Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 2D: Nonuniform Transmission Lines

2D.1Nonuniform Transmission Lines

A nonuniform transmission line can represent many physical phenomena in an inhomogeneous medium whose medium properties are functions of position. A one-dimensional equivalent ideal transmission line will have the per unit values of series inductance L′ and the shunt capacitance C′ as functions of one spatial coordinate.

Let

(2D.1a)$$L^{\prime }=L^{\prime }\left(z\right),$$

(2D.1b)$$C^{\prime }=C^{\prime }\left(z\right).$$

If ε and μ are homogenous in the transverse plane, then

(2D.2)$$\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}=\frac{1}{\sqrt{\mathrm{\epsilon }\mathrm{\mu }}}=u_{\text{p}},$$

where u_p is the phase velocity.

The first-order coupled differential equations are

(2D.3a)$$-\frac{\partial V}{\partial z}=L^{\prime }\left(z\right)\frac{\partial I}{\partial t},$$

(2D.3b)$$-\frac{\partial I}{\partial z}=C^{\prime }\left(z\right)\frac{\partial V}{\partial t}.$$

In the phasor form,

(2D.4a)$$-\frac{\partial \tilde{V}}{\partial z}=L^{\prime }\left(z\right)\text{j}\mathrm{\omega }\tilde{I},$$

(2D.4b)$$-\frac{\partial \tilde{I}}{\partial z}=C^{\prime }\left(z\right)\text{j}\mathrm{\omega }\tilde{V}.$$

From Equations 2D.3a and 2D.3b, we can ...