Walt Maxwell (W2DU) made much of conjugate matching in antenna systems, he wrote of his volume in the preface to (Maxwell 2001 24.5):

It explains in great detail how the antenna tuner at the input terminals of the feed line provides a conjugate match at the antenna terminals, and tunes a non-resonant antenna to resonance while also providing an impedance match for the output of the transceiver.

Walt Maxwell made much of conjugate matching, and wrote often of it as though at some optimal adjustment of an ATU there was a system wide state of conjugate match conferred, that at each and every point in an antenna system the impedance looking towards the source was the conjugate of the impedance looking towards the load.

This is popularly held to be some nirvana, a heavenly state where transmitters are “happy” and all is good. Happiness of transmitters is often given in online discussion by hams as the raison d’être for ATUs, anthropomorphism over science.

(Maxwell 2001 24.5) states

To expand on this definition, conjugate match means that if in one direction from a junction the impedance has the dimensions R + jX, then in the opposite direction the impedance will have the dimensions R − jX. Further paraphrasing of the theorem, when a conjugate match is accomplished at any of the junctions in the system, any reactance appearing at any junction is canceled by an equal and opposite reactance, which also includes any reactance appearing in the load, such as a non-resonant antenna. This reactance cancellation results in a net system reactance of zero, establishing resonance in the entire system. In this resonant condition the source delivers its maximum available power to the load. …(1)

Let us look at a very simple example in SimSmith.

The scenario is:

- a Thevenin source at 1MHz with a source impedance of 50+j0Ω;
- a nominal half wave of RG59 transmission line; and
- an adjustable load impedance.

This should not be taken to imply that ham transmitters are commonly well represented as a Thevenin source.

The load impedance has been adjusted for a nearly perfect match at the source.

Above is the SimSmith model. The load R and X were adjusted for extremely low |Γ| at the source. |Γ| at the source is extremely low (0.0000173), Return Loss is 95dB, this is a match better than instruments could ever measure. We have achieved an almost perfect conjugate match at the interface between source and T1.

So let us now examine the impedance looking both ways at the load to T1 interface.

SimSmith has an internal feature to calculate the impedance looking backwards into an element, and it is used to calculate the impedance looking back from the load into element L. It is shown under the generator element as L_revZ.

So at the load to T1 interface:

- Z looking into the load is 38.75+j1.813Ω; and
- Z looking into T1 is 58.43-j1.534Ω.

They are not conjugates of each other, not nearly, in fact the mismatch is characterised by Return Loss (in terms of the load Z) is just 14dB (or VSWR=1.5).

In this very simple configuration, a near perfect match at the source does not result in a similar quality match at the other node in the system.

(Maxwell 2001 24.5) relies on a quotation:

If a group of four-terminal networks containing only pure reactances (or lossless lines) are arranged in tandem to connect a source to its load, then if at any junction there is a conjugate match of impedances, there will be a conjugate match of impedances at every other junction in the system. (Everitt 1937 243) and (Everitt and Anner 1956 407)

The problem is that Maxwell silently dropped from his statement (1) above the requirement that networks and lines must be lossless, and the example calculated here shows that Maxwell’s proposition does not apply to real world networks that have loss.

Recourse to simple linear circuit analysis will reveal that lossy networks do not have the property Everitt ascribed to lossless networks.

Walt Maxwell’s conjugate mirror

does not apply in the real world, and the concept is of limited use in understanding real world antenna systems.

When you see people sprouting the Walt’s conjugate mirror

you can expect that they have not read widely or thought about the subject much.

- Duffy, O. Mar 2013. The failure of lossless line analysis in the real world. VK1OD.net (offline).
- Everitt, W L. 1937 Communications Engineering, 2nd ed. New York: McGraw-Hill Book Co.
- Everitt, W L, and Anner, G E. 1956 Communications Engineering, 3rd ed. New York: McGraw-Hill Book Co.
- Maxwell, Walter M. 2001. Reflections II. Sacramento: Worldradio books.

]]>

…so you are telling me that I could measure this Prev>Pfwd with a directional wattmeter like my Bird43… I have never seen it and I don’t believe it.

For clarification, I did not discuss Prev or Pfwd in respect of the three scenarios (other than to say Pref cannot exceed Pfwd).

