Readers may find it relevant and interesting.

]]>I have tried several times to reconcile built and tuned antennas and NEC models with Healey and failed, leading me to think of the problem and devise a good approximation that did reconcile (for me).

This article attempts to reconcile the example given at Designing a Gamma Match – Simsmith design tool and confirmation of as-built antenna, an example where the two built antennas reconcile well with the ARRL published design article and NEC model.

The example antennas are 4 element 144MHz Yagis built around 1970. They were originally designed with a 50Ω split dipole feed, or the option of a folded dipole with 4:1 half wave coax balun.

Gamma dimensions:

- dipole 12mm dia;
- gamma arm 12mm dia;
- gamma arm spacing 38mm;
- gamma arm length 160mm.

From this we can calculate using RF Two Wire Transmission Line Loss Calculator Zo=218Ω and length=28°.

Impedance at the tapping point:

\(Z_2=\frac{Z_1}{\cos^2 \theta}\\\) \(Z_2=\frac{50}{\cos^2 28}=64\)Using RF Two Wire Transmission Line Loss Calculator:

\(Z_2’=80+\jmath 103\)Using RF Two Wire Transmission Line Loss Calculator:

\(X_p=\jmath 115\)Since the tubes are equal size, the step up from Fig 3 is 4:1.

\(Z_{in}=78.5 + \jmath 246.1\\\)The calculated reactance does not reconcile with the Xc=86Ω to tune the built antenna.

The reactance would be nulled by a series Xc=246.1Ω leaving Z=78.5Ω which does not reconcile with the measured Z=50Ω.

This attempt to reconcile the example antennas failed.

- Healey, D. Apr 1969. An examination of the gamma match In QST Apr 1969.

- Lossless;
- Characteristic Impedance Z0=1+j0Ω; and
- load impedance other than 1+j0Ω, and such that Vf=1∠0 and Vr=0.447∠-63.4° at this point.

The ratio Vr/Vf is known as the reflection coefficient, Γ. (It is also synonymous with S parameters S11, S22… Snn at the respective network ports.)

Above is a phasor diagram of the forward and reflected voltages at the load.

There are times where the normalised voltage at a point is of interest, ie the resultant Vf+Vr or 1+Γ.

Above is the construction of 1+Γ.

Above is the construction on the Smith Chart, the length of 1+Γ is scaled from the “Radially scaled parameters” scales, and the angle read by projecting the phasor to the “Angle of transmission coefficient in degrees” scale. V=1+Γ=1.26∠-18.4°.

Now let’s consider what the forward and reflected voltages looking into 90° of this transmission line with this load at the far end.

Again taking the forward voltage as the reference (Vf=1∠0), we can determine that the reflected voltage will be of the same magnitude (by virtue of the lossless line property), but it is delayed 90° for the path to the load, and another 90° for the return path to the source.

Above is a phasor diagram of the forward and reflected voltages at the source (using the forward voltage at this point, the source, as the reference). The red arrow shows the rotation of the Vr phasor to the new Vr’ phasor (both relative to their own Vf). We are always plotting reflected voltage relative to forward voltage at the same point.

So. these plots are of forward and reflected voltage phasors, linearly scaled in magnitude and at some angle. Because the reflected phasor adds tail to head to the forward phasor, you might think of the magnitude of the reflected voltage as radially scaled from the ‘pivot’ point or centre.

We can map Γ to impedance and vice versa.

- \(Z_l=Z_0\frac{1+\Gamma}{1-\Gamma}\)
- \(\Gamma=\frac{Z_l-Z_0}{Z_l+Z_0}\)

Note that Γ and Z0 are both complex quantities, and they can each be expressed in polar form R∠θ where θ is the phase of each… but the phase of one is not the same as the phase of the other… something that seems to confuse lots of hams.

Admittance is simply the inverse of impedance, ie \(Y=\frac1{Z}\).

Whilst we might often measure Γ fairly directly, it would be convenient to map Γ to impedance and admittance using graph scales.

