- [Introduction]
- [MFJ-945E portable]
- [QRPprojects ZM-4 Z-match]
- [Alinco EDX-2 automatic]
- [MFJ-974HB balanced-line]
- [SWR]
- [References]

Last page update: 3 June 2018

©1999-2018 F. Dörenberg, unless stated otherwise. All rights reserved worldwide. No part of this publication may be used without permission from the author.

INTRODUCTION

What antenna tuners do & don't, can and can't (ref. 1, 2):

- Antenna tuners do not tune the antenna, in the sense of shifting the resonance frequency of an antenna to the operating frequency of the transmitter.
- Antenna tuners cannot change the feedpoint impedance of an antenna.
- Antenna tuners cannot change the impedance along the feedline between the tuner and the antenna.
- Hence, an antenna tuner cannot change the SWR along that feedline between the tuner and the antenna.
- An antenna tuner can, however, change the load impedance that appears at its connector to the transmitter. Hence, it changes the matching at the transmitter.
- Conversely, an antenna tuner can and does change the source impedance that appears at its connector to the feedline to the antenna. Hence, it changes the matching at the feedpoint of the antenna.

The function of a
"tuner" is to provide impedance matching for the purpose of maximizing power transfer between the
transmitter/transceiver and the antenna __system__ ( = antenna + feedline).

A tuner/coupler is only a
"local impedance transformer". It provides (or
improves) impedance matching only at the point where it is inserted into the
feedline. All other impedance mismatches elsewhere along the feedline (and
associated losses) to the antenna
are __not__ corrected!

More precisely: maximum power transfer requires __conjugate__ impedance
matching between the source and the load:

Simply put: cancel out the non-resistive (i.e., inductive, capacitive) part of its load impedance, and transform the resulting purely resistive impedance to 50 ohm ( = "matching"). So, it would be more appropriate to call a tuner/coupler an "impedance matching unit".

In principle, a "tuner" can be inserted anywhere along the feedline between the transmitter and the antenna.

If it is placed directly **at the antenna**, it is often referred to as an
"antenna coupler". Here, the "tuner" matches the impedance of the antenna to the characteristic impedance of the coax-feedline between the tuner and the transmitter. This minimizes feed-line losses. Obviously, if the tuner is located at the antenna, it must have remote control. Typically such remote tuners/couplers are automatic: the transmitter outputs a low-power carrier, and the tuner does its thing.

When the tuner is located closer to, or **at the transmitter**, it may
very well be able to provide perfect impedance matching between the transmitter,
and the "feed-line + antenna" system. The feed-line may be balanced (ladder
line, twin-lead), or unbalanced (coax). Whereas the transmitter may be very
happy in this configuration and see SWR 1:1, feed-line losses may actually be
(very) high: a tuner that is not inserted at the antenne feedpoint does not
correct the mismatch between the feedline and the antenna.

All elements placed between the transmitter and the antenna cause power loss: feedline, tuner, balun,... Power loss in a lossy item means dissipation, i.e., heat generation. Note that power loss in tuners is not limited imperfect components: in compact tuners, a coil that is placed close to the metal housing will cause induction heating (and also reduces the "Q" of the tuner). In general:

The larger the impedance mismatch that the tuner/coupler has to correct, the larger the tuner/coupler losses. These losses may be a very significant part of the transmitter output power!

Listed and described below, are the "tuners" that I have acquired over the years.

MFJ-945E MOBILE TUNER

Until mid-2011, I have been using a simple manual antenna tuner directly at the transceiver. I have used it with coax, and with 300/450 Ω ladder line (sometimes via 1:1 and 4:1 baluns at the antenna).

## HF + 6m antenna tuner MFJ-945E

QRPprojects ZM-4 Z-MATCH TUNER

In December of 2008, I built a small QRP antenna tuner kit (10 W max): the ZM-4 from QRPproject in Germany (ref. 2C). It has both a balanced and an unbalanced antenna input, and an LED indicator instead of an SWR-meter. It is a very compact Z-Match. Ref. 2D.

## Front and back of my ZM-4

If you want to print the labels that I designed: a printable file is provided at the end of ref. 2C.