I did discuss line voltage measurements you can make with a simple RF volt meter

which was in the article’s reference quote. But, let’s discuss what you might measure by inserting a 50Ω Bird43 directional wattmeter in the Load case 2 scenario.

Above is a calculated plot of the expected Pfwd and -Prev readings, Prev is shown negated so you can add it by eye with Pfwd to obtain the net power Power (blue line).

Note that Pref is never greater then Pfwd, there is no implied breach of the law of conservation of energy.

So, what is the implied ρ?

Above is a plot of calculated ‘actual’ ρ and ρ(50), ρ wrt Zo=50+j0Ω, which would be derived from the Bird43 measurements.

The ‘actual’ ρ is a smooth exponential curve, a consequence of transmission line loss. The ρ(50) curve oscillates, and the error is due to the fact that its reference impedance is not the transmission line actual Zo. You will also see that whereas ρ>1.0 at the load, ρ(50) is never greater than 1.0. When Zref is purely real, ρ(Zref)<=1.0.

Even though ρ(50) is not a good estimator of ‘actual’ ρ, the blue Power line in the previous chart is correct.

Further, that given the shape of the blue Power line in the previous chart, it is clear that it cannot be derived from the ‘actual’ ρ line in the second chart, many of the operations based on ρ depend on Zo being purely real (ie a Distortionless Line).

The same response can be expected of any directional wattmeter, VSWR meter, antenna analyser or VNA calibrated for a purely real impedance such a 50+j0Ω.

]]>Most tools and most derivations of SWR will produce negative SWR reports because they are more interested in mathematics than in measurements you can make with a simple RF volt meter.

…, this article explores the expected voltage on a practical transmission line under two mismatch scenarios, voltage that ought be measurable with a simple RF voltmeter.

Textbooks on transmission lines often introduce the concept of standing waves by presenting a plot of voltage along a mismatched lossless transmission line.

Above is a plot of calculated line voltage vs displacement from the load, -ve is towards the source.

In this case the fully developed standing wave pattern repeats identically from load to source because the line is lossless, and in this case the VSWR can easily be calculated as the maximum voltage to the minimum voltage, all the maxima and minima are the same, so VSWR is unambiguously 20:1 or 20.

Drilling down to a more fundamental metric, we can plot the complex reflection coefficient along the line.

Above is a plot of the phase and magnitude of the complex reflection coefficient Γ. Note that the amplitude is constant (0.9048), but the phase varies linearly along the line.

Metrics derived from the magnitude of Γ (|Γ|) such as rho (ρ), Return Loss (-20logρ), and VSWR will be constant along the line… so to speak of any of those quantities is unambiguous, they each apply anywhere along the line.

This is a pure mathematics exercise with no relevance to the real world because lossless lines do not exist in the real world. It is true though that sometimes approximating practical lines as lossless in certain scenarios may be a quite adequate approximation, but approximations do not determine fact.

Let us look at calculated values on a 1λ electrical length of practical transmission line at 100kHz. The choice of frequency is to expose effects that might be difficult to measure at higher frequencies where the loss per wavelength becomes lower.

The model uses Heaviside’s RLGC model for a transmission line, it is a lossy line having non-zero R and G terms. The line Zo has a significant imaginary component.

Load case 1 is a significantly mismatched purely resistive load.

Above is a plot of calculated line voltage vs displacement from the load, -ve is towards the source.

Note the difference to the chart for a lossless line. This is the same load, same nominal Zo, but now a practical lossy line.

A voltage standing wave is evident, as is the effect of line loss. The curve is a composite of an exponential decay in voltage due to line loss and superposed, the the reflected wave voltage which of course is also affected by line loss.

Given the definition of VSWR as the ratio of a voltage maximum to voltage minimum, what would you give as VSWR for this case?

VSWR is often calculated as a property of a point along a line (notwithstanding its definition), and the formula is given in most text books as VSWR=(1+ρ)/(1-ρ), and this would yield a curve for VSWR along the line. Likewise for Return Loss.

Whilst this is a calculated plot, it is the voltage that one would expect to measure along such a line under these source and load conditions. It is not simply a pure mathematics exercise with no relevance to the real world, it is a good mathematical model of the real world transmission line.

