The expansion of P=real((Vf+Vr)*conjugate(If+Ir)) gives rise to four terms.

This article looks at the components of that expansion for a mismatched line for a range of scenarios.

- Lossless Line;
- Distortionless Line; and
- practical line.

We will override the imaginary part of Zo and the real part of γ (the complex propagation coefficient) to create those scenarios. The practical line is nominally 50Ω and has a load of 10+j0Ω, and models are at 100kHz.

A Lossless Line is a special case of a Distortionless Line, we will deal with it first.

A Lossless Line has imaginary part of Zo equal to zero and the real part of γ equal to zero.

Above is a plot of the four components of power and their sum at distances along the line (+ve towards the load).

Term2 and Term3 exactly cancel at all points along the line, and the total power at any point is simply Term1+Term4. Term1 is often called Pfwd and -Term4 is often called Prev.

Pfwd and Pref are constant along the line, it has zero loss.

Above, a plot of standing wave voltage and VSWR calculated from ρ (the magnitude of the complex reflection coefficient Γ). The standing wave pattern is uniform by virtue of zero line loss, and the calculated VSWR is correct.

A Distortionless Line has imaginary part of Zo equal to zero, for the model we will set the real part of γ equal to that of the practical line.

Above is a plot of the four components of power and their sum at distances along the line (+ve towards the load).

Term2 and Term3 exactly cancel at all points along the line, and the total power at any point is simply Term1+Term4. Term1 is often called Pfwd and -Term4 is often called Prev.

Pfwd and Pref are not constant along the line, it has non-zero loss.

Above, a plot of standing wave voltage and VSWR calculated from ρ (the magnitude of the complex reflection coefficient Γ). The standing wave pattern is not uniform as a result of line loss, and the calculated VSWR is a bad estimator (bad extrapolation due to significant line loss).

A practical line has non-zero imaginary part of Zo and non-zero real part of γ, both set to those of the practical line.

Above is a plot of the four components of power and their sum at distances along the line (+ve towards the load).

Term2 and Term3 do not exactly cancel at all points along the line, and the total power at any point is Term1+Term2+Term3+Term4. Term1 is often called Pfwd and -Term4 is often called Prev.

Because Term2 is not equal to -Term3, we can not simply say that P=Pfwd-Prev.

Pfwd and Pref are not constant along the line, it has non-zero loss.

Above, a plot of standing wave voltage and VSWR calculated from ρ (the magnitude of the complex reflection coefficient Γ). The standing wave pattern is not uniform as a result of line loss, and the calculated VSWR is a bad estimator (bad extrapolation due to significant line loss).

Lossless lines have behavior that is quite simple to predict. Lossless lines do not exist in the practical world, nevertheless the case remains an interesting one as an approximation of practical lines.

Distortionless Lines with loss become more complicated. Distortionless Lines with loss are very very rare in the practical world, nevertheless the case remains an interesting one as an approximation of practical lines.

Practical lines are the stuff of the real world. Much of the convenience of Lossless Line analysis is not strictly available.

Nevertheless, Lossless Line and Distortionless Line analysis techniques may provide an adequate approximation for practical lines in some circumstances.

As said, distortionless or lossless analysis techniques may be adequate approximations. The matter of adequate is a judgement by the analyst considering the purpose and hand, needed accuracy, cost etc.

Lets discuss key issues:

- assumption of distortionless behavior, ie that Term2=-Term3;
- assumption of lossless behavior, ie that Term2=-Term3 and real(γ)=0;
- extrapolation of VSWR from Prev/Pfwd.

If we want to perform distortionless analysis on a practical line, that P is approximately Pfwd-Prev, we need Term2+Term3 to be small compared to Term4.

We can compare the value of Term2+Term3+Term4 to Term4 to derive an uncertainty in Prev=P-Pfwd when Term2 and Term3 are ignored.

Above is a plot of the limit of uncertainty as a function of |Xo|/Ro and ρ (ρ=|Vr/Vf|).

To use lossless analysis, the limits for distortionless analysis apply and attenuation must be negligible over the length of line analysed.

To reasonably accurately extrapolate VSWR from Prev/Pfwd (-Term4/Term1 or ρ^2) at some point, the limits for distortionless analysis apply and attenuation must be very negligible over region of line subject to extrapolation (eg +/- λ/4).

