What’s the Frequency, Rollo?

Function Like Bessel

(Exam level: CPBE)
In
the Dec. 7 issue of RWEE, we asked:
What is a Bessel
function and what is a typical application?
a.
A numerical description of
propagation through a solid body, most often used in coax design.
b.
A thermal transfer
derivative, most often used to enumerate fluid cooling such as around the new
solidstate components in liquidcooled transmitters.
c.
A solution to a particular
type of equation, used in broadcasting mainly as a determinant of FM modulation
levels.
d.
The delta change in
free air temperature, most often used in broadcasting to enumerate nonlinear
coax expansion and the potential for shearing.
e.
The delta change in wire
temperature as a function of uneven harmonic currents, most often used in
broadcasting to enumerate the capacity of neutral power conductors.


Certification Corner
is a series of questions intended to help you get in the certification examtaking
frame of mind.
Our editor has asked us to bury the answer to our latest
question in the body of our article to carry our readers into the text with the
hope that either knowledge related to the subject will be reinforced or some
new information will be absorbed. So please read on and we’ll get to the
answer, I promise.
In the meantime, let’s be mischievous and start
this column with the more important half of this question: What is
a typical application of a Bessel function?
In broadcasting, the answer to this subquestion is to
establish FM modulation levels.
By contrast,
determining modulation levels on AM is rather a straightforward affair. With a
precision envelope diode, an audio generator and a good scope, one can quickly
ascertain what audio input level produces 100 percent transmitter modulation.
Since system signaltonoise (S/N) on AM is a function of modulation level and
asymmetrical modulation is permitted, precisely identifying 100 percent
negative (carrier cutoff) and 125 percent positive (the maximum permitted) is
important.
In FM, the situation is a little more complicated.
S/N is set mainly as a function of the receiver’s ability to differentiate then
attenuate or eliminate the carrier noise and amplitude component (as well as
other amplitude nonlinearites) from the FM modulation. Simply put, there is no
“loudness war” on FM.
Frequency Modulation is a frequency
deviation of the carrier from its atrest frequency; in simple terms, the
greater the deviation, the greater the amplitude of the retrieved audio. For
single tones, the actual modulating frequency can be seen as a series of
sidebands at intervals equal to the tone frequency away from the carrier.
FM BASICS
Let’s define a few important terms that
we’ll need later:
Modulation index: Ratio of the frequency
deviation to the frequency of the modulating wave in a frequencymodulation
system when using a sinusoidal modulating wave.

Fig. 1: Bessel function graph


Deviation ratio: Ratio of the maximum
frequency deviation to the maximum modulating
frequency of a frequencymodulated system.
The FM envelope
we transmit is made up of pairs of sideband frequencies of which the ultimate
number is set by the modulation index. Suffice to say that the totality of
these sidebands is best described mathematically by Bessel functions (see Fig.
1). Annotating the math is a little beyond our scope and space available, but
see my references at the end of this story for some better sources for the math
discussion.
The math is essentially a complex angular velocity
calculation, and the Bessel function describes the relationship and amplitude
of the carrier and sidebands. What we have is essentially a sine wave
modulating a sine wave and when the former is much lower than the latter,
signal at the carrier will come out to zero at certain values of modulating
frequency and amplitude, i.e. the carrier will “null.” This phenomenon is best
observed via a spectrum analyzer (see Fig. 2). The industry vernacular for this
drop to zero is a “Bessel null.”

Fig. 2: The null at carrier indicates that we are at
a ‘Bessel null’ at this modulation index.


By the way, Bessel functions, although first defined by the
mathematician Daniel
Bernoulli, get their name from Friedrich Bessel
because he perfected the generalized math which is used
so widely today.
Now let’s get
back to the original question. Our FM standard dictates a 5to1 deviation ratio.
Thus if the maximum desired modulating frequency is 15 kHz, it produces a total
of 75 kHz deviation at 100 percent modulation.
Now let’s look at one of the special “Bessel null”
audio input frequencies we have calculated, say 13.586 kHz. The carrier
amplitude will null at 100 percent (75 kHz) modulation. We can double check our
findings/adjustments at other points by calculating a multitude of these
frequencies based on the Bessel function. Fig. 3 is a nice compendium of select
useful input frequencies.
So the correct answer to the question is “c.” A Bessel
function is a solution to a particular type of equation used in broadcasting
mainly as a determinant of FM modulation levels.

Fig. 3:
Input audio frequencies in kHz for Bessel nulls at 100 percent
modulation


This definite
Bessel null is valuable, as it allows an onsite confirmation that our FM
modulation monitors retain calibration as uncalibrated or suspect test
equipment is essentially worthless.
By way of reminder, do not forget that your
modulation monitor’s meter or bar display is not responding equally on all
frequencies. Our present standards, still with us from the time of Major
Armstrong, call for 75 microsecond preemphasis of the input audio. This is
done to optimize system S/N in higher frequencies (above about 2 kHz). The
preemphasis boost of 6 dB per octave parallels the increase in amplitude noise
as frequency increases. So the same input amplitude from the upper third
register of a busy piccolo in a Souza march (say, “The Thunderer” or “Stars and
Stripes Forever”) will have higher modulation levels than Barry White’s amorous
exhortations down on the bass end.
Note that the next SBE certification exams will be held at
NAB 2012 on April 17; the closing date for that exam cycle is March 23.
References
“Electronic and Radio Engineering” by
Dr. Frederick Terman, Fourth Ed. Page 586 and on; “NAB Engineering Handbook,
Fourth Edition,” Warren Bruene section on FM, Page 434 and on.
Charles “Buc” Fitch, P.E.,
CPBE, AMD, is a frequent contributor to Radio World. Missed some SBE
Certification Corners or want to review them for your next exam? See the
“Certification” tab under Columns.
Show
Me Some Skin

Question for next time
(Exam level: CBRE)
What is
skin effect in alternating current and what is its relationship to current flow?
a.
Skin effect
is the tendency of all current flow on printed circuit boards to concentrate on
the area against the nonconductive surface and create a capacitor.
b.
Skin
effect is the tendency for electrolytic capacitors to change value when touched
due to the requirement that the positive plate always be on the outside.
c.
Skin
effect is the tendency of current to flow through mainly the epitaxis layer of
the skin when experiencing an electric shock.
d.
Skin effect is the tendency of an alternating electric
current to
distribute itself within a conductor with the current density being largest
near the surface of the conductor, decreasing at greater depths.
e.
Skin
effect is the penchant of RF to want to flow through the conductive character
of a coaxial line.


