(source: Electronics World, Apr. 1968)
By WILLIAM M. STUTZ / Sr. Lab Technician, T.R.G. (Div. of Control Data
Corp.)
The newer filters, such as Butterworth, Chebyshev, and elliptic
types, are more accurate and easier to design.
WITH the many advances being made today in the field of semiconductors,
it is almost impossible to keep abreast of developments. Therefore, it
is not strange to find that familiar old passive circuits have advanced
too. One particular area, filters, has changed radically from what many
remember from technical school and college days. The old filters, such
as constantK and Mderived types, have been replaced by the Butterworth,
Chebyshev, and elliptic types. These newer designs have much to recommend
them. Their responses are more accurate over a wide range and are easier
to design. (Fig. 1 ) The constantK and Mderived types were traditionally
designed section by section. This method was involved since each section
had to be matched while trying to optimize the response. The results of
such designs were approximate to say the least. Today the filter is designed
as a whole, which simplifies the matching and allows us to optimize the
response. Both problems actually reduce to almost "cookbook" simplicity
in most cases. The third problem in filters, realization, now becomes
the main problem.
Fig. 1. Response curves of the three filters described in text.
The actual mathematics involved in a filter design today is very abstract
and is usually best left to the mathematicians. The filter's properties
are expressed by a transfer equation that is a function of frequency.
The shape of the response is then set, but actually that is all that is
permanent. The frequency and impedance information contained in such a
transfer equation may be extracted. The remaining information is said
to be "normalized ". A normalized number or quantity may then
be used under a large number of circumstances as opposed to the unique
value or quantity simply by supplying the missing information to denormalize
the quantity. This mathematical trick allows us to catalogue the elements
of a specific filter type and to design from them a filter of any particular
bandwidth or impedance value that we require.
Originally, the term "realization" meant the mathematical realization
process. This requires at least a mathematician and at most a digital
computer. The tern now means the physical as well as mathematical processes
of determining the actual L and C component values to be used. I prefer
the terms "denormalization" and "realization" to
distinguish the two.
Transformers are sometimes added to the filter for impedance matching
or for balancetounbalance conversion.
The various filter circuits can be made in two general forms. These are
the ladder network and the lattice or bridge networks, Fig. 2. The ladder
network is an unbalanced network, that is, it is a threeterminal network.
This type of network is used where the input is singleended such as for
unbalanced antenna lines or singleended r.f. or a.f. amplifiers.
The three basic filter circuits may all be built as a ladder, in fact,
this is the most common configuration.
The lattice or bridge type is a balanced or fourterminal network. This
makes the circuit useful only when the source is balanced with respect
to ground such as in 3(R)ohm balanced lines or any other area where the
source and loud are balanced. This circuit is sometimes used with crystals
to form a crystallattice network.
Fig. 2. (A) Unbalanced and (B) balanced arrangements.
Table 1. Unloaded minimum "Q' " for two filter types.
Physical capacitors and inductors are not perfect. i.e., the best capacitors
have sonic leakage and the best coils have some resistance. flow, there,
may a filter be made which was designed assuming ideal elements? As a
rule of thumb, it has been found that the more the filter requirements
tighten, the less dissipation is allowable, and the closer the elements
must be to the ideal. This is related to the elements by a term called
the minimum "Q ". The ratio of reactance to resistance of a
coil is called "Q ". When this ratio is small, the coil acts
as an RL circuit and not like an inductor. This will put a large insertion
loss in the filter passband and distort the passband shape, usually at
the edges first. Because of this problem we must have a certain type of
filter and a certain number of poles. Since the bandwidth may be vastly
different for different designs, we normalize this information and allow
it to be added later.
Table 1 is a list of the normalized unloaded "Q" required as
a function of the number of poles and type of filter.
If the design is a lowpass type, these "Q's" are the minimum
required for each element of the filter. If a band pass design is used,
the normalized "Q" must be multiplied by the total filter "Q ".
This, then, is the minimum "Q" required.
To illustrate how all of these aspects dovetail, let us design a filter
for the band from 2 to 30 MHz with a v.s.w.r. of no greater than 1.5:1,
an attenuation at least 35 dB at twice the3 dB bandwidth at 50 ohms impedance.
A v.s.w.r. of 1.5 represents a ripple of 0.177 dB (see Fig. 3) so a ripple
of 0.1 dB is acceptable. The shape factor will be the limiting factor
for the minimum number of elements we may use.
A good choice is a 5element Chebyshev, 0.1 dB ripple filter. See Fig.
4 below.
The normalized elements are found to be: L1 =1.15 µH, C2 =1.39 µF, L3=
1.97 pH, C4 =1.39 µF, and L5 =1.15 H.
Denormalizing by the following
formulas gives us values for L' and C':
We get L' =0.327 pH, C2' =156 pF, L3' =0.560 µH, C4' =156 pF, and L5'=
0.327 uH. This gives us the lowpass prototype filter from which we can
get the final bandpass design. This is achieved by resonating the lowpass
elements at the geometrical mean frequency of the band we wish to filter,
or
f_{o} =__/ [f1 X f2 =__/ [2X30MHz] = 7.74 MHz
The additional elements required are thus found to be: C1' =1400 pF,
L2'= 2.7 AFL C3' =725 pF, L4' =2.7 pH, and C5' =1400 pF.
The filter is of the ladder type and is shown along with its response
in Fig. 4.
In practice, this filter would have to be built and aligned. To do this,
certain changes are made. One is in the capacitors, two capacitors would
be used, one fixed and the other a variable unit which could trim the
fixed unit to the precise value required. The inductors would either be
variable units or precision toroids: both could be pre adjusted on a "Q"meter
to the exact value and then installed.
Fig. 3. The amount of loss due to ripple for various v.s.w.r.'s.
Fig. 4. The filter employed along with its frequency response.
