AN001: Impedance / Frequency Scaling

 

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

I PROVIDE THIS FREE SOFTWARE "AS IS". ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO ANY WARRANTY OF NON-INFRINGEMENT, THE IMPLIED WARRANTIES OF MERCHANTABILITY, SATISFACTORY QUALITY, REASONABLE CARE AND SKILL, AND FITNESS FOR A PARTICULAR PURPOSE ARE EXPRESSLY DISCLAIMED.

 

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