## The inside of the ZM-4

ALINCO EDX-2 AUTOMATIC ANTENNA COUPLER

Early 2011 I decided to have a go at an "Automatic Antenna Tuner" (ATU) antenna coupler. My rig is an Alinco DX70TH, so I started looking for the matching Alinco EDX-2 tuner. Ref. 4. The EDX-2 supposedly "tunes" just about any "element" that is over 10 feet (3 m) long, from 3.5-30 MHz (80-10m) band, and anything over 40 feet (12 m) from 1.8-30 MHz (160-10m). Note that the ability of the coupler to "load" a random piece of metal at a certain frequency does not imply whatsoever, that the "antenna" will radiate efficiently. The EDX-2 is very similar in design and construction to the Icom AH-4. The effect of the automatic tuning is fun to watch - on the S-meter of my old MFJ-tuner (yes, in bypass mode).

## The EDX-2 with the installation hardware that is included

## The inside of the EDX-2

The 4-conductor control cable between the EDX-2 and the transceiver has the perfect length for capturing interference from the antenna while transmitting. That would not be good! So I looped the control cable several times through two FT140-43 ferrite rings: one at the EDX-2, one at the transceiver.

I have also installed a current choke at the EDX-2 end of the coax - *just in case*.
I used a large (!) clamp-on ferrite of material type 31- supposedly significantly better below 5 MHz than the #43 material. The core
of this "split round cable assembly" (or "Round Cable Snap-It") is about
2.2" (≈5.6 cm) tall and across.

## EDX-2 installed on my terrace - ferrite ring on the control cable, large ferrite clamp on the coax

## FT140-43 ferrite ring on the control cable, at the transceiver

To make antenna experiments easier, and to be able to quickly disconnect in case of a thunderstorm, I have wired two banana jacks to the terminals of the coupler/tuner:

## Banana jacks for quicly disconnecting an antenna or swapping antennas

### Note: the ground/earth terminal is at the bottom of the box, the antenna-wire terminal at the top....

These days, I connect dipole antennas to this tuner/coupler via a section of 300 ohm twin-lead. To make it easy to connect/disconnect antennas durng experiments, I use a regular 220 Volt household plug instead of banana plugs. Do not use this method near a 110/220 Volt power outlet!

The tuning process is quite simple, see the timing diagrams below:

- T
_{0}: the operator activates the "tune" function on the DX-70 transceiver. The transceiver asserts the active-low ( = pull-down) START signal to the EDX-2. - In the EDX-2, the START input is pulled up to +5 VDC. Pull-down in the DX-70 is open-collector.
- T
_{1}= T_{0}+ 15 msec: the EDX-2 responds by asserting the active-low KEY signal to the EDX-2. - In the DX-70, the KEY input is pulled up to +5 VDC. Pull-down in the EDX-2 is open-collector.
- T
_{2}= T_{1}+ 2 msec: in response to the active KEY signal, the DX-70 sends a carrier (3-10 watt). - T
_{3}= T_{2}+ T_{tune}: the EDX-2 tunes (typ. 3-8 sec). upon finishing the tuning process, the EDX-2 relinquishes the KEY signal to the transmitter. - T
_{4}= T_{4}+ 8 msec: the transmitter relinquishes the START signal and stops transmitting the carrier. - If the transmitter does not send (sufficient) RF, the RDX-2 relinquishes the KEY signal 330 msec after activating that signal.

## Timing diagrams of the EDX-2 control signals

In 2012, I acquired a very compact QRP HF/VHF/UHF transceiver: a Yeasu FT-817ND. I really wanted to use it with the EDX-2. However, the FT-817 does not have a "tune" button or function! Performing a manual „tune“ takes 5-11 manipulations on the FT-817, depending on the mode that the transceiver is in. Very annoying! But don't despair! I use the Ham Radio Deluxe (HRD) digi-mode software package (I still use the V5.11 freeware version) to fully control the FT-817 via a CAT-interface. HRD has a tune-function ("tools" ─► "tune up").

## xxxxx

However, HRD has no way to assert a START signal to the EDX-2 or to respond to the KEY signal from the EDX-2. So, no automatic tuning cycle. Not a big problem. There is an easy way to manually assert the START signal to the EDX-2. The EDX-2 control cable is disconnected from the DX-70 transceiver, and a push button switch is installed across the START wire and ground. An LED with pull-up resistor is connected to the KEY wire of the control cable, so we can see what the EDX-2 is doing. See the schematic below. I just activate the tune function of HRD and immediately hit the push button. That's all. Not fully automatic, but hey.... it works like a charm!

