E9: Antennas and Transmission Lines

Before we can talk about fancy antennas, we need the vocabulary that describes any antenna with numbers. This group is about those numbers, and about the all-important idea of measuring gain against a reference.

The isotropic radiator: a perfect imaginary yardstick

To say one antenna has "gain," you must compare it to something. The cleanest reference is the isotropic radiator. This is a hypothetical, lossless antenna that radiates equally in every direction, a perfect glowing point that sends the same strength up, down, sideways, everywhere. It cannot actually be built; it exists only on paper. But because it is perfectly even and perfectly lossless, it makes an ideal yardstick. When gain is measured against it, we write the units as dBi ("decibels over isotropic").

The other common reference is a real antenna: the half-wave dipole. Gain measured against a dipole is written dBd ("decibels over dipole"). Here is the single most useful conversion in this whole group: a half-wave dipole itself has 2.15 dB of gain over isotropic. So to convert, dBi = dBd + 2.15, and dBd = dBi − 2.15. For example, an antenna with 6 dBi gain has 6 − 2.15 = 3.85 dBd compared to a dipole.

Gain does not create power, it concentrates it

A common beginner misunderstanding: an antenna with "gain" does not somehow make more power than you put in. Think of a flashlight reflector. The bulb puts out the same number of watts whether you point the reflector or not, but the reflector concentrates the light into a beam so it looks much brighter in one direction. An antenna does exactly this with radio energy. So for a lossless antenna with gain versus an isotropic radiator fed the same power, the total radiated power is the same; the directional antenna simply aims it.

ERP and EIRP: how strong your signal really is in the favored direction

When you account for everything, the power coming out of the radio, the losses in the feed line and other gear, and the gain of the antenna, you get a single honest number for how strong your signal is in the direction the antenna favors. The term for "total radiated power taking into account all gains and losses" is effective radiated power (ERP). When the gain is referenced to a dipole (dBd), the result is ERP. When the gain is referenced to isotropic (dBi), the result is called effective isotropic radiated power (EIRP). The math is identical; only the reference for the antenna gain differs.

How to work an ERP problem. Add up all the gains and subtract all the losses to get a single net figure in dB, then convert that to a power multiplier. The multiplier is 10 raised to the power of (net dB ÷ 10). Multiply by your transmitter output watts and you are done.

Worked example (E9A02). A repeater has 150 watts output, 2 dB feed line loss, 2.2 dB duplexer loss, and 7 dBd antenna gain. Net dB = 7 − 2 − 2.2 = 2.8 dB. Multiplier = 10 raised to (2.8 ÷ 10) = 10 raised to 0.28, which is about 1.91. ERP = 150 × 1.91 = about 286 watts.

Worked example (E9A06). 200 watts output, 4 dB feed line loss, 3.2 dB duplexer loss, 0.8 dB circulator loss, 10 dBd gain. Net dB = 10 − 4 − 3.2 − 0.8 = 2 dB. Multiplier = 10 raised to 0.2 = about 1.585. ERP = 200 × 1.585 = about 317 watts.

Worked example (E9A07, EIRP). 200 watts, 2 dB feed line loss, 2.8 dB duplexer loss, 1.2 dB circulator loss, 7 dBi gain. Notice the gain is in dBi, so the answer is EIRP. Net dB = 7 − 2 − 2.8 − 1.2 = 1 dB. Multiplier = 10 raised to 0.1 = about 1.259. EIRP = 200 × 1.259 = about 252 watts.

Radiation resistance and antenna efficiency

An antenna turns electrical power into radio waves, but it also wastes a little as heat in the wire and the ground. We separate these with two ideas. Radiation resistance is the imaginary resistance that "represents" the power actually radiated as useful signal, the power that escapes. Loss resistance represents the power wasted as heat. The total of the two is the total resistance.

Antenna efficiency is then defined as radiation resistance divided by total resistance. If most of the resistance is radiation resistance, the antenna is efficient; if a lot is loss resistance, it is wasteful. This matters most for short antennas and ground-mounted verticals, where ground losses can be large.