This example explores a scenario where Distortionless Line analysis might be of acceptable accuracy.

This example again shows the same line type and load, but frequency increased to 10MHz where |Xo|/Ro=0.016.

Term2 and Term3 do not exactly cancel at all points along the line resulting in a small sinusoidal variation in the Sum of the terms.

Above, a plot of standing wave voltage and VSWR calculated from ρ (the magnitude of the complex reflection coefficient Γ). The standing wave pattern is not quite uniform as a result of line loss, but the calculated VSWR is not such a bad estimator in the region of the calculated point.

None of this speaks to other error in Zo. Modelling, measuring and calculating with Zo different to the actual Zo of the DUT gives rise to its own error.

This article speaks of the case where Xo is taken as zero (Distortionless Line) and loss taken as zero (Lossless Line), but of course using the wrong value for Ro also leads to error.

]]>SimSmith uses different transmission line modelling to what was used in that article, but a SimSmith model of RG58A/U allows illustration of the principles and it will deliver similar results.

Let’s explore the voltage maximum and minimum nearest the load to show that VSWR calculated from the magnitude of reflection coefficient is pretty meaningless in this scenario.

Above is the basic model. I have created two line sections, one from the load to the first voltage maximum, and another to the first voltage minimum where I have placed the source. I have set Zo to the actual Zo of the line as calculated by SimSmith (56.952373-j8.8572664Ω), *effZ* as SimSmith calls it, so the Smith chart relates to the real transmission line.

You will note that the load is outside the chart, it is because of the load value and the chart reference (56.952373-j8.8572664Ω). It also happens that the complex reflection coefficient Γ is 1.06∠98°, that’s fine.

See how the path follows a smooth spiral inwards due to the line attenuation, reaching the point of the first voltage maximum.The voltage maximum occurs where 1+Γ reaches a maximum, it also corresponds to the point that is the greatest distance from the left hand extremity of the chart. At this point, Γ1=0.810∠1.7° and 1+Γ1=1.810∠0.761°.

Continuing on, we reach a voltage minimum where 1+Γ is minimum, in this case Γ2=0.501∠-173° and 1+Γ2=0.5064∠6.925°.

Just a small side task, let’s calculate the two way voltage gain of that last 421m section of line from the spiral. It is |Γ2|/|Γ1|=0.501/0.810=0.6185, and the one way voltage gain OWVG is 0.6185^0.5=0.7865.

Ok, now let’s calculate the ratio of the nearby voltage maximum to the voltage minimum, the VSWR. VSWR=|1+Γ1|/|1+Γ2|*OWVG=1.810/0.5064*0.7865=2.811. (This reconciles with the displayed value for V at each end of T1.)

You will note that SimSmith calculates the VSWR at the input end to be 3.009, and half way between the minimum and maximum it shows 4.708 (see graphic), and 9.5 at the voltage maximum.

None of the SimSmith calculated VSWR figures give a hint of the VSWR as measured above. The problem is not SimSmith’s calculations, it is that the assumptions on which calculating VSWR do not apply in this scenario (as discussed at On negative VSWR), the user does need to understand transmission lines to know what figures are valid in the scenario at hand.

- Setting generator Zo to the cable Zo is done by inserting the formula
*G.Zo=T1.effZ;*into the Plt box. - Calculating 1+Γ can be done with a hand calculator, but you may find this online calculator convenient:
- There appear to be defects in SimSmith’s handling of a complex chart reference (eg some arcs appear wrongly scaled).
- An eagle-eye questioned that the length of T1 is not exactly 90°. We commonly talk of the distance between voltage maximum and minimum being 90°, but that is exactly correct
**only**when Zo is purely real.

Distortionless Lines (and Lossless Lines are a special case of Distortionless Lines) have purely real Zo.

Practical transmission lines almost never have purely real Zo, Zo usually has a non-zero imaginary part, even if very small.

The maths of transmission line behavior in terms of known Zo, Zload, and propagation constant γ is sound, solutions exist for the complex reflection coefficient Γ along the line.

ρ, the magnitude of Γ, is often derived but we should remember that in discarding the phase of Γ we know less.

However, ρ leads to some interesting inferences but **onl**y in the case where Zo is purely real.

ρ<=1.

