# From lossless transmission line to practical – Zo and γ

On the concept of that P=Pfwd-Prev discussed the expression for power at a point on a line in terms of the travelling wave voltage and current components.

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.

## The 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.

### Lossless Line

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.

### Distortionless Line

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).

### Practical line

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).

### Scenario conclusions

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.

### Assumption of distortionless behavior, ie that Term2=-Term3

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|).

### Assumption of lossless behavior, ie that Term2=-Term3 and real(γ)=0

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

### Extrapolation of VSWR from Prev/Pfwd

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).

#### Example

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. 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 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.

## Caveats

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.