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# Power Up With Matched Impedances

The amazing transmission line

SBE certification is the emblem of professionalism in broadcast engineering. To help you get in the exam frame of mind, Radio World Engineering Extra poses a typical question in each part of this series. Although similar in style and content to the exam questions, these are not from past exams nor will they be on future exams in this exact form.

Oh, to Be 50 Again
Question from the June 12 issue
(Exam level: CBRE)

Your radio station transmitter is a perfect 50 ohm source connected to pure 50 ohm line that feeds an ideal 50 ohm antenna. What are the current and voltage characteristics along the transmission line?

a. Since the line is merely a big capacitor, current will lead voltage.
b. Since the line is mainly an inductor, voltage will lead current.
c. Voltage and current obey wavelength concepts and are offset by sine and cosine values as a function of whether the line length is more than 180 degrees or less than 180 degrees of a wavelength at the load respectively.
d. Copper losses will reduce the voltage proportionately but the current will diminish asymptotically as it is liberated only in the load.
e. The current and voltage will be the same at all points on the transmission line.

In previous Certification Corner columns, we have promulgated that a large part of our job as broadcast engineers is power conservation. It is a major goal to get as much of the power we buy from the utility into the antenna and onto our listeners.

Maximum power transfer always occurs when the generator matches the load in impedance. In our hypothetical question here, we outline the perfect case (with the line and antenna in this case representing the load).

Part of the purpose of this column is to get your brain (we have some of the best in the world in our profession, and yours is probably one of them) into the exam-taking frame of mind. An element of that is to help you to read the question and then ask yourself:

• What am I actually being asked;
• What useful/important information am I being provided; and
• Which is the best answer of the choices I am given.

Although the circumstances in our question assume an ideal match between transmitter, line and antenna, that’s not always the case. One aspect of broadcast engineering is creating the perfect match by design and proper installation.

POWER TRANSFORMER
What is an antenna system but a special sort of transformer to couple RF energy into electromagnetic waves that can travel long distances wirelessly? When the antenna “output impedance” matches the electromagnetic characteristics of free space (around 377 ohms), the maximum energy will become radio waves. To make this match, impedance transformation networks are implemented, which use lump constants of capacitive and inductive reactance. At AM frequencies these are discrete components. At FM, quite often the reactive aspect of coaxial components is utilized.

We’re not going to divert today to network design or any of the related graphic aids such as Smith charts (we’ll get to these in the future), but merely note that if improperly used coax can take on reactive qualities as alluded to in answer (a) and (b). However, this reactance presents a large problem in a series circuit such as your main run to the antenna. The goal is to get the power to the antenna, not present a resonant circuit. Answers (a) and (b) are out.

We have to have some Buc gobbledygook in these columns somewhere, and that is what answer (c) is. As above, all reactance has a phase component in it someplace and that’s why the uninitiated might think (c) was our best answer.

Your distinguished editor and I have constantly reinforced that a good grounding in basic electrical science is probably the best preparation for success in our field. Here in answer (d) is a classic, mini case of that. The transmitter, coax line and antenna are essentially a two-wire series circuit. In this situation, voltage drops to zero around the circuit, but current must remain constant, according to Kirchoff’s laws. Hence (d) is easily dismissed.

Now onto answer (e) and at this point, possibly our most correct answer. One of our greatest leading lights in communications engineering, Major Howard Armstrong, was an extremely literate and perceptive gentleman. During his long and tenuous court battle with RCA to recognize his patents, he said something prescient about that legal experience:

Lawyers reduce ideas to words and then argue over the words.

Well, engineers struggle for precision in ideas and descriptions as well, but we’re paid a lot less for it and we’re usually a lot more genial and mannerly in our discussions. Our distinguished editor and your humble writer have had “discussions” about “the words” and whether answer (e) is accurate.

In a perfectly matched coaxial transmission system with RF conducted through it, the values of voltage and current are constant at all points. Obviously this is alternating current, so the instantaneous values change but the average values remain constant. So, let the two of us settle our “discussion” with an agreement that generally the wording (e) is the most correct answer.

While there may be a slight amount of loss to calculate in the real (non-ideal) case, this is one of the most amazing concepts of electromagnetics: In a system where the generator output impedance is matched to the transmission line impedance, which in turn is matched to the load (antenna) impedance, essentially all the power is transferred into the load. Transmission lines have unique electromagnetic properties that allow us to completely ignore the capacitive and inductive reactance. Thus they operate to move high-frequency energy from generator to load without concern for the operating frequency.

WHAT MAKES A TRANSMISSION LINE?
As we have previously written, the characteristic impedance of a transmission line is based on the ratio of the inner to the outer diameter. The classic formula for impedance of a line is:

Zline = square root of L/C, where L is the amount of inductance per unit length and C is the capacitance per unit length. The unit comes out as ohms if you do the math.

In a line with a uniform cross section, these values of inductance and capacitance can be easily calculated from the type of materials used (typically copper), the distance between the inner and outer conductors and the insulating material that separates them (often air or a plastic foam). Now you know the rest of the story on how these remarkable lines are built.

LESS THAN PERFECTION
EQUALS REFLECTION

Our perfect system might exist somewhere, but I’ve never encountered it. We get close, but a perfect match is elusive. Mismatches in coaxial transmission systems normally produce reflections of the energy you’ve introduced. One way of expressing this mismatch is by calculating the Standing Wave Ratio (SWR). Our constant values will be affected by the phase and amplitude of the reflected wave as it returns to the generator (our transmitter). This reflected wave, depending on the length of the transmission line, will add or subtract algebraically with the generator wave to produce a voltage that varies with the distance on the transmission line. We define the Standing Wave Ratio as simply the ratio of Emax / Emin.

As a practical point, since the values of the forward and reflected waves at FM frequencies are usually ascertained from a directional coupler on the transmitter, these are the values at that point. Since there is loss (albeit small in comparison to the total power) the calculated SWR at this point will, in fact, be less than the actual value. Ideally the SWR should be determined at the far end, (the antenna load) from the peak value of the reflected power measured there.

For more on this subject, see “Transmission Lines” by Robert A. Chipman, part of the Schaum’s Outline series, Chapter 8 starting at page 156, and “Electronic and Radio Engineering” by Dr. Frederick K. Terman, Chapter 4 Transmission Lines, centered around page 96.

The next SBE certification exams will be given Feb. 7–17, 2014, in the local chapters. Applications must be in by Dec. 31, 2013. Remember a dream is just a dream … a goal is a dream with a plan and a deadline!So get yourself and your confreres motivated to become certified or advance a grade today. Do not pass over this opportunity to learn and advance.

Charles “Buc” Fitch, P.E., CPBE, AMD, is a frequent contributor to Radio World. Missed some Certification Corners or want to review them for your next exam? See the “Certification” tab under Columns at radioworld.com.

MR. SMITH CHECKS HIS CHART

Question for next time
(Exam level: CPBE)

A Smith chart is a graphic display of the orthogonal curvilinear coordinates of the normalized impedance components on the voltage reflection coefficient plane. With that in mind, what is the first step necessary for meaningful results when using this chart?

a. Annotate the cross field axis (susceptance, admittance, purveyance, persistence).
b. Assign the value of normalization (typically 50 ohms for ‘1’).
c. Determine if you will be calculating a double or triple tuner.
d. Determine if you will be using real or imaginary numbers.
e. Decide whether negative or positive will be on the upper quadrant.

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