Attenuation is uniform with displacement.

We can talk of the notion of forward and reverse power (Pfwd and Prev) as components of power at some point along the line, irrespective of the ratio V/I at that point (ie independent of Z at that point).

We can speak of Return Loss being Pfwd/Prev calculated from ρ as ρ^-2, or more commonly in dB as -20*log(ρ).

We can speak of Mismatch Loss based on ρ as -10*log(1-ρ^{2}), but only when the Thevenin equivalent source impedance is exactly the same as Zo (and of course by virtue of that, purely real).

Additionally we can infer VSWR in that region of the line based on ρ (VSWR=(1+ρ)/(1-ρ)), but **only** when line loss is very low.

Within those restrictions, calculated values are sane.

When Zo is approximately real and approximately equal to that of the real line, we can do all those things listed under “When Zo is purely real”, but there will be some error, and if the results seem silly, it is not the formulas but the misapplication that is the problem.

The error will be less where:

- the imaginary part of Zo is relatively very small (-Xo/Ro<<<1); and
- ρ is very small (ρ<<1).

If results look insane, better review the basis for your approximation.

Measuring instruments are almost always calibrated for purely real Zo or Zref, and as such and within the limits of accuracy, all those things listed under “When Zo is purely real” can be done with good accuracy… but the measurer must keep in mind that the measurements are not in terms of the adjacent things and that itself is a source of error.

An article on the practical pitfalls with only a small imaginary part to Zo is given at On Witt’s calculation of Matched Line Loss from Return Loss. Notwithstanding the failure, lots of ham grade antenna analysers have a “Matched Cable Loss” function that falls foul of the discrepancy.

]]>The Superposition Theorem is an important tool in linear circuit analysis, and is used to find the combined response of independent sources. Superposition applies to voltages and currents, but **not** power.

To find the real power P at a point in a transmission line, we need to find the real part of the product of voltage and current considering their phase difference. We can write P=real(V*conjugate(I)).

The voltage at a point is that due to superposition of the voltages due to forward and reflected waves, V=Vf+Vr. Likewise for current, I=If+Ir.

So now we can write P=real((Vf+Vr)*conjugate(If+Ir)). When this is expanded, there are four terms, a VfIf term, a VrIr term and two cross products.

Practice is to call the VfIf term, Term1, forward power Pfwd, and magnitude of the VrIr term Term4 reverse (or reflected) power Prev, and to ignore the VfIr and VrIf cross product terms Term2 and Term3, but that is not always safe as they may each contribute significant positive (forward) and negative (reverse) power at various points along the line.

Lets look at the power calculated from voltages and currents for the example at 100kHz where Zo=50.71-j8.35Ω and Zload=5+j50Ω.

Above, the four component terms are plotted along with the sum of the terms.

Note that at the load the magnitude of Term4, the so-called Prev, is greater than Term1, the so-called Pfwd but this does not lead to negative real power (ie power flowing from load to source) because P=Term1+Term2+Term3+Term4 (Sum curve) and it positive at all displacements and monotonically decreasing from source to load (ie there is no hidden power gain in the maths). For the same reason, Return Loss calculated from Term1 and Term4 alone is questionable.

A property of Distortionless Lines (and Lossless Lines are a special case of Distortionless Lines) is that Zo is a purely real number.

When Zo is real, the phase difference between Term2 and Term3 is 180°, and since their amplitudes are equal, they cancel leaving P=Term1+Term2 and we can say P=Pfwd-Prev.

In the real world, practical transmission lines are not Distortionless, but the case for real Zo can often be used as a good approximation where:

- the imaginary part of Zo is relatively small (meaning that Term2 and Term3 almost cancel); or
- ρ is small (meaning that the magnitudes of Term2 and Term3 are small).

Measurement instruments are almost always calibrated for a purely real reference impedance.

Measurements in that context should always deliver results consistent with purely real Zo, but keep in mind that what is indicated is not strictly what is happening on the adjacent real world transmission line.

In the context of instruments calibrated for real Zo, P=Pfwd-Prev is sound, ReturnLoss=20*log(1/ρ) is sound.

]]>Return Loss is defined as the ratio Pfwd/Prev, often given in dB.

