Characteristic impedance
Suppose, though, that we had a set of
parallel wires of infinite length, with no lamp at
the end. What would happen when we close the switch? Being
that there is no longer a load at the end of the wires, this
circuit is open. Would there be no current at all?
Despite being able to avoid wire resistance
through the use of superconductors in this "thought
experiment," we cannot eliminate capacitance along the
wires' lengths. Any pair of conductors separated by
an insulating medium creates capacitance between those
conductors:
Voltage applied between two conductors
creates an electric field between those conductors. Energy
is stored in this electric field, and this storage of energy
results in an opposition to change in voltage. The reaction
of a capacitance against changes in voltage is described by
the equation i = C(de/dt), which tells us that current will
be drawn proportional to the voltage's rate of change over
time. Thus, when the switch is closed, the capacitance
between conductors will react against the sudden voltage
increase by charging up and drawing current from the source.
According to the equation, an instant rise in applied
voltage (as produced by perfect switch closure) gives rise
to an infinite charging current.
However, the current drawn by a pair of
parallel wires will not be infinite, because there exists
series impedance along the wires due to inductance. Remember
that current through any conductor develops a
magnetic field of proportional magnitude. Energy is stored
in this magnetic field, and this storage of energy results
in an opposition to change in current. Each wire develops a
magnetic field as it carries charging current for the
capacitance between the wires, and in so doing drops voltage
according to the inductance equation e = L(di/dt). This
voltage drop limits the voltage rate-of-change across the
distributed capacitance, preventing the current from ever
reaching an infinite magnitude:
Because the electrons in the two wires
transfer motion to and from each other at nearly the speed
of light, the "wave front" of voltage and current change
will propagate down the length of the wires at that same
velocity, resulting in the distributed capacitance and
inductance progressively charging to full voltage and
current, respectively, like this:
The end result of these interactions is a
constant current of limited magnitude through the battery
source. Since the wires are infinitely long, their
distributed capacitance will never fully charge to the
source voltage, and their distributed inductance will never
allow unlimited charging current. In other words, this pair
of wires will draw current from the source so long as the
switch is closed, behaving as a constant load. No longer are
the wires merely conductors of electrical current and
carriers of voltage, but now constitute a circuit component
in themselves, with unique characteristics. No longer are
the two wires merely a pair of conductors, but rather
a transmission line.
As a constant load, the transmission line's
response to applied voltage is resistive rather than
reactive, despite being comprised purely of inductance and
capacitance (assuming superconducting wires with zero
resistance). We can say this because there is no difference
from the battery's perspective between a resistor eternally
dissipating energy and an infinite transmission line
eternally absorbing energy. The impedance (resistance) of
this line in ohms is called the characteristic impedance,
and it is fixed by the geometry of the two conductors. For a
parallel-wire line with air insulation, the characteristic
impedance may be calculated as such:
If the transmission line is coaxial in
construction, the characteristic impedance follows a
different equation:
In both equations, identical units of
measurement must be used in both terms of the fraction. If
the insulating material is other than air (or a vacuum),
both the characteristic impedance and the propagation
velocity will be affected. The ratio of a transmission
line's true propagation velocity and the speed of light in a
vacuum is called the velocity factor of that line.
Velocity factor is purely a factor of the
insulating material's relative permittivity (otherwise known
as its dielectric constant), defined as the ratio of
a material's electric field permittivity to that of a pure
vacuum. The velocity factor of any cable type -- coaxial or
otherwise -- may be calculated quite simply by the following
formula:
Characteristic impedance is also known as
natural impedance, and it refers to the equivalent
resistance of a transmission line if it were infinitely
long, owing to distributed capacitance and inductance as the
voltage and current "waves" propagate along its length at a
propagation velocity equal to some large fraction of light
speed.
It can be seen in either of the first two
equations that a transmission line's characteristic
impedance (Z0) increases as the conductor spacing
increases. If the conductors are moved away from each other,
the distributed capacitance will decrease (greater spacing
between capacitor "plates"), and the distributed inductance
will increase (less cancellation of the two opposing
magnetic fields). Less parallel capacitance and more series
inductance results in a smaller current drawn by the line
for any given amount of applied voltage, which by definition
is a greater impedance. Conversely, bringing the two
conductors closer together increases the parallel
capacitance and decreases the series inductance. Both
changes result in a larger current drawn for a given applied
voltage, equating to a lesser impedance.
Barring any dissipative effects such as
dielectric "leakage" and conductor resistance, the
characteristic impedance of a transmission line is equal to
the square root of the ratio of the line's inductance per
unit length divided by the line's capacitance per unit
length:
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REVIEW:
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A transmission line is a pair of
parallel conductors exhibiting certain characteristics due
to distributed capacitance and inductance along its
length.
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When a voltage is suddenly applied to one
end of a transmission line, both a voltage "wave" and a
current "wave" propagate along the line at nearly light
speed.
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If a DC voltage is applied to one end of
an infinitely long transmission line, the line will draw
current from the DC source as though it were a constant
resistance.
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The characteristic impedance (Z0)
of a transmission line is the resistance it would exhibit
if it were infinite in length. This is entirely different
from leakage resistance of the dielectric separating the
two conductors, and the metallic resistance of the wires
themselves. Characteristic impedance is purely a function
of the capacitance and inductance distributed along the
line's length, and would exist even if the dielectric were
perfect (infinite parallel resistance) and the wires
superconducting (zero series resistance).
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Velocity factor is a fractional
value relating a transmission line's propagation speed to
the speed of light in a vacuum. Values range between 0.66
and 0.80 for typical two-wire lines and coaxial cables.
For any cable type, it is equal to the reciprocal (1/x) of
the square root of the relative permittivity of the
cable's insulation.
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