Phillip Smith thought so, and though it took him many years to evolve the ‘Smith Chart’ (1931-66), he eventually produced the now familiar circle chart that was used as a graphical computer for a long time, and now more often for visualisation of transmission line problems rather than the computational tool.

The Smith Chart is widely used as a display presentation on network measurement.

Lets overlay the first phasor diagram with a set of Resistance and Reactance scales that correspond the the values of radially scaled Γ.

There are also four scales around the perimeter of the circular chart, study them carefully.

Note that our example used Z0=1 and the chart above uses Zref=1 (the impedance at the centre of the chart).

In this graphic, the phasors are plotted for explanatory purposes, but it is usual to simply plot an impedance as a point with an X at its location.

It would not be very practical to seek stationery with Z0 to suit arbitrary applications as they arise, so impedances are normalised for plotting on the standard Smith Chart. The normalised impedance \(Z’=\frac{Z}{Z_0}\) and so real world impedance values are normalised to plot their points, and solution values scaled from the chart are denormalised to obtain the real world impedances.

Note that computer software and instrument displays often render denormalised or real world values for convenience.

Above is an example from Simsmith of a computer generated graphic of this transform, and in this case, the path is shown to further explain what is happening. This display uses Z0=1 because that is the problem context. The chart is rendered wrt Z0, which if it were say 50Ω, would result in denormalised display scales

As noted, we can map Γ to impedance.

- \(Z_l=Z_0\frac{1+\Gamma}{1-\Gamma}\)

If we substitute \(Z=\frac1Y\) we can develop the admittance map relative to the impedance map.

\( Z_l=Z_0\frac{1+\Gamma}{1-\Gamma}\\\frac1{Y_l}=\frac1{Y_0}\frac{1+\Gamma}{1-\Gamma}\\

Y_l=Y_0\frac{1-\Gamma}{1+\Gamma}\\

Y_l=Y_0\frac{1+\Gamma}{1-\Gamma}\cdot 1 \angle 180°\\\)

So the to transform the impedance map \(\frac{1+\Gamma}{1-\Gamma}\) to the admittance map \(\frac{1-\Gamma}{1+\Gamma}\), we rotate it 180°. Note the peripheral scales are not rotated, just the real and imaginary scales.

Above is the admittance map of the Smith chart, note the outer scales are the same as in the impedance mapped chart. Note in the upper half of the chart, B<0, ie negative.

The foregoing explains why rotating a point 180° about the chart centre inverts Z or Y.

A simple technique for using the basic Z Smith chart for admittance was to rotate the chart so infinite conductance (short circuit) was at the left end etc. Note that B<0 (ie negative) in the upper half of the chart.

Smith did create a version of his chart with both Z an Y scales.

Above, Smith’s combined chart also showing here the radial scales. This combined chart can be more convenient when working with mixed series and shunt elements.

The normalised admittance \(Y’=\frac{Y}{Y_0}=Y Z_0\) and so real world admittance values are normalised to plot their points, and solution values scaled from the chart are denormalised to obtain the real world admittances.

The previous discussion was based on forward and reflected waves expressed as equivalent voltages. As you might expect, it is also possible to use currents. The relationships are:

- \(V_r=\Gamma V_f\)
- \(I_r=- \Gamma I_f\)

The last can be rewritten \(\Gamma=\frac{I_r}{-I_f}\), so we can construct the phasor Γ by adding If and Ir head to tail starting at the right side of the chart (Γ=1 point, when Γ=1 Il=If-Ir=0).

Having constructed the phasor Γ, we can use the impedance and admittance overlays already discussed.

So the same chart can be used, plot Γ the same way for voltage or current, and the rest of the chart applies.

There are times where the normalised current at a point is of interest, ie the resultant If+Ir or 1-Γ.

Above is the construction of 1-Γ. To read the angle of 1-Γ you could use a parallel rule to project it from the chart centre and read the angle from the “Angle of reflection coefficient in degrees” scale.

Above, the voltage and current phasors combined and calculation of Z reconciles with the original statement of the scenario.

