Passive filters are made of resistors, capacitors and inductors. There are
several conceptual and computational approximations that work fairly well.
Table of Approximations
resistors as having a fixed impedance, capacitors as having a high impedance at
low frequencies and low impedance at high frequencies. Inductors have low
impedance at low frequencies and high impedance at high frequencies. For a
simple RC high pass filter like that at the right, one can approximate them by
assuming there is little attenuation as long as XC < XR
and attenuation when XR < XC. Note that XC = XR
when wRC = 1, or wt = 1.
For a low pass filter it is just the reverse (with R and C switched around);
little attenuation when XR < XC and attenuation when XC < XR
The numerical approximations follow the above conceptual approximation. Here one say that for a high pass filter, when wt is small (say <1/2 or 1/3)
where fC is the cutoff frequency in Hz. (fC = 1/(2pt) where t = RC). Therefore if fC is 2000 Hz and f = 500Hz, |H| = 500/2000 = 1/4 is a good approximation. (It is off by only a few %.) Even when f/fC = 1/2 , the approximation is |H| = 0.5 and the exact value is 0.45 to two figures. This is good enough for most cases.
If f/fC = wt is > 1, say > 2 or 3, then one approximates |H| = 1.
For f/fC = 2 the actual value is 0.89 instead of 1. Again close enough for many applications. If f/fC = 3, |H| = 0.95, and |H| = 1 is an even better approximation.
Only in the region 1/2 < f/fC < 2 do you have to worry about the deviations and the worst deviation is when f/fC = 1 when both approximations yield |H| = 1, while the true value is 0.7.
For low pass filters it is just the reverse. Here
and for f/fC <1/2, |H| is approximately 1 and for f/fC >2, |H| is approximately fC/f. (Note the inversion, |H| = 1/[f/fC ]!).
For a second order Low Pass filter, one can approximate the transfer function in the following way.
For an nth order low pass filter
where fc is the cutoff frequency (-3dB point). In practice these depend on the details of the filter. For a Butterworth type they work very well since for an nth order low pass filter
For a high pass filter