# Omega and Gama Matching

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Omega’s and Gamma’s

### Impedance Limits

The best matching system for feeding a grounded Marconi element,
unless we are very lucky or very careful in planning the installation, is a
simple gamma capacitor and shunt feed conductor with the tap point adjusted to
find 50 ohms resistive.

While losses are not significantly increased when
using an Omega match with reasonably sized components, matching range is not
significantly extended either! At best, the Omega match saves us a few trips up
and down the tower while searching for the precise shorting strap position
between the gamma wire or cage and the tower. If we are already at the top of a
tower with the gamma tap point, and if at that point we find resistance (after
reactance is cancelled by adjustment of the gamma) is too high, an Omega will
not help. The Omega can only match loads LESS than 50 ohms
resistive by stepping the resistance up! It can NOT step or transform antenna
feed resistance downwards when using capacitors, and neither the Omega or Gamma

By adding an inductor, we can greatly extend the matching range.
The extra time, expense, and complexity of adding a high-Q inductor would not
offset the slight reduction in effort to simply find the correct shorting point
for gamma and omega capacitor systems.

#### Gamma Match

The gamma match capacitor can only cancel reactance, it can not
modify the “real part” (resistance) presented to the feed line. It is
the most simple form of matching, and has the lowest operating Q and loss of any
system (when it is useable). Adjustment of resistance (real part) requires
adjusting the diameter, spacing, or length of the gamma section.

The voltage rating of the capacitor is simple to find. It
is  Y*I*X where Y is the safety factor and adjustment for peak voltage
(RMS*1.414), I is the RMS current from the feed line, and X is the reactance (in
ohms) of the capacitor. Y is normally a factor of four to allow for peak
conversion, SWR, component flaws, and transients.

The current rating is simply the RMS current plus a safety
factor. Component physical size as well as Q control the current rating. See
component ratings.

Looking at a Smith Chart, we see a single line (yellow)
following a portion of the resistance circle with a resistance value of 1 (1 is
normalized to 50 ohms). The gamma match can not
deviate from that resistance circle, and so the resistance must be adjusted or
transformed  by other
means. We normally adjust the gamma system’s tap point by using a movable
shorting strap to change R. Just moving a short is enough, it is not necessary to remove the extra
“unused” portion above or beyond the shorting strap.

Let’s consider a 1500-watt transmitter and 50-ohm
transmission line, with a 1:1 VSWR. The matched line current is found using
.  The result is 5.5a RMS current.

With 1500 watts and 50 ohms we have 5.5 amperes RMS. Assuming
the match requires 300pF at 1.8MHz we have:

4*295* 5.5 = 6490 volts. It’s easy to see why arcing is a
problem when capacitance is low! Very few air variables would take such a high
voltage, and insulators (especially when exposed to moisture) also become a
problem. This is typical of a shunt-fed element when the
matching section is a thin long wire.

We can decrease reactance by increasing the
effective diameter of gamma conductors, and this will increase required
capacitance value. One way to decrease reactance is to use a cage of wires
spaced as far apart as possible. Halving the reactance (doubling the capacitance value) will
cut peak voltage in half, with no other changes. This has the effect of quadrupling
the power rating when using the same breakdown voltage ratings in components!

A thicker gamma conductor also lowers operating (loaded) Q. This
is a series-resonant system, operating Q is set by the amount of reactance in
series with the load resistance (50 ohms). Reducing operating Q  has the effect of increasing bandwidth
while simultaneously  reducing loss and increasing power rating.

Note that bandwidth
increased while efficiency increased.
This happens in many cases.
Popular folklore tells us narrow antennas are efficient, but that is true only
in a few specific cases. In most cases, bandwidth by itself tells us nothing about
system efficiency!