The ground: losses and "ground gain"

• For a ground-mounted quarter-wave vertical, the return currents flow through the soil, and poor soil wastes power. Installing a ground radial system (a fan of wires laid out from the base) gives those currents a low-loss path and improves efficiency.
• What actually determines those ground losses on HF? Mainly the soil conductivity, how well the dirt conducts. Rich, damp, salty soil is far better than dry sand or rock.
• Surprisingly, the ground can also help. Ground gain means an increase in signal strength from ground reflections in the environment of the antenna. The wave that bounces off the ground can add to the direct wave at low angles, boosting your signal toward the horizon.

A couple of leftover facts

• The feed point impedance of an antenna is affected by its height above ground (among other things). It is not changed by your transmission line length in any fundamental sense, nor by an antenna tuner back at the rig, nor by your power level.
• The Fresnel zone is the football-shaped region around the line-of-sight path that needs to be clear for a good microwave link. The higher the frequency, the smaller this zone, so of a list of bands, the highest frequency (5.8 GHz) has the smallest first Fresnel zone.

A radiation pattern is a polar (circular) plot showing how strongly an antenna radiates in each direction. Reading these plots is a guaranteed exam skill, and three questions point you to Figure E9-1 and three to Figure E9-2. Let us learn the vocabulary, then read the figures.

Azimuth versus elevation patterns

There are two slices you can plot. An azimuth pattern is the bird's-eye, top-down view: it shows direction around the compass (north, east, south, west). An elevation pattern is the side view: it shows angle above the horizon, which tells you how high or low the antenna shoots. In Figure E9-2, the pattern shown is an elevation pattern (it shows the takeoff angle), and its peak response is at an elevation angle of 7.5 degrees, a nice low angle good for working distant stations.

Beamwidth

The 3 dB beamwidth (also called the half-power beamwidth) is the width of the main lobe measured between the two points where the signal has fallen to half its peak power, which is 3 dB down. You find the peak, drop down 3 dB on each side, and read the angle between those two points. For the pattern in Figure E9-1, that angle is 50 degrees.

Front-to-back and front-to-side ratios

These tell you how well an antenna rejects signals it is not pointed at, which is exactly what you want for cutting interference.

• The front-to-back ratio compares the strength of the main lobe (the front) to the lobe pointing in the exact opposite direction (the back), 180 degrees away. In Figure E9-1 it is 18 dB; in Figure E9-2 it is 28 dB.
• The front-to-side ratio compares the main lobe to the lobe pointing 90 degrees to the side. In Figure E9-1 it is 14 dB.

How to read these off the chart: the rings on the plot are marked in dB. Read the dB value where the main lobe touches the outer edge, then read the dB value where the back (or side) lobe reaches, and take the difference. The gridlines do the subtraction for you.

The far field

Patterns are only meaningful far enough away from the antenna that the wave has "settled down." The far field is defined as the region where the shape of the radiation pattern no longer varies with distance. Up close (the near field), the pattern is messy and changes shape; far away, it locks into the fixed shape you plot.

Modeling antennas on a computer

You can predict an antenna's pattern and impedance with software before you ever cut wire. The standard math technique is the Method of Moments. Its principle: a wire is modeled as a series of segments, each having a uniform value of current. The computer solves for the currents in all the little segments at once, then computes the resulting pattern.

One practical caution: you must use enough segments. The disadvantage of using fewer than about 10 segments per half-wavelength is that the computed feed point impedance may be incorrect. Too coarse a model gives you wrong numbers.

This group is a tour of real antennas you can string up in a backyard, plus how phasing two verticals steers the pattern, and how the ground beneath everything changes the result.

Phased vertical pairs

Take two quarter-wave verticals and feed them with controlled timing (phase), and you can sculpt the combined pattern. Memorize these three classic combinations, they appear word for word:

• 1/2-wavelength apart, fed 180 degrees out of phase: a figure-eight oriented along the axis of the array (the strong directions point along the line connecting the two antennas, called end-fire).
• 1/2-wavelength apart, fed in phase: a figure-eight broadside to the axis of the array (the strong directions point out the sides, perpendicular to the line connecting them).
• 1/4-wavelength apart, fed 90 degrees out of phase: a cardioid pattern (heart-shaped, strong in one direction with a deep null in the other).

A memory aid: "in phase, broadside; 180 out, end-fire; the 90-degree quarter-wave case gives the one-way cardioid."