The Smith Chart really is a chart of Γ with some convenient mapping scales overlayed, with some interesting transforms conveniently available by following the constant VSWR, constant R, constant X, constant G, and constant B arcs.

This article is NOT about “doing Smith Charts”. “Doing Smith Charts” is the traditional way of teaching Smith Charts, teaching the mechanics of solving some simple example transmission line problems (eg a single stub tuner)… without needing or imparting an understanding of the Smith Chart.

If you didn’t witness this approach in college, Youtube abounds with contemporary examples which seem the product of “I didn’t understand <subject> at all, so I thought I would do a video on it to learn about it.” The Γ phase missionaries are an example, belief over science, and popularity determines fact.

This article has explained the essence of the Smith Chart, a foundational concept that must be understood to truly understand the Smith Chart at large, and from which more elaborate problem solving is easily developed on classic transmission line theory.

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- Design / build project: Guanella 1:1 ‘tuner balun’ for HF – #1
- Design / build project: Guanella 1:1 ‘tuner balun’ for HF – #2
- Design / build project: Guanella 1:1 ‘tuner balun for HF’ – #3
- Design / build project: Guanella 1:1 ‘tuner balun for HF’ – #4
- Design / build project: Guanella 1:1 ‘tuner balun for HF’ – #5
- Design / build project: Guanella 1:1 ‘tuner balun for HF’ – #6
- Design / build project: Guanella 1:1 ‘tuner balun for HF’ – #7

- This article describes a measurment of common mode impedance Zcm of the packaged balun.

The prototype fits in a range of standard electrical boxes. The one featured here has a gasket seal (a PTFE membrane vent was added later).

Above, the exterior of the package with M4 brass screw terminals each side for the open wire feed line, and an N(F) connector for the coax connection. N type is chosen as it is waterproof when mated.

The interior shows the layout. The wires use XLPE high temperature, high voltage withstand, moderate RF loss insulation. Two short pieces of 25mm electrical conduit serve to position the balun core against the opposite side of the box, and a piece of resilent packing between lid and core holds the assembly in place.

The fixture described is not perfect, and it is challenging to measure common mode impedance of the packaged balun. The fixture here tries to use the shortest possible connections. The VNA is calibrated at its SMA jack.

Above, a N short circuit plug is screwed onto the N jack on the balun, and a single 0.5mm wire is zip tied to the knurled collar for connection to that end of the balun.

0.5mm wires are connected to each of the two screw terminals, and twisted together to make the connection to the other end of the balun.

Note that each end of the balun has its conductors shorted.

The twisted pair will be clamped to the outside of the VNA Port 1 SMA jack, and the single wire inserted into that jack… all done with the shorted possible wires.

Above, the prepared conductors.

Above, the twisted pair will be clamped to the outside of the VNA Port 1 SMA jack (using a modified clothes peg), and the single wire inserted into that jack… all done with the shortest possible wires.

The fixture is not perfect, more on that later.

Above, the measurement of impedance using s11 reflection.

Of most interest to me is the frequency range over which impedance is high. The above plot has markers at the lower and upper frequencies where |Zcm|>2000Ω. It remains above 1000Ω from 1.8MHz to 30MHz, so it is somewhat effective over that range.

Above is the R+jX plot with the markers displayed.

As mentioned, the fixture is not perfect. It is likely that the intrinsic self resonant frequency measured is a little low, and the response generally is squeezed a little left. The upper 2kΩ bound may well be over 20MHz.

]]>Where the antenna system incorporates ferrite elements, a possible / likely explanation is that loss in a ferrite core has been extreme and raised core temperature to the Curie temperature at which it quickly loses its magnetic properties.

In that scenario, theoretically, the complete temperature curve would look like this.

The initial rate of temperature increase here is 5°/s, and we can safely assume that almost all of the power absorbed by the core is stored as heat energy, little energy is lost the the air when the temperature difference is very small.

The core will never reach Tmax, temperature increase is terminated when the Curie temperature is reached and the core will be at 130° or more… likely to be sufficient to cause damage to wire insulation and enclosure.

There is something seriously wrong!

