Report/ratio of standing wave

The report/ratio of standing waves ( ROS ) and the standing-wave ratios ( TOS ) express the quality of the adaptation of antenna, with a line of coaxial transmission, or two-wire.

Definitions

In a line of transmission an incidental wave , of amplitude $V\_i$, and a reflected wave coexist, of amplitude $V\_r$.

The extrema of the resulting wave are the following:

• the maximum is reached when the incidental wave and the considered wave produce constructive interferences. There is thus $V\_\; \backslash \; mathrm\; \{max\}\; =\; V\_i\; +\; V\_r$;
• reciprocally, the minimum is reached when the two waves produce destructive interferences. There is thus $V\_\; \backslash \; mathrm\; \{min\}\; =\; V\_i\; -\; V\_r$.

The ROS (in English, SWR or more precisely VSWR) is defined as being the report/ratio of the extrema:

$\backslash \; mathrm\; \{ROS\}\; =\; \backslash \; frac\; \{V\_\; \backslash \; mathrm\; \{max\}\}\; \{V\_\; \backslash \; mathrm\; \{min\}\}\; =\; \backslash \; frac\; \{V\_i\; +\; V\_r\}\; \{V\_i\; -\; V\_r\}$

One also defines the coefficient of reflection Γ as being the report/ratio of the amplitudes (complex) considered and incidental:

$\backslash \; Gamma\; =\; \backslash \; frac\; \{\backslash \; underline\; \{V\}\; \_r\}\; \{\backslash \; underline\; \{V\}\; \_i\}$

Γ is complex: it takes account of the various phases. However, one generally handles ρ, the module of Γ:

$\backslash \; rho\; =\; |\backslash \; Gamma|\; =\; \backslash \; frac\; \{V\_r\}\; \{V\_i\}$

One can rewrite $V\_\; \backslash \; mathrm\; \{min\}$ and $V\_\; \backslash \; mathrm\; \{max\}$ using ρ:

• $V\_\; \backslash \; mathrm\; \{max\}\; =\; V\_i\; (1\; +\; \backslash \; rho)$;
• $V\_\; \backslash \; mathrm\; \{min\}\; =\; V\_i\; (1\; -\; \backslash \; rho)$.

From where a new expression of the ROS according to ρ:

$\backslash \; mathrm\; \{ROS\}\; =\; \backslash \; frac\; \{1\; +\; \backslash \; rho\}\; \{1\; -\; \backslash \; rho\}$

This formula makes it possible to pass from the module of Gamma (ρ) to the ROS.

The standing-wave ratio (TOS) is as for him equal to 100ρ, or if one wants, the expression of ρ like a percentage. By definition, it is the value of the amplitude of the considered wave expressed like a percentage of that of the incidental wave. One will be able to thus add the suffix " %".

To pass directly from the TOS to the ROS: Since ROS = (1 + ρ)/(1 - ρ) and that ρ = TOS/100, one will have:

ROS = (1 + TOS/100)/(1 - TOS/100) and after simplification,

$\backslash \; mathrm\; \{ROS\}\; =\; \backslash \; frac\; \{100\; +\; \backslash \; mathrm\; \{TOS\}\}\; \{100\; -\; \backslash \; mathrm\; \{TOS\}\}$

Example: So in a system antenna/line of transmission, 35% of the incidental tension are considered (thus a TOS of 35%), then the ROS (or the SWR or the VSWR) will be:

ROS= (100+35)/(100-35) = 2,08

--- By isolating term TOS algebraically, one will also obtain:

$\backslash \; mathrm\; \{TOS\}\; =100\; \backslash \; frac\; \{\backslash \; mathrm\; \{ROS\}\; -\; 1\}\; \{\backslash \; mathrm\; \{ROS\}\; +\; 1\}$

Example: If the ROS is 3,5:1, then the TOS will be of 55,6%

Connection with the impedances

Let us consider a broadcasting transmitter, of impedance Zs output, supplying an antenna, whose impedance of radiation is Rr through a line of transmission characterized by a characteristic Impédance Zc. So that a maximum of energy is radiated by the antenna, it is necessary that two conditions are met:

• Zs = Zc;
• Zc = Rr; when a line of transmission is thus finished on a load of impedance equal to its characteristic impedance, it is said that it is adapted .

The ROS is defined in the following way:

• ROS = Zc/Rr when Zc is higher or equal to Rr;
• ROS = Rr/Zc if Zc is lower than Rr; the ROS is thus always equal to or higher than the unit.

When Zs = Zc and that the ROS is worth 1, all the energy provided by the transmitter (with share losses in the line) is accepted by the antenna and transform in electromagnetic waves. On the other hand, if ROS > 1, part of energy is returned towards the transmitter; that can damage the stage of exit of the transmitter if the powers concerned are raised and ROS >> 1.

When Zs, Zc and Rr are not equal, one uses aerial matching transformers which will be placed between the exit of the amplifier of emission and the line (if Zs is not equal to Zc) and between the exit of the line and the antenna (to adapt Zc and Rr). The aerial matching transformers will be Transformateur S or lines quarter of wave.

Interpretation

When ROS > 1, circulate simultaneously in the line a direct wave, transmitter towards the antenna, and a wave reflected, antenna towards the transmitter. The superposition of these two waves in the line causes the appearance of standing waves: at certain places of the line, the amplitudes of the two waves are added, one has bellies (strong amplitude); in other places, the amplitudes are withdrawn, the amplitude of the resulting wave is minimum, it is what one calls the nodes .

See too

• Adaptation of impedances

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