Load case 1 is a significantly mismatched load with a dominant imaginary part.

Above is a plot of calculated line voltage vs displacement from the load, -ve is towards the source.

In broad form, the plot is similar to the last, and again we would expect to be able to measure along such a line under these source and load conditions. It is not simply a pure mathematics exercise with no relevance to the real world, it is a good mathematical model of the real world transmission line.

Let’s drill down on Γ.

Above is a plot of the phase and magnitude of the complex reflection coefficient Γ. It is different to the case of the lossless line presented earlier as the magnitude of Γ varies along the line due to line loss.

But wait, look at the magnitude of Γ at the load, it is greater than 1.0. That has not given rise to an unrealistic voltage standing wave as plotted, in fact it is not very different to the purely resistive load case where Γ at the load was less than 1.0.

What is different is that for formula VSWR=f(ρ) will have a singularity where ρ=1 (VSWR=∞), and for ρ>1, calculated VSWR will be negative. Some will insist that this proves that ρ must never equal or exceed 1.0, but the problem is actually that that is a constraint of Distortionless Lines (of which Lossless Lines are a special case).

Another justification often given to argue that ρ must be less than 1.0 is that it implies that reflected power is greater than forward power, violating the law of conservation of energy. Sounds impressive, but it is due to flawed analysis of power on a line with complex Zo.

Some authors insist that negative VSWR is invalid and hide that the underlying conditions gave rise to negative VSWR.

Above is a calculation from the popular SimSmith (v16.9) package. Note that the cursor is at the load which plotted on a Smith chart centred on the complex Zo from the T1 section is outside the outer circle. It is reported in the cursor data, and in the element calculations as having VSWR=39.96. In fact, it is easy to see that ρ>1, and that VSWR should be negative, it has been sanitised / corrupted.

Above is a calculation of various metrics from the scenario Zload and Zo using accepted formulas, and it yields a negative VSWR, a warning that the scenario is one where common Distortionless Line assumptions for ρ, Return Loss, and VSWR are likely to be unsafe.

You might well ask what you can do with a cooked VSWR value, you cannot derive the value of ρ from it (ie there is no inverse function), you are in a fools paradise where you have a nice positive value that means nothing and masquerades as something that it is not.

]]>I have long held the view that these things are most useful when accompanied by a capable PC client that performs flexible text book presentations of data.

Considering buying one, my first step was to perform a desk evaluation of a popular PC client, which seems to be nanovna-saver.

Before downloading it, I examined the first screenshot on the github page.

It gives evidence that the author does not follow industry standard convention for transmission line terms and theory.

In the results shown above (s11) impedance is 39.105+j39.292Ω and some transformations of that value.

Above is a cross check of key values.

The obvious and blinding error is that nanovna-saver reports the Return Loss as a negative value, the author does not subscribe to the industry standard meaning of Return Loss… very hammy. The value reported appears to be |s11| in dB.

Like most ham tools, they choose to talk in parallel R and X rather than admittance components (conductance and susceptance), but in fact the quantity labelled parallel X should be 1/B=1/-j0.01279=j78.19 which should be stated as X=78.19Ω. It does appear that the author has calculated the equivalent inductance, not the quantity stated to the left.

Above the plot title infers that S11 and Return Loss are synonymous.

They are not, ReturnLoss=-10*log(|s11|) dB (s11 is a complex quantity, it has magnitude and phase).

So, the graphics cannot be directly published (respectably) because of the hammy meaning attributed to Return Loss.

One might ask whether this lack of compliance with industry standard terms and the untidiness in labeling is more widespread, whether there is the rigorous attention to detail necessary in such software.

It has been my experience that authors of this type of stuff are resistant to correction. I will sit this one out for a while.

]]>I took a baseline measurement with an AA-600 after some refurbishment work in Jan 2018, and was able to compare a current sweep to that baseline.

Above, a wide Return Loss sweep of the Diamond X-50N with feed line compared to the baseline (the thin blue line).

By and large they are almost identical, save small departure around 435MHz.

Above is a comparison of the Return Loss at low values. Antscope does not display mathematically correct plots when the data goes off scale (as in this case), this plot is mathematically correct and allows better comparison of the important out of band Return Loss.

