6.1) Phase Relationships Along the Length of a Mismatched Transmission Line
With a transmission line that is long in relation to the applied signal frequency, there are many revolutions of phase along the length of the line and this produces maximum sum INCIDENT and REFLECTED voltage measurements that form a graph similar to figure 6.1. It is easy to imagine an engineer in the early days of radio going along an open transmission line with a voltmeter and making measurements at regular intervals in order to find the VSWR. Then maybe plotting out the results and arriving at the generally accepted but wrong conclusion that the impedance of the line was continuously varying with distance. It is not possible to make these measurements along a modern coaxial cable with a voltmeter in the same way, but it is possible with the Voltage/Current Detector using diode linearity correction and amplitude normalisation. The length of the coaxial cable is not changed, but the frequency of the signal applied is changed in steps to give appropriate phase changes. This will give you conclusive proof of what is really happening in a coaxial cable. These measurement have been performed by us and a successful graph made in order to prove the principal using a coaxial cable. A simpler method is to build up a graph of the simulated voltages along a badly matched transmission line as shown in figure 6.1 by using a matrix of calculations. Excel is adequate for this exercise and figure 6.1 was in fact generated with Excel in linearity-spreadsheet-01.xls
An Input Voltage is entered into the spreadsheet and in the case shown above it was 1.0 Volts. The phase shift along the sample (One Way) was selected at 90° which gives calculation points at 10° intervals along the length of the line. The termination was selected as OPEN and so the graph finishes with the termination point on the right hand side of the graph at double the selected Input Voltage.
Each calculation point has 5 columns which are used for the FORWARD and REFLECTED phases and voltages with a SUM column on the right hand side. The formula y = A sin θ is used repeatedly to calculate the INCIDENT and REFLECTED voltages at 10° points along the transmission line in relation to the input voltage and phase.
Each column has 37 lines calculating possible INCIDENT and REFLECTED phases and the resulting voltages in 10° steps, the 37 row being used to check that the calculation returns to the starting point. The phase difference of the INCIDENT and REFLECTED columns remain constant on all of the rows and this is the phase difference set by the return length of the line and the type of termination connected. Each row has the INCIDENT and REFLECTED voltages added in the Sum column and the Final Maximum Sum Voltage is selected from these. The maximums for each point are then used to construct the graph above. The formula used to find the instantaneous voltages is :-
y = A sin θ Formula 6.2
Where y is the resulting instantaneous voltage
A is the Input Voltage
It would have been simpler to directly calculate the vector addition of the INCIDENT and REFLECTED voltages but this would not demonstrate so well that the selected SUM is the MAXIMUM SUM.
The maximum sum INCIDENT and REFLECTED current measurements can also be performed in a similar way. This mimics an engineer in the early days of radio going along an open transmission line with a current detector. In order to detect current in an open wire, it is possible to use a toroidal transformer, a loop of wire or a current clamp. All of these measurement devices will have the characteristic of shifting the phase of the current by 180° in relation to the voltage measurements and so will not indicate peaks and troughs at the same points as the voltmeter. As we have already seen the reason for this measurement difference is, when the voltage measurements are made, the INCIDENT and REFLECTED voltages will flow in the same direction through the voltmeter to ground and therefore sum. Whereas, when the current measurements are made, the currents in the 'transformer' flow in opposite directions and subtract. The Voltage/Current Detector uses a ferrite transformer and so will produce a vector difference current when used with a coaxial cable and replicate the results found on an open wire. This measurement will also help demonstrate how the current and voltage are always in phase for a signal going in one direction. It can also be seen that the INCIDENT and REFLECTED signals behaviours are quite independent and should be studied individually!
Once the principle of what happens in an OPEN or SHORTED transmission line is understood, all becomes clear. There is no longer any need to think of varying impedance along a line, how a signal passes a short or an open or even for the mythical 'standing waves'. Simple logic can replace all of the confusion and you can move on to grasp more difficult concepts.
6.2) Complex Impedance
In paragraph 4.2, it was shown that the termination had no effect on the INCIDENT power. In paragraph 5.3, it was shown that the termination has no effect on the INCIDENT voltage. If the INCIDENT power and voltage remain constant, then the INCIDENT current must also remain constant because P = V x I. The INCIDENT impedance must then also remain constant because Z = V / I and is therefore the characteristic impedance of the cable (50 Ohms in this case) regardless of the termination at the end of the two metre cable (providing that the measurement is made more than a few degrees of phase from the termination). This is logical because the measured signal has not yet reached the termination and cannot therefore be effected by any future conditions.
