1.14 Rise Time/Fall Time
The typical logic signals we will encounter are voltages that step between two values. The one value is approximately the power supply voltage V and the second value is approximately the zero reference. For most of the discussions in this book we use the power supply voltage and zero volts. Logic is usually controlled by a clock signal that is a square wave. The rise and fall time of this clock signal is usually less than 10% of the clock cycle. A 100 MHz clock rate might have rise and fall times of 1 ns. The rise and fall time of the logic signals should be in this same range.
The Fourier spectrum of a square wave consists of sine waves at the fundamental frequency and at all odd harmonics of that frequency. The amplitude of the fundamental harmonic is the voltage V1 = V/π volts rms. The amplitude of the nth harmonic is V/πn volts rms. A 10-MHz square wave is made up of sine waves at 10, 30, 50, 70, … MHz. The amplitudes of these sine waves are V/3.14, V/9.42, V/15.75, V/21.98, … When these sine wave voltages are added together, a square wave of voltage V is the result.
In a practical circuit, a square wave has a finite rise or fall time. The harmonics (sine waves) that make up this square wave will have amplitudes that fall off proportional to frequency out to a frequency of 1/πτr, where τr is the rise time or fall time, whichever is the smallest. Beyond this frequency, the harmonic amplitudes fall off proportional to the square of frequency. If the 10-MHz square waves have a rise time of 10 ns then the harmonics above 30 MHz are not a serious problem.
It is impractical to analyze logic signals using sine waves. Logic signals are not perfect square waves and the length of the transmission lines is an important factor that is often not considered. The rise time (fall time), however, does provide key information that will be very useful in later discussions. It is important to understand the ideas of harmonic analysis as this is often the only tool we have to solve certain types of problems.