Passive Filters

Passive filters are made of resistors, capacitors and inductors. There are several conceptual and computational approximations that work fairly well.

Table of Approximations


Conceptual Approximation

Think of 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 X< XR and attenuation when XR < XC. Note that X= 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 X< XR .


Numerical Approximations

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 ]!).

 


Second and Higher Order Filters.

For a second order Low Pass filter, one can approximate the transfer function in the following way. 


Active filters and LCR filters

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