"Long" and "short" transmission
lines
In DC and low-frequency AC circuits, the
characteristic impedance of parallel wires is usually
ignored. This includes the use of coaxial cables in
instrument circuits, often employed to protect weak voltage
signals from being corrupted by induced "noise" caused by
stray electric and magnetic fields. This is due to the
relatively short timespans in which reflections take place
in the line, as compared to the period of the waveforms or
pulses of the significant signals in the circuit. As we saw
in the last section, if a transmission line is connected to
a DC voltage source, it will behave as a resistor equal in
value to the line's characteristic impedance only for as
long as it takes the incident pulse to reach the end of the
line and return as a reflected pulse, back to the source.
After that time (a brief 16.292 �s for the mile-long coaxial
cable of the last example), the source "sees" only the
terminating impedance, whatever that may be.
If the circuit in question handles
low-frequency AC power, such short time delays introduced by
a transmission line between when the AC source outputs a
voltage peak and when the source "sees" that peak loaded by
the terminating impedance (round-trip time for the incident
wave to reach the line's end and reflect back to the source)
are of little consequence. Even though we know that signal
magnitudes along the line's length are not equal at any
given time due to signal propagation at (nearly) the speed
of light, the actual phase difference between start-of-line
and end-of-line signals is negligible, because line-length
propagations occur within a very small fraction of the AC
waveform's period. For all practical purposes, we can say
that voltage along all respective points on a low-frequency,
two-conductor line are equal and in-phase with each other at
any given point in time.
In these cases, we can say that the
transmission lines in question are electrically short,
because their propagation effects are much quicker than the
periods of the conducted signals. By contrast, an
electrically long line is one where the propagation time
is a large fraction or even a multiple of the signal period.
A "long" line is generally considered to be one where the
source's signal waveform completes at least a quarter-cycle
(90o of "rotation") before the incident signal
reaches line's end. Up until this chapter in the Lessons
In Electric Circuits book series, all connecting lines
were assumed to be electrically short.
To put this into perspective, we need to
express the distance traveled by a voltage or current signal
along a transmission line in relation to its source
frequency. An AC waveform with a frequency of 60 Hz
completes one cycle in 16.66 ms. At light speed (186,000
m/s), this equates to a distance of 3100 miles that a
voltage or current signal will propagate in that time. If
the velocity factor of the transmission line is less than 1,
the propagation velocity will be less than 186,000 miles per
second, and the distance less by the same factor. But even
if we used the coaxial cable's velocity factor from the last
example (0.66), the distance is still a very long 2046
miles! Whatever distance we calculate for a given frequency
is called the wavelength of the signal.
A simple formula for calculating wavelength
is as follows:
The lower-case Greek letter "lambda" (λ)
represents wavelength, in whatever unit of length used in
the velocity figure (if miles per second, then wavelength in
miles; if meters per second, then wavelength in meters).
Velocity of propagation is usually the speed of light when
calculating signal wavelength in open air or in a vacuum,
but will be less if the transmission line has a velocity
factor less than 1.
If a "long" line is considered to be one at
least 1/4 wavelength in length, you can see why all
connecting lines in the circuits discussed thusfar have been
assumed "short." For a 60 Hz AC power system, power lines
would have to exceed 775 miles in length before the effects
of propagation time became significant. Cables connecting an
audio amplifier to speakers would have to be over 4.65 miles
in length before line reflections would significantly impact
a 10 kHz audio signal!
When dealing with radio-frequency systems,
though, transmission line length is far from trivial.
Consider a 100 MHz radio signal: its wavelength is a mere
9.8202 feet, even at the full propagation velocity of light
(186,000 m/s). A transmission line carrying this signal
would not have to be more than about 2-1/2 feet in length to
be considered "long!" With a cable velocity factor of 0.66,
this critical length shrinks to 1.62 feet.