## A simple manual control interface for the EDX-2

## My EDX-2 manual tuning starter

MFJ-974HB BALANCED-LINE TUNER

In 2013, I expanded my collection of antenna tuners/couplers with an MFJ-974HB "balanced line tuner":

SWR

Some words about SWR....

Antennas are connected to a transmitter via an RF transmission line. This is typically a coax cable, "open wire" ladder-line, window line, or a twin-lead cable. Such transmission lines can be modeled as a series of infinitesimally small sections of four components:

- series inductance per unit of length, denoted
*L*_{0}. - loss resistance per unit of length,
*R*_{0}, in series with inductance*L*_{0}. - parallel capacitance per unit of length, denoted
*C*_{0}. - leakage
__conductance__per unit of length,*G*_{0}, parallel to the capacitance*C*_{0}. This is the reciprocal of the loss resistance in the dielectric between the two conductors of the transmission line.

## Unbalanced transmission line (coax) - modeled as an infinite series of distributed components

## Balanced transmission line (twin-lead, window line, ladder line) - modeled as an infinite series of distributed components

A transmission line has a so-called *characteristic impedance*, denoted *Z*_{0}.
If the leakage conductance *G* is
small enough and the
loss resistance *R* is also small enough, then *Z*_{0}
is __only__ related to the ratio of two parameters that are distributed along
the transmision line: *L*_{0} and *C*_{0}:

Note that this is independent of frequency and independent of the length of the transmission line!

Coax cable is typically dimensioned and constructed such that *Z*_{0} is 50 ohms
(or 75 ohms for TV and satellite receiver coax). For twin-lead cable and
window-line, standard *Z*_{0} is 300 or 450 ohms.
The *Z*_{0} of ladder-line also depends on the wire spacing, wire diameter,
and wire insulation material. Such line is often dimensioned
for 450, 600 ohm, or more.

What happens, if a transmitter inserts a signal into the input end of the transmission line, but the transmission line is __not__ terminated with a
load impedance *Z*_{L} that is equal to the line's characteristic impedance *Z*_{0}.
That is, *Z*_{L} ≠ *Z*_{0}. This impedance mismatch
causes an impedance *discontinuity* at the output end of the transmission
line. Part (or even all) of the input signal ("wave") is **reflected** at
that discontinuity - just like a mirror. The forward "wave" (also referred to as
"incident wave") and the reflected wave travel in opposite directions. They
combine into an interference pattern.

For simplicity and illustration purposes, let's assume that the input signal is a
constant sinewave. The forward and reflected waves have the same frequency
(wavelength). The resulting pattern is then called a **standing wave**. Unlike
the forward and reflected waves, the standing wave does not travel along the transmission line:
it is stationary ( = **standing** still).

## Example: standing wave that is formed by 100% reflection of a constant sinewave

### (blue = forward travelling wave, red = reflected travelling wave, black = resulting standing wave)

The wavelength of the standing wave is the same as that of the forward wave and
of the reflected wave. Hence, the amplitude pattern of the standing wave repeats
itself every full wavelength. As explained below, we are actually only interested in
the *absolute* amplitude of the standing wave. Its pattern repeats
itself every *half* wavelength.

The animation above is for 100% reflection of the forward wave. In this case (and only in this case), the reflected wave has exactly the same amplitude as the forward wave. The forward and reflected wave combine in a constructive and destructive manner ( = superposition). As a result, the amplitude of the standing wave is twice that of the forward wave, and the absolute minimum amplitude of this standing wave is zero:

Obviously, the reflection cannot be more than 100% if the load is passive. If it is *less* than 100%, then the reflected wave has
an amplitude that is *less* than the amplitude of the forward wave. The
resulting standing wave now has a __maximum__ voltage amplitude |*V*_{max}| that is less than twice
the amplitude of the forward wave. Likewise, this standing wave has a __minimum__ voltage amplitude |*V*_{min}|
that is non-zero, but never larger than the amplitude of the forward wave.
The opposite extreme case is 0% reflection. In this case, there is no standing
wave, and the forward power wave is completely absorbed in the load. This is
what we want!

The **standing wave ratio** (**SWR**) is the ratio of these maximum and
minimum voltage
amplitudes of the standing wave. More precisely, this is the *Voltage* SWR (**VSWR**, often pronounced
as "vizwar"). Likewise, there is a standing wave pattern of the forward and
reflected *currents*. This is the *Current* SWR, or **ISWR**. It has the same value
as the VSWR:

SWR is always in the range [1,∞] because |*V*_{min}| cannot
exceed |*V*_{max}| and cannot be less than zero (since it is an abolute value).

