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Dear Andy, LF Group,
Well, this should put an end to anyone's insomnia
:-)
For your particular circuit, the "missing"
zero is at minus 40MHz. The poles and zeroes in a
transfer function have to be symmetrical about the real axis of the complex
frequency plane, so have to occur in conjugate pairs if they are not on the real
axis. In this case there are two zeros at s = 0 +/- j*2pi*40Meg. But this does
not mean that a bandpass frequency response has to be symmetrical
about the centre frequency - it is quite OK for there to be a rejection notch on
one side of the passband and not on the other. It does mean the response will be
the same for positive and negative frequencies, i.e. symmetrical about zero
frequency.
One classic way of designing a bandpass filter
with a symmetrical response is to start with a low-pass ladder filter (i.e.
series inductors, shunt capacitors) response with the desired bandwidth, ripple,
etc., then to perform "lowpass to bandpass transformation", which results in
each shunt C being replaced with a parallel LC with the original value of
C, and each series L being replaced by a series LC with the original value
of L. The other component in each LC has a value such that each LC is
resonant at the desired centre frequency. This gives a response with the same
bandwidth and shape as the original low-pass, but with perfect geometrical
symmetry about the centre frequency, i.e. if centre frequency = fo, the response
at k*fo is the same as at fo/k . This process also works if you start with
an elliptic low-pass, in which case the series arm capacitors in the low-pass
elliptic become parallel LCs and shunt arm inductors become series LCs. This
will again give a geometrically symetrical response, with any zeros (notches)
above the passband mirrored by corresponding zeroes below the passband. The
trouble with this type of design is that for passbands that are a small fraction
of the centre frequency, the component values vary over an impractically wide
range (very small shunt inductors, very large series inductors and so on). There
are a whole range of tricks for making the design more practical, as Mike
pointed out.
The initial circuit Andy has designed is a "coupled
resonator" filter, which is the classic way of designing a narrow-band
bandpass filter. This has the advantage that the starting point is a number of
identical LC resonators (or other types of resonator), and the required
bandwidth and response shape is obtained by changing the coupling capacitors (or
it could be inductors, irises etc.), and the load resistances. Again there are
lots of tricks for making the design more practical, but you only have to worry
about getting the right coupling. The trouble with this is that it relies on a
"narrow band approximation", where the mathematical synthesis assumes (so I'm
told...) that the coupling impedance does not vary with frequency. This is
a good approximation in a narrow bandwidth, but gives rise to increasing
deviation from the assumed frequency response as you move further from the
centre frequency. Thus, the type of filter in Andy's design is inherently
asymmetrical, as can be seen from the frequency response plots - the high
frequency side has lower attenuation. The limitation of this type of design
comes when the passband becomes unacceptably lop-sided with wide bandwidths. On
the other hand, the transformed low-pass type can have any combination of centre
frequency and bandwidth, and will still give a response which is
perfectly symmetrical when plotted on a log frequency axis.
Andy's modification results in a circuit with a
combination of asymmetrical band-pass and notch filter responses - provided the
desirable notch reponse does not turn out to be horribly sensitive to slight
component value changes, or to depend on the precise interaction of several
components, this is OK. You will probably also find that stop-band
attenuation is reduced due to the by-passing effect of the added capacitor -
from the plots, a few dB attenuation have been lost at the HF end. But
this might be perfectly acceptable. Another possible problem is the depenence on
the unloaded Q of the components. In the circuit shown on Andy's web page, there
is no loss, so infinite Q. It might be found that the notch is much less
pronounced if some loss resistance is added in series with each
inductor.
Cheers, Jim Moritz
73 de M0BMU
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