Exact Calculation of Standing Waves on a Mismatched Transmission Line

Author: R.J.Edwards G4FGQ © 9th December 2003

           <------ D metres ------>¦<------ L-D metres ------>
 
 GENERATOR o-----------------------D-------------------------o LOAD=ZL
 Fixed 100v              <-- SLIDING PROBES -->

 Vin=100 Volts -->           Volts across line           Load volts
 Input current -->           Current in line             Load current
  Zin=Rin+jXin -->           Zin=Rin+jXin -->            ZL=RL+jXL
   Input power -->           Forward power flow -->      Power in load
          Rho1 -->           Rho2 -->                    Rho3
 Angle of Rho1               Angle of Rho2               Angle of Rho3
          SWR1               SWR2                        SWR3
                         <-- Zin=Rin+jXin
                         <-- Reverse power flow

This program models making measurements of volts and amps along a trans-line. The input impedances looking in both directions from D are also measured. From this data various other parameters of interest are calculated. Exact classical transmission line formulae are used. Results are exact to the number of digits displayed. For SWR2, slide D in fine increments to find max and min of V and I.

The model uses a common type of underground phone line. A twisted pair of 0.93 mm wires are spaced 1.8mm between centres insulated with air + paper with permittivity=1.9 At HF Zo=120 ohms. At 1KHz Zo=450 ohms. Of particular interest is the angle of Zo which nears -45 degrees at power and audio frequencies Such lines were used for carrier telegraphy and telephony, speech, music and now carry internet traffic over local lines. Test frequencies range from 10 Hz to 30 MHz.

How the Program Models V and I Measurements at D Metres Along Line Length L
The sending generator's internal resistance is zero so the input impedance at D looking back to the generator is that of the short-circuited line of length D. Line D is now a new generator loaded by the impedance looking into L-D as terminated by load ZL. The open-circuit volts of this new generator is now calculated. The measured probe voltage is that developed across the input impedance of line L-D looking towards the load ZL. It is then easy to calculate the current at D flowing towards the load as measured by the current probe. Notice that the through-power must be calculated from the current flowing into the RESISTIVE COMPONENT of the impedance looking towards ZL. The author defines this as the "Forward Power"; "Reverse Power" being defined as the SAME current flowing into the RESISTIVE COMPONENT of the impedance seen from D looking back towards the generator. But program users may have their own definitions of forward and reverse power as calculated conventionally from values of Rho or SWR. Note that the current probe does not indication the direction of current flow at D.

This program, based on only one type of line, is mainly of educational value. value. It can be used for research into the practical meaning of measurements made on any line defined by its impedance Zo = Ro+jXo and its propagation constant Po = Ao+jBo, where Ao is attenuation in nepers and Bo is the phase delay in radians, terminated with ANY load ZL = RL+jXL. The model's general behaviour vs frequency and lengths L and D is identical to ALL uniform lines.

Rho is the magnitude of (Zin-Zo)/(Zin+Zo) at any point D, where Zin is the complex input impedance of the line looking towards the load. Conventionally calculated SWR = (1+Rho)/(1-Rho) which always has a real value. But remember, when the angle of Rho is discarded incorrect conclusions may be drawn. The ordinary SWR/Rho meter does in fact loose 1/2 of the total information contained in Rho.

What does Rho>1 mean? The greatest value is 2.414 which is approached at audio and power frequencies by setting the load to ZL=0+jZo. So what is reflected power?

What do negative values of SWR mean? Use an inductive load because it is of of opposite sign to the -ve value of Xo. Negative values of SWR do not always coincide with high values of Rho. Initially plot volts, amps, power vs D-L.

Does the concept of power-waves have any value or meaning? When through-power is plotted graphically vs. distance D what relationship does it have to voltage and current waves? Has the SWR meter outlived its imagined usefulness? TLI?

Run this Program from the Web or Download and Run it from Your Computer
This program is self-contained and ready to use. It does not require installation. Click this link rho_swr then click Open to run from the web or Save to save the program to your hard drive. If you save it to your hard drive, double-click the file name from Windows Explorer (Right-click Start then left-click Explore to start Windows Explorer) and it will run.