Long wires and rhombics

• As you make an unterminated long-wire antenna longer, additional lobes form, with the major lobes increasingly aligned with the axis of the antenna (they swing to point more along the wire itself).
• Adding a terminating resistor to a rhombic or long-wire antenna changes the pattern from bidirectional to unidirectional. The resistor absorbs the energy that would otherwise reflect back and radiate the other way, so the antenna fires mostly one direction.

Folded dipoles and feed point impedance

A folded dipole is a half-wave dipole with an additional parallel wire connecting its two ends, basically a skinny wire loop the shape of a stretched-out dipole. The extra conductor raises the feed point impedance: the center feed point of a two-wire folded dipole is about 300 ohms (compared to roughly 72 ohms for a plain dipole). That 300-ohm value is a natural match for 300-ohm twin-lead.

Named wire antennas you must recognize

• Off-center-fed dipole (OCFD): feeding a dipole between the center and one end instead of at the middle is done to create a similar feed point impedance on multiple bands, which lets one antenna work several bands.
• G5RV: a wire antenna center-fed through a specific length of open-wire line connected to a balun and coaxial feed line. A very popular multi-band wire.
• Zepp: an end-fed half-wavelength dipole (it was first trailed behind Zeppelin airships, hence the name).
• Extended double Zepp: a center-fed 1.25-wavelength dipole, longer than a normal dipole for extra gain broadside.

How ground and height bend the pattern

• A vertically polarized antenna over seawater versus ordinary soil: radiation at low angles increases. Salt water is a superb reflector, so it greatly enhances the low-angle signal prized for long-distance contacts.
• For a horizontally polarized antenna, as you raise it higher above ground, the takeoff angle of the lowest elevation lobe decreases, the antenna shoots lower, again favoring distant stations.
• That same horizontal antenna over a long downhill slope: the main lobe takeoff angle decreases in the downhill direction, as if the hillside tilts the beam lower that way.

Now we get into the high-performance antennas, the Yagi beam and the parabolic dish, plus the tricks for making a too-short antenna work and the meaning of antenna Q.

The Yagi and its elements

A Yagi has one driven element (the one connected to the feed line) plus parasitic elements that are not connected to anything, they pick up energy from the driven element and re-radiate it with a delay, reinforcing the signal one way and canceling it the other.

• The driven element is approximately 1/2 wavelength long, like a dipole.
• The parasitic elements are deliberately made longer or shorter than resonance to provide control of phase shift, which is how the beam is aimed. A reflector is slightly longer and goes behind; a director is slightly shorter and goes in front.
• Most two-element Yagis with normal spacing use a reflector rather than a director because it gives higher gain at that spacing.

Circular polarization from two Yagis

To get circular polarization (handy for satellites), you arrange two Yagis on the same axis and perpendicular to each other, with the driven elements at the same point on the boom, fed 90 degrees out of phase. One Yagi horizontal, one vertical, fed a quarter-cycle apart, makes the combined field rotate.

The parabolic dish

A parabolic reflector (dish) focuses energy like a satellite TV dish. A key fact: when you double the operating frequency, the gain of an ideal dish increases by 6 dB. (Higher frequency means the dish is electrically larger, and gain rises fast.)

Loading an electrically short antenna

An antenna shorter than resonance looks capacitive (it has leftover capacitive reactance). A loading coil fixes this. Its function: to resonate the antenna by cancelling the capacitive reactance, the coil's inductive reactance offsets the antenna's capacitive reactance, bringing it to resonance.

• Best location for the coil: near the center of the vertical radiator. Placing it there (rather than at the base) keeps more of the antenna carrying current, improving radiation.
• The coil should have a high ratio of reactance to resistance in order to maximize efficiency, you want it to provide reactance without wasting power as heat.
• Top loading (adding a capacitance hat at the top) gives improved radiation efficiency, because it raises the current over more of the antenna.
• As you load a short antenna with coils, the SWR bandwidth decreases (the usable range gets narrower).

Antenna Q and radiation resistance

• Q measures how "sharp" or narrowband the antenna is. As Q increases, SWR bandwidth decreases. High-Q antennas are touchy and work over only a narrow range.
• The radiation resistance of a base-fed whip decreases as you go below its resonant frequency, which is exactly why short verticals are inefficient and need loading.