Return Loss is usually calculated as 20*log(1/ρ), it yields negative calculated Return Loss for ρ>1. It would be a mistake to doctor the result to hide the negative return loss as it is a strong hint that the results may be invalid.

An important consideration here is the validity of the concept of Pfwd and Prev.

The forward and reverse waves are subject to superposition, but it is voltages and currents that superpose, not power.

The relation that P=Pfwd-Prev is valid only when Zref is purely real (see Power in a mismatched transmission line).

So, to speak in Return Loss is really only valid when Zref or Zo is purely real or approximately so.

Measurement instruments are almost always calibrated for a purely real Zref.

Almost all practical low loss transmission lines have approximately real Zo above 50MHz, and most are good down to 1MHz. Transmission lines using poor conductors, or composite conductors with thin cladding may depart significantly below 50MHz.

On negative VSWR – a worked example discussed various aspects, but did not touch on Return Loss.

Above is the plotted Return Loss (wrt Zo=50.71-j8.35Ω). The negative value plotted near to the load flags the underlying problem that Zo is not real, and therefore the calculated Return Loss may be invalid.

]]>This article exposes an example at 100kHz where Zo=50.71-j8.35Ω and Zload=5+j50Ω.

If we were to use a probe to directly measure the magnitude of line voltage, we would expect the following.

Above, the standing wave plot. At first appearance it might look like a classic standing wave plot, but it is not… there is a tiny difference in the shape at the right hand side.

If you put a ruler to it, you might estimate the VSWR over that region to be around 2.8. But, note that you cannot **directly** measure VSWR at a point, it is not a point property in the general case.

Above is a plot of the calculated magnitude and phase of the complex reflection coefficient Γ, displacement is negative from load to source.

There is nothing particularly anomalous about this data, except that ρ, the magnitude of the complex reflection coefficient Γ, is just a little greater than unity at the load (0m displacement).

There is a point where ρ=1. If you try to calculate VSWR=(1+ρ)/(1-ρ) there is a singularity at ρ=1 and it yields the following plot.

Above, notwithstanding that VSWR is not a point property, a plot of calculated VSWR at points along the line using the lossless line formula. The singularity where ρ=1 causes extreme values of VSWR either side of it.

The chart shows calculated VSWR way higher that implied by visual interpretation of the standing wave plot given earlier.

]]>Considering the meaning of VSWR: the ratio of the voltage maximum on a long transmission line to the adjacent voltage minimum, calculated negative VSWR might seem an aberration, invalid even. Note that nothing in this definition makes VSWR a property of a dimensionless point on a line.

VSWR can be measured directly by sampling voltage along a transmission line with a voltage probe. That said, it is almost never done and VSWR is **inferred** from other measurements, usually point measurements.

A transmission line is free to carry waves in two directions, and the ratio of voltage to current for each of those waves is the characteristic impedance Zo.

When a long line is terminated in some impedance Zl, the wave components Vf and Vr, If and Ir must reconcile with Vload and Iload, the variable that accommodates this is the complex reflection coefficient Γ (see Telegrapher’s Equation). Zo of a lossless line is always a purely read number, it no imaginary part.)

If such a line is lossless, it is easy to see that as we travel back towards the source, the phase of the forward and reflected voltage components vary (the magnitude remains constant, it is a lossless line), and so at some points the voltage will be a maximum of |Vf|+|Vr| and at some other points the voltage will be a minimum of |Vf|-|Vr|.

In such a case, we can see that the ratio of Vmax to Vmin is given by (Vf+Vr)/(Vf-Vr) or (1+|Γ|)/(1-|Γ|). ρ is sometimes used to represent |Γ|, so you will also see VSWR=(1+ρ)/(1-ρ).

The last function is invertible, we can write ρ=(VSWR-1)/(VSWR+1);

In the lossy line case things are different on two important counts:

- Zo is a complex value with a small but sometimes significant imaginary part;
- loss means that directly measured voltage maxima and minima will not reconcile with value ρ derived at an arbitrary point, ρ varies along the line.

In some scenarios, application of the lossless line based formulas above will result in significant error, it is incumbent on the user to prove suitability of the results to the scenario.

These errors are mostly ignored until something that seems on the surface quite wrong is encountered, for example a negative VSWR result… it **must** be dodgy? Some propose that it can be fixed by modifying the formula to prevent a negative result… but that is taking a formula that is not valid for the scenario and doctoring the result to ‘fix’ the sign problem to make it look more pleasing but ignoring the strong hint that the formula is not suited to the scenario. It is a kind of fool’s paradise.

