Maxwell's equations for the transverse electromagnetic (TEM) waves on multiconductor transmission lines reduce to the telegrapher's equations. The general form of the telegrapher's equations in the frequency domain are given by:

where, boldface lower-case and upper-case symbols denote vectors and matrices, respectively.

For the TEM mode, the transverse distribution of electromagnetic fields at any instant of time is identical to that for the static solution. So, you can derive the four parameters for multiconductor TEM transmission lines, the resistance matrix

Unfortunately, all lines do not support pure TEM waves; some multiconductor systems inherently produce longitudinal field components. In particular, waves propagating in the presence of conductor losses or dielectric inhomogeneity (but not dielectric losses) must have longitudinal components. However, if the transverse components of fields are significantly larger than the longitudinal components, the telegrapher's equations and the four parameter matrices obtained from a static analysis still provide a good approximation. This is known as a quasi-static approximation. Multiconductor systems in which this approximation is valid are called quasi-TEM lines. For typical microstrip systems, the quasi-static approximation holds up to a few gigahertz.

In contrast to the static (constant)

where

On the other hand, the conductance matrix

where Go models the shunt current due to free electrons in imperfect dielectrics and Gd models the power loss due to the rotation of dipoles under the alternating field1.

All matrices in the previous section are symmetric. The diagonal terms of L and C are positive nonzero. The diagonal terms of R

The elements of admittance matrices are related to the self/mutual admittances (as those inputted by U Element):

where Y stands for C, G

A diagonal term of an admittance matrix is the sum of all the self and mutual admittances in its row. It is larger in absolute value than the sum of all off-diagonal terms in its row or column. Admittance matrices are strictly diagonally dominant (except for a zero matrix).

To illustrate the physical processes of wave propagation and reflection in transmission lines,2 consider the line with simple terminations excited with the voltage step as shown in Propagation of a Voltage Step in a Transmission Line.

At the time t=t

At t=t

At t=t

A summary of the process in Propagation of a Voltage Step in a Transmission Line is:

- Signals from the excitation sources spread out in the termination networks and propagate along the line.
- As the forward wave reaches the far-end termination, it reflects, propagates backward, reflects from the near-end termination, propagates forward again, and continues in a loop.
- The voltage at any point along the line, including the terminals, is a superposition of the forward and backward propagating waves.

System Model for Transmission Lines shows the system diagram of this process.

The model reproduces the general relationship between the physical phenomena of wave propagation, transmission, reflection, and coupling in a distributed system. It can represent arbitrarily distributed systems such as transmission lines, waveguides, and plane-wave propagation. The model is very useful for system analysis of distributed systems, and lets you write the macrosolution for a distributed system without complicated mathematical derivations.

W

Transmission lines along with terminations form a feedback system (as shown in System Model for Transmission Lines). Since the feedback loop contains a delay, the phase shift and the sign of the feedback change periodically with frequency. This causes the oscillations in the frequency-domain responses of transmission lines, as those in Star-Hspice Simulation Results(b).

An important special case occurs when the line terminates in another line. The system diagram of a line-to-line junction is shown in System Model for a Line-to-Line Junction. It can be used to solve multilayered plane-wave propagation problems, analyze common waveguide structures, and derive generalized transmission and reflection coefficient formulas and scattering parameter formulas.

The propagation functions, W