04.Multiport circuit parameters and transmission lines

In similar fashion the current wave equation can be found:

The solution for these two wave equations given below in terms of the arbitray functions F1 and F2 can be verified by substitution back into Eqs. (4.46) and (4.47):

The most useful function for F1 and F2 is the exponential function exp[j ωt š ˇz ], where ˇ D ω/v. The term Z0 is the same characteristic impedance in Eq. (4.41), which for the telegrapher’s equations is

L1

Z0 D D Lv D 4.51

C Cv

The variables L and C are given in terms of Henries and Farads per unit length and are thus distinguished from L and C used in lumped element circuit theory.

4.5THE TRANSMISSION LINE EQUATION

The transmission line equation was determined in Section 4.3 for a cascade of lumped element matched circuits. It is the input impedance of a transmission line terminated with a load, ZL, and it can also be found directly from analysis of a transmission line itself. The transmission line is characterized by its mechanical length, L, and its characteristic impedance, Z0. The characteristic impedance of a transmission line is a function only of the geometry and dielectric constant of the material between the lines and is independent of its terminating impedances. The input impedance of the transmission line depends on L, Z0, and ZL. When terminated with a nonmatching impedance, a standing wave is set up in the transmission line where the forwardand backward-going voltages and currents are as indicated in Fig. 4.10. At the load,

which is a measure of velocity of energy flow. For low-loss media, vgv D c2/ε, where c is the velocity of light in a vacuum. The negative-going wave of course has a phase velocity of ω/ˇ.

This traveling wave corresponds to the solution of the lossless telegrapher’s equations. The total voltage at any position, z, along the transmission line is the sum of the forwardand backward-going waves:

The total current at any point z is by Kirchhoff’s law the difference of the of the two currents:

At the input to the line where z D $, the ratio of Eqs. (4.58) and (4.59) gives the input impedance:

A few special cases illustrates some basic features of the transmission line equation. If z D 0, Zin 0 D ZL no matter what Z0 is. If ZL D Z0, then Zin z D Z0 no matter what z is. For a quarter wavelength line, Zin z D &/4 D Z20/ZL. The input impedance for any length of line can be readily calculated from Eq. (4.62), or by using the Smith chart.

4.6THE SMITH CHART

The Smith chart, as shown in Fig. 4.13, is merely a plot of the transmission line equation on a set of polar coordinates. The reflection coefficient is really an alternate way of expressing the input impedance relative to some standard value (Z0), which is typically 50 '. The reflection coefficient, , has a magnitude

64 MULTIPORT CIRCUIT PARAMETERS AND TRANSMISSION LINES

FIGURE 4.13 The Smith chart.

between 0 and 1 and a phase angle between 0° and 360°. The equations describing the radii and centers of the circles of the coordinates of the Smith chart are found by solving the normalized version of (4.60):

Solution of the real part of Eq. (4.64) gives the center of the resistance circles asr/ 1 C r , 0 with a radius of 1/ 1 C r . Solution of the imaginary part gives the center of the reactance circles as 1, 1/x with a radius of 1/x [2, pp. 121–129].

The voltage between the plates is the integral of the electric field:

a

V D Exdx D aEx 4.72

0

The magnetic field in the y direction will produce a current in the conductor that will travel in the z direction according to Ampere’s` law:

b

I D Hydy D Hyb 4.73

0

Substitution of Eqs. (4.72) and (4.73) into Eq. (4.44) gives

Comparison of this with Eq. (4.70) indicates that

-a

A similar substitution of Eqs. (4.72) and (4.73) into Eq. (4.45) and comparison with Eq. (4.71) indicates that

εb

so that the characteristic impedance for the parallel plate guide is

La -

Z0 D D 4.77

C b ε

4.7.3Coaxial Transmission Line

Coaxial transmission line comes in the form of rigid, semirigid, and flexible forms. The end view of a coaxial line, which is shown in Fig. 4.16, consists of an inner conductor and the outer conductor, which is normally grounded. The electric field points from the outer to the inner conductor, and the longitudinal current on the center conductor produces a magnetic field concentric to the inner conductor. The potential between the two conductors is a solution of the transverse form of Laplace’s equation in cylindrical coordinates where there is no potential difference in the longitudinal z direction. The notation for the divergence and curl operators follows that given in [3]:

r

H2

H1

FIGURE 4.17 The continuity of the magnetic field along the center conductor.

The outward normal vector of the center conductor, rO, is shown in Fig. 4.17. The surface current on the center conductor is determined by the boundary condition for the tangential magnetic field:

The later result occurs because the magnetic field is zero inside the conductor. The total current flowing in the center conductor is

Coaxial Dielectric Loss. The differential form of Ampere’s` law relates the magnetic field to both the conduction current and the displacement current. In the absence of a conductor,

By taking the curl of Eq. (4.88), the Helmholtz wave equation for H can be found. Solution of the wave equation would give the propagation constant, :

where εr is the relative dielectric constant and k0 is the propagation constant in free space. A lossy dielectric is typically represented as the sum of the lossless

70 MULTIPORT CIRCUIT PARAMETERS AND TRANSMISSION LINES

(real) and lossy (imaginary) parts:

The revised propagation constant is found by substituting this into Eq. (4.89). The result can be simplified by taking the first two terms of the Taylor series expansion since ε00r − ε0r:

The power loss is proportional to exp 2˛z .

Coaxial Conductor Loss. The power loss per unit length, P$, is obtained by taking the derivative of the power at a given point along a transmission line:

For a low-loss conductor where the dielectric losses are negligible, Eq. (4.87) becomes, with the help of Ohm’s law,

where . is the metal conductivity. This would be the same as Eq. (4.88) if

With this substitution the wave impedance becomes the metal surface impedance:

At the surface there will be a longitudinal electric field of ZmJs directed in the zO direction. Thus, in a lossy line, the fields will no longer be strictly TEM. This longitudinal electric field produces energy flow into the conductor proportional