So this raises the question of just how much power does it take to increase the temperature of a FT240-43 core by 5°/s.

The Heat Capacity of ZnMn ferrite is around 1050J/kg/K, and the mass of the FT240-43 is around 0.12kg… so the energy to raise it 5° is 1050*0.12*5=630J. The average power \(P=\frac{E}{t}=\frac{600}{1}=630 \text{ W}\)

Is it feasible to supply 630W of RF to an antenna balun or transformer from a transmitter capable of 1500W (or more). Certainly, and especially if there is a fault in the antenna system.

In this case, the heat input is extreme and the core reaches Curie temperature in just 30s. More subtle failure may occur where the average core heating power us just 10 or 20W and the temperature rises slowly past the safe temperature for the enclosure and wire insulation.

]]>Let’s walk through an example.

Above is an example for discussing the Gamma Match. In this case, the assumed feed point impedance of a simple split dipole feed is 17+j2Ω, and the challenge is to design a practical Gamma Match to match it to 50+j0Ω.

The design tool assumes that the connections to the open circuit stub are at the feed point, ie that the gap in the gamma arm outer is at the inboard end. There are other ways to build a gamma match and the model may not suit them without tweaking.

It can be approximated as three distinct steps of impedance transformation:

- impedance transformation by virtue of the parallel dipole conductor and gamma conductor, the diameters of each and their spacing are used in the calculation;
- shunt short circuit stub formed by the gamma arm and dipole conductor; and
- series open circuit stub.

Step 1 is the transformation due to two parallel conductors with mutual inductance and self inductance, immersed in an approximately uniform electric field. The ratio of the current in both (in common) to the current in the gamma arm sets up Step 1 of the transformation. Choose practical values for Dood, Gdia, and Gspa to get a transformed G of <0.02, the example above uses 0.0117.

Step 2 is the transformation due to the shunt short circuit stub. Choose practical values for Ciod and Coid (the dimensions of the coaxial line so formed) and adjust its length so the the magenta arc just intersects the R=50 circle.

Step 3 is the transformation due to the series open circuit stub. Set the VF2 and stub length to land on the chart centre, 50+j0Ω.

This can be an iterative process, especially for instance if the length of the open circuit stub is greater than the length of the short circuit stub. Optimisation may involve changing some of the Step 1 parameters to stage for easier transformations in Steps 2 and 3.

]]>This article provides an updated Simsmith model that incorporates the necessary calculations (ie without depending on external calculators).

Much is written about the virtue of the Gamma Match, and near as much about how they work, and the difficulty in design and implementation.

Designing a Gamma match using a Smith chart showed a design method for a simple Gamma Match using a Smith chart as the design tool.

This article visits the implementation on a pair of antennas that I built 50 years ago, and are still in use today (albeit with some small preventative maintenance once during that interval). The basic antenna is a four element Yagi for 144MHz copied from an ARRL handbook of the time, probably based on NBS 688. It was designed to deliver a split dipole feed point impedance of 50+j0Ω.

I built them using a Gamma Match, partly to get some familiarity, but mostly to implement a Gamma Match that was reliable, weatherproof and lasting… features that are alien to most implementations I had seen at that point.

Both antennas were constructed and the Gamma Match adjusted for VSWR<1.1 using a Bird 43 directional wattmeter. The dimensions of each (including the key gamma dimensions) are the same, not surprising, but a confirmation of repeatability. See Novel Gamma Match Construction for more discussion.

Above is a dimensioned drawing of the construction.

Above, the 1970 build pictured in 2023.

The design will be reviewed using the method set out at Designing a Gamma match using a Smith chart.

The Simsmith model defines some key parameters in the Parameters element:

- Dood: outside diameter of the dipole tube;
- Gdia: outside diameter of the gamma tube;
- Gspa: centre to centre spacing of dipole and gamma tubes;
- Ciod: outside diameter of inside conductor of capacitive stub;
- Coid: inside diameter of outside conductor of capacitive stub; and
- VF2: velocity factor of capacitive stub.

The Parameters element calculates some intermediate values and populates fields of the following elements.

