Low Pass RC Filter

For the low-pass filter of Fig. 1, and with as defined earlier. Note that now the sign of the imaginary component in the denominator is positive and $ f_{c} $ appears in the denominator of the frequency ratio rather than in the numerator, as in the case of $ f_{c} $ for the high-pass filter. In terms of magnitude and phase, An analysis similar to that performed for the high-pass filter will result in Note in particular that the equation is exact only for frequencies much greater than $ f_{c} $, but a plot of Eq. (4) does provide an asymptote that performs the same function as the asymptote derived for the high-pass filter. In addition, note that it is exactly the same as Eq. (2), except for the minus sign, which suggests that the resulting Bode plot will have a negative slope [recall the positive slope for Eq. (2)] for increasing frequencies beyond $ f_{c} $. A plot of Eq. (4) appears in Fig. $ 2 $ for $ f_{c}=1 \mathrm{kHz} $. Note the 6- $ \mathrm{dB} $ drop at $ f=2 f_{c} $ and the 20- $ \mathrm{dB} $ drop at $ f=10 f_{c}$. At $ f \gg f_{c} $, the phase angle $ \theta=-\tan ^{-1}\left(f / f_{c}\right) $ approaches $ -90^{\circ} $, whereas at $ f \ll f_{c}, \theta=-\tan ^{-1}\left(f / f_{c}\right) $ approaches $ 0^{\circ} $. At $ f=f_{c}, \theta= $ $ -\tan ^{-1} 1=-45^{\circ} $, establishing the plot of Fig. 3. Note again the $ 45^{\circ} $ change in phase angle for each tenfold increase in frequency. Even though the preceding analysis has been limited solely to the R-C combination, the results obtained will have an impact on networks that are a great deal more complicated. One good example is the high and low-frequency response of a standard transistor configuration. Some capacitive elements in a practical transistor network will affect the low-frequency response, and others will affect the high-frequency response. In the absence of the capacitive elements, the frequency response of a transistor would ideally stay level at the mid-band value. However, the coupling capacitors at low frequencies and the bypass and parasitic capacitors at high frequencies will define a bandwidth for numerous transistor configurations. In the low-frequency region, specific capacitors and resistors will form an R-C combination that will define a low cutoff frequency. There are then other elements and capacitors forming a second R-C combination that will define a high cutoff frequency. Once the cutoff frequencies are known, the -3-dB points are set, and the bandwidth of the system can be determined.