- [Introduction]
- [Unknown inductance]
- [Unknown capacitance]
- [Ferrite & iron powder cores]
- [Filters]
- [Crystals]
- [Directional coupler]
- [Coax switch]
- [Common-mode chokes]
- [Antenna systems]
- [SWR]
- [References]

Last page update: 23 April 2017

INTRODUCTION

*A FOOL WITH A TOOL IS STILL A FOOL!WER VIEL MISST, MISST VIEL MIST!"He who measures a lot, measures a lot of crap"*

Ca. 2007, I bought a "miniVNA". It is very small, and (relatively) inexpensive **V**ector **N**etwork **A**nalyzer
(ref. 1).
I got it primarily for measuring antennas and antenna systems. For obvious
reasons, I refer to it as "the little blue box":

It covers 0.1 to 180 MHz. The miniVNA is controlled by software that runs on a PC. It communicates with the unit via USB (which also powers the unit), and presents the measurement data graphically to the user. VNAs in this price range clearly have significant limitations (ref. 1D). But for most amateur radio purposes, it is quite adequate. This miniVNA is basically "first generation". Subsequent models have expanded range and can be operated via wireless. A top-level block diagram of the miniVNA is shown below.

## Simplified block diagram of the miniVNA

The Device Under Test (DUT) port is the signal generator output port. The gain and phase detectors in the VNA compare the excitation signal from the signal generator against one of two signals:

- the signal that is reflected by a single-port device (or system) that is attached to the DUT port. This is done via the directional coupler at the DUT port. Not all VNAs on the market have this, and require an external directional coupler for certain types of measurement.
- the signal at the DET port that output by a two-port device that is excited by the DUT port, or that is captured via coupling to a single-port device.

Measurements that involve only the DUT port, are done in "antenna" mode. Measurements that involve both the DUT port and the DET port are done in "transmission" mode.

### (miniVNA analysis mode selections in the original IW3HEV-IW3IJZ software)

### (miniVNA analysis parameter display selections in the original IW3HEV-IW3IJZ software)

All VNAs must be calibrated to the extent possible, before using them! Check
the manual! Also note that relatively (!) inexpensive VNAs such as the miniVNA
cannot determine the sign of the reactance part *Xs* of the measured
complex impedance *Z* = *Rs* + j*Xs*.
I.e., it cannot tell an inductance from a capacitance.

The sections below provide a brief description of some basic measurements that can be done with a VNA such as the miniVNA.

I use the following freeware:

- On my Windows PCs and laptops: the original software by IW3HEV/IW3IWZ/G3RXQ/IK3ZGB, as well as more recent Java-based vna/J by Dietmar Krause (DL2BSA).
- On my Android tablet: the nice and flexible Blue VNA application (both via USB & Bluetooth) by Dan Toma (YO3GGX).

There are other applications (e.g., qVNAmax for Linux, and 2012 VB software by PA7N), but I have no experience with them.

DETERMINING UNKNOWN INDUCTANCE

An electromagnetic coil is simply an inductor that is wound so as to have the shape of a coil, helix, or spiral. The current through the coil generates a magnetic field that interacts with the coils itself. For most applications, an ideal coil only has inductive reactance and no losses. A coil is a coil is a coil - it doesn't know (or care) how it is used and for what purpose. Its characteristics and behavior do not depend on the application.

## A coil

And here are some BIG coils.

## One of the antenna tuning coils of the 1 megawatt VLF "Goliath" transmitter (1940s)

### (these enormous variometers comprised a fixed coil (3.5 m diameter, 11.5 ft), into which a slightly smaller coil (3.2 m diameter, 10.5 ft) could be inserted hydraulically with a precision of 0.1 mm! The coils were 5 m tall (16 ft) and weighed about 5000 kg (11k lbs)

## Tuning coil of the VLF transmitter at Rugby/UK (mid-1920s)

In antenna systems, coils - if any - are primarily used in two ways:

- As part of a "
**trap**". This is a parallel circuit of an inductor coil and a capacitor. At the resonance frequency of this circuit, its impedance becomes quite high. A trap is inserted in the radiating element(s) of an antenna, somewhere between the feedpoint and the tip of that radiator. Around the resonance frequency of the trap, the part of the radiator*beyond*the trap is basically disconnected. In this case, only the part of the radiator between the feedpoint and the trap is "active". This is used in some multi-band antennas: the full length of the antenna is used on one band, the shortened length is used at a higher band. - Note that far enough below the resonant frequency, a trap basically acts like just a loading coil.
- As a
**loading**coil. This is a coil used by itself. Short antennas have a capacitive reactance at the feedpoint. This can be compensated by introducing inductance somewhere in the radiating element(s). One way to do that is with a "lumped" inductance: an inductor coil. Contrary to popular belief, a loading coil does not add "missing electrical length"! - Note that no coil is perfect: there is always loss resistance and stray (parasitic) capacitance. That is, any coil by itself also has a resonance frequency, and the coil will act as a "trap". This is why loading coils should be operated far away enough from that self-resonance frequency.

I am not a expert (real or self-anointed) of coils. So, rather than writing a lot of rubbish here, I refer to the list of references

The impedance of an (ideal) inductor is:

This formula can be re-written as:

where *f* in MHz and *L* in μH. At the frequency where |Z| = 50 ohm, this simplifies to:

Hence, with a simple |Z| measurement, we can determine an unknown inductance value. Actually, it is "estimate" rather than "determine": VNAs such as the miniVNA are only reasonably accurate around 50 Ω.

**Note**: in all test set-ups, wiring should be kept as short as practicable.

## Test set-up for impedance-based measurement of L_{x}

### (miniVNA software in "antenna" mode, |Z| parameter displayed)

Outside the range of 20-200 Ω, miniVNA accuracy is very poor. Resonance-based methods are more accurate. The standard formula for the resonance frequency of an (ideal) LC-circuit is:

This equation can be re-arranged as:

which can be re-written as:

where *L* is in μH, *C* is in pF, and *f _{res}* is in MHz. So, if we
use a known capacitance

*C*and measure the resonance frequency

*f*, we can determine the unknown inductance

_{res}*L*.

For the special case where *C* = 253 pF (e.g., 220 pF and 33 pF in parallel), the
equation simplifies to:

## Test set-up for parallel-resonance measurement of L_{x}

### (miniVNA software in "antenna" mode, "loss / dB" and "phase" parameters displayed)

## Alternative test set-up for parallel-resonance measurement of L_{x}

### (miniVNA software in "transmission" mode, "loss / dB" and "phase" parameters displayed)

Note that in __parallel__-resonance configuration, the LC-impedance at the resonance frequency is __high__. In the __
series__-resonance configuration, it is __low__.

## Test set-up for series-resonance measurement of L_{x}

### (miniVNA software in "transmission" mode, "loss / dB" and "phase" parameters displayed)

The resonance frequency is easily identified in the loss/phase plot:

## Phase and loss plot for the LC-circuit, miniVNA in "transmission" mode

**Note** the -30 dB “loss offset” setting on the left hand side of the
plot. This was done so as to lower the "loss" curve to where it was fully
visible.

**Note**: accuracy of determination not only depends on the miniVNA
accuracy, but also on that of the reference capacitor. Over thirty years of
professional engineering experience has taught me that anything within ±20% is
within engineering accuracy, hihi! Capacitors typically have poor tolerances with respect to their nominal value. When new, ceramic disk caps typically have +80/-20% tolerance, milar
polyester ±5 or ±20%, tantalum ±10 or ±20%, metalized polypropylene typ. ±5,
±10, or ±20%, electrolytic typ. ±20%, silver mica ±0.1 to %±1%. Polystyrene capacitors are
available with ±1 and ±2% tolerance. I normally use styroflex capacitors (a
special form of styrene). Special capacitors are available down to ±0.01%
tolerance.

A full-sweep plot shows some phase "dips" and "reversals" at (much) higher frequencies. They may be related to resonances due to stray capacitance of the coil, wiring of the test setup, etc.:

## Phase plot of parallel LC-circuit

### (miniVNA in "antenna" mode, 1-180 MHz sweep)

Obviously, you can use a conventional/classical "dipmeter" to determine the resonance "dip". But the miniVNA can also be configured as a "dipper / sniffer":

## Dipper/Sniffer set-up for parallel-resonance measurement of L_{x}

### (miniVNA software in "transmission" mode, "loss / dB" and "phase" parameters displayed)

Note that you cannot use "dipper/sniffer" method to determine the Q of the coil: the phase-vs-frequency plot depends on the coupling (i.e., distance, orientation) between the test coil and the “sniffer coil”. As with the classical dipmeter, the resonance frequency also tends to shift with the level of coupling.