The results obtained for Vr and Ir in paragraph 5.3 can also be used in Formula 2.1 to find the reverse looking impedance at the input of the cable using:-
Z = V/I Formula 2.1
The reverse looking impedance also calculates at 50 Ohms.
This approach does not agree with some classical transmission line theory. The results of the measurements of the cable vector sum voltage and vector difference current at any point along a cable which is terminated into an OPEN or a SHORT can be used to calculate the complex impedance at this point from a more traditional stand point using the following definition and formula taken from, TRANSMISSION LINES AND NETWORKS. INTERNATIONAL STUDENTS EDITION by Walter C. Johnson, 1963 by McGraw-Hill Book Co. Singapore ISBN 0-07-Y85348-7, Page 96.
The impedance of a transmission line at any point is defined as the complex ratio of E to I at that point.
Where Z is the impedance.
E is the complex sum voltage.
I is the complex sum current.
These calculations will give an impedance (note that Johnson does not call it characteristic impedance) of nearly zero Ohms ( a short) for the OPEN Termination measurement and approaching infinity (an OPEN) for the SHORT Termination when the transmission line length is a quarter wave (90°).
On page 34 Walter C. Johnson gives another formula for transmission line impedance as shown Figure 6.3 which has a very different basis.
Where Zo is the characteristic impedance.
R is the resistance in Ohms/m.
L is the inductance in Henrys/m.
G is the conductance in Siemens/m.
C is the capacitance in Farads/m.
jω is the angular velocity in radians/s.
We have a conflict in statements here where formula 6.2 gives an impedance based on the the frequency of the applied signal, the properties of the cable and the termination characteristics, whereas formula 6.3 gives an impedance based purely on the the frequency of the applied signal and the properties of the cable. The classical approach given in 6.2 will give a varying impedance which will be be indicative of the phase relationships along the length of the transmission line and may assist some engineers to visualise what is happening. It will however, confuse the majority of engineers and will create nonsense questions like how does the signal pass through a 'short' or an 'open' or why does the signal not reflect from the first 'short' or 'open' that it meets? It looks very much as though Johnson fully understands that the Characteristic Impedance does not vary along the length of line but is avoiding disagreement with old existing theories by referring simply to the Impedance. This is therefore an avoidance of getting into a controversy which does not help engineers to understand the reality of what is happening along the length of the cable.
From the results that we have seen in this study it would appear that it is best to ignore the old 'standing wave' theory and consider the individual FORWARD and REFLECTED signals as being separate. It would also be wise to remember that there are many misconceptions that have arisen due to the fact that the analysis of Tx lines was not finished completely and nobody seems to have publicly recognised that the voltage measurements are a vector sum voltage and the current measurements are a vector difference current. The following list shows misconceptions that have arisen from the resulting belief in so called 'standing waves'. There may be more and it would be helpful if you could notify us of any that we have missed out.
1) The current and voltage in a transmission line in one direction are not 180° out of phase. They are in phase, it is the current meter and voltmeter principals of operation that are out of phase.
2) Electrons or charged particles do not travel slowly in a Tx line or their charges travel independently. The charged particles travel at near the speed of light and carry their charges with them.
3) A mismatched Tx line will not have a varying characteristic impedance along its length. The characteristic impedance is a property of the physical cable/s and is engineered to be nearly a constant across its working frequencies, only the phase difference and attenuation of the FORWARD and REFLECTED signals change along the length of the line.
4) There are no voltage peaks in a mismatched Tx line at 180° intervals that could reach a theoretical infinite voltage and could cause damage and stresses in the Tx line as some believe. A voltmeter across the line will not indicate more than twice the original input voltage and this is in fact two separate occurrences of the same input voltage that the voltmeter interprets as double the voltage because the voltmeter current has doubled.
5) An OPEN circuit does not reflect current with a 180° phase reversal. It is the current meter that has the 180° phase reversal. An OPEN does not cause a current or voltage phase reversal.
6) The reflected power in and antenna transmission line does return into the Tx amplifier output stage. The power amplifier is likely to be quite well matched to the Tx line and virtually all of the reflected power will enter the output stage.
It has taken me 14 years of study, calculation and building specialised test equipment to completely understand how a Tx line really performs and to be completely confident that 'standing waves' in a transmission line are a myth. This belief has existed for over 100 years and it will probably take more than another 100 years for the reality of the true behaviour of transmission lines to become completely accepted by the engineering community. If you now understand the reality, please spread the word but if you are not in agreement with me please use Contact Us and explain why.
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First published in 2004 and repeatedly updated.
W J Highton 6/4/2014
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