16. Transmission-Line Equations

Consider the TEM mode in a parallel-plate waveguide shown in the figure below.

Parallel-Plate Transmission LIne

The only non-zero field components are
       

       

Making use of this information as well as , we can rewrite Faraday's and Ampere's laws:

       

(1)

       

(2)

into
       

(3)

       

(4)

Furthermore, we can relate the electric potential difference V between the two plates (as indicated in the figure above) to the electric field via
       

(5)

and the electric current I flowing on the two plates to the magnetic field via the introduction of the surface current K,
       

(6)

       

(7)

The closed line integral in (6) is carried out around the loop shown in the figure above. Then, (3) and (4) become
       

(8)

       

(9)

For parallel plates, the capacitance per unit length and inductance per unit length are
       

(10)

       

(11)

Together with (8) and (9) we obtain the two transmission-line equations:
       

(12)

       

(13)

These two equations are also known as telegrapher's equations

Wave Equations and Solutions

If we differentiate (12) with respect to z and substitute (13) for the resulted right-hand side, then we can show that the Helmholtz equation governs the electric potential between the two plates.
       

(14)

Similarly, differentiating (13) with respect to z and substitute (12) for the resulted right-hand side leads to
       

(15)

A simplest solution to (14) is of the form:
       

(16)

provided the corresponding dispersion relation
       

(17)

is satisfied.

In general, along a transmission line, the general solution to (14) includes voltages travelling in both positive and negative z-direction, hence is of the form:
       

(18)

which is subject to the dispersion relation (17). The corresponding current can be obtained from (12):
       

(19)

where
       

(20)

and
       

(21)

is the characteristic impedance of the transmission line. Also
       

(22)

is the characteristic admittance of the transmission line. For a parallel-plate transmission line,
       

(23)

and for a coaxial line where
       

(24)

and
       

(25)

are the inductance and capacitance per unit length, the characteristic impedance is
       

(26)

This page was last updated on March 9, 1998.