**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 solid-state components in liquid-cooled 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 non-linear 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 exam-taking 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 sub-question 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 signal-to-noise (S/N) on AM is a function of modulation level and asymmetrical modulation is permitted, precisely identifying 100 percent negative (carrier cut-off) 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 non-linearites) from the FM modulation. Simply put, there is no “loudness war” on FM.

Frequency Modulation is a frequency deviation of the carrier from its at-rest 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 frequency-modulation 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 frequency-modulated 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 5-to-1 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 pre-emphasis of the input audio. This is done to optimize system S/N in higher frequencies (above about 2 kHz). The pre-emphasis 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 non-conductive 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.