Keep in mind that true |*V*_{max}| and |*V*_{min}| are only found if the
transmission line is at least half a wavelength long. A line that is shorter than that,
will only have a __local__ minimum and/or maximum. This local minimum
may not be as small as the global minimum of a sufficiently long transmission
line. Likewise, the local maximum may not be as large as the global maximum of a
sufficiently long line. But based on the
definition of SWR, this does not affect
the SWR.

For a complex load impedance *Z _{load}* =

*R*+ j·

_{load}*X*, the above equation can be expanded as:

_{load}The reader may verify that for a purely resistive load equal to the
characteristic impedance of the feed line, i.e., *Z _{load}* =

*R*=

_{load}*Z*, we obtain SWR = 1.

_{0}We have seen that SWR is related to "reflection" at the load end of the
transmission line. In fact, it is directly related to the **reflection
coefficient**, usually denoted **Γ** (capital letter Gamma).

In general, *Z*_{load} is a complex impedance, so the the
parameter Γ is a complex number that has a magnitude |Γ| which is often denoted **ρ**
(rho), and a phase angle. |Γ| is always in the range of [0,+1].

When a forward wave with amplitude *V*_{forward} is
reflected at the load-end of the transmission line, then the reflected wave has
an amplitude *V*_{reflected} = Γ*·V*_{forward}. In
general, Γ is a complex number. If the
transmission line is terminated with *Z*_{load} = *Z*_{0},
then there is no impedance mis-match and no reflection, hence Γ = 0. Termination with a short circuit (i.e., *Z*_{load} =
0) results in Γ = -1. That is, 100% reflection, with opposite polarity. Termination with an open circuit (*Z*_{load} = ∞) results in Γ =
+1. That is, 100% reflection, with same polarity.

When a forward wave with amplitude *V*_{forward} is
reflected at the load-end of the transmission line, then the reflected wave has
an amplitude *V*_{reflected} = Γ*·V*_{forward}. In
general, Γ is a complex number. If the
transmission line is terminated with *Z*_{load} = *Z*_{0},
then there is no impedance mis-match and no reflection, hence Γ = 0. Termination with a short circuit (*Z*_{load} =
0) results in Γ = -1. That is, 100% reflection, with opposite polarity.
Termination with an open circuit (*Z*_{load} = ∞) results in Γ =
+1. That is, 100% reflection, with same polarity.

Note: if the *transmitter output* impedance is not equal to *Z*_{0},
then there is an impedance mismatch at the *input* side of the transmission line.
This discontinuity also causes reflections, with its own reflection coefficient value Γ_{input}.
Likewise, reflections may be caused by local deformation of the transmission
line (e.g., sharp folding over of a coax cable, change in wire spacing of a
ladder line, etc.)

The relationship between (V)SWR and Γ is as follows:

Conversely:

In other words:

Since Γ is a voltage ratio, it is also the square root of a power ratio:

So, (V)SWR can be determined by measuring the ratio of reflected power and foward power. This requires a measurement instrument that can distinguish between these two directional power flows, and can measure both simultaneously.

Another parameter that is often used is **return loss** (*RL*), which
is expressed in dB:

Note that the __better__ the impedance match at the load-end of the transmission
line, the __smaller__ the reflection and |Γ|, so the __higher__ *RL *
! Yes, this is
rather counter-intuitive. But keep in mind: *RL* represents reduction ( =
loss) of the reflection, not loss (of the forward wave) caused by reflection! *RL* is always in the range of [0,
+∞], based on the definition and |Γ|
always being in the range of [0,+1].

If we change the sign of *RL*, we obtain the so-called **scattering-parameter**
S_{11 }:

S_{11} is one of the four S-parameters (S_{11}, S_{12}. S_{21}, S_{22})
that are used in network analysis to characterize two-port networks such as
transmission lines. Ref. 5C, 5D.

For a transmission line with characteristic impedance *Z*_{0},
the SWR referenced to *Z*_{0} is only determined by two parameters:
this *Z*_{0} and the load impedance *Z*_{L}:

The ratio is inverted as necessary, such that the SWR is never less than 1, which represents perfect matching.