Above is the solution of the example 144MHz 4el gamma matched Yagi.

The lengths of the gamma arm and series sections reconcile well with the built and adjusted antennas.

The Simsmith file is here: GammaArm144Yagi.7z.

]]>One of the very common reports is of something unexpected happening while adjusting an ATU, after perhaps 30s of power applied, VSWR suddenly becomes unstable, changing for some unknown reason, and attempts to find optimal settings of the ATU fails.

A likely cause of this is non-linear behavior of the ferrite core in a balun in the system.

Let’s talk about that.

A simple model that gives useful insight is to consider the case of a toroid core in still air, being heated by constant applied RF power giving rise to core loss.

Core temperature rises quickly initially, then more slowly as the core heats up and loses more and more heat to the surrounding air.

We can write and expression for core temperature T: \(T=T_{max}\left(1-\mathrm{e}^{-\frac{t}{\tau}}\right)\) where τ is the thermal time constant and Tmax is the final temperature if things continued without disruption.

Above is an example where τ=20s and temperature rise is 100°.

Note that at the beginning (t=0), the slope of the line is pretty constant, but as temperature increase, slope decreases until eventually it is almost zero.

The slope can be calculated by differentiating the expression above: \(\frac{dT}{dt}=\frac{T_{max}\mathrm{e}^{-\frac{x}{\tau}}}{\tau}\).

At t=0, \(\frac{dT}{dt}=\frac{T_{max}}{\tau}\), \(T_{max}=\tau \frac{dT}{dt}\).

From the graph, by eye we can estimate the slope near t=0 to be about 5°/s and τ=20 (the time to reach 63% of Tmax) and calculate that Tmax is 5*20=100°.

Let’s now look at a practical device using ferrite toroid using a FT240-43 core initially at 30°.

Ferrite materials are subject to Curie effect, above a certain temperature (known as the Curie temperature) they rapidly lose their magnetic properties (temporarily or permanently).

For #43 material, that is specified at 130° or higher.

We can also measure the thermal time constant τ for the FT240 core, and find τ=2000.

We apply RF power and find that after 30s, the indicated VSWR changes without external change, the core has reached its Curie temperature and is losing its magnetic properties.

Because the time observed, 30s, is very small compared to τ=2000, we can reasonably estimate the slope of the response near t=0 as \(\frac{T_{rise}}{t}=\frac{100}{20}=5 \text{°/s}\) and go on to calculate \(T_{max}=\tau \frac{dT}{dt}=2000 \cdot 5=10000 \text{°}\).

Theoretically, the complete temperature curve would look like this.

The core will never reach Tmax, temperature increase is terminated when the Curie temperature is reached and the core will be at 130° or more… likely to be sufficient to cause damage to wire insulation and enclosure.

There is something seriously wrong!

The observation of this behavior is a warning that something is not right, and system components should be carefully inspected for faults and possible consequent damage (eg overheating of a ferrite core as a result of some other fault).

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The thinking is often “my balun is rated at 5kW, how can I damage it with a transmitter that is only 100W?”

This article discusses heat dissipation from a small sealed plastic enclosure. This is not an unusual problem, it exists widely in industry and is solved routinely for enclosures ranging from small ones like these to large switchboard enclosures.

Above is a balun enclosure located under the soffit of the building, so outside in free air and where the sun will not shine on it.

This box is ABS, has a surface area of 0.034m^2, and will probably withstand 80° before there is risk of deformation.

Allowing for a maximum ambient temperature of 40° for this location, a temperature rise of 80-40=40°=40K can be accommodated.

If the box surface is of uniform temperature, we can expect that the enclosure can dissipate around 5W/m^2/K from internal sources in still air, so about 7W. (Some manufacturers publish data from which a base figure outside in still air in the range 3 to 8 W/m^2/K can be derived.)

Of course dissipation would be higher in a breeze, lower if subject to external heat (such as insolation), and lower if the box surface temperature is not uniform.

Because of the way the core assembly is clamped to the base of the box, I would rate it at 5W because of less than uniform temperature distribution.

