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Beyond Design: When Do Traces Become Transmission Lines?

At low frequencies, traces and components on a PCB behave simply as lossless lumped elements—as taught in Circuit Theory 101. But as the frequency increases, the copper trace and adjacent dielectric(s) become a transmission line, the skin effect forces current into the outer regions of the conductor and frequency dependant losses impact on the quality of the signal. The PCB trace now behaves as a distributed system with parasitic inductance and capacitance characterized by delay and scattered reflections. The behavior we are now concerned about occurs in the frequency domain rather than the familiar time domain. This is the real world of high-speed design.

Ideally, square wave signals are just that—perfect square waves with an evenly, sloping rising and falling edge. However, in the real world, things are quite different. Figure 1 illustrates the rising edge of a square wave, in the ideal case (low frequency), compared to the real world (high frequency). The transmission line effects create under and overshoot resulting in ringing in the signal. If this ringing crosses the voltage input high threshold (VIH), at the receiver, then it may cause false triggering.

The Fourier theorem states that every function can be completely expressed as the sum of sines and cosines of various amplitudes and frequencies. The Fourier series expansion of a square wave is made up of a sum of odd harmonics. Figure 2 shows the conversion of a square wave from the time domain to the frequency domain and the resultant amplitudes of the frequency components. If the waveform has an even mark-to-space ratio, then the even harmonics cancel. The high-frequency content of a square wave is significantly affected by the rise time of the waveform. Also, as the frequency increases, the amplitude decreases. In the real world, one needs to consider the maximum bandwidth of a signal, including harmonics, rather than assume the perfect square wave fundamental frequency model.

Surprisingly, even at very low frequency, an old-fashioned telegraph line is a transmission line simply because the wire length is comparable to the signal rise time. In recent years, edge rates have become much faster, to the point where short traces, on a PCB, are a small multiple of the edge rates propagating through them. As such, one should consider these PCB traces to be transmission lines and analyze their signal integrity.

In general, all drivers whose trace length (in inches) is equal to or greater than the rise time (in nanoseconds) should be considered critical and treated as high-speed transmission lines. It is the signal rise/fall time, rather than the signal clock frequency, that determines the critical signal speed. However, a steep rise/fall time may be slowed by loading the signal line with a damping/back-matching resistor close to the source.

Impedance is the key factor that controls the stability of a design—it is the core issue of both the signal and power integrity methodology. At low frequencies, a PCB trace is almost an ideal circuit with little resistance, and without capacitance or inductance. Current follows the path of least resistance. But at high frequencies, alternating current circuit characteristics dominate causing inductance and capacitance to become prevalent.

To read this entire column, which appeared in the October 2017 issue of The PCB Design Magazine, click here.