So, contrary to popular belief:

for a given
combination of characeristic impedance of the transmission line and the load
impedance,

SWR(*Z*_{0}) is a constant.

It can __not__ be changed by changing the length of the feedline!

So, why can *measured* SWR change when the feedline
length is changed? An SWR meter only measures something at the point were it is
inserted in the transmision line. So, obviously, it cannot measure the voltages
or currents all along the transmission line. There are two basic types of
SWR meters:

- an impedance bridge circuit that is only balanced when the measured impedance matches the reference impedance in the bridge.
- current transformers.
- directional couplers.

There are several possible reasons for this, e.g.:

- The SWR instrument mis-reads due to RF interference.
- Feedline loss
- The SWR meter is an impedance bridge, so it can only
measure the ratio of the
*local*impedance and*Z*_{0}. - IThe SWR meter senses forward and reflected current
with a
*directional coupler*, but the directivity discrimination of the coupler is not perfect. Directivity should typically be at least 15 dB, to reduce the power measurement error to less than about 1 dB. - The instrument does not measure at least not for the actual
*Z*_{0}of the transmission line. - The actual
*Z*_{0}of the transmission line is not exactly the*Z*_{0}for which the SWR meter is designed and calibrated. - Note that, e.g., "50 ohms" coax is not "50.0 + j·0", but often around 52 ohms with some non-zero reactance.
- Note that the "50 ohms" output of a solid-state transmitter or transceiver is rarerly 50 ohms. This may explain some discrepancies between SWR-readings by the internal SWR-meter of transmitters/transceivers and external meters.

Depends on whether we are only interested in keeping the transmitter happy by loading its output with the proper impedance (e.g, 50 ohms), independent of the transmission line SWR, or we also want to match the antenna to the tranmission line, in order to minimize transmission line losses.

The only point along the transmission line where conventional SWR meters .

SWR is highest closest to the load, and only "improves" as the distance from the load increases, creating the false impression of a matched system.

Note: it is very **important** to always make clear which SWR we are talking
about: SWR(*Z*_{0}) or
"measured SWR", which is also referred to as "true SWR", and SWR(True).

Conventional SWR-meters measure the ratio of
the __local__ impedance (at the antenna/output connector of the meter) , and the reference impedance that the meter is designed
for (and hopefully calibrated to), typ. 50 ohm. Obviously, it cannot do a
remote measurement of the load-impedance *Z*_{load}.
Likewise, it cannot measure the maximum and minimum voltage (or current) along
the transmission line. So, such an instrument can only
measure the *Z*_{load} / *Z*_{0} ratio, __if__ it
is placed at the antenna feedpoint! Note that a VNA does not measure a local impedance: it measures the reflection
coefficient. The magnitude of that coefficient is directly related to the SWR.

Note: whereas (by definition) SWR is a constant that is independent of the length of
the transmission line, the __impedance__ along that line is __not__ constant, __unless__ the
impedance *Z*_{load} of the load ( = antenna feedpoint) is equal to *Z*_{0}.

So, how *does* the impedance observed at the transmitter end of coax
cable depend on the electrical length of that cable and termination
impedance? Let *Z*_{input} be the impedance that appears at the input
end ( = transmitter end) of a transmission line that has a characteristic
impedance *Z*_{0}, and that is terminated with a load
impedance *Z*_{load} at the oposite end ( = antenna end). The
general expression for *Z*_{input} as a function of *Z*_{0}
and *Z*_{load} is (ref. 5A):

where *Z*_{load} is a complex impedance ( = resistance +
reactance):

and the phase angle expresses the electrical distance (in radians) from the load-end of the transmission line:

So, except in a *very* small number of special cases:

the impedance that
appears at the transmitter-end of the transmission line

varies as a function
of

the *Z*_{0} of that line, the length of that line, and the load-impedance *Z*_{load}

If you set *Z*_{load} = *Z*_{0} in the above
equation for *Z*_{input}, it is easy to see that the resulting *Z*_{input}
= *Z*_{0}:

Or, to put it differently:

for a __any__
length of transmission line,

the impedance that
appears at the transmitter-end of that line,

is only equal to the impedance at
the load-end of the line,

if the load-impedance *Z*_{load} is equal to the *Z*_{0}
of the line

What does that mean? Example: if a coax with *Z*_{0} = 50 ohm is terminated with a 50 ohm
load resistor ( = "matched line"), then the 50 ohm of the load appears at the opposite end of the cable *independent of the
length of the cable*. The local impedance at any point along the entire
feedline is 50 ohm:

## Local impedance along a correctly terminated transmission line

### (source: adapted from ref. 5B)

With antennas, this hardly ever the case! If *Z*_{L} ≠ *Z*_{0}, then the transmission line
acts as an impedance transformer: the local impedance depends on the distance
(in wavelengths) from the load-end. This means that the __impedance__ at the
input-end changes when the cable length is changed. These
changes may be very large. "Impedance" SWR-meters may interpret this change as an SWR change.