Note that a similar sized PVC box might not be safe above 60° (depending on its formulation), so temperature rise 60-40=20°=20K, leading to maximum dissipation of 3.5W on the same rationale.

Ferrite cores have quite high heat capacity, it may take more than half an hour for the core to reach even half of its final temperature given constant power input. This means that there is thermal averaging and the core temperature will depend on power input averaged over many minutes, perhaps even half and hour, depending on core size.

So, it is the power averaged over minutes, perhaps even as long as 30min that applies to the problem.

The case of a transmitting balun is more challenging than the common electrical enclosure in industry as power dissipated is dependent on many external factors.

Several approaches are in common use:

- ignore it;
- copy others, hope for success;
- depend on manufacturer’s power rating;
- calculate power dissipation from manufacturer’s loss specification and guestimate safe dissipation.

None of these properly take account of your own scenario.

Calculation of temperature rise based on manufacturer’s power rating or published device loss is fraught with problems.

The simplest method that has some reliability is the progressively increase power whilst observing exterior temperature of the enclosure. A simple temperature test is that if you can touch the thing for 10s comfortably, without getting burnt, it does not exceed 60°. This might not be very convenient if it is hoisted aloft and some other method might need to be devised.

Keep in mind the very long thermal time constant and averaging, and that average to peak ratio is dependent on modulation type and on/off duty cycle. If you have proven that it stands 1kW PEP SSB telephony for 5min, that does not prove it will withstand a RTTY contest hour on hour.

Realise also that the mechanism is one of temperature rise approximately proportional to power dissipated. If you observe a barely acceptable box temperature on a cold day, the temperature rise is high and the same power scenario may be damaging on a hot day.

Enclosures subject to high mechanical forces, like suspending the spans of a dipole from eyebolts fixed to the enclosure, may deform at lower temperatures.

Under fault conditions, dissipation may change for the worse and one of the consequences of a fault in an antenna system is balun damage. If you observe changes in antenna system VSWR, or ATU settings, investigate the cause.

If a station is capable of peak power of say 1kW, and the balun enclosure is capable of dissipating say 5W, that is 0.5% of PEP. Whilst that might work for a given modulation type, transformer / choke loss profile, antenna impedance / balance… it might not provide much wiggle room if something goes wrong with the antenna system.

Baluns that are marginally rated for your own scenario might just be an expendable component when things change.

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Above is an air cored solenoid of about 20µH connected between Port 1 and Port 2 of a NanoVNA-H4 which has been calibrated. The whole lot is sitting on an inflated HDPE bag to isolate the DUT from the test bench.

Note that the calibration used a 250mm RG174 jumper from Port 1 to Port 2, and it is not in place for the measurement, so a correction will be made later.

Measurements will be made at 2MHz so that lead lengths are relatively insignificant.

The 20µH inductor has a reactance of 250Ω at 2MHz, and some series resistance of the order of a couple of ohms.

We would expect:

- severe mismatch at Port 1, so relatively high |s11|;
- little conversion of RF energy to heat in the coil, so relatively low loss; and
- quite a reduction in power reaching Port 2, so reduced |s21|.

Above is a screenshot showing |s11| and |s12| which are the inputs to the calculation process to find Loss, Insertion Loss, and Mismatch Loss.

The expected loss in the missing 250mm of RG174 mentioned earlier is 0.009dB at 2MHz, it is so small it will be ignored. There is no need to compensate for the phase difference of s21 as the phase is not used in the calculation.

For convenience, this online calculator will be used.

Above, the adjusted measurement data is entered and results calculated. Not that the angles of |s11| and |s21| were not entered, they are not used in this calculation.

As expected:

- severe mismatch at Port 1, |s11|=-0.6960dB and MismatchLoss=8.295dB;
- little conversion of RF energy to heat in the coil (and possibly some RF radiation), Loss=1.2dB;
- quite a reduction in power reaching Port 2, |s21|=-9.6dB giving and InsertionLoss=9.512dB.

The main contribution to InsertionLoss is MismatchLoss.

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