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:
- REVIEW:
- A transmission line is a pair
of parallel conductors exhibiting certain characteristics due to
distributed capacitance and inductance along its length.
- 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.
- 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.
- 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).
- 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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