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Softrock Lite 6.2
Adventures in Electronics and Radio
Elecraft K2 and K3 Transceivers
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Measuring FM Deviation
Via Bessel Nulls
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What is a Bessel Null and how does it
relate to a Z90?
One use of a Z90, in addition to its obvious use as a ham
band monitor, is to calibrate FM modulation of a transmitter or a signal
generator via the Bessel Null technique.
The principle behind Bessel Nulls is that in a phase or
frequency modulation system, the relationship of the carrier level, the
modulating frequency and the deviation is defined by a mathematical relationship
known as a Bessel function, named after their discoverer, the German
mathematician Friedrich Bessel. Bessel lived nearly 100 years before FM was
invented but his mathematical techniques have proven useful in describing many
physical phenomena. For more details, see Agilent's Application Note 1390.
At the moment, it may be found at
http://cp.literature.agilent.com/litweb/pdf/5988-5677EN.pdf, but Agilent
periodically rearranges its web site so this address may not last indefinitely.
For our purpose, it is sufficient to understand that for
certain values of deviation and modulating frequencies, the carrier will reduce
to zero. More practically, a deep null, on the order of 30 or 40 dB can be seen
as these values are approached. Appropriately enough, these are called Bessel
nulls, and the more useful values are:
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Null |
Modulation Index β |
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1 |
2.40 |
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2 |
5.52 |
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3 |
8.66 |
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4 |
11.79 |
|
5 |
14.93 |
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6 |
18.07 |
Modulation index β is defined as:
In other words, if we modulate an FM transmitter with a 1
KHz audio tone, and slowly increase the deviation, we see the carrier drop in
level, and disappear when the deviation is 2.40 KHz and then increase, to
disappear again when the deviation is 5.52 KHz, etc.
If we instead modulate the transmitter with a 3 KHz tone,
the first null is seen with 3.0 KHz * 2.4 = 7.2 KHz deviation, etc.
To clarify one point, the total power in an FM
system remains constant, regardless of deviation and modulating frequency.
However, the total power is spread out amongst the carrier and the modulating
sidebands in a complex fashion, described by the Bessel functions. At a Bessel
null, 100% of the power is in the sidebands, with the carrier reduced to zero.
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Let's see how this works in practice. I connected a Boonton
102D signal generator to a Z90. The particular frequency I used is 4915 KHz, but
this is not important to the concept. In order to resolve the carrier, I've set
the Z90's span to 10 KHz. |
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Signal generator with modulation turned off. Carrier level
is approximately 40 dB over the base line.
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I then selected FM modulation, 1 KHz internal source and set the deviation
to be an indicated 3 KHz. The result shows the carrier and several modulating
sidebands.
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With 1 KHz modulating frequency and 3 KHz deviation, many
sidebands are seen. The carrier is approximately 12 dB reduced from the no
modulation stage.
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I gradually reduced the deviation control, watching the carrier level on the
Z90's screen. |
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First Bessel Null. 1 KHz modulation frequency.
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At this deviation setting, the carrier is reduced by 30 dB from the unmodulated
state. This is "close enough" to zero for our purpose.
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The modulation setting on my 102D at the level
corresponding to the null condition shown above. The meter multiplier is 0-3
KHz, so the displayed value is 2.50 KHz deviation.
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The theoretical deviation at the first null is 2.40 KHz, and my 102D reads 2.50
KHz, so it's about 4% high. The 102D's specification is ±10%, so it's well
within tolerance.Let's try a different combination
- 3 KHz modulating frequency. The 1st Bessel null is found at modulation index
β= 2.40. Since β is defined as deviation/modulating frequency, then
deviation = β * modulating frequency
With 3 KHz modulation, we should see the carrier null at a
deviation of 2.40 * 3.0 KHz = 7.20 KHz.
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First Bessel null, 3 KHz modulating frequency, 7.2 KHz
deviation
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At this carrier null, the 102D's meter reads 7.6 KHz (0-10
KHz scale)
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The error between indicated deviation and actual deviation is 5.5%, again well
within the 102D's ±10% FM modulation indication accuracy. |
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