Note that the VSWR=f(ρ) and ρ=f(VSWR) expressions given earlier are mathematically sound even if the meaning of that VSWR seems of no use. For example, if VSWR is calculated to be -20, the calculation ρ=(-20-1)/(-20+1)=1.1 is sound and that value of ρ could occur in practice.

The VSWR formulas given in the lossless lines section above depend on that assumption and may have significant errors when applied to some scenarios of lossy lines.

The specific case of negative VSWR arises from scenarios where ρ>1.

This will never happen when Zref is purely real. If you make measurements with an instrument calibrated for a purely real Zref, you will never correctly observe ρ>1. This is true even if such an instrument is inserted in a line section where ρ>1 within the line section (ie when measured wrt the actual line Zo).

If you are working with modelling tools that use complex values of Zo and Zload, and ρ is greater than 1 in the specific scenario, use Γ by all means but realise that as soon as you derive or use values for VSWR you have discarded important information and the result may have error, perhaps significant error.

]]>**Inductance** of a conductor is the property that a change in current in a conductor causes a electro motive force (emf or voltage) to be induced in a conductor.

We can speak of **self inductance** where the voltage is induced in the same conductor as the changing current, or **mutual inductance** where the changing current in one conductor induces a voltage in another conductor.

At low frequencies, the current is distributed uniformly inside the conductor, and its self inductance can be calculated readily (formula is in most good text books). For example, the self inductance of a 2mm diameter round copper conductor is about 1370nH/m. Note that this includes the effect of flux within the copper conductor, **internal inductance**, it contributes 50nH/m.

At low frequencies, the current is distributed uniformly inside the conductor, and its self inductance can be calculated readily (formula is in most good text books). For example, the self inductance of a 10mm diameter thin round copper conductor is about 998nH/m. There is no magnetic flux inside the tube as there is no current flowing there to create flux.

If we locate a round conductor concentrically or coaxially within a hollow tubular conductor (shield), there is not only the self inductance L1 and L2 of each conductor respectively at play, but the mutual inductances M12 and M21. M12 and M21 are equal to each other, and by virtue of the fact that all of the flux of L2 is shared with L1, M12 and M21 are equal to L2.

If we consider the series path of current flowing in the inner conductor and returning via the outer conductor, we have L=L1-M21+L2-M12 and given M12=M21=L2, we can write L=L1-L2 which tells us that the inductance is due to the flux inside the shield, there is no flux outside the the shield.

If the example 2mm and 8mm conductors were arranged coaxially, the return circuit inductance would be L1-L2=1370-998=372nH.

The effect of self inductance is to cause current to concentrate in the area where self inductance is least. This effect is more pronounced as frequency increases and because tends to flow mainly near the surface it is commonly known as skin effect. We can think of the current density as decaying exponentially with increase depth, and although it is more complicated than that, this is an adequate model for this discussion. We speak of the skin depth δ as the depth that carries 63% of the total current and often make the assumption that the total current is carried in a depth of 3δ, it that there is insignificant current at greater depth.

Skin effect implies that at sufficiently high frequency, the current on the inner conductor flows mainly near the outer surface. A consequence is that the internal inductance approaches zero.

Skin effect implies that at sufficiently high frequency, the current on the outer conductor flows mainly near the inner surface.

With well developed skin effect, the outer conductor behaves almost like two independent / isolated tubes being the inner and outer surfaces with negligible current flowing in the region between those ‘layers’.

Coaxial cable is usually used in TEM (Transferse Electro Magnetic) mode, magnetic flux is circular within the coax, and radial electric field exists between the outside surface of the inner conductor and inside surface of the outer conductor (there is no magnetic flux due to inside currents outside the coax). Other modes are possible at some frequencies, but they cause higher loss and are usually discouraged.

In TEM mode in the presence of well developed skin effect, a current I flowing at a point along the coax on the outer surface of the inner conductor is accompanied by an equal current flowing in the opposite direction on the inner surface of the outer conductor. This is a really important attribute to consider when analysing a system.

