"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 µs 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.
- REVIEW:
- 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.
- 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.
- 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.
- A rule-of-thumb for transmission line
"shortness" is that the line must be at least 1/4 wavelength before it
is considered "long."
- 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.
- 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.
- 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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