=Z_{L} \| Z_{C}=\frac{\frac{j \omega L}{j \omega C}}{j \omega L+\frac{1}{j \omega C}}=\frac{j \omega L}{1-\omega^{2} L C} .
Note that when \omega^{2} L C=1, the impedance of the circuit is infinite! The frequency
\omega_{0}=\frac{1}{\sqrt{L C}}
is the resonant frequency of the L C circuit. If this resonant frequency were selected to be equal to 60 \mathrm{~Hz}, then the series circuit would show an infinite impedance to 60-Hz currents, and would therefore block the interference signal, while passing most of the other frequency components. We thus select values of L and C that result in \omega_{0}=2 \pi \times 60. Let L=100 \mathrm{mH}. Then
C=\frac{1}{\omega_{0}^{2} L}=70.36 \mu \mathrm{F}
The frequency response of the complete circuit is given below:
H_{V}(j \omega)=\frac{\mathbf{V}_{o}(j \omega)}{\mathbf{V}_{i}(j \omega)}=\frac{R_{L}}{R_{S}+R_{L}+Z_{\|}}=\frac{R_{L}}{R_{S}+R_{L}+\frac{j \omega L}{1-\omega^{2} L C}}
and is plotted in Figure 6.24.
Comments– It would be instructive for you to calculate the response of the notch filter at frequencies in the immediate neighborhood of 60 \mathrm{~Hz}, to verify the attenuation effect of the notch filter.