One example of a conundrum with gamma capacitors is found in yagi
antennas. If we increase capacitor dielectric thickness by decreasing inner rod size, the voltage across the capacitor can actually increase at a
faster rate than insulation thickness increases! The same is true if we increase
gamma capacitor length in an effort to restore capacitance after increasing
outer conductor diameter. The effect thicker insulation and higher voltage
rating, if done incorrectly, can be to make little change or perhaps even decrease
gamma capacitor power rating!! Always use a large diameter gamma
rod, and increase the dielectric thickness ONLY by increasing the outer rod to
accommodate dielectric thickness changes. At the same time, the inner rod should
be increased to avoid any requirement of making the gamma capacitor longer. Avoid sharp points or edges on the rods, just as you would avoid sharp
points in any HV system….especially inside the “capacitor”.

The above example of decreased power rating is especially
important to Amateurs using coaxial cables as capacitors. Voltage is NOT
constant along the length of a long coaxial capacitor. Maximum voltage in the
component is always HIGHER than the actual voltage
across the terminals of the “capacitor”, and it is higher than the
voltage calculated by the current through the
capacitor! Coaxial capacitors or linear stubs used as reactive elements always
have significantly lower operating Q, higher power loss, and operate under more
electrical stress than a well-designed lumped component. Stubs and linear
as much, even though there is a lot more heat energy! Just don’t let the smaller
temperature rise fool you into thinking the system has less power loss.

### Omega Match

The Omega Match is really just a form of the simple L-network.

In this system, C2 parallels the shunt terminal and primarily
sets the resistance transformation C1 is in series with the feed line, and
primarily cancels reactance. If we look at this network on a Smith Chart, we see
the resistance range is somewhat expanded over a conventional Gamma Match.

We also can see it is not the panacea often claimed. Look at the
“r” circles on the above chart. “r=1” is normalized to 50
ohms. With that in mind “r=2” is 100 ohms, while “r=0.5”
becomes 25 ohms. The Omega’s
resistive matching range, using 1000pF vacuum capacitors, would only from 50
ohms downwards. Voltage ratings are still a problem when the
system requires capacitor C1 to have a small capacitance value (high reactance), and now we have two
capacitors that have to handle essentially the same high voltages when C1’s
operating capacitance value is small!

This arrangement once again requires antenna shunt wire impedance
to be lower than feed line impedance. The network, since it only
uses capacitors, must be used with inductive loads or
the feed line terminal will be reactive.

One useful application of this circuit is in matching very
short non-resonant vertical element, sometimes called a “Hairpin Monopole”.
Well-designed capacitors, such as vacuum
capacitors, have such high Q they are essentially lossless. With
vacuum capacitors and very good connections, losses in the matching system are
very low even with very high currents. We shouldn’t be misled into thinking we
have a “free-lunch” magical short antenna! We “pay for our meal” in greatly
increased conductor losses in the antenna conductor. The conductors not only
carry common-mode radiation currents, they also must carry significant
circulating currents involved with the feed-system. These high currents causes
using a reasonable-Q lumped inductor. Bandwidth and radiation resistance for a
given element size is often significantly less than conventional series-feed
when using a hairpin monopole with Omega loading and matching, although the
system has the advantage of being easily tuned to new frequencies. The
Omega/hairpin vertical also allows the feedpoint resistance to be adjusted to
nearly any value we might require.

C1 is primarily for power factor (reactance) canceling. It is rated exactly as in the gamma,
voltage and current being determined by the required amount of operating reactance and the line current.

C2 primarily sets resistance transformation. The component’s
electrical requirements are found only by knowing the load impedance
and the reactive voltage and current in that branch of the system. It handles a portion of the
input current, as well as a portion of the shunt current, so the current in C2
is always higher than the current from the source flowing through C1. The
required voltage rating of C2 is also always higher than C1.

Assuming we have 10-1000pF variables, the normal matching range
covers inductive reactance ranging from 80 to 4000 ohms with a resistance
between 1 and 50 ohms. Lower reactances are only able to be tuned when the
resistance is near 50 ohms. As you can see, resistive matching range is wide but
limited to values below 50 ohms.

