Finite-length transmission lines
A transmission line of infinite length is
an interesting abstraction, but physically impossible. All transmission
lines have some finite length, and as such do not behave precisely the
same as an infinite line. If that piece of 50 Ω "RG-58/U" cable I
measured with an ohmmeter years ago had been infinitely long, I actually
would have been able to measure 50 Ω worth of resistance between the
inner and outer conductors. But it was not infinite in length, and so it
measured as "open" (infinite resistance).
Nonetheless, the characteristic impedance
rating of a transmission line is important even when dealing with
limited lengths. An older term for characteristic impedance, which I
like for its descriptive value, is surge impedance. If a
transient voltage (a "surge") is applied to the end of a transmission
line, the line will draw a current proportional to the surge voltage
magnitude divided by the line's surge impedance (I=E/Z). This simple,
Ohm's Law relationship between current and voltage will hold true for a
limited period of time, but not indefinitely.
If the end of a transmission line is
open-circuited -- that is, left unconnected -- the current "wave"
propagating down the line's length will have to stop at the end, since
electrons cannot flow where there is no continuing path. This abrupt
cessation of current at the line's end causes a "pile-up" to occur along
the length of the transmission line, as the electrons successively find
no place to go. Imagine a train traveling down the track with slack
between the rail car couplings: if the lead car suddenly crashes into an
immovable barricade, it will come to a stop, causing the one behind it
to come to a stop as soon as the first coupling slack is taken up, which
causes the next rail car to stop as soon as the next coupling's slack is
taken up, and so on until the last rail car stops. The train does not
come to a halt together, but rather in sequence from first car to last:
A signal propagating from the source-end
of a transmission line to the load-end is called an incident wave.
The propagation of a signal from load-end to source-end (such as what
happened in this example with current encountering the end of an
open-circuited transmission line) is called a reflected wave.
When this electron "pile-up" propagates
back to the battery, current at the battery ceases, and the line acts as
a simple open circuit. All this happens very quickly for transmission
lines of reasonable length, and so an ohmmeter measurement of the line
never reveals the brief time period where the line actually behaves as a
resistor. For a mile-long cable with a velocity factor of 0.66 (signal
propagation velocity is 66% of light speed, or 122,760 miles per
second), it takes only 1/122,760 of a second (8.146 microseconds) for a
signal to travel from one end to the other. For the current signal to
reach the line's end and "reflect" back to the source, the round-trip
time is twice this figure, or 16.292 µs.
High-speed measurement instruments are
able to detect this transit time from source to line-end and back to
source again, and may be used for the purpose of determining a cable's
length. This technique may also be used for determining the presence
and location of a break in one or both of the cable's conductors,
since a current will "reflect" off the wire break just as it will off
the end of an open-circuited cable. Instruments designed for such
purposes are called time-domain reflectometers (TDRs). The basic
principle is identical to that of sonar range-finding: generating a
sound pulse and measuring the time it takes for the echo to return.
A similar phenomenon takes place if the
end of a transmission line is short-circuited: when the voltage
wave-front reaches the end of the line, it is reflected back to the
source, because voltage cannot exist between two electrically common
points. When this reflected wave reaches the source, the source sees the
entire transmission line as a short-circuit. Again, this happens as
quickly as the signal can propagate round-trip down and up the
transmission line at whatever velocity allowed by the dielectric
material between the line's conductors.
A simple experiment illustrates the
phenomenon of wave reflection in transmission lines. Take a length of
rope by one end and "whip" it with a rapid up-and-down motion of the
wrist. A wave may be seen traveling down the rope's length until it
dissipates entirely due to friction:
This is analogous to a long transmission
line with internal loss: the signal steadily grows weaker as it
propagates down the line's length, never reflecting back to the source.
However, if the far end of the rope is secured to a solid object at a
point prior to the incident wave's total dissipation, a second wave will
be reflected back to your hand:
Usually, the purpose of a transmission
line is to convey electrical energy from one point to another. Even if
the signals are intended for information only, and not to power some
significant load device, the ideal situation would be for all of the
original signal energy to travel from the source to the load, and then
be completely absorbed or dissipated by the load for maximum
signal-to-noise ratio. Thus, "loss" along the length of a transmission
line is undesirable, as are reflected waves, since reflected energy is
energy not delivered to the end device.
Reflections may be eliminated from the
transmission line if the load's impedance exactly equals the
characteristic ("surge") impedance of the line. For example, a 50 Ω
coaxial cable that is either open-circuited or short-circuited will
reflect all of the incident energy back to the source. However, if a 50
Ω resistor is connected at the end of the cable, there will be no
reflected energy, all signal energy being dissipated by the resistor.
This makes perfect sense if we return to
our hypothetical, infinite-length transmission line example. A
transmission line of 50 Ω characteristic impedance and infinite length
behaves exactly like a 50 Ω resistance as measured from one end. If we
cut this line to some finite length, it will behave as a 50 Ω resistor
to a constant source of DC voltage for a brief time, but then behave
like an open- or a short-circuit, depending on what condition we leave
the cut end of the line: open or shorted. However, if we terminate
the line with a 50 Ω resistor, the line will once again behave as a 50 Ω
resistor, indefinitely: the same as if it were of infinite length again:
In essence, a terminating resistor
matching the natural impedance of the transmission line makes the line
"appear" infinitely long from the perspective of the source, because a
resistor has the ability to eternally dissipate energy in the same way a
transmission line of infinite length is able to eternally absorb energy.
Reflected waves will also manifest if the
terminating resistance isn't precisely equal to the characteristic
impedance of the transmission line, not just if the line is left
unconnected (open) or jumpered (shorted). Though the energy reflection
will not be total with a terminating impedance of slight mismatch, it
will be partial. This happens whether or not the terminating resistance
is greater or less than the line's characteristic
impedance.
Re-reflections of a reflected wave may
also occur at the source end of a transmission line, if the
source's internal impedance (Thevenin equivalent impedance) is not
exactly equal to the line's characteristic impedance. A reflected wave
returning back to the source will be dissipated entirely if the source
impedance matches the line's, but will be reflected back toward the line
end like another incident wave, at least partially, if the source
impedance does not match the line. This type of reflection may be
particularly troublesome, as it makes it appear that the source has
transmitted another pulse.
- REVIEW:
- Characteristic impedance is also known
as surge impedance, due to the temporarily resistive behavior
of any length transmission line.
- A finite-length transmission line will
appear to a DC voltage source as a constant resistance for some short
time, then as whatever impedance the line is terminated with.
Therefore, an open-ended cable simply reads "open" when measured with
an ohmmeter, and "shorted" when its end is short-circuited.
- A transient ("surge") signal applied
to one end of an open-ended or short-circuited transmission line will
"reflect" off the far end of the line as a secondary wave. A signal
traveling on a transmission line from source to load is called an
incident wave; a signal "bounced" off the end of a transmission
line, traveling from load to source, is called a reflected wave.
- Reflected waves will also appear in
transmission lines terminated by resistors not precisely matching the
characteristic impedance.
- A finite-length transmission line may
be made to appear infinite in length if terminated by a resistor of
equal value to the line's characteristic impedance. This eliminates
all signal reflections.
- A reflected wave may become
re-reflected off the source-end of a transmission line if the source's
internal impedance does not match the line's characteristic impedance.
This re-reflected wave will appear, of course, like another pulse
signal transmitted from the source.
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