The difference between VSWR and the S₁₁ reflection coefficient

A traveling signal along an electrical transmission line reflects back toward the source when one or more events occurs:
One or both of the conductors is open at the far end (receiving end).
The conductors are shorted at the far end (very common).
There is a discontinuity in the characteristic impedance or an electrical fault at some point along the transmission line.
The transmission line is not terminated in its characteristic impedance at either end.

That said, there are numerous ways to characterize signal reflections in transmission lines. Two of the most fundamental ways which are sometimes confused are the voltage standing-wave ratio, or VSWR, and the reflection coefficient, normally represented on vector network analyzers as the S₁₁ S parameter.

The VSWR is equal to the maximum voltage on the transmission line divided by the minimum voltage. The voltage fluctuations come about as a result of the constructive and destructive interference between voltage components from the forward power and the reflected power summing together. VSWR is actually a measure of how well or how badly the impedance of a load matches the characteristic impedance of a transmission line or waveguide. Impedance mismatches are what cause the standing waves along the transmission line.

The ratio part of VSWR arises because maximum and minimum voltages are expressed as a ratio. For example, the VSWR value 1.2:1 means standing waves along the transmission line gives the ac voltage a peak value 1.2 times that of the minimum ac voltage along that line, assuming the line is at least one-half wavelength long. When we are talking about the ratio of the maximum amplitude to minimum amplitude of the transmission line’s currents, electric field strength, or the magnetic field strength, rather than the ratio of the voltages, the more general term is the standing wave ratio. The SWR for all these entities is identical, neglecting transmission line loss.

VSWR serves as a measure of the degree to which the impedance of a load matches the characteristic impedance of a transmission line carrying radio frequency signals. Impedances match when the source impedance is the complex conjugate of the load impedance. The optimum scenario is for the imaginary part of the complex impedance of both the source and load to be zero, that is, pure resistances, equal to the characteristic impedance of the transmission line. A mismatch between the load impedance and the transmission line reflects part of the forward wave sent toward the load back towards the source. The source then sees a different impedance than it expects.

The mismatch causes the standing waves along the transmission line which magnify transmission-line losses. A matched load would mean a VSWR of 1:1 implying no reflected wave. An infinite VSWR represents complete reflection with all the incident power reflected back toward the source.

That brings us to the reflection coefficient. As we’ve noted, a wave in a transmission line is partly reflected when the line is terminated with an impedance unequal to its characteristic impedance. The reflection coefficient is often written as S₁₁ in S parameters or as Greek capital gamma, Γ.

S₁₁=Γ =V_r /V_f = (Z_L-Z₀)/(Z_L+Z₀)
where V_r is the complex magnitude of the reflected wave, V_f is the complex magnitude of the forward wave, Z_L is the complex load impedance, and Z₀ is the complex source impedance.  Thus Γ is a complex number that describes both the magnitude and the phase shift of the reflection. The simplest cases with Γ measured at the load are Γ=-1, meaning there is a complete negative reflection (when the line is short-circuited); Γ=0, meaning there is no reflection and the line is perfectly matched; and Γ=1 meaning  complete positive reflection and the line is open-circuited.

Further, the VSWR directly corresponds to the magnitude of Γ. This becomes evident by considering that at some points along the transmission line, the forward and reflected wave interfere constructively with the resulting amplitude V_max given by the sum of the wave amplitudes:

|V_max| = |V_r |+|V_f| = |ΓV_f|+|V_f| = (|Γ|+1) |V_f|

At other points, forward and reflected waves are completely out of phase and the amplitudes cancel:

|V_min| = |V_f |-|V_r| = |V_f|-|ΓV_f| = (1-|Γ|) |V_f|

Then VSWR = |V_max|/|V_min|  = (|Γ|+1)/(1-|Γ|)

The old way of measuring VSWR and reflection coefficients is with a slotted line. It consists of a precision transmission line with a movable insulated probe inserted into a longitudinal slot cut into the line. In a co-axial slotted line, the slot is cut into the outer conductor of the line. A probe is inserted past the outer conductor, but not far enough to touch the inner conductor. In a rectangular waveguide, the slot is usually cut along the center of the broad wall of the waveguide.

The probe samples the electric field inside the transmission line. For accuracy, the probe must disturb the field as little as possible, so the probe diameter and slot width are kept small (usually around 1 mm). In waveguide slotted lines the slot must be positioned where the current in the waveguide walls is parallel to the slot. Then the slot won’t disturb the current as long as it is not too wide. For the dominant mode this is on the center-line of the broad face of the waveguide, but for some other modes it may need to be off-center. This is not an issue for coax lines because they operate in the TEM (transverse electromagnetic) mode and the current is everywhere parallel to the slot.

There are two parts to the disturbance to the field inside the line caused by the insertion of the probe. The first comes from the power the probe has extracted from the line and manifests as a lumped equivalent circuit of a resistor. Minimizing the distance the probe is inserted into the line also minimizes the amount of power extracted. The second part of the disturbance is from energy stored in the field around the probe and appears as a lumped equivalent of a capacitor. This capacitance can be cancelled out with an inductance of equal and opposite impedance. Lumped inductors are impractical at microwave frequencies. so instead, an adjustable stub with an inductive equivalent circuit is used to tune out the probe capacitance. The result is an equivalent circuit of a high impedance shunting across the line which has little effect on the transmitted power in the line.

Of course, slotted lines can only carry out a measurement at one frequency at a time so a fair amount of manual labor must go into producing a plot of a parameter versus frequency. Consequently, VNAs have replaced slotted lines for work at lower RF frequencies. VNAs used at millimeter wave frequencies still tend to be pricey, so slotted lines can still be found in labs that do work at those frequencies.