Using Transmission Line Equations and Parameters

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. v is the voltage vector across the lines and i is the current vector along the lines.

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 R , the inductance matrix L , the conductance matrix G , and the capacitance matrix C , from a static analysis. The telegrapher's equations and the four parameter matrices from a static analysis completely and accurately describe TEM lines.

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.

Using Frequency-Dependent Resistance and Conductance Matrices

In contrast to the static (constant) L and C matrices, which provide good accuracy for a wide range of frequencies, the static (DC) R is only good for a very limited frequency range mainly due to the skin effect. A good approximate expression of the resistance matrix R with the skin effect is:

where Ro is the DC resistance matrix and Rs is the skin effect matrix. The imaginary term depicts the correct frequency response at high frequency; however, it may cause significant errors for low frequency applications. In the W Element, this imaginary term can optionally be excluded.

On the other hand, the conductance matrix G is often approximated as:

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.

Determining Matrix Properties

All matrices in the previous section are symmetric. The diagonal terms of L and C are positive nonzero. The diagonal terms of R o , R s , G o , and G d are nonnegative (can be zero). Off-diagonal terms of impedance matrices L, R o , and R s are nonnegative. Off-diagonal terms of admittance matrices C, G o , and G d are nonpositive. Off-diagonal terms of all matrices can be zero.

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

where Y stands for C, G o , or G d . The elements of the impedance matrices L, R o , and R s are the same as self/mutual impedances.

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).

Understanding Wave Propagation on Transmission Lines

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 1 , a voltage step from the source e 1 attenuated by the impedance Z 1 is propagating along the transmission line.

At t=t 2 , the voltage wave arrives at the far end of the transmission line, gets reflected, and is propagating in the backward direction. The voltage at the load end is the sum of the incident and reflected waves.

At t=t 3 , the reflected wave arrives back at the near end, gets reflected again, and is again propagating in the forward direction. The voltage at the source end is the sum of the attenuated voltage from the source e 1 , the backward wave, and reflected forward wave.

Figure 18-1: Propagation of a Voltage Step in a Transmission Line

Propagating a Voltage Step in a Transmission Line

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.

Figure 18-2: System Model for Transmission Lines

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 vr and W vb are the forward and backward matrix propagation functions for voltage waves; T 1 , T 2 and 1 , 2 stand for the near- and far-end matrix transmission and reflection coefficients.

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).

Handling Line-to-Line Junctions

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.

Figure 18-3: System Model for a Line-to-Line Junction

The propagation functions, W vr and W vb , describe how a wave is affected by its propagation from one termination to another, and are equal for the forward and backward directions: W vr and W vb . Coupling between the conductors of a multiconductor line is represented by the off-diagonal terms of the propagation functions. As a wave propagates along the line, it experiences delay, attenuation and distortion (see Propagation Function Transient Characteristics (unit-step response)). Lines with frequency-dependent parameters, and, therefore, all real lines, do not contain the frequency-independent attenuation component.

Figure 18-4: Propagation Function Transient Characteristics (unit-step response)