Understanding Coaxial RF Transmission Lines by Measurement and Calculation Page 9

In paragraph 4.2, it was shown that the termination had no effect on the forward power. In paragraph 5.3, it was shown that the termination has no effect on the forward voltage. If the forward power and voltage remain constant, then the forward current must also remain constant because P = V x I. The forward 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 V_r and I_r in paragraph 5.3 can be used in Formula 2.1 to find the reverse looking impedance at the input of the cable using:-

This approach does not agree with classical transmission line theory. The results of the measurements at the cable input into the OPEN or SHORT Terminations 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.

Z = E / I

Where Z is the impedance.

E is the complex sum voltage.

I is the complex sum current.

These calculations will give an impedance of nearly zero Ohms ( a short) for the OPEN Termination measurement and infinity ( an open) for the SHORT Termination when the transmission line length is a quarter wave (90°). We have a conflict in approach here and both methods are valid in their own way. The classical approach will indicate phase relationships along the length of the transmission line and may assist some people to visualise what is happening. It will however, confuse the majority 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?

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 FORWARD 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 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 FORWARD 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 FORWARD 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 FORWARD 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 FORWARD 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 :-

It would have been simpler to directly calculate the vector addition of the FORWARD and REFLECTED voltages but this would not demonstrate so well that the selected SUM is the MAXIMUM SUM.

The maximum sum FORWARD and REFLECTED current measurements can also be performed in a similar way. This has to mimic 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 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. The reason for this measurement difference is, when the voltage measurements are made, the FORWARD 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 also indicate a 180° phase reversal of 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 FORWARD and REFLECTED signals behaviours are quite independent and should be studied individually!

Once the principle of what happens to 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.