The theme of this section is the distributed parameter transmission line shown in Fig. 14.1.1. Over any finite axial length of interest, there is an infinite set of the basic units shown in the inset, an infinite number of capacitors and inductors. The parameters L and C are defined per unit length. Thus, for the segment shown between z +
z and z, L
Fig 14.1.1 Incremental length of distributed parameter transmission line. In the limit where the incremental length
0, this distributed parameter transmission line serves as a model for the propagation of three types of electromagnetic fields.
First, it gives an exact representation of uniformly polarized electromagnetic plane waves. Whether these are waves in free space, perhaps as launched by the dipole considered in Sec. 12.2, or TEM waves between plane parallel perfectly conducting electrodes, Sec. 13.1, these fields depend only on one spatial coordinate and time. Second, we will see in the next section that the distributed parameter transmission line represents exactly the (z, t) dependence of TEM waves propagating on pairs of axially uniform perfect conductors forming transmission lines of arbitrary cross-section. Such systems are a generalization of the parallel plate transmission line. By contrast with that special case, however, the fields generally depend on the transverse coordinates. These fields are therefore, in general, three dimensional. Third, it represents in an approximate way, the (z, t) dependence for systems of large aspect ratio, having lengths over which the fields evolve in the z direction (e.g., wavelengths) that are long compared to the transverse dimensions. To reflect the approximate nature of the model and the two- or three-dimensional nature of the system it represents, it is sometimes said to be quasi-one-dimensional.
We can obtain a pair of partial differential equations governing the transmission line current I(z, t) and voltage V(z, t) by first requiring that the currents into the node of the elemental section sum to zero
and then requiring that the series voltage drops around the circuit also sum to zero.
Then, division by
results in the transmission line equations.
The remainder of this section is an introduction to some of the physical situations represented by these laws.
Plane-Waves
In the following sections, we will develop techniques for describing the space-time evolution of fields on transmission lines. These are equally applicable to the description of electromagnetic plane waves. For example, suppose the fields take the form shown in Fig. 14.1.2.
Then, the x and y components of the laws of Ampère and Faraday reduce to
Fig 14.1.2 Possible polarization and direction of propagation of plane wave described by the transmission line equations. These laws are identical to the transmission line equations, (4) and (5), with
With this identification of variables and parameters, the discussion is equally applicable to plane waves, whether we are considering wave transients or the sinusoidal steady state in the following sections.
Ideal Transmission Line
The TEM fields that can exist between the parallel plates of Fig. 14.1.3 can either be regarded as plane waves that happen to meet the boundary conditions imposed by the electrodes or as a special case of transmission line fields. The following example illustrates the transition to the second viewpoint.
Fig 14.1.3 Example of transmission line where conductors are parallel plates.
Example 14.1.1. Plane Parallel Plate Transmission Line
In this case, the fields Ex and Hy pictured in Fig. 14.1.2 and described by (7) and (8) can exist unaltered between the plates of Fig. 14.1.3. If the voltage and current are defined as
Equations (7) and (8) become identical to the transmission line equations, (4) and (5), with the capacitance and inductance per unit length defined as
Note that these are indeed the C and L that would be found in Chaps. 5 and 8 for the pair of perfectly conducting plates shown in Fig. 14.1.3 if they had unit length in the z direction and were, respectively, "open circuited" and "short circuited" at the right end.
As an alternative to a field description, the distributed L-C transmission line model gives circuit theory interpretation to the physical processes at work in the actual system. As expressed by (1) and hence (4), the current I can be a function of z because some of it can be diverted into charging the "capacitance" of the line. This is an alternative way of representing the effect of the displacement current density on the right in Ampère's law, (7). The voltage V is a function of z because the inductance of the line causes a voltage drop, even though the conductors are pictured as having no resistance. This follows from (2) and (5) and embodies the same information as did Faraday's differential law (8). The integral of E from one conductor to the other at some location z can differ from that at another location because of the flux linked by a contour consisting of these integration paths and closing by contours along the perfect conductors.
In the next section, we will generalize our picture of TEM waves and see that (4) and (5) exactly describe transverse waves on pairs of perfect conductors of arbitrary cross-section. Of course, L and C are the inductance per unit length and capacitance per unit length of the particular conductor pair under consideration. The fields depend not only on the independent variables (z, t) appearing explicitly in the transmission line equations, but upon the transverse coordinates as well. Thus, the parallel plate transmission line and the generalization of that line considered in the next section are examples for which the distributed parameter model is exact.
In these cases, TEM waves are exact solutions to the boundary value problem at all frequencies, including frequencies so high that the wavelength of the TEM wave is comparable to, or smaller than, the transverse dimensions of the line. As one would expect from the analysis of Secs. 13.1-13.3, higher-order modes propagating in the z direction are also valid solutions. These are not described by the transmission line equations (4) and (5).
Quasi-One-Dimensional Models
The distributed parameter model is also often used to represent fields that are not quite TEM. As an example where an approximate model consists of the distributed L-C network, suppose that the region between the plane parallel plate conductors is filled to the level x = d < a by a dielectric of one permittivity with the remainder filled by a material having a different permittivity. The region between the conductors is then one of nonuniform permittivity. We would find that it is not possible to exactly satisfy the boundary conditions on both the tangential and normal electric fields at the interface between dielectrics with an electric field that only had components transverse to z.
