Square waves are useful in testing components and circuit at higher frequencies. Ideal square waves are often pictured in textbooks with steep sides (representing fast rise and fall times) and a flat top (for well-controlled amplitude behavior). When a square wave is needed for experimentation, one may have the expectation that the test equipment, typically a function generator, will provide a square wave that looks like those images from the textbook or in a test-equipment-manufacturer's data sheet. But this is not always the reality.

As most engineers know, square waves are formed by the summation of all odd harmonics of a fundamental frequency. Unfortunately, function generators do not have infinite bandwidths and cannot include all harmonics of a fundamental frequency in shaping the rise and fall times of a square wave. Thus, a real-world square wave will have less-than-perfect rise/fall times. This then begs the question, "When is a square wave square?" The answer will depend on an engineer's requirements and what is considered adequate for a given test application. The answer will further depend on a statistical distribution and how one suitably specifies the square-wave performance of a function generator. When a function generator is selected by its specified performancesuch as 10-, 20-, or 50-MHz outputs for sine waves, square waves, and pulsed outputsthe actual performance may fall short of what is needed for a particular test application. At the function generator's maximum specified frequency, F_{max}, the sine-wave output will look like Fig. 1, while the square-wave output will look more like Fig. 2.

But each measurement application is different, and the requirements of different applications will dictate whether function-generator performance represented by Fig. 3 (which is at two-thirds F_{max}) is good enough, or if the performance represented in Fig. 4 (in which the sine wave contains third-harmonic frequencies) or in Fig. 5 (in which the sine wave contains fifth-harmonic frequencies) is needed. It is simply a matter of how many odd-harmonic frequencies must be included to create the necessary rise and fall times in the square wave, and how this related to the output bandwidth of the function generator.

Some circuit design, especially in high-speed analog and digital circuits, uses the 20% to 80% points of a rising edge as the definition of rise-time, τ_{R}. In this article, the more traditional 10% to 90% points are assumed as the definitions for rise and fall time, as shown in Fig. 6. For a sine wave, the amount of phase rotation in going from its 10% to 90% points, defining the rise time (τ_{R}) is shown to be 2 x 53.13 = 106.26. This is determined by:

θ = sin^{-1} (0.8) (1)

The ratio of this rise time to the period (T) of the sine wave is:

106.26/360 = 0.295T (2)

This leads to the relation:

τ_{R} = 0.295T (for sine waves) (3)

Since period T is related to frequency by T = 1/F, we have Eq. 4:

F = 0.295/τ_{R} (4)

Any given rise time for τ_{R} has the corresponding sine-wave frequency of 0.295/t_{R}.

The way to make use of this relationship is to make a decision as to how many odd harmonics one desires in defining the "squareness" of a square wave or pulse. By comparing the figures above, if it is adequate to include only the third harmonic (as shown in Fig. 4), then a function generator must have the capability of delivering a sine wave (produce a sine wave with enough bandwidth) that is three times higher in frequency than the fundamental frequency. Similarly, to produce a waveform containing the fifth harmonic (as shown in Fig. 5), the instrument must be capable of producing a sine wave with bandwidth that is five times that of the fundamental frequency. However, most test equipment specifications don't indicate the output bandwidth. Often, it will be possible to find the sampling rate and number of bits of digital resolution on a data sheet, but the key specification (if it is included at all) is the square-wave rise time. With that value and Eq. 4, it is possible to determine if a generator is capable of providing the output waveforms required for a test application, by equating the rise time to the third-, fifth-, seventh (and so forth) -harmonic of the square wave.

It may help to provide an example. For a test application requiring a 4-MHz square wave and waveform as shown in Fig. 4, what type of generator specifications are needed? At first, a 10-MHz function generator would appear to provide more than adequate bandwidth. However, if the upper limit of the function generator is not much more than 10 MHz, it will not have the necessary third-harmonic bandwidth of 3 x 4 MHz = 12 MHz. Such a function generator might indicate a typical rise time, τ_{R}, of 28 ns. Using Eq. 4, this translates to an upper frequency of 10.5 MHz, which is well short of the 12 MHz necessary.

Unfortunately, a design engineer needing a 10-MHz square wave may not be satisfied by the output signals provided by a 10-MHz function generator. More likely, that engineer will need a function generator specified for an output frequency of 30 MHz and which specifies a rise time in the neighborhood of 9.5 ns or faster.

Perhaps what is needed from suppliers of function generators is a bit more clarity in expressing the actual performance of their instruments. For example, instead of stating that an instrument produces 30-MHz square-wave and sine-wave outputs, specify the performance levels as 30-MHz sine waves and 11-MHz square waves. The numbers may not be as impressive, but they would be closer to the truthand in the end, will leave far more test engineers disappointed with the performance of their instruments.

**Philip Arnold** Electrical Engineer

(818) 781-7782

FAX: (818) 988-5951

e-mail: Philip_arnold@ieee.org