Can the magnitude of the complex reflection coefficient (ρ) be greater than 1

Let’s examine a number of transmission line loss calculators on the following scenario:

  • line type Belden 8259 (RG58A);
  • length 1m; and
  • frequency 0.1MHz,
  • Zload=5+j50Ω;

TLLC

Parameters
Transmission Line Belden 8259 (RG58A/U)
Code B8259
Data source Belden
Frequency 0.100 MHz
Length 1.000 m
Zload 5.00+j50.00 Ω
Yload 0.001980-j0.019802 S
Results
Zo 50.71-j8.35 Ω
Velocity Factor, VF -2 0.651, 2.360
Length 0.18 °, 0.001 λ, 1.000 m
Line Loss (matched) 4.66e-3 dB
Line Loss 4.69e-2 dB
Efficiency 98.93 %
Zin 5.09+j50.32 Ω
Yin 0.001989-j0.019673 S
VSWR(50)in 19.84
Γ, ρ∠θ, RL, VSWR, MismatchLoss (source end) -1.715e-2+j1.064e+0, 1.065∠90.9°, -0.54 dB, -31.98, **
Γ, ρ∠θ, RL, VSWR, MismatchLoss (load end) -2.403e-2+j1.065e+0, 1.066∠91.3°, -0.55 dB, -31.44, **
Vout/Vin 9.968e-1+j7.507e-4, 9.968e-1∠0.0°
Iout/Iin 1.003e+0-j3.277e-4, 1.003e+0∠-0.0°
S11, S21 5.378e-4-j1.707e-6, 9.995e-1-j3.174e-3
Y11, Y21 1.923e+0-j5.643e+0, -1.923e+0+j5.643e+0
NEC NT NT t s t s 1.923e+0 -5.643e+0 -1.923e+0 5.643e+0 1.923e+0 -5.643e+0 ‘B8259 , 1.000 m, 0.100 MHz
k1, k2 1.487e-5, 2.744e-10
C1, C2 4.701e-1, 2.744e-1
dB/m @1MHz: cond, diel 0.014866, 0.000274
γ 5.369e-4+j3.220e-3
Loss model source data frequency range 1.000 MHz – 1000.000 MHz
Correlation coefficient (r) 0.999924

 

First thing to note is the Loss model source data frequency range in red. This flags that the calculations are being made outside the frequency range of the loss data on which the model was based, we are extrapolating and that is always at a risk of increased error.

The magnitude of ρ at the load is given as 1.066, though not much greater than 1.0, it is greater.

Note also that using the traditional formula for VSWR(ρ), VSWR at the load is -31.44… yes it is negative though that might not seem to make sense in terms of the definition of VSWR as a ratio that must be positive. In this case, the calculator uses the common VSWR(ρ) formula (VSWR=(1+ρ)/(1-ρ)) and users in the know will understand the cause implications of the negative VSWR.

Lets calculate VSWR for Zo=50+j0Ω, VSWR(50), at the load end.

Screenshot - 07_01_16 , 16_27_28

VSWR(50) at the load is is 20.05. This is the VSWR that would occur on a transmission line with =50+j0Ω, so it is somewhat hypothetical, but it IS what would be indicated by an accurate VSWR meter calibrated for 50+j0Ω (as they tend to be). VSWR(50), and VSWR for any real Zo (meaning Xo=0) will ALWAYS be a positive number equal to or greater than 1.0. It is also true that ρ wrt to any real Zo will be in the range 0.0 to 1.0, NEVER greater than 1.0.

TLW

Screenshot - 07_01_16 , 16_12_51

Above, TLW calculation of the same scenario. No warnings about the frequency being used.

TLW uses a different value of Zo, and that flows into the calculated results.

Screenshot - 07_01_16 , 16_36_06

Calculation of the load end VSWR=-74.48 from Zload and Zo reconciles with TLW, and we can see that ρ=1.03.

W9CF

Screenshot - 07_01_16 , 12_12_38

Above, W9CF calculation of the same scenario. No warnings about the frequency being used.

W9CF uses a different value of Zo, and that flows into the calculated results.

W9CF gives the load end VSWR as -125.62.

Screenshot - 07_01_16 , 12_16_28

Calculation of the load end VSWR=-124.90 from Zload and Zo reconciles reasonably with W9CF, and we can see that ρ=1.02.

Tldetails

Screenshot - 07_01_16 , 12_14_21

Above, Tldetails calculation of the same scenario. No warnings about the frequency being used.

Tldetails uses a different value of Zo, and that flows into the calculated results.

Tldetails gives the load end VSWR as >999 (ie a very large POSITIVE number).

Screenshot - 07_01_16 , 12_15_24

Calculation of the load end VSWR=-39.13 from Zload and Zo does not reconcile with Tldetails, and we can see that ρ=1.04.

Summary

The four different calculators produce different results. In all cases they come to a different value of Zo and that is critical to the following results.

Tldetails calculates VSWR at the load that does not reconcile with its calculated Zo and the known Zload.

TLLC, W9CF and TLW all reconcile calculated VSWR at the load with their calculated Zo and known Zload.

TLLC, W9CF and TLW all calculate a negative VSWR at the load, and that implies ρ>1,0, not much greater but it is greater.

When Zo is real (ie Xo=0):

  • VSWR must be in the range 1.0 to infinity (incl), it cannot be negative;
  • ρ must be in the range 0.0 to 1.0 (incl), it cannot be greater than 1.0.

Instruments are usually calibrated for real Zo, and such an instrument will NEVER accurately incidate ρ>1, or Reflected Power > Forward Power.