I have also used my fancy dipmeter with digital frequency read-out
(ref. 3) to
determine resonance frequencies. The dipmeter’s excitation coil
was coaxially aligned with my test coil. Inserting a fiberglass fishing pole (my dipole is made up of two such poles) into the coil
core did not cause the resonance frequency *f*_{res}
to shift. A good thing!

## Test setup with my dipmeter

The "sniffer/dipper" method can also be used to measure coils *in-situ*:

The techniques discussed above can be also be used to characterize unknown parallel LC-circuits, such as antenna "trap" filters that are used in many antenna designs.

If *L*_{x} and *C*_{x} are the unknown parallel elements of the L/C trap, *C*_{p} is the additional known parallel capacitor, *f*_{res1} is the resonance frequency of the trap without *C*_{p}, and *f*_{res2} is the resonance frequency of the
"trap plus *C*_{p}", then:

After determining *f*_{res1} and *f*_{res2} with one of the methods discussed above, we can solve for *C*_{x}:

Once *C*_{x} is found, the standard resonance formula can be used to determine *L*_{x}.

DETERMIINING UNKNOWN CAPACITANCE

Based on duality of inductive and capacitive reactance, basically the same methods described above can also be used to determine an unknown inductance. First the (coarse) impedance method. The impedance of an (ideal) inductor is:

This equation can be re-written as:

where *f* in MHz and *C* in pF. At the frequency where |Z| = 50 ohm, this simplifies to:

Hence, with a simple |Z| measurement, we can determine an unknown capacitance value. As stated above for the inductance determination with this method: it is "estimate" rather than "determine", as VNAs such as the miniVNA are only reasonably accurate around 50 Ω (definitely not outside the 20-200 Ω range).

## Test set-up for impedance-based measurement of C_{x}

Then the resonance-based methods. Again, starting from the LC-resonance equation:

This equation can be re-arranged as:

which can be re-written as:

where *L* is in μH, *C* is in pF, and *f _{res}* is in MHz.
So, if we use a known inductance

*L*and measure the resonance frequency

*f*, we can determine the unknown capacitance

_{res}*C*.

For the special case where *L* = 235 μH (I wish you good luck finding
such a coil off-the-shelf, but you can make one...), the
equation simplifies to:

## Test set-up for parallel-resonance measurement of C_{x}

### (miniVNA software in "antenna" mode, "loss / dB" and "phase" parameters displayed)

## Alternative test set-up for parallel-resonance measurement of C_{x}

### (miniVNA software in "transmission" mode, "loss / dB" and "phase" parameters displayed)

## Test set-up for series-resonance measurement of C_{x}

### (miniVNA software in "transmission" mode, "loss / dB" and "phase" parameters displayed)

As for the inductance measurement, the resonance frequency is easily identified in the loss/phase plot. Also, the "sniffer/dipper" method or a dipmeter can be used to determine the resonance frequency.

FILTERS

The miniVNA can be used to determine the characteristics of passive filters (LC-circuits and crystal filters). Typical parameters are the corner-frequencies and skirt-steepness of low-pass, high-pass, band-pass, and band-stop filters. Other parameters are insertion loss and ripple. This is very straightforward for filters with a 50 Ω input and output:

## Test set-up for impedance-based measurement of C_{x}

### (miniVNA software in "transmission" mode, "loss / dB" and "phase" parameters displayed)

For filters with I/O port impedance other than 50 Ω, (passive) adapter networks should be used. They can be transformer based (you will need "Un-Un" transformers), or a simple resistor network:

## Test set-up for filter measurements with resistor adapter-networks

With resistor networks, the "loss offset" in the analyzer GUI must be set to 2x the attenuation of resistor network. See examples above. Low-inductance resistors should be used, such as "metal film". For 10:1 adapters, oscilloscope probes may be used (they are 500-to-50).When using transformer adapters, the miniVNA must be calibrated with the input and output adapter connected back-to-back (i.e., without the filter).

For narrow crystal filters, the sampling speed ( = number of samples/sec) of the analyzer sweeps may have to be reduced. Also: some crystal filters require a small capacitor (e.g., 10 or 20 pF) to be installed across the input and across the output. Check the data sheet!

