Page 143 - Analog and Digital Filter Design

1 40 Analog and Digital Filter Design

Component values can now be found by choosing an arbitrary value capacitor,
C. Let C = lOOpF.
1 1
~1~ ~4 =- - =1.126kR
20C - 2.4439.44 .lo-’
1
=
~2 = ~3 = - =1.567kR
w,,C 6380.10-’

Letting R = lOkR gives R5 = 454R. This is too low, so let R = 33 kR. Now
R5 = 1500R.
w, = the denormalized zero frequency of 35,877.5 rads. Let gain A = 1.

(z, )? (35,877.5)
R6= .AR= 6380 .33kIl.Hence R6= 1kR.

Frequency Dependent Negative Resistance (FDNR) Filters

Frequency dependent negative resistance (FDNR) circuits can be used to make
an active filter based on a passive ladder filter design. In applications where an
elliptical lowpass filter is required and an active filter is possible, FDNR filters
can be used as an alternative to a biquad filter. For example, a third-order ellip-
tic lowpass filter requires a biquad design with four op-amps, ten resistors, and
three capacitors. The same design using an FDNR requires two op-amps, eight
resistors, and four capacitors. An obvious advantage is the reduction of op-amps
from four down to two. Halving the number of op-amps required for the filter
halves the supply current, assuming that the same type of op-amp would be
required in both circuits.

However, there is a catch. In order for the circuit to work as specified, the source
impedance should be zero. This can be compensated for by simply reducing the
value of a series resistor in the design (more on this later). The greater problem
is the output load. The load must be high impedance for the circuit to work
properly. Of course, in multistage filters such as a seventh-order elliptic filter,
a biquad design would require three biquad stages connected in series (twelve
op-amps). A similar FDNR filter would require six op-amps, seven including a
buffer at the output.

The most significant advantage of doubly terminated lossless ladder circuits is
the low sensitivity to component tolerances. However, inductors are bulky and
are difficult to obtain. Low value inductors for radio applications are reason-
ably easy to find, but audio frequency applications require much larger values.
High-value inductors often have to be specially wound in order to obtain the
required inductance.