It is worth remembering that the AA-600 operates on second harmonic above 200MHz, and third harmonic above 400MHz, so the measurements become a little noisier.

Importantly, the out of band Return Loss is almost identical and this would not be so if feed line loss had degraded (eg due to water ingress), so there is no evidence to suggest that the feed line had degraded.

Above is a narrower sweep around the normal operating frequencies. There is a small degradation in Return Loss which is probably attributable to temperature differences of more than 20° between measurements.

So, the comparison with the archived baseline gives no cause of concern, the antenna system is probably unchanged.

Well, in fact I have done just that at 80 frequencies over a wide range, in-band and out-of-band, if you like. That captures much more information than VSWR measurement at one or a few frequencies.

The traditional ham approach is the measure VSWR at the operating frequency and focus on that, but that is unlikely to be very sensitive to some types of transmission line degradation (eg increased loss).

Analysis of the derived Return Loss figures in-band and especially out-of-band gives much more insight.

I have a clear window that shows if water has leaked down the cable. It should not leak down the inside because it is closed cell foam, but it should indicate if the birds pick a hole in the jacket… and possibly the copper.

]]>Note that the measurements are of a particular implementation and should not be taken to imply generally to 5/8λ verticals, but the solution method can be applied more generally. Lets assume that the measurement is not affected by common mode current.

The answer to the last question first is that a series inductor will not bring the VSWR much below 3. It is a common belief that a 5/8λ vertical can be matched simply with a series inductor.

There are many ways to match the measured antenna, and there are articles on this site describing some of them, but a simple and effective method in this case is the single stub tuner.

Above is a graphical solution using Simsmith. The section of line nearest the measurement load is -ve length, it is to back out the effect of the line section into which measurements were made (antenna feed point is at the cursor, 139-j191Ω). The next line section is the series section, followed by the S/C stub. In this case the series section and stub use RG213 to reduce loss. Total matching system loss is a little under 0.3dB, and the stub can easily be weatherproofed with hot glue and heat shrink tube.

One could use RG58, an exercise for the reader is to assess the loss of that option.

Obviously the length of the measurement section plays into the solution, and using its length to the mm in the model gives a more accurate result.

]]>Distortionless Linesfrom time to time, often in the vein of

they don’t exist, so why discuss them?

The concept derives from the work of Heaviside and others in seeking a solution to distortion in long telegraph lines.

The problem was that digital telegraph pulses were distorted due to different attenuation and propagation time for different components of the square waves.

Heaviside proposed that transmission lines could be modelled as distributed resistance (R), inductance (L), conductance (G) and capacitance (C) elements.

In each incremental length Δx, there is incremental R, L, G and C.

Characteristic impedance Zo and complex propagation coefficient γ can be derived from the model, and it becomes apparent that only under certain conditions is the attenuation and phase velocity independent of frequency.

Heaviside determined that condition was that G/C=R/L, so this is the condition for a Distortionless Line.

That same condition means that Zo is a purely real number, and if Zo is purely real, then the line is Distortionless.

Note that Lossless Line is a special case of Distortionless Line, and necessarily Zo is purely real.

It is true that fabrication of a Distortionless Line is a considerable challenge, though some techniques might deliver an approximately Distortionless Line over a limited frequency range.

Whilst some might dismiss the concept of Distortionless Line as impractical, they probably apply Distortionless Lines freely in their analytical techniques without understanding what they are doing.

Let us look at some common operations.

Waves add, the E and H fields add vectorially, as do their V and I equivalents.

It happens that in the special case of purely real Zo (ie Distortionless Line) that the directional powers add, but they add by virtue of expansion of the expression for adding V and I (which simplify when Zo is real).

Most use of a Smith chart involves normalising plotted values to some real Zref. In so doing, an assumption is made that constant VSWR circles are wrt that Zref, and solutions (eg matching) based on that are conditional on the assumed real (Distortionless) Zref.

The clean separation of real and imaginary Z and Y components and the arcs of constant R, X, B, G on a Smith chart are conditional on real Zo, as is the assumption that all practical values of Z map inside the ρ=1 circle. (The assumption that ρ<=1 depends on real Zo.) The loss of this property quite restricts its usefulness.

