Stop Band Filters

Stop-band filters can also be constructed using a low-pass and a high-pass filter. However, rather than the cascaded configuration used for the pass-band filter, a parallel arrangement is required, as shown in Fig. 1. A low-frequency $ f_{1} $ can pass through the low-pass filter, and a higher-frequency $ f_{2} $ can use the parallel path, as shown in Figs. 1 and 2. However, a frequency such as $ f_{o} $ in the reject-band is higher than the low-pass critical frequency and lower than the high-pass critical frequency, and is therefore prevented from contributing to the levels of $ V_{o} $ above $ 0.707 V_{\max } $. 
Stop-band filter.
Fig. 1: Stop-band filter.
Stop-band characteristics.
Fig. 2: Stop-band characteristics.
Since the characteristics of a stop-band filter are the inverse of the pattern obtained for the pass-band filters, we can employ the fact that at any frequency the sum of the magnitudes of the two waveforms to the right of the equals sign in Fig. $ 3 $ will equal the applied voltage $ V_{i} $.
Fig. 3: Demonstrating how an applied signal of fixed magnitude can be broken down into a pass-band and stop-band response curve.
For the pass-band filters of Figs. $ 4 $ and $ 5 $,
Fig. 4: Series resonant pass-band filter.
Fig. 5: Parallel resonant pass-band filter.
therefore, if we take the output off the other series elements as shown in Figs. $ 6 $ and $ 7 $, a stop-band characteristic will be obtained, as required by Kirchhoff's voltage law.
Fig. 6: Stop-band filter using a series resonant circuit.
Fig. 7: Stop-band filter using a parallel resonant network.
For the series resonant circuit of Fig. 6, at resonance
$$V_{o_{\min }}=\frac{R_{l} V_{i}}{R_{l}+R}$$
For the parallel resonant circuit of Fig. 7, , at resonance,
$$V_{o_{\min }}=\frac{R V_{i}}{R+Z_{T_{p}}}$$
The maximum value of $ V_{o} $ for the series resonant circuit is $ V_{i} $ at the low end due to the open-circuit equivalent for the capacitor and $ V_{i} $ at the high end due to the high impedance of the inductive element. For the parallel resonant circuit, at $ f=0 \mathrm{~Hz} $, the coil can be replaced by a short-circuit equivalent, and the capacitor can be replaced by its open circuit and $ V_{o}=R V_{i} /\left(R+R_{l}\right) $. At the high-frequency end, the capacitor approaches a short-circuit equivalent, and $ V_{o} $ increases toward $ V_{i} $.

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