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### Standing Waves

In the previous section, we introduced the voltage reflection coefficient that relates the forward to reflected voltage phasor.

Let us look at the physical meaning of these variables.

(a)
$\Gamma _L$ is the voltage reflection coefficient at the load (z=0),
(b)
$z$ is the axis in the direction of wave propagation,
(c)
$\beta$ is the phase constant,
(d)
$Z_0$ is the impedance of the transmission line.
(e)
$\tilde {V}_0^+$ is the phasor of the forward-going wave at the load,
(f)
$\Gamma _L \tilde {V}_0^+$ is the phasor of the reflected going wave at the load,
(g)
$\tilde {V}_0^+ e^{-j \beta z}$ is a forward voltage anywhere on the line,
(h)
$\Gamma _L \tilde {V}_0^+ e^{j \beta z}$ is a reflected voltage anywhere on the line,
(i)
$\tilde {V}(z)$ is the total voltage phasor on the line, the sum of forward and reflected voltage.
(j)
$\tilde {I}_0^+$ is the phasor of the forward-going current at the load,
(k)
$-\Gamma _L \frac {\tilde {V}_0^+}{Z_0}$ is the phasor of the reflected current at the load,
(l)
$\tilde {I}_0^+ e^{-j \beta z}$ is the phasor of a forward current anywhere on the line,
(m)
$-\Gamma _L \frac {\tilde {V}_0^+}{Z_0} e^{j \beta z}$ is the phasor of a reflected current anywhere on the line,
(n)
$\tilde {I}(z)$ is the phasor of the total current on the line, the sum of forward and reflected voltage.

The magnitude of a complex number can be found as $|z|=\sqrt {z z^*}$.Therefore the magnitude of the voltage anywhere on the line is $|\tilde {V}(z)|=\sqrt {\tilde {V}(z) \tilde {V}(z)^*}$. We can simplify this equation as shown in Figure eq:SWsw1.

The Equation eq:SWsw1 is written in terms of $z$. We set up the load at $z=0$, and the generator at $z=-l$. The positions of the maximums and minimum total voltage on the line will be at some position at the negative part of z-axis $z_{max}=-l_{max}$, and the minimums will be at $z_{max}=-l_{min}$.

The magnitude of the total voltage on the transmission line is given by Eq.eq:SWsw1. We will now visualize how the magnitude of the voltage looks on the transmission line.

Observe waves in the app below.

Is the black wave moving left or right?

left right

Is the black wave a forward, reflected or the sum of the two?

Forward Reflected The sum of the two

Observe waves in the app below.

Is the black wave moving left or right? The wave is not moving left or right, it is standing in place, so it is called a standing wave. Is the black wave a forward, reflected or the sum of the two?

Forward Reflected The sum of the two

### Voltage Standing Wave Ratio (VSWR) - pron: ”vee-s-uh-are”

The ratio of voltage minimum on the line over the voltage maximum is called the Voltage Standing Wave Ratio (VSWR) or just Standing Wave Ratio (SWR).

Note that the SWR is always equal or greater than 1.