Antennas rarely present exactly 50 ohms, the impedance our coax and radios expect. A matching system transforms the antenna's impedance to match the feed line so power flows efficiently and SWR stays low. This group is a catalog of the standard methods.

The named matching systems

• Gamma match: matches coax to an antenna by connecting the shield to the center of the antenna and the center conductor a fraction of a wavelength to one side. It includes a series capacitor whose purpose is to cancel unwanted inductive reactance. The gamma match is also one way to shunt feed a grounded tower at its base.
• Beta or hairpin match: this one requires the driven element to be insulated from the boom, and it needs the driven element feed point to be capacitive (the driven element electrically shorter than 1/2 wavelength) so the hairpin's inductance can tune it out.
• Stub match: uses a short length of transmission line connected in parallel with the feed line at or near the feed point. The stub adds the reactance needed to cancel the mismatch.

The quarter-wave Q-section (impedance transformer)

A quarter-wavelength piece of transmission line is a magic impedance transformer. The line impedance you need is the geometric mean (square root of the product) of the two impedances you are matching: Z = square root of (Z1 × Z2).

Worked example (E9E06). To match a 100-ohm feed point to a 50-ohm line, you need a Q-section of square root of (100 × 50) = square root of 5000 = about 70.7 ohms. Of the available choices, 75 ohms is the suitable standard cable.

Describing the mismatch and dividing power

• The parameter that describes the interaction of a load and a transmission line (how much signal reflects back) is the reflection coefficient.
• A Wilkinson divider is used to divide power equally between two 50-ohm loads while maintaining a 50-ohm input impedance, handy for feeding two antennas from one line while keeping everything matched.
• Using multiple driven elements connected through phasing lines lets you control the antenna's radiation pattern, the same phasing idea from E9C, now built as a fed array.

A feed line is a circuit element in its own right. This group covers how fast signals travel in it, how long it "looks" electrically, and the surprising impedances that shorted and open lines present. Expect several questions here.

Velocity factor and electrical length

Velocity factor is the velocity of a wave in the transmission line divided by the velocity of light in a vacuum. Because the insulation slows the wave, velocity factor is always less than 1 (typical coax is around 0.66 to 0.8). What has the biggest effect on it? The insulating dielectric material, the plastic or foam between the conductors.

Because the wave moves more slowly in a coaxial cable than in air, the electrical length of a coax is longer than its physical length. In other words, a wave takes the same time to cross a short slow cable as it would to cross a longer stretch of free space, so electrically the cable "counts" as longer. To find physical length for a given electrical length, multiply the free-space length by the velocity factor.

Worked example (E9F06). Find the physical length of an air-insulated parallel line that is electrically 1/2 wavelength at 14.10 MHz. A full wavelength in free space is 300 ÷ frequency in MHz = 300 ÷ 14.10 = about 21.3 meters, so a half wavelength is about 10.6 meters. Air-insulated line has a velocity factor near 1.0, so the physical length is essentially 10.6 meters.

Coax versus open-wire, and dielectrics

• Parallel-conductor (open-wire) line compared to plastic-dielectric coax has lower loss, mostly air for insulation means very little wasted power.
• Foam-dielectric coax compared to solid-dielectric coax of the same type: all of these are true, it has a higher velocity factor, lower loss per unit length, but a lower safe maximum operating voltage.
• Microstrip is precision printed circuit conductors above a ground plane that provide constant-impedance interconnects at microwave frequencies, essentially a flat transmission line etched onto a circuit board.

The stub tricks: what shorted and open lines look like

Here is a set of facts the exam loves. A short piece of line, depending on length and whether the far end is shorted or open, can imitate a capacitor, an inductor, a very high impedance, or a very low impedance. Memorize this table:

Line length	Far end SHORTED	Far end OPEN
1/8 wavelength	Inductive reactance	Capacitive reactance
1/4 wavelength	Very high impedance	Very low impedance
1/2 wavelength	Very low impedance (repeats the load)	Very high impedance (repeats the load)

Two patterns make this easy. First, a quarter-wave line inverts whatever is on its end: a short becomes very high impedance, an open becomes very low impedance. Second, a half-wave line repeats whatever is on its end: a short stays low, an open stays high. And for the eighth-wave cases, just remember shorted-and-short equals inductive, open-and-short equals capacitive. So: shorted 1/2-wave gives very low impedance; shorted 1/4-wave gives very high impedance; shorted 1/8-wave gives inductive reactance; open 1/8-wave gives capacitive reactance; open 1/4-wave gives very low impedance.

