RC Low Pass Filters

The R-C filter, incredibly simple in design, can be used as a low-pass or a high-pass filter. If the output is taken off the capacitor, as shown in Fig. 1, it will respond as a low-pass filter. If the positions of the resistor and capacitor are interchanged and the output is taken off the resistor, the response will be that of a high-pass filter. A glance at Fig. 2 reveals that the circuit should behave in a manner that will result in a high-level output for low frequencies and a declining level for frequencies above the critical value. Let us first examine the network at the frequency extremes of f = 0 Hz and very high frequencies to test the response of the circuit.
At f = 0 Hz, and the open-circuit equivalent can be substituted for the capacitor, as shown in Fig. $3$, resulting in $\mathrm{V}_{o}=\mathrm{V}_{i}.$ At very High frequencies, the reactance is and the short-circuit equivalent can be substituted for the capacitor, as shown in Fig. $4$, resulting in $\mathrm{V}_{o}=0$ V. A plot of the magnitude of $V_{o}$ versus frequency will result in the curve of Fig. 5. Our next goal is now clearly defined: Find the frequency at which the transition takes place from a pass-band to a stop-band. For filters, a normalized plot is employed more often than the plot of $ V_{o} $ versus frequency of Fig. 5. A normalized plot in the filter domain can be obtained by dividing the plotted quantity such as $ V_{o} $ of Fig. 5 with the applied voltage $ V_{i} $ for the frequency range of interest. Since the maximum value of $ V_{o} $ for the low-pass filter of Fig. 1 is $ V_{i} $, each level of $ V_{o} $ in Fig. 5 is divided by the level of $ V_{i} $. The result is the plot of $ A_{V}=V_{o} / V_{i} $ of Fig. 6. Note that the maximum value is 1 and the cutoff frequency is defined at the $ 0.707 $ level. At any intermediate frequency, the output voltage $ \mathbf{V}_{o} $ of Fig. 1 can be determined using the voltage divider rule: or and The magnitude of the ratio $ V_{o} / V_{i} $ is therefore determined by and the phase angle is determined by For the special frequency at which $ X_{C}=R $, the magnitude becomes which defines the critical or cutoff frequency of Fig. 6.The frequency at which $ X_{C}=R $ is determined by and The impact of Eq. (1) extends beyond its relative simplicity. For any low-pass filter, the application of any frequency less than $ f_{c} $ will result in an output voltage $ V_{o} $ that is at least $ 70.7 % $ of the maximum. For any frequency above $ f_{c} $, the output is less than $ 70.7 \% $ of the applied signal. Solving for $ \mathbf{V}_{o} $ and substituting $ \mathbf{V}_{i}=V_{i} \angle 0^{\circ} $ gives and The angle $ \theta $ is, therefore, the angle by which $ \mathbf{V}_{o} $ leads $ \mathbf{V}_{i} $. Since $ \theta= $ $ -\tan ^{-1} R / X_{C} $ is always negative (except at $ f=0 \mathrm{~Hz} $ ), it is clear that $ \mathbf{V}_{o} $ will always lag $ \mathbf{V}_{i} $, leading to the label lagging network for the network of Fig. 1. At high frequencies, $ X_{C} $ is very small and $ R / X_{C} $ is quite large, resulting in $ \theta=-\tan ^{-1} R / X_{C} $ approaching $ -90^{\circ} $. At low frequencies, $ X_{C} $ is quite large and $ R / X_{C} $ is very small, resulting in $ \theta $ approaching $ 0^{\circ} $. At $ X_{C}=R $, or $ f=f_{c},-\tan ^{-1} R / X_{C}=-\tan ^{-1} 1=-45^{\circ} $. A plot of $ \theta $ versus frequency results in the phase plot of Fig. 7. The plot is of $ \mathbf{V}_{o} $ leading $ \mathbf{V}_{i} $, but since the phase angle is always negative, the phase plot of Fig. 8 $V_{o}$ lagging $ \left.\mathbf{V}_{i}\right) $ is more appropriate. Note that a change in sign requires that the vertical axis be changed to the angle by which $ \mathbf{V}_{o} $ lags $ \mathbf{V}_{i} $. In particular, note that the phase angle between $ \mathbf{V}_{o} $ and $ \mathbf{V}_{i} $ is less than $ 45^{\circ} $ in the pass-band and approaches $ 0^{\circ} $ at lower frequencies. In summary, for the low-pass $ R-C $ filter of Fig. 1: The low-pass filter response of Fig. 2 can also be obtained using the $ R-L $ combination of Fig. 9 with In general, however, the $ R-C $ combination is more popular due to the smaller size of capacitive elements and the nonlinearities associated with inductive elements. The details of the analysis of the low-pass $ R-L $ will be left as an exercise for the reader.