Matching to 50 Ω can be measured with the following set-up:

## Test set-up for impedance-based measurement of C_{x}

### (miniVNA software in "antenna" mode, "loss / dB" and "phase" parameters displayed)

A_{L} OF FERRITE AND IRON-POWDER CORES

The miniVNA can also be used to determine the A_{L} value of an unknown ferrite or iron-powder core.
This value expresses how many wire turns have to be wound onto the core, to
obtain a certain inductance. So, if we know the number of turns, and measure the
resulting inductance, we can determine the A_{L} value. And with the
A_{L} value, we can identify the type of core (if the core manufacturer
is known). Here are some A_{L} values for commonly used ferrite and
iron-powder cores (note the large tolerances!):

## A_{L} values of some common ferrite and iron-powder toroidal cores (rings)

For whatever reason, the A_{L} is defined differently for
ferrite and for iron-powder cores. The difference is a factor of 10. For ferrite
cores:

For iron-powder cores:

As stated above, all we need to do is wind a couple of turns on the core, and measure the resulting
inductance with one of the methods described above for unknown inductances. Then apply the appropriate A_{L}
formula.

## Test set-up for series-resonance measurement of L

### (miniVNA software in "transmission" mode, "loss / dB" and "phase" parameters displayed)

## Alternative test set-up for resonance-based measurement of L

### (miniVNA software in "transmission" mode, "loss / dB" and phase parameter displayed)

CRYSTALS

A mechanically vibrating quartz crystals can be represented by an equivalent electrical circuit (ref. 4C):

Hence, measuring the resonance characteristics of a crystal is straightforward:

## Test set-up for the impedance-based measurement of C_{x}

### (miniVNA software in "transmission" mode, "loss / dB" parameter displayed)

The plot below shows the typical resonance curve of a crystal. Note that there are multiple resonances! The lowest resonance frequency is always a series-resonance. It has the lowest damping. The next higher resonance frequency, with higher damping, is a parallel-resonance.

## "Loss / dB" plot of a 10.250 MHz crystal

### (miniVNA in "transmission" mode)

The "Q" of the crystal can be determined by
zooming in the frequency sweep-range on the lowest frequency, and determining
the -3 dB bandwidth. "Q" is the resonance frequency divided by that bandwidth.
Typical crystal Q-values range from 10^{4} to 10^{6}. This is orders of magnitude higher than
typical LC-oscillators.

DIRECTIONAL COUPLERS & POWER-DIVIDERS - INSERTION LOSS & DIRECTIONAL ATTENUATION

A directional coupler is a 3-port or 4-port device. It has an input port, and output port, and one or two coupled ports. Insertion loss between the input and output port is the "through" loss between these ports, plus the "coupling loss" due to the load that is transformed from the coupled port(s) to the through-path between input and output port. The latter amount depends on the coupling ratio: the ratio between the power applied at the input port and the resulting power that appears at the coupled port. All ports must be properly terminated! The miniVNA can be used to measure the coupling (damping / isolation) between any two ports.

Shown below is a simple power divider. The coupled port is transformer-coupled to the 1:1 through-path between the input and output port. My -30 dB divider is described on this page.

## Test set-up for the insertion-loss measurement the 1:1 through-path

### (miniVNA software in "transmission" mode, "loss / dB" parameter displayed)

## Test set-up for the coupling ratio between the 1:1 through-path and the coupled port

### (miniVNA software in "transmission" mode, "loss / dB" parameter displayed)

COAX SWITCH - INSERTION LOSS & ISOLATION

I wanted to share my antenna between two solid-state transceivers, so I needed a coax switch. One transceiver has an output power of 5 Watt, the other 100 Watt. Clearly, I don't want to blow up the receiver input of the QRP transceiver when the coax-switch connects the antenna to the 100 Watt rig. For the HF-bands, a good coax switch has at least 60-70 dB cross-talk damping (isolation), and less than 0.1 dB insertion-loss.

We can measure this with an antenna/network analyzer, such as the miniVNA that I have. Note that for the insertion-loss measurement, we must account for the loss caused by the connectors between the switch and the miniVNA. Note that coax connectors really depends on production quality ( = price). This is particularly important on VHF and above. N-type connectors typically have negligible insertion loss, whereas cheap "oriental" PL-259/SO-239 connectors may have 0.2 dB insertion loss and frequency-dependent impedance transition.