#### Cp/Cs Omega Match

The parallel-C input, series-C output is also a form of the
L-network. The output, once again, must be the lower impedance while the
feed line must have the higher impedance. Once again, this system only matches inductive

It is not a very
useful circuit, except in specific applications. The only advantage of this arrangement
is C2 operates at significantly less voltage. The voltage across C2 is always no greater than the transmission
line voltage, larger receiving-type air variables will operate at 1500-watt and
higher power levels.  Because C2 is under significantly less
electrical stress, It is the better choice if you
are very close to having a match and only need adjust resistance upwards
slightly for perfect match.

C1 primarily sets reactance, while C2 primarily sets
resistance.

Assuming we have 10-1000pF variables, the normal matching range
is reactances up 8000 ohms inductive and resistances between 38 and 50 ohms. As
you can see, resistive matching range is very limited. C2 would need to be a
very large capacitor (very low reactance) to have a wider resistance range.

#### Pi Matching

In cases where we can not find a 50-ohm tap point, we can add an
inductor. In this case C1 can be a fixed or variable capacitor capable of
handling modest voltage and current, such as a snubber-mica or receiving-type
air variable.

L1 has to be fairly large to match high resistance and
reactance ratios, at least 100uH on 160 meters for the extremes shown in the
Smith Chart below. Assuming a 10-1000pF capacitor for C2, a 500pF capacitor for
C1, and a 100uH adjustable inductor for L1 we would have the following matching
range:

Matching range is greatly extended, and even covers capacitive

### Power Ratings

Heating is a long term problem that accumulates with time.
Failure occurs when the heat causes a component to physically change from
excessive heat. Heating is related to power dissipated by a simple formula
The
resistance is normally determined by knowing the component’s Q (not the
operating Q) and the current through the component.

The series resistance that dissipates energy as heat is found by
dividing reactance by component Q. With 88 ohms and a Q
of  5000 (typical of a good large air capacitor) Equivalent Series Resistance is
88/5000 or .0176 ohms. Heat is given by .0176 squared times 3.11 amperes. Heat
is .0176*3.11*3.11 or .170 watts, not bad for an air variable capacitor. (Multi-layer chip capacitors can have Q’s in the tens or hundreds of
thousands, as can vacuum capacitors. Simple ceramic disc and small mica
capacitors generally have Q’s in the upper hundreds to many thousands.)

Current through C2 is found by dividing 274 volts by R-ant. If R-ant were
15
ohms the current would be 274/15 or 18.27 amperes. Voltage would be C2’s reactance
of 88 ohms times 18.26 amperes times the safety and peak factor of 4. Peak voltage would be
6500 volts for an 88-ohm reactance capacitor. This is a good example of why
“T” antenna tuners arc with low load impedances while working fine at
higher impedances, and handle more power when tuned to use maximum possible
capacitance.

Arcing is an instantaneous voltage problem. In
general a solid dielectric (insulation) punches through and fails almost
instantly. Voltage rating is reduced with every arc, even brief unnoticeable
arcs. As a general rule solid dielectrics suffer non-recoverable damage from any
arc, even a very brief arc (such as a momentary low-current static discharge.

If the dielectric is a vacuum, liquid, or gas, and if the
component’s conductors or dielectric do
not physically distort or change from heating, the insulation can “heal”
and full performance is often restored. With an air capacitor, ionized air must circulate out of the
capacitor.

If an arc causes a physical change, such as a hole, carbon path,
sharpened or raised edge, breakdown voltage is almost always permanently
reduced. One exception would be when an arc removes debris or sharp points, such
as melting copper whiskers in a vacuum capacitor (copper can actually grow tiny
whiskers in a high vacuum). Sharp points, even microscopic sharp points, greatly
reduce breakdown voltage. If a sharp point is reduced and rounded by an arc,
voltage breakdown will increase. If we actually melt the plate, voltage rating
is reduced. Surfaces must be micro-polished, we cannot sand a capacitor plate
to restore full breakdown levels if a capacitor is arc-damaged (variable
capacitor plates are commonly polished by tumbling them in a very soft abrasive like walnut
shells).

