### 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_{C }< X_{R}
and attenuation when X_{R} < X_{C}. Note that X_{C }= X_{R}
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 X_{R} < X_{C} and attenuation when X_{C }< X_{R}
.

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 f_{C} is the cutoff frequency in Hz. (f_{C} = 1/(2pt)
where t = RC). Therefore if f_{C} 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/f_{C} = 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/f**_{C}** = ****wt****
is > 1, say > 2 or 3, then one approximates |H| = 1.**

For f/f_{C} = 2 the actual value is 0.89 instead of 1. Again close
enough for many applications. If f/f_{C} = 3, |H| = 0.95, and |H| = 1 is
an even better approximation.

Only in the region 1/2 < f/f_{C} < 2 do you have to worry about
the deviations and the worst deviation is when f/f_{C} = 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/f**_{C}** <1/2,
|H| is approximately 1** and for **f/f**_{C}**
>2, |H| is approximately f**_{C}**/f.**
(Note the inversion, |H| = 1/[f/f_{C} ]!).

### 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 n^{th} order low pass filter

where
f_{c} 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 n^{th} order low pass filter

For
a high pass filter