** **

An interesting side effect of the
Internet is the instant feedback that it fosters. Like many
engineers, I have published designs and fun projects on the
net to share with other interested folks and without fail I
get questions like: "How can I change your 'A' band receiver
to cover 'B' band frequencies?" or "How can I change your
'A' frequency filter to work at 'B' frequency?"

It used to be that anyone who ever
designed a filter from one of the classic filter design
texts knew how to do this sort of scaling, but with the
advent of so many computer programs available this skill has
been largely forgotten. My intent here is not to present
anything that is 'new' per say, but to document the process
and caveats of this frequency scaling procedure.

All the old filter books listed the
filter design tables in terms of a one ohm, one radian per
second normalized values. The process was then to select the
proper normalized coefficients for the proper filter shape
and order then scale the values to the desired frequency /
impedance level required by the final circuit.

The same procedure can also be used to
scale existing circuits to new requirements. The basic
scaling formulas are,

Where,

Rckt_old is the original circuits
resistance level

Rckt_new is the new desired circuits
resistance level

Fref_old is the original circuits
operating frequency

Fref_new is the new desired circuits
operating frequency

Rold is the original resistor
value

Rnew is the new resistor value with
resistance scaling applied

Cold is the original capacitor
value

Cnew is the new capacitor value with
scaling applied

Lold is the original inductance
value

Lnew is the new inductance value with
scaling applied

Length_old is the original resistor
value

Length_new is the new resistor value
with frequency scaling applied

** **

**Spreadsheet makes scaling
easy**

These formulas are easy to convert to
spreadsheet form and one can be downloaded with this design
idea.

*Here a frequency scaling factor
from the original value of 1 to 2 is applied. Since all the
terms have been normalized the actual scaling factor applied
can easily be seen.*

**Caveats**

As with all things care must be taken
in the application of these formulas. If the circuit barely
worked at the original frequency and or resistance level it
is highly likely that it won't work at all with the new
scaling applied (that's just Mr. Murphy's in action).

For the scaling to work well, all the
circuits second order parameters must still be second order
after the scaling. If for instance a 1 MHz LC filter is
scaled to 1 GHz it's highly likely that the second order
parameters of element Q's at 1 MHz are going to be vastly
lower at 1 GHz hence the circuits dependence on Q is going
to be much more pronounced at 1 GHz than it was at 1 MHz.

If the circuit changes modes of
operation with the scaling it won't work. For example, if a
1 MHz crystal oscillator operating in the fundamental mode
is scaled to operate at 100 MHz, it's likely that the
resonator will be a third or fifth overtone. Hence the mode
of the circuits operation will have to be different and
direct scaling probably won't work because the circuit
topology also needs to change too.

Another instance to be careful of is
gain bandwidth limits. This comes into play when for
instance a 1 kHz active filter is scaled higher in
frequency. If the loop gain of the active element isn't high
enough at the new frequency the scaling won't work as
expected, but instead the center frequency and the circuit Q
will be lower than expected.

Many discrete amplifiers, especially
for RF frequencies make extensive use of RF chokes to block
the RF energy from biasing networks. Scaling these values
can lead to squegging [1] oscillations that were not present
in the original design.

Scaling Length is for use with
microstrip structures and the like. The scaling formula
works as long as the velocity of propagation (Vp) is the
same for the old circuit to the new. Making a jump from
Teflon PCB material to FR-4 and only applying direct
frequency scaling won't work because of the materials vastly
different Vp.

With all these downsides where does
scaling work well you might ask?

Well it works very well when the
frequency or circuit resistance levels are changed by small
amounts. In many cases I have the need to take a perfectly
good filter and 'tweak' it's cutoff by 15 or 20% for a new
application, scaling in this instance usually works
perfectly the first time. Also when changing a 50 ohm
network to 75 ohms at video frequencies, scaling works very
well. Usually up to an octave change works -- anything after
that care should be used or you can expect a call from Mr.
Murphy.

[1] Squegging is a particularly nasty
form of oscillation that is usually highly non-linear and
usually very low in frequency (audio rate). It is caused
predominately by resonant circuits or unaccounted for phase
shifts in biasing networks and because of it's highly
non-linear, rail-rail limit cycles - many times causes the
afflicted circuit to actually burn up before your very
eyes!

Download the Excel
spreadsheet here: spreadsheet.zip

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