Capacitors in parallel with a known resistance. RMS sinewave voltage across
the component
is given by

where P= maximum possible level of PEP applied.  R=resistance the capacitor
parallels.

Capacitors arc and fail from instantaneous voltage
peaks, not
from average or RMS voltage. We multiply the RMS voltage across C1 times 1.414
factor. Multiplying RMS voltage by four normally works for outdoor mounted
air-capacitors, as long as they are kept debris and moisture free.

A 50-ohm feed line with 1500 watts of absolute peak power (1500
watts CW or FM carrier or 1500 watts PEP SSB or AM) applied results in 274 volts RMS times
Use a 1.1kV or higher voltage variable. This would be about .030″ air gap
in a typical construction air capacitor with normal manufacturing irregularities
(generally very safe to assume .01″ spacing for every 300 volts). With 50-ohm matched lines and a
1500-watt PEP (or CW carrier) transmitter, smaller transmitting or larger
receiving air variables
will work.

#### Heating

Current through resistance causes heating. The resulting heat
over some period of time can cause physical changes that physically alter the
component. In the short term, this can change the electrical parameters such as
capacitance or voltage breakdown.

To find electrical energy converted to heat, we have to know the
equivalent dissipative parallel resistance and voltage across the component or
the equivalent series resistance (ESR) and current through the component. I
generally work with current and ESR.

The current is a function of reactance and voltage across the
capacitor, just like current through a resistance is found from voltage across
the value of resistance. Assuming we have 1500 watts and 50 ohms resistive load
directly across the capacitor, we have 274 RMS volts across the reactance of C1.
This voltage, over the reactance in ohms, gives us the RMS (heating) current
flowing through the capacitor.

Use the formula I = E/X.  With a 1000pF
capacitor (88 ohms on 160m) we have 274/88 or about 3.11 amperes.

Note: This is not exact because capacitors have complex
series-impedances, but it is close enough for our questions as long as we have
reasonably good capacitors. One common exception where this simplification will
not work is with long coaxial stubs. Long coaxial stubs have a considerable
amount of distributed series inductance. This causes a voltage increase as we
move away from the feedpoint in an open stub, or a current increase as we move
out on a shorted stub.  This increase in voltage or current increases loss,
decreases bandwidth, and increases effective loss resistance. Not only that, a
stub (or linear loading) has higher distributed resistance because it has long
conductors, rather than thick compact conductor area like a conventional
component.

#### Loss and Component Failure

We often gauge system “loss” by the temperature of the
component. We often assume a “cool” feeling component has low loss and
a hot feeling component has very high loss. Using “touch or feel” temperature
without considering
size and ability to transfer heat is a good way to estimate component life, but
it is not a good way to estimate system efficiency or component losses. Unless we carefully consider
the complex issue of physical size and ability to transfer or dissipate heat to
surrounding objects or air, we will have no idea how much power is actually lost
in the component. Small components not only get hotter, they get hotter much
faster than large components!

A physically long coaxial capacitor has a large surface area to
dissipate heat. Even a very lossy coaxial capacitor will have significantly less
temperature rise than a small very low loss compact disk capacitor, because the
capacitor concentrates all the heat into one very tiny area. We might form the
impression we have an inefficient system because the small capacitor drifts in
capacitance value or fails from heat, yet it is usually the cool-running stub that has
much higher power loss!

This doesn’t mean we shouldn’t use a coaxial stub as a reactance
if they are reliable, but it does mean we have to be careful assuming it is the
best solution for loss.

#### Bandwidth

Bandwidth tells us nothing about efficiency, unless we
understand the entire system in detail.

In general the largest bandwidth occurs with the fewest reactive
components. Any unnecessary reactance will increase system operating Q. One
exception is when multiple networks with low loaded (operating) Q are used to
obtain a large phase shift, rather than a few components that would require high