In the presence of well developed skin effect, the combination of self inductance and mutual inductance of each of the outer surface of the inner conductor and inner surface of the outer conductor result in no external magnetic or electric fields, and the constraint that a current I flowing at a point along the coax on the outer surface of the inner conductor is accompanied by an equal current flowing in the opposite direction on the inner surface of the outer conductor.

These if you like define what is going on inside the coax as a private scope that is affected only by the transmission line characteristics, load impedance and source. Calculators such as RF Transmission Line Loss Calculator provide solutions to this problem.

In the presence of well developed skin effect, the outside surface of the outer conductor can carry currents independent of what his happening inside the coax. The fields due to these currents are entirely external to the outside surface of the outer conductor, and the outer conductor is free to interact with other sources of magnetic and electric fields.

In the presence of well developed skin effect, the inner and outer surfaces of the outer conductor are effectively isolated, but they connect to each other at the ends of the coaxial structure.

One can analyse the configuration by considering that there is a node formed at the shield end, and connected to that node are the inner and outer surfaces of the shield and any other external conductors. At the node, Kirchoff’s Current Law applies.

Two scenarios will be analysed to demonstrate an approach to the problems.

An ideal resistor for this discussion is one that is electrically so small that we can ignore distributed capacitance and inductance and phase change of current through the resistor.

If a ideal resistor is attached to the cut of of a coax cable and excited from the other end, the current flowing from the inner conductor to the resistor flows entirely to the inner surface of the outer conductor, there is no residual to flow to the outer surface of the outer conductor. This is independent of whether there are standing waves on the inside of the coax (ie whether the load resistor equals Zo).

Nothing above prevents current flowing on the outer surface of the outer conductor due to other excitation, but none will flow into the inner of the coax.

The configuration is a coaxial cable directly attached to the centre of a practical half wave dipole. For the purpose of discussion, the dipole is not perfectly symmetric and the current flowing into one leg is not exactly equal to that from from the other leg.

Lets call the current from the centre conductor to one dipole leg I1, and the current from the other leg I2. I2 flows into the node formed by the connection of the inner surface of the other conductor, other surface of the outer conductor and the dipole leg. By Kirchoff’s Current law, the current flowing into the outer surface of the outer conductor at the node is I2-I1, and that current (often termed common mode current) gives rise to external fields (including radiation).

Likewise, voltage induced into the outer surface of the outer conductor by external source will flow into the same node and divide between the attached dipole leg and inner surface of the outer conductor. The current flowing to the dipole leg causes a current in the other dipole leg providing the complementary differential current in the interior of the coax, eventually delivering energy to the remote load.

]]>(Holbrook and Dixon 1939) explored the subject measuring the voice characteristics of many talkers (as there is variation amongst talkers) to come up with an average characteristic.

Whilst in its day, obtaining instantaneous samples of voice was a challenge, it is trivial today and if you can’t believe the numbers given, try your own experiment (but realise it is for your own voice rather than the general population).

Many modern PC sound applications are capable of the measurement, I will demonstrate it with the feed Windows application Audacity with the stats.ny addin.

Above is a screenshot of a 6s recording of my voice made without stopping for breath. The statistics window shows a peak of -8.9dBFS and RMS of -27.4dBFS, giving a peak voltage to RMS voltage ratio of 18.5dB.

On repeated trials it is within tenths of a dB. If you try the experiment, keep your voice level constant, don’t stop for breath, don’t pause as you might in reading sentences as all these will result in an overestimate of instantaneous peak voltage to RMS voltage. Make sure the peak is well less than 0dBFS, otherwise you will underestimate instantaneous peak voltage to RMS voltage.

(Holbrook and Dixon 1939) gave the graph above which characterises the ratio of instantaneous peak to RMS voltage of voice telephony for different numbers of channels in a multiplex and different expectation of overload or clipping.

My measured instantaneous peak voltage to RMS voltage of 18.5dB reconciles well with (Holbrook and Dixon 1939) approximate operating limit (the dashed line) for n=1 channels.

Remember that PEP is 3dB less than the instantaneous peak voltage indicates, so in the measurement PEP/Pav=18.5-3.0=15.5dB. The chart suggests PEP/Pav=18.0-3.0=15dB.