The Smith chart is a clever circular graph that turns ugly impedance calculations into geometry you can do with a pencil and compass. You will not have to solve one on the exam, but you must know what it is, what its parts are called, and what it is used for. Several questions reference Figure E9-3.

What the Smith chart is for

The Smith chart lets you find impedance along transmission lines and is commonly used to determine impedance and SWR values in transmission lines. A very common practical use is to determine the length and position of an impedance matching stub. So whenever a question offers a transmission-line impedance or matching task, the Smith chart fits; it does not compute antenna gain, radiation patterns, or propagation.

The coordinate system: resistance circles and reactance arcs

The chart's grid is built from two families: resistance circles and reactance arcs. (Stated another way, the two families represent resistance and reactance.) Every point on the chart is one complex impedance, a resistance value combined with a reactance value. The arcs represent points of constant reactance, and the circles represent points of constant resistance.

Reading Figure E9-3

• The large outer circle on which the reactance arcs terminate is the reactance axis.
• The only straight line on the chart is the resistance axis (it runs straight across the middle).

Normalizing and extra features

• You normalize a Smith chart by reassigning the prime center's impedance value, that is, you decide what impedance the exact center represents (usually 50 ohms), and all other values scale relative to it. This lets one chart serve any system impedance.
• When designing matching networks, people often add a third family of circles: constant-SWR circles. As you move along a transmission line, your impedance point travels around one of these constant-SWR circles.
• The wavelength scales around the rim are calibrated in fractions of transmission line electrical wavelength, so you can step along the line by reading off how many wavelengths you have moved.

On the low bands, the challenge is not radiating power, it is hearing weak signals through heavy noise. Special receiving antennas trade away efficiency for a clean directional pattern. This group covers the Beverage, the direction-finding loop, and the math term RDF.

Why receiving antennas can be "inefficient"

On 160 and 80 meters, atmospheric noise is so high that directivity is much more important than losses. A receiving antenna that throws away signal but rejects noise from unwanted directions can hear better than a big efficient antenna that picks up noise from everywhere. That single idea explains the whole group.

The Beverage antenna

A Beverage is a very long wire strung low to the ground, pointed at the station you want to hear.

• For good performance it should be at least one wavelength long, the longer the better.
• Its termination resistor (a resistor to ground at the far end) functions to absorb signals from the reverse direction, turning the bidirectional wire into a one-direction antenna.
• You know the resistor value is correct when you see minimum variation in SWR over the desired frequency range, the sign that the wire is properly terminated and not reflecting.

Direction-finding loops and the sense antenna

Small loop antennas are prized for direction finding (RDF, radio direction finding) because of their sharp nulls, directions of near-zero response. A null is easier to pinpoint than a broad peak.

• The challenge with a small wire loop is that it has a bidirectional null pattern, two nulls 180 degrees apart, so by itself it cannot tell you which of two opposite directions the signal came from.
• A sense antenna solves this: it modifies the pattern of a DF antenna to provide a null in only one direction (turning the figure-eight into a cardioid), resolving the ambiguity.
• An electrostatic shield around the loop eliminates unbalanced capacitive coupling to the antenna's surroundings, improving the depth of its nulls, making the direction sharper.
• A single-turn terminated loop such as a pennant antenna produces a cardioid pattern, and a cardioid is useful for direction finding because it has a single null.
• To boost the tiny output of a multiple-turn receiving loop, increase the number of turns and/or the area enclosed by the loop.

Receiving directivity factor (RDF)

One more meaning of the letters RDF: receiving directivity factor is the peak antenna gain compared to the average gain over the hemisphere around and above the antenna. It is a way to score how well a receiving antenna concentrates on one direction while rejecting noise from everywhere else, the higher the RDF, the quieter and more directional the antenna.