RC High Pass Filters

 As noted early in Section (R-C low pass filters), a high-pass $ R $ - $ C $ filter can be constructed by simply reversing the positions of the capacitor and resistor, as shown in Fig. 1. At very high frequencies the reactance of the capacitor is very small, and the short-circuit equivalent can be substituted, as shown in Fig. 2. The result is that $ \mathbf{V}_{o}=\mathbf{V}_{i} $. At $ f=0 mathrm{~Hz} $, the reactance of the capacitor is quite high, and the open-circuit equivalent can be substituted, as shown in Fig. 3. In this case, $ \mathbf{V}_{o}=0 \mathrm{~V} $. A plot of the magnitude versus frequency is provided in Fig. 4, with the normalized plot in Fig. 5. At any intermediate frequency, the output voltage can be determined using the voltage divider rule: or and The magnitude of the ratio $ \mathbf{V}_{o} / \mathbf{V}_{i} $ is therefore determined by and the phase angle $ \theta $ by For the frequency at which $ X_{C}=R $, the magnitude becomes as shown in Fig. 5. The frequency at which $ X_{C}=R $ is determined by and For the high-pass $ R-C $ filter, the application of any frequency greater than $ f_{c} $ will result in an output voltage $ V_{o} $ that is at least $ 70.7 \% $ of the magnitude of the input signal. For any frequency below $ f_{c} $, the output is less than $ 70.7 \% $ of the applied signal. For the phase angle, high frequencies result in small values of $ X_{C} $, and the ratio $ X_{C} / R $ will approach zero with $ tan^{-1}\left(X_{C} / R \right) $ approaching $ 0^{\circ} $, as shown in Fig. 6. At low frequencies, the ratio $ X_{C} / R $ becomes quite large, and $ tan ^{-1}\left(X_{C} / R \right) $ approaches $ 90^{\circ} $. For the case $ X_{C}=R $, $ tan ^{-1}\left(X_{C} / R \right)= tan^{-1} 1=45^{\circ} $. Assigning a phase angle of $ 0^{\circ} $ to $ \mathbf{V}_{i} $ such that $ \mathbf{V}_{i}=V_{i} \angle 0^{\circ} $, the phase angle associated with $ \mathbf{V}_{o} $ is $ \theta $, resulting in $ \mathbf{V}_{o}=V_{o} \angle \theta $ and revealing that $ \theta $ is the angle by which $ \mathbf{V}_{o} $ leads $ \mathbf{V}_{i} $. Since the angle $ \theta $ is the angle by which $ \mathbf{V}_{o} $ leads $ \mathbf{V}_{i} $ throughout the frequency range of Fig. 6, the high-pass $ R $-$C$ filter is referred to as a leading network. In summary, for the high-pass $ R$-$C $ filter: For $f < fc$, $V_o < 0.707V_i$
whereas for $f > fc$, $V_o > 0.707V_i$
At $f_c$, $V_o$ leads $V_i$ by $45^\circ$
The high-pass filter response of Fig. $ 5 $ can also be obtained by interchanging their positions, as shown in Fig. 7.