So: what happens if a coax is terminated with an impedance that is not a pure 50 ohm resistor? The answer is....... that depends! It depends on that
termination impedance, and on the length of the coax. To be more precise: not
the *physical* length of the line, but the *electrical* length. The latter is a
function of the Velocity Factor (VF) of the line, and the frequency of the
transmitted signal. VF depends
on the dimensions of the line and the type of dielectric. In a coax cable, the
dielectric is the material between the center conductor and the
shield. VF is typ. around 0.66 for coax with solid polyethylene, and 0.8 - 0.88
with foam
polyethylene. The wavelength in the cable is equal to the "free space"
wavelength of the signal, multiplied by the VF. Example: a frequency of 10 MHz
is equivalent to a free-space wavelength of 30 meters. In a coax cable with a VF
of 0.8 ( = 80%), a 10 MHz signal has a wavelength of 30 x 0.8 = 24 meters.
Conversely, a cable with VF=0.8 and a physical length *L*_{phys} of 1 wavelength λ, has an
electrical length *L*_{elec} = λ / 0.8 = 1.25 *L*_{phys}. Note that it is tacitly assumed
that the VF is frequency-independent constant, which is a simplification.

Let's
look at a very specific *electrical* length - whole multiples of ½λ (i.e.,
½λ, λ, 1½λ,...). That is, in the general formula for *Z*_{input }
: *k* = ½, 1, 1½, ... so φ = π, 2π, 3π, ... Hence, cos(φ) = 1 or -1,
and sin(φ) = 0. As sin(φ) = 0, the resulting *Z*_{input} is the
same for cos(φ) = 1 and for cos(φ) = -1:

So, clearly:

for a transmission-line with an *electrical* length that is a whole
multiple of 1/2 λ,

the impedance *Z*_{input} that appears at the transmitter-end of the cable

is the same as the impedance *Z*_{load} that is connected at
the load-end of the cable,

independent of *Z*_{0} !

So, an "open" (*Z*_{load} = ∞) at the load-end is
transformed to an "open" at the input-end (*Z*_{input} = ∞).
Likewise, a
"short" (*Z*_{load} = 0) appears as a "short" (*Z*_{input}
= 0), and any other impedance also as that same impedance (*Z*_{input}
= *Z*_{load}). Remember that the *electrical* length of a given cable is a function of
the frequency. This 1:1 impedance transformation of a fixed length segment of
cable __only__ occurs for the harmonic and sub-harmonic frequencies for which the *electrical*
length of that cable segment is ½λ, λ, 1½λ,...

Let's look at another special case: the *electrical* length of the transmission line is
an odd multiple of ¼λ. That is, in the general formula for *Z*_{input }
: *k* = ¼, ¾, 1¼, ... so φ = ½π, 1½π, 2½π, ... Hence, cos(φ) =
0 and sin(φ) =
1 or -1. As cos(φ) = 0, the resulting *Z*_{input} is the same
for sin(φ) =
1 and for sin(φ) = -1:

To put this into words:

for a transmission-line with an *electrical* length that is an
odd multiple of 1/4 λ,

the impedance *Z*_{input} that appears at the transmitter-end of
the cable

is the ratio of the square of *Z*_{0}, and the impedance *Z*_{load}

So, an "open"at the load-end (*Z*_{load} = ∞) is
transformed to a "short" at the input-end (*Z*_{input} = 0).
Likewise, a
"short" (*Z*_{load} = 0) appears as an "open" (*Z*_{input}
= ∞). Any other load-impedance is transformed per the ratio shown in the formula above.

