Smith chart radially scaled parameters - Page 1
Forum section: electronics / rf-microwave

author: jmw
author_rank: Frequent Contributor
timestamp: December 10, 2019, 09:42:40 pm
content: (Note: using MathJax, make sure your browser is compatible)I was curious about the "radially scaled parameters" part of printed Smith charts that have several scales printed in nomographic form. Some of them are easy to understand, others I had decipher from scans of Smith's 1944 paper.My question is to what extent are the below quantities still used by present-day microwave engineers? The reflection and transmission coefficients are fundamental, and VSWR and return loss are commonly cited as figures of merit. On on other hand, I haven't seen VSWR cited in dB and I can't find any reference to "standing wave loss coefficient", or "standing wave peak" in Pozar, etc. Makes me think some of these are not how people characterize systems anymore, or they are called something else.For any one who stumbles upon this later, a reference of these scales and how they are related:REFLECTION COEFFICIENT, E OR IThis is the easiest. It's the magnitude of the complex reflection coefficient \$|\Gamma|\$, measured from the center of the chart to the point of interest.REFLECTION COEFFICIENT, PThe reflection coefficient in terms of power. Power is proportional to the square of voltage, so this is \$|\Gamma|^2\$.RETURN LOSSThe decibel form of the reflection coefficient, \$20\log|\Gamma|=10\log|\Gamma|^2\$.VSWR and dBSThe voltage standing ratio, \$\mathrm{VSWR}=(1+|\Gamma|)/(1-|\Gamma|)\$. dBS is decibel form of VSWR and computed as \$20\log\mathrm{VSWR}\$.TRANSMISSION COEFFICIENT, E OR IThis is magnitude of the transmission coefficient, \$|1+\Gamma|\$. This is measured from the origin (the left-most point on the circle) instead of the center.TRANSMISSION COEFFICIENT, PThe transmission coefficient in terms of power. Since the power reflected is \$|\Gamma|^2\$, the fraction of transmitted power is \$1-|\Gamma|^2\$.REFLECTION LOSSThe decibel form of the power transmission coefficient, \$10\log (1-|\Gamma|^2)\$.STANDING WAVE LOSS COEFFICIENTThe standing wave ratio expressed in terms of power, \$(1+|\Gamma|^2)/(1-|\Gamma|^2)\$.STANDING WAVE PEAK (CONSTANT POWER)This scale is \$\sqrt{\mathrm{VSWR}}\$, the square root of the VSWR scale. For a fixed level of power transmission, the magnitude of the voltage wave is proportional to the square root of the power. The ratio of the maximum to minimum voltage on the line is the VSWR, so the ratio of the peak voltage to the magnitude of the voltage wave is the square root of the VSWR.ATTENUATIONThis scale is obtained from taking \$10\log|\Gamma|\$, and allows the calculation of the effects of a lossy line or inline attenuator on the reflection coefficient. If the line has a one-way attenuation of \$A\ \mathrm{dB}\$, then the magnitude of the transmission coefficient is \$|T|=10^{-A/20}\$. The attenuation is seen twice by the reflected wave, so the reflection coefficient seen with the attenuator is \$|\Gamma'|=|T|^2|\Gamma|=10^{-A/10}|\Gamma|=|\Gamma| - 2A\ \mathrm{dB}\$. For example, to compute the effect of a 3 dB attenuation, plot the reflection coefficient on this scale, then move 3 units in the direction of the generator (to add the effect of the attenuator), or in the direction of the load (to remove the effect of the attenuator). The resulting point is plotted back on the reflection coefficient scale.