Reflections of signals on conducting lines

This can happen, for instance, if two lengths of dissimilar transmission lines are joined.

There is a relationship between the measures of reflection coefficient and standing wave ratio.

Since the reflected current is equal in magnitude to the incident current, it must also be so that These two voltages will add to each other so that after the step has been reflected, twice the incident voltage appears across the output terminals of the line.

the reflected step arrives at the generator end and the condition of double voltage and zero current will pertain there also as well as all along the length of the line.

the step transient will be absorbed in the generator internal impedance and there will be no further reflections.

The same situation pertains if a very short transmission line is inserted between the generator and the open circuit.

and noticeable end-to-end delay is inserted, the generator – being initially matched to the impedance of the line – will have

Again, all of the energy must be reflected back up the line and the reflected voltage must be equal and opposite to the incident voltage by Kirchhoff's voltage law: and As the reflection travels back up the line, the two voltages subtract and cancel, while the currents will add (the reflection is double negative - a negative current traveling in the reverse direction), the dual situation to the open circuit case.

By definition, terminating in the characteristic impedance has the same effect as an infinitely long line.

The magnitude of the reflection will be smaller than the magnitude of the incident wave if the terminating impedance is wholly or partly resistive since some of the energy of the incident wave will be absorbed in the resistance.

), may be calculated by replacing the output of the line with an equivalent generator (figure 4) and is given by[3] The reflection,

is a complex function but the above expression shows that the magnitude is limited to The physical interpretation of this is that the reflection cannot be greater than the incident wave when only passive elements are involved (but see negative resistance amplifier for an example where this condition does not hold).

In this case, Since then showing that all the incident wave is reflected, and none of it is absorbed in the termination, as is to be expected from a pure reactance.

, in the reflection given by A discontinuity, or mismatch, somewhere along the length of the line results in part of the incident wave being reflected and part being transmitted onward in the second section of line as shown in figure 5.

, that it is transmitted in the forward direction: Another kind of discontinuity is caused when both sections of line have an identical characteristic impedance but there is a lumped element,

Even a simple overvoltage pulse entering a cable system as uncomplicated as the power wiring found in a typical private home can result in an oscillatory disturbance as the pulse is reflected to and from multiple circuit ends.

These ring waves as they are known[7] persist for far longer than the original pulse and their waveforms bears little obvious resemblance to the original disturbance, containing high frequency components in the tens of MHz range.

The points where the waves are in phase are anti-nodes and represent a peak in amplitude.

Nodes and anti-nodes alternate along the line and the combined wave amplitude varies continuously between them.

[9] The incident wave can be characterised in terms of the line's propagation constant

is measured in the reverse direction back up the line and the voltage is increasing closer to the source.

Likewise the reflected voltage is given by The total voltage on the line is given by It is often convenient to express this in terms of hyperbolic functions Similarly, the total current on the line is The voltage nodes (current nodes are not at the same locations) and anti-nodes occur when Because of the absolute value bars, the general case analytical solution is tiresomely complicated, but in the case of lossless lines (or lines that are short enough that the losses can be neglected)

The voltage equation then reduces to trigonometric functions and the partial differential of the magnitude of this yields the condition, Expressing

For terminations that are not purely resistive the spacing and alternation remain the same but the whole pattern is shifted along the line by a constant amount related to the phase of

at anti-nodes and nodes is called the voltage standing wave ratio (VSWR) and is related to the reflection coefficient by for a lossless line; the expression for the current standing wave ratio (ISWR) is identical in this case.

VSWR and the positions of the nodes are parameters that can be directly measured with an instrument called a slotted line.

One use is that VSWR and node position can be used to calculate the impedance of a test component terminating the slotted line.

[11][12] VSWR is the conventional means of expressing the match of a radio transmitter to its antenna.

and the expression reduces to trigonometric functions There are two structures that are of particular importance which use reflected waves to modify impedance.

This produces a purely imaginary impedance at its input, that is, a reactance By suitable choice of length, the stub can be used in place of a capacitor, an inductor or a resonant circuit.

A time-domain reflectometer ; an instrument used to locate the position of faults on lines from the time taken for a reflected wave to return from the discontinuity.
Fig. 1. Step voltage disturbance V u(t) is injected into the input of the line, v i travels down the line and is reflected back at the far end as v r .
Fig. 2. Equivalent circuit of generator feeding a line.
Fig. 3. Open circuit generator
Fig. 4. Equivalent circuit of an incident wave on a transmission line arriving at an arbitrary load impedance.
Fig. 5. Mismatch of transmission line characteristic impedances causes a discontinuity (marked with a star) in the line parameters and results in a reflected wave.
Fig. 6. Lumped components or networks connected to the line also cause a discontinuity (marked with a star).
Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Black dots represent electrons, and the arrows show the electric field.