For
purely resistive loads, the calculations are easy. Example: for a "50 ohm"
transmission-line, a 100 ohm resistive load is transformed to 50^{2} / 100 = 25
ohm (resistive) at the input-end of the line. Conversely, a 25 ohm resistive load
is transformed to 2500
/ 25 = 100 ohm (resistive) at the input-end. In both cases, SWR = 2. However, unlike the case where the
line is *correctly* terminated, the local impedance along the feedline is now __not__
resistive - except at the 1/4 and 1/2 electrical wavelength points, and multiples
thereof. Between the latter points, the local impedance comprises both
resistance and reactance (inductive or capacitive, depending on the distance
from the end of the transmission line):

## Local impedance along a 50 Ω transmission line terminated with 100 Ω (SWR = 2)

### (source: adapted from ref. 5B)

## Local impedance along a 50 Ω transmission line terminated with 25 Ω (SWR = 2)

### (source: adapted from ref. 5B)

The two figures above show that the pattern repeats itself every 1/2
electrical wavelength. Of course, for loads with a reactive component, the
calculation is a bit more complicated, as obvious from the general formula for *Z*_{input}.

THE EFFECT OF LOSSES

The above discussion only covers the simplistic and (typically unrealistic) case WITHOUT LOSSES !!!

In due time, I will expand this section to cover that as well...
In the mean time, have a look at references 5G-5N.

Total power loss in the transmission line __system__ consists
of:

- Transmission line loss:
- Matched Line Loss (MLL): this is the power loss
in a length of transmission line when it is perfectly matched by the load. It is typically
characterized as dB for a given
__physical__length, typically dB/100 ft or dB/100 m. It depends on the general line type (coax, twin-lead, wondow, ladder line), the specific model and manufacturer. MML is also frequency dependent: the higher the frequency, the higher the loss. See the attenuation graph below. - Additional loss due to SWR.
*This loss can be much higher than the MML*! - Device insertion loss. Each device that is inserted in the transmission line between transmitter and antenna load causes frequency-dependent power loss:
- Impedance matching unit (antenna coupler or
"tuner"). In general, the larger the mismatch to be matched, the larger the
loss.
*This loss can be very significant (i.e., more than 50%)*! - Impedance transformers (baluns, ununs, common-mode chokes,...).
- Instruments (SWR/power meters, current/voltage couplers,...)
- Connectors. This loss is typically in the range of 0.1 to 0.3 dB per connector. Primary causes are:
- Reflected losses, due to impedance mismatch between
connector and
*Z*_{0}. - Dielectric losses, due to dissipation in the dielectric materials of the connector.
- "Copper" losses, due to dissipation in the conducting surfaces of the connector (base metal such as copper, brass, steel; plating such as gold, silver, nickel).
- Lightning arrestors.

