# VSWR Converter Description

#### Description

When an incident signal propagates along a transmission line into a load, some part of the its power will be reflected back down the line. The power transfer efficiency depends on the impedance match between the load and the line. When the impedances are equal, the whole signal power is delivered to the load. In real life there is always some degree of impedance mismatch and as a result back reflections. The figure below illustrates this schematically. Reflections may lead to several unwanted effects, including:

1. Destructive interferences due to multiple reflections. These are manifested as amplitude and phase ripple in the overall system frequency domain transfer function ($$S_{21}$$), resulting in amplitude flatness degradation and dispersion.
2. In high power applications like transmitter to antenna interface, reflections have much power that may damage the previous (source) stage.
3. Loss due to the wasted reflected energy. Typically such mismatch loss is relatively small, $$<1dB$$.
It is common to characterize the impedance mismatch extend by one of the following parameters:
1. VSWR (Voltage Standing Wave Ratio) – defined as the ratio between the partial standing wave maximum and minimum amplitude along the transmission line. VSWR is the least intuitive measure of impedance mismatch, however it is pretty popular because of early adoption. Expressing it in term of the transmitted and reflected power yields $$VSWR=\dfrac{1 + \sqrt{P_r/P_i}}{1-\sqrt{P_r/P_i}}$$. VSWR values range from 1 for a perfect impedance match to infinity for a total reflection.
2. Return Loss – the power of the reflected signal expressed in dB relative to the incident power, $$RL=10\log_{10}\dfrac{P_i}{P_r}$$. Return Loss values range from infinity for a perfect impedance match to 0dB for a total reflection.
3. Mismatch Loss – the transmitted signal power loss expressed in dB relative to the incident power, $$ML=10\log_{10}\dfrac{P_i}{P_i - P_r}$$. Mismatch Loss values range from 0dB for a perfect impedance match to infinity for a total reflection.
4. Power Reflected - the power of the reflected signal expressed as a percentage relative to the incident power, $$PR=\dfrac{P_r}{P_i}$$. Power Reflected values range from 0% for a perfect impedance match to 100% for a total reflection.
5. Voltage Reflection Coefficient – the amplitude ratio of the reflected signal to the incident signal, $$|\Gamma|=\dfrac{\sqrt{P_r}}{\sqrt{P_i}}$$. Voltage Reflection Coefficient values range from 0 for a perfect impedance match to 1 for a total reflection.

Here is a great video explanation and demonstration of waves reflections and the concept of VSWR:

#### Equations

VSWR Return Loss (dB) Mismatch Loss (dB) Power Reflected (%) Voltage Reflection Coefficient, $$\Gamma$$
VSWR - $$VSWR = \frac{1 + 10^{-\frac{RL}{20}}}{1 - 10^{-\frac{RL}{20}}}$$ $$VSWR = \frac{1 + \sqrt{1 - 10^{-\frac{ML}{10}}}}{1 - \sqrt{1 - 10^{-\frac{ML}{10}}}}$$ $$VSWR = \frac{1 + \sqrt{\frac{PR}{100}}}{1 - \sqrt{\frac{PR}{100}}}$$ $$VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$
Return Loss (dB) $$RL = -20\log_{10}\frac{VSWR - 1}{VSWR + 1}$$ - $$RL = -10\log_{10}(1 - 10^{-\frac{ML}{10}})$$ $$RL = -10\log_{10}\frac{PR}{100}$$ $$RL = -20\log_{10}|\Gamma|$$
Mismatch Loss (dB) $$ML = -10\log_{10}(1 - (\frac{VSWR - 1}{VSWR + 1})^2)$$ $$ML = -10\log_{10}(1 - 10^{-\frac{RL}{10}})$$ - $$ML = -10\log_{10}(1 - \frac{PR}{100})$$ $$ML = -10\log_{10}(1 - |\Gamma|^2)$$
Power Reflected (%) $$PR = 100\cdot(\frac{VSWR - 1}{VSWR + 1})^2$$ $$PR = 100\cdot10^{-\frac{RL}{10}}$$ $$PR = 100\cdot(1 - 10^{-\frac{ML}{10}})$$ - $$PR = 100|\Gamma|^2$$
Voltage Reflection Coefficient, $$\Gamma$$ $$|\Gamma| = \frac{VSWR - 1}{VSWR + 1}$$ $$|\Gamma| = 10^{-\frac{RL}{20}}$$ $$|\Gamma| = \sqrt{1 - 10^{-\frac{ML}{10}}}$$ $$|\Gamma| = \sqrt{\frac{PR}{100}}$$ -