Characteristic impedance is the ratio of voltage to current for a wave that is propagating in single direction on a transmission line. This is an important parameter in the analysis and design of circuits and systems using transmission lines. In this section, we formally define this parameter and derive an expression for this parameter in terms of the equivalent circuit model introduced in Section 3.4.
Consider a transmission line aligned along the
axis. Employing some results from Section 3.6, recall that the phasor form of the wave equation in this case is
Equation 3.7.1 relates the potential phasor
to the equivalent circuit parameters
. An equation of the same form relates the current phasor
to the equivalent circuit parameters:
satisfy the same linear homogeneous differential equation, they may differ by no more than a multiplicative constant. Since
is potential and
is current, that constant can be expressed in units of impedance. Specifically, this is the characteristic impedance, so-named because it depends only on the materials and cross-sectional geometry of the transmission line – i.e., things which determine γγ – and not length, excitation, termination, or position along the line.
are complex-valued constants whose values are determined by boundary conditions; i.e., constraints on
at some position(s) along the line. Also, we will make use of the telegrapher’s equations (Section 3.5):
We begin by differentiating Equation 3.7.4 with respect to
, which yields
Now we use this this to eliminate
in Equation 3.7.6, yielding
Solving the above equation for
Comparing this to Equation 3.7.5, we note
We now make the substitution
As anticipated, we have found that coefficients in the equations for potentials and currents are related by an impedance, namely,
The characteristic impedance
(ΩΩ) is the ratio of potential to current in a wave traveling in a single direction along the transmission line.
Take care to note that
is not the ratio of
in general; rather,
relates only the potential and current waves traveling in the same direction.
Finally, note that transmission lines are normally designed to have a characteristic impedance that is completely real-valued – that is, with no imaginary component. This is because the imaginary component of an impedance represents energy storage (think of capacitors and inductors), whereas the purpose of a transmission line is energy transfer.
- “Characteristic impedance” on Wikipedia.
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