If you have seen figures of Pav/PEP of 20% (-7dB) or PEP/Pav=5 (7dB) bandied around for uncompressed SSB telephony without experimental evidence or explanation, you might question the credibility of the source.

This is a measurement at source of my voice and the recorded data shows that no clipping took place.

Different voices may produce different results.

Lower Peak/Average is almost always a sign of non-linearity (eg peak clipping, compression etc) and warrants further tests of system linearity.

- B D Holbrook and J T Dixon. Oct 1939. Load Rating Theory for Multi-Channel Amplifiers” in Bell System Technical Journal, Vol. 18.

There are common some key properties that are relevant:

- where loss is high, core loss tends to dominate;
- the specific heat of ferrite is typically quite high;
- the capacity to dissipate heat is related to many factors.

Ferrite materials have loss at HF and above that warrants consideration.

Even though the effective RF resistance of conductors is much higher than their DC resistance, the wire lengths are short and conductor loss is usually not very high.

Core loss will commonly be much larger that conductor loss and so dominate.

The specific heat of ferrite is typically towards 800K/kgK, almost as high as aluminium so ferrite absorbs a lot of heat energy to raise its temperature.

When heated by a constant source of power, temperature will rise exponentially as a result of the combination of mass, specific heat, and loss of heat from the core as temperature increases. We can speak of a thermal time constant being the time to reach 63% of the final temperature change, and for large ferrite toroids (eg FT240) that may be over 2000s.

Factors include the temperature difference between the core and ambient and if you like, the thermal resistance between core and ambient. Ambient temperature may be high if the device is installed in a roof space. Incident heat from the sun increases the challenge.

Maximum core temperature depends on maximum operating temperature of the enclosure (PVC), wire insulation maximum temperature, fasteners (eg nylon screws or P clips), and Curie temperature all weigh in.

Thermal resistance is higher where the core is contained in a closed enclosure.

Lets say a EFHW transformer using a FT240-43 is housed in a small sealed PVC box mounted outside in fee air. The transformer uses a 2t primary winding as per a plethora of articles on the ‘net.

Above is a core loss profile for the transformer where the load is such that the impedance looking into the primary is 50+j0Ω. At 3.5MHz, core loss is 34%.

Lets say that the core can dissipate 10W continuously without damage or compromise. In that case, with core loss of 34%, the transformer could be rated for 10/0.35=28.6W continuous or average RF power input. One would confirm this continuous rating with a bench test measuring temperature until it stabilised. Thermographs are a good means of documenting the heat rise.

In applications where the transmitter was active only half the time, an ICAS (Intermittent Amateur and Commercial Service) rating would be appropriate, we would rate it as 28.6/0.5=57.2W ICAS.

Note that as we ‘increase’ the power rating, consideration must be given to voltage breakdown which is an instantaneous mechanism, there is no averaging like heat effects.

Now some modes have average power (ie heating effect) less than the PEP, so we could factor that in. Average power of SSB telephony develops a Pav/PEP factor for compressed SSB telephone of 10%, so we can calculate a SSB telephony (with compression) PEP ICAS rating as 57.2/0.1=572W.

So this is a pretty ordinary ordinary transformer which we have been able to rate at 570W SSB ICAS exploiting the low average power of such a waveform.

Above is a core loss profile for the transformer where the load is such that the impedance looking into the primary is 50+j0Ω. At 3.5MHz, core loss is 8.5%.

Lets say that the core can dissipate 10W continuously without damage or compromise. In that case, with core loss of 8.5%, the transformer could be rated for 10/0.085=118W continuous or average RF power input. Again, one would confirm this continuous rating with a bench test measuring temperature until it stabilised. Thermographs are a good means of documenting the heat rise.

In applications where the transmitter was active only half the time, an ICAS (Intermittent Amateur and Commercial Service) rating would be appropriate, we would rate it as 118/0.5=236W ICAS.

Lets calculate the SSB compressed telephony rating. we can calculate a SSB telephony (with compression) PEP ICAS rating as 236/0.1=2360W.

Even more important at this power level is assessment of the voltage withstand.

So, when you see claims of power rating, read the details carefully to understand whether they are applicable to your scenairo. The last scenario about might be find for 1500W SSB compressed telephony, but not suitable for 500W of FT8.

An exercise for the reader: calculate the power rating for A1 Morse code (assume Pav/PEP=0.44).

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