None of this is to suggest that a Smith chart with Zref including a significant imaginary part is invalid… just many things we have learned to do with a ‘real’ Smith chart may not be valid (ie within acceptable error limits), and the scales are no longer as simple and useful as the chart devised by Philip Smith.

ρ, Return Loss, and VSWR are derived from the complex reflection coefficient Γ. Calculating Γ wrt some nominal real Zo involves some error, and raises the question of whether application of those to a real scenario (where Zo is different, eg not real) is within acceptable error limits.

S parameter techniques assume additive power, and that may not be appropriate for some problem solutions. The analysis tools effectively coerce a Distortionless Lines treatment.

Practitioners use analytical techniques that depend on an assumption that Zo is real (ie Distortionless Line) widely, and often without acknowledging it or even thinking about the implications.

Virtual Distortionless Lines are everywhere, and worth understanding when Distortionless Line analysis is being applied and whether it is appropriate.

]]>This article digs a little further with analyses at both 100kHz and 10MHz.

A plot was given of the components and sum of terms of the expression for power at a point 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.

Term1 is often known as Pfwd and -Term4 is often known at Prev, and when Zo is real, Term2=-Term3 and they cancel, and in that circumstance P=Pfwd-Prev.

These are calculated using the actual value of Zo, Zload and propagation constant.

Above is a plot of impedance along the line.

We can use the impedance along the line to calculate the expected result if measurements were made along the line with an instrument calibrated for Zref=50+j0Ω. We will obtain a different values for Γ and ρ as they will not related to the actual line but to the Zref in use.

Above is a plot of actual ρ on the line, and ρ wrt 50+j0Ω (ρ50). You will note that ρ is a smooth exponential curve as determined by the line attenuation, whereas ρ50 varies cyclically and seems inconsistent with expected behavior of a transmission line.

Because ρ50 varies in this way, so will VSWR50 and ReturnLoss50. All of these metrics are of very limited value because Zref is so different to Zo.

We can calculate the expected reading of ‘Directional’ Power (as would be displayed on a directional wattmeter.

Above, the blue line is the actual power along the line and it varies cyclically because for this line, under standing waves more power is lost per unit length in regions of high current that those of high voltage.

An important attribute is that where Zref is real:

- Pfwd and Prev are each meaningful if Zref=Zo; and
- where Zref is not equal to Zo, Pfwd and Prev each are of no stand alone relevance to the actual line, but P does equal Pfwd-Prev.

Let’s plot the components and sum of terms of the expression for power at a point along the line.

Lets look at the power calculated from voltages and currents for the example at 10MHz where Zo=50.01-j0.8025Ω and Zload=5+j50Ω.

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

Term1 is often known as Pfwd and -Term4 is often known at Prev, and when Zo is real, Term2=-Term3 and they cancel, and in that circumstance P=Pfwd-Prev.

These are calculated using the actual value of Zo, Zload and propagation constant.

Above is a plot of impedance along the line.

We can use the impedance along the line to calculate the expected result if measurements were made along the line with an instrument calibrated for Zref=50+j0Ω. We will obtain a different values for Γ and ρ as they will not related to the actual line but to the Zref in use.

Above is a plot of actual ρ on the line, and ρ wrt 50+j0Ω (ρ50). You will note that ρ is a smooth exponential curve as determined by the line attenuation, whereas ρ50 varies cyclically and seems inconsistent with expected behavior of a transmission line.

Because ρ50 varies in this way, so will VSWR50 and ReturnLoss50. All of these metrics are of somewhat limited value because Zref is a little different to Zo.

We can calculate the expected reading of ‘Directional’ Power (as would be displayed on a directional wattmeter.

Above, the blue line is the actual power along the line and it varies cyclically because for this line, under standing waves more power is lost per unit length in regions of high current that those of high voltage.

An important attribute is that where Zref is real:

- Pfwd and Prev are each meaningful if Zref=Zo; and
- where Zref is not equal to Zo, Pfwd and Prev each are of no stand alone relevance to the actual line, but P does equal Pfwd-Prev.

Whilst it is convenient to treat Zo of practical transmission lines as a purely real quantity, it isn’t and the error may be significant.

The departure from ideal Zo is typically worst at lower frequencies, and may be very small, perhaps insignificantly so above 100MHz.

]]>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.