## Cross-talk (isolation) measurement

## Insertion-loss measurement measurement

## Specification of the above coax switch (Jetstream JTSC-2M)

COMMON-MODE CHOKES

The common-mode attenuation of 1:1 current "chokes" is determined by measuring the loss (damping) on the coax shield (braid), between the two sides of the "choke". On HF, 25-30 dB common-mode damping is "good". The test set-up for some standard common-mode chokes is shown below.

## Common-mode attenuation of a W2DU-style choke with 24 beads of #77 ferrite material

### (attenuation is about 20-22 dB above 7 MHz)

ANTENNA SYSTEMS

Measurement of antennas (i.e., at the feedpoint) and of antenna systems (antenna + feedline(s) + balun(s) + choke(s) + antenna tuner) is easy. However, quite often, interpreting the data is not! I am not going into detail here, but ref. 4J may provide some practical examples.

**Note**: antenna resonance-frequencies and SWR-minimum almost __never__
coincide exactly! Per definition, at resonance, the impedance is purely
resistive: reactance is zero. The SWR minimum occurs where the impedance is
nearest to 50 Ω. Most antennas do not have an impedance of 50 Ω at resonance!

**Note**: there is no requirement to operate an antenna at a resonance frequency.
Doing so only makes matching to a feedline easier.

**Note**: obviously, you can measure an antenna __system__ by connecting the
VNA at the end of the feedline. However, unless the feedline is __exactly__ half a wavelength long
(including accounting for the velocity factor) such that it replicates the
complex impedance at the opposite end, this does not tell you very much about
what the impedance (Rs and Xs) is at the antenna. This also means that an
antenna coupler ("tuner") that can match the feedpoint-impedance of the
antenna, may not be able to do so at the end of the feedline (and vice versa).

The first plot shows a nice SWR-dip at/near resonance just above 7 MHz. At a slightly lower frequency, around 6.8 MHz, there is a small dip. Where does it come from? In this particular case, the antenna has a radial that is too long with respect to the antenna's resonance frequency. Sometimes such dips occur due to coupling with objects near the antenna.

## SWR dip of an antenna at/near the resonance frequency - secondary dip below it

### (source: personal communications with Gerd Koetter (DO1MGK, SK))

The next plot shows the SWR-sweep of a multi-band antenna. Two of the dips (around 8 and 25 MHz) could be deepened by adding radials of the appropriate length.

SWR

Some words about SWR....

The transmission line (feed line) of the antenna system is terminated by the antenna ( = load). If the antenna's feedpoint impedance is not equal to the feedline's characteristic impedance *Z*_{0} (i.e., there is a mismatch), then part of the energy that enters the input-end of the transmission line, is reflected at the load-end of that line. This results in standing waves on that line, as the forward and reflected waves combine
along the entire length of the line.

As a result, there is (at least) one point along the line that has maximum voltage amplitude |Vmax|,
and (at least) one other point that has minimum voltage amplitude |Vmin|. The SWR (actually the Voltage SWR, VSWR) is the ratio of these voltage
amplitudes: SWR = |Vmax| / |Vmin|. Likewise, there are points with maximum and minimum *current* along the
line. The ratio of these currents is the Current SWR or ISWR. It has the same
value as the VSWR.

The coax cable is a transmission line, just like a ladder-line or a twin-lead cable.
A transmission line has a so-called *characteristic impedance*, denoted *Z*_{0}.
For coax this is typ. 50 ohms (or 75 ohms for TV and satellite receiver coax). For
commercial twin-lead cable, standard *Z*_{0} is 300 or 450 ohms.
The *Z*_{0} of ladder-line depends on the wire spacing
and wire diameter, and is often dimensioned
for 450 or 600 ohm.

SWR is only determined by the characteristic impedance *Z*_{0} of the transmission
line, and the load impedance *Z*_{L} (assuming constant-*Z*_{0} cable and proper
termination at the input-end):

SWR = *Z*_{0} / *Z*_{L} (if *Z*_{L}
≤ *Z*_{0})
or *Z*_{L} / *Z*_{0} (if *Z*_{L} ≥ *Z*_{0})

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

So, contrary to popular belief:

SWR is constant along the entire length of the transmission line!