When an electrical source is connected to a
load via a "short" transmission line, the load's impedance
dominates the circuit. This is to say, when the line is
short, its own characteristic impedance is of little
consequence to the circuit's behavior. We see this when
testing a coaxial cable with an ohmmeter: the cable reads
"open" from center conductor to outer conductor if the cable
end is left unterminated. Though the line acts as a resistor
for a very brief period of time after the meter is connected
(about 50 Ω for an RG-58/U cable), it immediately thereafter
behaves as a simple "open circuit:" the impedance of the
line's open end. Since the combined response time of an
ohmmeter and the human being using it greatly exceeds
the round-trip propagation time up and down the cable, it is
"electrically short" for this application, and we only
register the terminating (load) impedance. It is the extreme
speed of the propagated signal that makes us unable to
detect the cable's 50 Ω transient impedance with an
ohmmeter.
If we use a coaxial cable to conduct a DC
voltage or current to a load, and no component in the
circuit is capable of measuring or responding quickly enough
to "notice" a reflected wave, the cable is considered
"electrically short" and its impedance is irrelevant to
circuit function. Note how the electrical "shortness" of a
cable is relative to the application: in a DC circuit where
voltage and current values change slowly, nearly any
physical length of cable would be considered "short" from
the standpoint of characteristic impedance and reflected
waves. Taking the same length of cable, though, and using it
to conduct a high-frequency AC signal could result in a
vastly different assessment of that cable's "shortness!"
When a source is connected to a load via a
"long" transmission line, the line's own characteristic
impedance dominates over load impedance in determining
circuit behavior. In other words, an electrically "long"
line acts as the principal component in the circuit, its own
characteristics overshadowing the load's. With a source
connected to one end of the cable and a load to the other,
current drawn from the source is a function primarily of the
line and not the load. This is increasingly true the longer
the transmission line is. Consider our hypothetical 50 Ω
cable of infinite length, surely the ultimate example of a
"long" transmission line: no matter what kind of load we
connect to one end of this line, the source (connected to
the other end) will only see 50 Ω of impedance, because the
line's infinite length prevents the signal from ever
reaching the end where the load is connected. In this
scenario, line impedance exclusively defines circuit
behavior, rendering the load completely irrelevant.
The most effective way to minimize the
impact of transmission line length on circuit behavior is to
match the line's characteristic impedance to the load
impedance. If the load impedance is equal to the line
impedance, then any signal source connected to the
other end of the line will "see" the exact same impedance,
and will have the exact same amount of current drawn from
it, regardless of line length. In this condition of perfect
impedance matching, line length only affects the amount of
time delay from signal departure at the source to signal
arrival at the load. However, perfect matching of line and
load impedances is not always practical or possible.
The next section discusses the effects of
"long" transmission lines, especially when line length
happens to match specific fractions or multiples of signal
wavelength.
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REVIEW:
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Coaxial cabling is sometimes used in DC
and low-frequency AC circuits as well as in high-frequency
circuits, for the excellent immunity to induced "noise"
that it provides for signals.
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When the period of a transmitted voltage
or current signal greatly exceeds the propagation time for
a transmission line, the line is considered
electrically short. Conversely, when the propagation
time is a large fraction or multiple of the signal's
period, the line is considered electrically long.
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A signal's wavelength is the
physical distance it will propagate in the timespan of one
period. Wavelength is calculated by the formula λ=v/f,
where "λ" is the wavelength, "v" is the propagation
velocity, and "f" is the signal frequency.
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A rule-of-thumb for transmission line
"shortness" is that the line must be at least 1/4
wavelength before it is considered "long."
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In a circuit with a "short" line, the
terminating (load) impedance dominates circuit behavior.
The source effectively sees nothing but the load's
impedance, barring any resistive losses in the
transmission line.
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In a circuit with a "long" line, the
line's own characteristic impedance dominates circuit
behavior. The ultimate example of this is a transmission
line of infinite length: since the signal will never
reach the load impedance, the source only "sees" the
cable's characteristic impedance.
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When a transmission line is terminated by
a load precisely matching its impedance, there are no
reflected waves and thus no problems with line length.
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