## Matched Line Loss (MLL) for various types of feedline (coax, window line, ladder line)

### (source: adapted from ref.XX)

## Additional transmission line loss due to SWR

### (source: adapted from ref.XX)

REFERENCES

**Ref. 1:**"Antenna Matchboxes", Richard Westerman (DJ0P/G5BMH/NJ0IP) [pdf]**Ref. 2:**Antenna tuners/couplers**Ref. 2A:**"What tuners do", Steve Hunt (G3TXQ) [pdf]**Ref. 2B:**"Antenna Tuners (Antenna Couplers)", Larry Benko (W0QE), June 2011 [pdf]**Ref. 2C:**"ZM-4 ATU for symmetrical and unsymmetrical Antennas", manual V1.3UK, QRPproject, Dl-QRP-AG [pdf]- I made a slight modification (per Olaf, DL7VHF) to the SWR bridge, in order to make sure that the LED can go off on 10 m. [pdf]
- The design of the front & back labels that I made is
here. I
printed the labels on a sheet of self-adhesive paper (actually a sheet with
several large labels for file binders). Before you print on adhesive paper,
print on regular paper and check the size of the labels: my file is set up for
the printer that I use. You may have to adjust the printing scale in the printer
settings, to get the correct scaling for
__your__printer. After printing, I applied a thin layer of clear matt acrylic coating (e.g., from arts & crafts supply store) to protect the paper. **Ref. 2D:**"The simple Z Match simplified", Lloyd Butler (VK5BR) [pdf]**Ref. 2E:**"Putting a Balun and a Tuner Together", Kevin Schmidt (W9CF) [pdf]**Ref. 2F:**"On the Quest for an Ideal Antenna Tuner", John (Jack) Belrose (VE2CV), in "QST", October 2004, pp. 35-39 [pdf]**Ref. 2G:**"Do you really need an antenna tuner?", Steve Ford (WB8IMY), in "QST", January 1994, pp. 70-72 [pdf]**Ref. 2H:**"S-Match™, the balanced universal ATU", Frits Geerlings (PA0FRI), December 2012 [pdf]**Ref. 2I:**"Baluns and Tuners", Martin Ehrenfried (G8JNJ) [pdf]**Ref. 3:**SWR**Ref. 6A:**"How important is low SWR?", Stan Gibilisco (W1GV), in "Ham Radio Magazine", August 1981, pp. 33-37 [pdf]**Ref. 6B:**"Too low SWR can kill you", Walt Maxwell (W2DU), adapted from QST, April 1973 [pdf]**Ref. 6C:**"Low SWR for the WRONG reasons", Walt Maxwell (W2DU), Chapter 5 of "Reflections 2", adapted from QST, April 1973 [pdf]**Ref. 6D:**"Low SWR for the RIGHT reasons", Walt Maxwell (W2DU), Chapter 6 of "Reflections 2", adapted from QST, April 1973 [pdf]**Ref. 4:**Remote/automatic tuners**Ref. 4A:**"EDX-2 Automatic Antenna Tuner", Alinco, Inc., operating/installation manual, 2 pp.**Ref. 4B:**"EDX-2 Service manual", Alinco, Inc.**Ref. 4C:**"EDX-2 von Alinco - der preiswerte unter den Automatiktunern", Alexander Barz (DL4EEC), Funkamateur, FA 4/03, pp. 348-349**Ref. 4D:**"Smartuners for stealth antennas", SGC Inc., 1997, 35 pp.**Ref. 4E:**"SG-230 Smartuner Installation and Operations Manual", SGC, Inc., 2000, 81 pp.**Ref. 5:**Feedline and impedances**Ref. 5A:**"Transmission Line Imput Impedance", Jim Styles, EECS 723 "Microwave Engineering" class notes, The University of Kansas, January 2005, 9 pp. [pdf]**Ref. 5B:**"My Feedline Tunes My Antenna", B. Goodman (W1DX, SK), in "QST", March 1956, pp. 49-51, 124 [pdf]**Ref. 5C:**"S-Parameters", Chapter 14, pp. 663-708 in "Electromagnetic Waves and Antennas", Sophocles J. Orfanidis, 2016, 1413 pp. [pdf]**Ref. 5D:**"RF engineering basic concepts: S-pameters", F. Caspers, pp. 67-93 in "Proceedings, CAS - CERN Accelerator School: RF for Accelerators, Ebeltoft, Denmark, 8-17 June [pdf]**Ref. 5E:**"Transmission Lines - Your Pipeline to the Outside World", Jim Fisk (W1HR), in "Ham Radio Horizon" (HRH), May 1977, pp. 32-39.**Ref. 5F:**"Transmission Line Details", Dan Maguire (AC6LA), standalone calculator Visual Basic program (no installation required), V2.0.1, 28 November 2014- Calculates impedance and reflection coefficient parameters (SWR, reflection coefficient) at both ends of a transmission line, and the details of power loss in the line (Matched Line Loss), electrical length, etc. Includes built-in specifications for approximately 100 different line types.
**Ref. 5G:**"Understanding SWR by Example", Darrin Walraven (K5DVW), in "QST", November 2006, pp. 37-41. [pdf]**Ref. 5H:**"The Facts about SWR and Loss", Steve Stearns (K6OIK), resented at ARRL Pacificon Antenna Seminar October 15, 2010. [pdf]**Ref. 5J:**"The Joy of Matching", Steve Stearns (K6OIK), presented at the FARS Membership Meeting August 28, 2015. [pdf]**Ref. 5K:**"Conjugate Match Myth", Steve Stearns (K6OIK), presented at ARRL Pacificon Antenna Seminar October 14-16, 2011. [pdf]**Ref. 5L:**"A Transmission Line Power Paradox and Its Resolution", Steve Stearns (K6OIK), presented at ARRL Pacificon Antenna Seminar October 10-12, 2014. [pdf]**Ref. 5M:**"Walter Maxwell’s teachings on system wide conjugate matching – a SimSmith example", blog by Owen Duffy (VK1OD), latest update: 19 October 2019.**Ref. 5N:**"The conjugate match stuff crashes out again", blog by Owen Duffy (VK1OD), latest update: 25 January 2020.

External links last checked: October 2015