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

Why do some people report SWR changes when the feedline length is changed?
Either because they have RF interference in their SWR instrument and it mis-reads,
or they are not measuring SWR. Some conventional SWR-meters measure the ratio of
the __local__ impedance 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*_{L}. So, such an instrument can only
measure the *Z*_{L} / *Z*_{0} ratio, __if__ it
is placed at the antenna feedpoint! The 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 constant along the entire length of
the transmission line, the __impedance__ along the line is __not__ constant, __unless__ the
impedance *Z*_{L} of the load ( = antenna feedpoint) is equal to *Z*_{0}.

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*_{L} 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, then the 50 ohm of the load appears at the opposite end of the cable, independent of the
length of the cable. Also, the local impedance at any point along the entire
feedline is also 50 ohm:

## Local impedance along a correctly terminated transmission line

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

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 if a coax is terminated with an impedance other
than a 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.

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's
look at a very specific *electrical* length - whole multiples of ½λ
(i.e., ½λ, λ, 1½λ,...):

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.

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,...

And for another specific *electrical* length - odd multiples of ¼λ
(i.e., ¼λ, ¾λ, ...):

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 load 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):

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

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

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

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

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, and beyond the scope of this discussion.

REFERENCES

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**Ref. 1:**network & antenna analyzers**Ref. 1A:**"Understanding the Fundamental Principles of Vector Network Analysis", Agilent Technologies, Application Note, 2012, 15 pp. [pdf]**Ref. 1B:**"Network Analyzer Basics", Agilent Technologies, 2000, 100 slide presentation [pdf]**Ref. 1C:**"Fundamentals of Vector Network Analysis - Primer", Rohde & Schwarz, V1.2, 46 pp. [pdf]**Ref. 1D:**"Introduction to Network Analyzer Measurements - Fundamentals and Background", National Instruments, 44 pp. [pdf]**Ref. 1E:**"Vector network analyzer comparisons" [miniVNA, MFJ259B, AIM, HP, TenTec], Rudy Severns (N6LF [pdf]**Ref. 1F:**"miniVNA", Jim Gerwitz (AC7FN), July 2008, 29 slide presentation on the miniVNA and its applications [pdf]**Ref. 1G:**"Reference manual for mRS antenna analyser type miniVNA 0.1-180 MHz", miniRadioSolutions, V2.24, 8 pp [pdf]**Ref. 1H:**"mini Radio Solutions miniVNA Network and Antenna Analyzer", in "QST", August 2008 [pdf]**Ref. 2:**Coil calculators:**Ref. 2A:**"Helical coil calculator" on pages of the Tesla Coil web-ring**Ref. 2B:**"K1QW Inductor Calculators"**Ref. 2C:**"ON4AA Single-layer Helical Round Wire Coil Inductor Calculator"**Ref. 3:**"DipIt - the ultimate dipmeter of the German QRP Club DL-QRP-AG [pdf]**Ref. 4:**measurements**Ref. 4A:**"Homebrew Your Own Inductors!", Robert Johns (W3JIP), QST, August 1997, p. 35. [pdf]**Ref. 4B:**"HF-Messungen mit dem Netzwerktester - Das Praxisbuch zum FA-NWT" [in German], Hans Nussbaum (DJ1UGA), Box 73 Amateurfunkservice GmbH, 2007, 144 pp.**highly recommended book!****Ref. 4C:**"Quartz Crystal Theory", Jauch Quartz G.m.b.H., 6 pp. [pdf]**Ref. 4D:**"Measuring crystal motional parameters with the MiniVNA", Joop Lous (PE1CQP)**Ref. 4E:**"Q Measurement with the AIM4170", July 2008, 13 pp. [pdf]**Ref. 4F:**"The Two Faces of Q", Wes Hayward (W7ZOI), January 2011, 21 pp. [pdf]**Ref. 4G:**"Q Factor Measurement on L-C Circuits", Jacques Audet (VE2AZX), in "QEX", Jan/Feb 2012, pp. 7-11 [pdf]**Ref. 4H:**"How to measure parallel-resonances and Q with our analyzer" [minVNA], Gerd Koetter (DO1MGK, SK) [pdf]**Ref. 4J:**"Tuning a 160 Meter Vertical" [with an AIM4170], Jay Terleski (WX0B) [pdf]**Ref. 5:**"My Feedline Tunes My Antenna", B. Goodman (W1DX, SK), in "QST", March 1956, pp. 49-51, 124 [pdf]

External links last checked: October 2015

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