Basic Transmission Lines
Electrical signals propagate in circuits with a finite velocity. In many cases, the time delays associated with signal propagation are negligible. However, for signals with high-frequency content that propagate significant distances, these delays cannot be ignored. Transmission line modeling is a relatively simple and intuitive way to analyze circuits with signals propagating on circuit board traces or cables whose length cannot be neglected.
Consider the simple circuit shown in Figure 1 that models a load resistance connected to a source and switch through a pair of wires. The switch closes at t = 0. Applying circuit theory, we expect the voltage across the resistor to change from 0 to VS at time t = 0. However, in real circuits, there is always a delay between the time the switch closes and the time the voltage at the load changes. This is because electromagnetic energy travels with a finite velocity.
In high-speed digital circuit designs, there are many situations where the time delay cannot be neglected. Depending on the properties of the dielectric in which the waves travel, the delay associated with circuit board traces or cables is typically on the order of 35-70 picoseconds per centimeter. A signal traveling across a large circuit board or a short cable may arrive several nanoseconds after it was initially sent.
Beyond the obvious timing implications for digital signals, there is an issue related to the fact that the source cannot initially supply the correct current before it has “seen” the load impedance. As a result, signal energy may bounce back and forth between the source and load before reaching the correct steady-state value. This can result in both signal integrity and electromagnetic compatibility problems.
In situations where the non-zero time delay is significant, engineers can predict and control its effects using uniform transmission line theory. Uniform transmission lines are conductor pairs with a uniform cross-section that carry electrical power or signals. Several common uniform transmission line configurations are shown in Figure 2.
The cross-sectional dimensions of uniform transmission lines must be small relative to the minimum wavelength contained in the signal. However, the length of a transmission line is unconstrained, and practical transmission lines can be hundreds of wavelengths long. Cross-country power lines, telephone wires, TV cables, and high-speed printed circuit board traces are all examples of transmission lines.
In schematics, transmission lines are typically represented by two parallel rectangles, as shown in Figure 5.3. The symbol resembles a pair of parallel wires, but it is used for any two-conductor transmission line, including microstrip traces and coaxial cables. If the transmission line is short and the propagation delay is negligible, it may not have any properties that need to be included in the circuit analysis. For longer conductor pairs, transmission line models can be combined with traditional circuit models to characterize the circuit's behavior.
Transmission line models recognize that the two conductors have physical properties that may impact their electrical behavior. For example, unless the conductors are superconducting, they will each have a certain amount of resistance. This resistance is not located at a particular point in the circuit but is distributed along the length of the transmission line. The total resistance of the wires is directly proportional to the length of the line. The distributed resistance of the transmission line is the sum of each conductor’s per-unit-length resistance and can be expressed in units of ohms per meter (Ω/m).
Current flowing in the conductor pair sets up a magnetic flux in the space surrounding the conductors. Any change in the current amplitude changes the magnetic flux, resulting in a voltage drop along the length of the line. This voltage can be expressed in terms of an inductance times the rate of change of the current, just as in circuit theory. The inductance is not located at one point. It is distributed along the length of the transmission line and can be expressed in units of henries per meter (H/m).
Transmission lines also have a distributed capacitance, expressed in farads per meter (F/m), due to electric-field coupling between the two conductors. If the dielectric material between the conductors is lossy, current leaking from one conductor to the other through the dielectric can be represented by a distributed conductance expressed in siemens per meter (S/m).
The equivalent circuit shown in Figure 4 models the effects of the distributed resistance, inductance, capacitance, and conductance. In this circuit representation, the distributed parameters of the transmission line are modeled using lumped elements. The lumped element model divides the transmission line into several electrically short (e.g., < λ/10) sections. The inductance of each section is represented by an inductor whose value is the inductance per unit length of the transmission line times the length of the section. A series resistor, a capacitor, and a parallel resistor represent the resistance, capacitance, and conductance of each section. Note that the series resistors in Figure 4 represent the sum of the resistances of both conductors. The inductors represent the inductance per unit length of the conductor pair. The model does not attempt to assign partial inductances and resistances to each conductor, since this would unnecessarily complicate the model and encourage misuse. It’s important to note that the reference voltage at the load end will always be undefined relative to the reference voltage at the source end if they are at electrically distant points.