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Calculating, rather than experimenting to find, resistor values for the NTX2 voltage divider

A previous blog post showed the voltage divider circuit I'm using to drive the Radiometrix NTX2 transmitter in GAGA-1. And another blog post gave a spreadsheet for working out the voltages based on resistor values.

What you really want, though, is a way to say "given this required frequency shift what should the resistor values be?". Recall the circuit diagram I was using.
Arduino                         Radio
7 ---27K--------------\
                       ---------- TX
8 ---22K----2K2-------/
Now, replace the specific resistances on the left with R1 and R2 and the input impedance to ground with 100K (the value from the datasheet):
Arduino                      Radio
7 ---R1--------------\
                      ----------100K---Gnd
8 ---R2--------------/
Pins 7 and 8 are connected to either 5V or Gnd to make the divider so you have two configurations:
Arduino                      Radio
5V  ---R1--------------\
                        ----------100K---Gnd
Gnd ---R2--------------/

Gnd ---R1--------------\
                        ----------100K---Gnd
5V  ---R2--------------/
In both cases you can see that one of R1 or R2 is in parallel with the 100K input impedance so the dividers look like (I've used || to indicate the resistances in parallel):
5V  ---R1--------------X----------R2||100K---Gnd

5V  ---R2--------------X----------R1||100K---Gnd
Where the X is the connection to the TX pin and hence where we need specific voltages. For a 425Hz shift the difference between the two voltages needs to be 3/5000*425 (since the NTX2 has a deviation of 5kHz driven by a 0 to 3V value). Thus you can derive the following formula for the difference between the two voltages.

This simplifies to the following when you assume that R1 and R2 are positive (they are resistors after all!)

And that's a rather simple formula relating the resistances. Here's a graph of it (all resistance values are in k).

The cross hairs there correspond to one 22K resistor and another at about 24K2. Of course, if the input impedance isn't precisely 100K this isn't going to work, but the general method will.

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