Simulating Digital Signal Behavior in a Transmission Line Model

From power generation and transmission to high-speed signaling lines in a PCB, transmission line behavior in an electrical system is an important point to consider with switching or oscillating signals. If your PCB traces exhibit transmission line behavior, then you’ll need to use a transmission line model to simulate its termination network and transient response. This helps ensure your signals are free of ringing or other signal integrity problems.

Transmission Line Models: Lossy vs. Lossless

Transmission lines can be modeled in two ways. An ideal transmission line is lossless, meaning there are no resistive losses in the line as the circuit propagates. Real transmission lines are lossy, meaning some of the voltage carried by the signal is converted to heat during propagation. Both of these models are cascaded models, meaning that the transmission line is modeled by placing multiples of the lossy or lossless circuit in series.

A lossless transmission line model ignores Ohmic losses due to resistance in the copper trace and substrate as the signal propagates, and each portion of the transmission line is treated as an LC circuit. This becomes important at lower speed/lower frequency signals as it determines the rate at which the transmission line impedance saturates to a fixed value with frequency.

Since copper has some small resistance, and because the dielectric between a PCB trace and its ground plane or return line is not a perfect insulator, there will be some signal loss as the signal propagates along the trace. This is included by placing a small resistor in series with the inductor, and a large resistor in parallel with the capacitor. Note that the parallel conductor is normally written in terms of its conductance G, which is just the inverse of the resistance of the substrate.

Lossy and lossless transmission line models

One should immediately note that 3 of the 4 parameters in a lossy transmission line are related to the material properties of the substrate. All of these parameters are expressed in per unit length of the trace.

• Conductance G: This is related to the conductivity of the substrate.

• Capacitance C: This depends on the dielectric constant of the substrate, the width of the trace, and the distance to the return line or ground plane.

• Inductance L: This depends on the relative magnetic permeability of the substrate, the area enclosed by the transmission line, and the width of the trace.

If you examine the impedance of the RLC network in the transmission line model, you can easily find that the trace impedance is equal to:

Trace impedance in the transmission line model

Note that, in the limit of high frequency, a lossy transmission line behaves as a lossless transmission line and the impedance is independent of frequency. Each of the parameters can be calculated by considering the geometry of the entire transmission line and dividing by the longitudinal length for each element. This provides a very good approximation for each parameter in the transmission line.

The Telegrapher’s Equations and Propagation Delay

The two equations that define the behavior of voltage and current on a trace are the Telegrapher’s equations:

Telegrapher’s equations

Here, x is the distance along the transmission line and t is time. Note that this assumes the cross sectional dimensions of the trace are much smaller than the wavelength for any signal travelling along the trace, thus transverse resonances and signal propagation along y and z can be ignored. This is valid up to tens of THz for a typical PCB trace. These two equations can be decoupled into their own wave equations:

Wave equations for voltage and current in a lossy transmission line model

These equations show that attenuation occurs in the circuit due to the (RC + GL) term. In the case of a lossless transmission line (R = G = 0), the above wave equations reduce to a lossless wave equation. These equations assume linear homogeneous isotropic media without dispersion.

Because electromagnetic signals move at the speed of light, the signal experiences some propagation delay as it travels from the source to the load. The propagation delay can be found by considering the effective refractive index seen by signals travelling along the trace, or it can be extracted directly from the wave equations for the voltage and current. The signal velocity can be written in terms of the circuit elements in the lossy transmission line model:

Signal velocity along a lossy transmission line

One can easily calculate the propagation delay from the signal velocity and trace length. Again, the lossless case is found by taking G = R = 0. One also finds that the signal velocity approaches the lossless velocity as frequency increases. The signal delay along the transmission line must be included in a SPICE-based simulation when modeling signal propagation.

Identifying Digital Signal Integrity Problems

A transmission line model can be used to simulate two important aspects of signal propagation. First, a transmission line model is an equivalent LC or RLC network, and it has some transient response when the signal switches. Adding the proper series termination resistor allows you to change the response from underdamped to perfectly damped. This is very important for eliminating ringing. Note that this slightly increases the signal rise/fall time seen at the load end of the transmission line.

Termination is important for suppressing signal reflections at the load end of the transmission line. This refers to impedance matching the load to the impedance of the transmission line. When there is an impedance discontinuity, the signal can reflect off the input port of the load, which induces another transient response as the signal travels back along the transmission line. Using a SPICE-based simulator allows you to examine the damping effect of a series termination resistor or the impedance matching provided by the termination network.

When modeling signal propagation, you’ll need to specify the load impedance in order to analyze the effect of reflections on signal integrity. Repeated ringing and a stepwise rise in the voltage at the load can arises due to back-and-forth reflections in a transmission line. These problems are eliminated by impedance matching the load to the transmission line.

Even if the line is terminated with a matching network, ringing can still arise during switching. Your goal should be to prevent ringing by ensuring your matching network provides the necessary level of damping to suppress ringing. When you examine an output from a SPICE-based simulator, you’ll be able to measure the damping constant and natural frequency of the RLC network in the transmission line. This then helps you design your matching network to suppress these signal integrity problems.

Example ringing signal at the load due to impedance mismatch and multiple reflections. This signal has 0.3 ns total propagation delay.

Now you can examine the temporal and frequency domain response of a transmission line model when you work with the OrCAD PSpice Simulator from Cadence. This unique package is built for circuit design and analysis in complex PCB designs, allowing you to build and analyze complex circuit models in your schematic and/or PCB easily.

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