# Transmission lines: departure from ideal Zo

The article On the concept of that P=Pfwd-Prev discussed the question of the validity of the concept of that P=Pfwd-Prev, exploring an example of a common nominally 50Ω coaxial cable at 100kHz. The relatively low frequency was used to accentuate the departure from ideal.

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

## 100kHz

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.

## 10MHz

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.

## Conclusions

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.