The circuit in Fig. 12.84a is called a notch filter. From a sketch of its Bode plot in Fig. 12.84b, we see that at the notch frequency, fn, the transfer function gain is zero, while at frequencies above and below fn the gain is unity. Let us design a notch filter to remove an annoying 60-Hz hum

Question

The circuit in Fig. 12.84a is called a notch filter. From a sketch of its Bode plot in Fig. 12.84b, we see that at the notch frequency, [latex]f_n[/latex], the transfer function gain is zero, while at frequencies above and below [latex]f_n[/latex] the gain is unity. Let us design a notch filter to remove an annoying 60-Hz hum from the output voltage of a cassette tape player and generate its Bode plot.

Accepted Answer

Figure 12.84c shows a block diagram for the filter implementation. The tape output contains both the desired music and the undesired hum. After filtering, the voltage [latex]\mathbf{V}_{ ext {amp }}[/latex] will have no 60-Hz component as well as some attenuation at frequencies around 60 Hz. An equivalent circuit for the block diagram including a Thévenin equivalent for the tape deck and an equivalent resistance for the power amp is shown in Fig. 12.84d. Applying voltage division, we find the transfer function to be

[latex]\frac{V_{ ext {amp }}}{V_{ ext {tape }}}=\frac{R_{ ext {amp }}}{R_{ ext {amp }}+R_{ ext {tape }}+\left(s L / / \frac{1}{C s} ight)}[/latex]

After some manipulation, the transfer function can be written as

[latex]\frac{\mathbf{V}_{ ext {amp }}}{\mathbf{V}_{ ext {tape }}}=\frac{R_{ ext {amp }}}{R_{ ext {amp }}+R_{ ext {tape }}}\left[\frac{s^{2} L C+1}{s^{2} L C+s\left(\frac{L}{R_{ ext {tape }}+R_{ ext {amp }}} ight)+1} ight][/latex]

We see that the transfer function contains two zeros and two poles. Letting s = jω, the zero frequencies, [latex]\omega_{z}[/latex], are found to be at

[latex]\omega_{z}=\pm \frac{1}{\sqrt{L C}}[/latex]

Obviously, we would like the zero frequencies to be at 60 Hz. If we arbitrarily choose C = 10 μF, then L = 0.704 mH.

The Bode plot, shown in Fig. 12.84e, confirms that there is indeed zero transmission at 60 Hz.

Question 12.39: The circuit in Fig. 12.84a is called a notch filter. From a ...

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