THE SINES of the times
Nov 1, 1999 12:00 PM,
Sine waves are dead, right? After all, we have TEF and all that other goodtest gear. Not so fast. TEF uses a sine wave as a source, as do so manyother pieces of test gear. Why is the sine wave so important? For onething, it can always be defined. The equation is always the same, and thewave shape never changes. The frequency and amplitude may change, but thewave shape remains the same. Also, we can reproduce a sine wave that ispure and free from distortion and noise.
At first glance a square wave might appear mathematically attractivebecause its peak, average and rms amplitudes are all the same value (seeFigure 1). The area under the curve is A = 2(x + y) where x is the one halfcycle (+ or -) amplitude (vertical), and y is one half of the frequency(horizontal).
It is also easy to draw. The square wave is made up of two vertical linesthe same length and one twice as long, and two equal horizontal lines and athird one equidistant between them and twice as long.
It is not, however, a signal that occurs in nature, and aural judgments ofits purity are difficult. A square wave is difficult for an amp toreproduce. Loudspeakers shudder at the thought of trying to follow thewaveform and nature also finds it impossible to recreate.
Random noise is a frequently used signal, but it theoretically has aninfinitely high peak value, and the extraction of phase information isquite difficult. With all else failing, the steady-state sine wave comes toour rescue.
So let us look at our old faithful signal, the sine wave. A sine wave looksrather complicated in form, and in fact, it is, but it can be drawn fairlyeasily (see Figure 1). Draw a horizontal line and mark it zero orreference. Next, take a cylindrical object such as a quarter and center iton the line. Place a straight edge parallel to the horizontal line andtangent to the bottom of the coin. Place a marker at the nine o’clockposition, which is also on the reference line. Rotate the coin clockwisethrough 360degree while it transverses along the straight edge, and we havemade a sine wave.
Now that we have one complete cycle of a sine wave, we can increase ordecrease its amplitude by stretching or shrinking the wave vertically. Toincrease or decrease the frequency, we shrink or stretch the wavehorizontally.
In audio and acoustics the normal practice is to measure the root meansquare (rms) amplitude value. This is used because it represents the samedissipation heating quantity as an identical DC amplitude would. Figure 1allows the easy visualization of the mathematical relationship of the sinefunction to this function. Sine waves are periodic waves, and the equationfor a sine wave is A = A subscript maxSin(t) where A is the instantaneousamplitude, A subscript max is the maximum amplitude, and t is time indegrees of rotation.
The peak and peak-to-peak value of any wave is easy to measure with acalibrated oscilloscope trace, but there are only a few other measuringdevices that can actually measure it. A peak-to-peak value is twice thepeak value if the waveform is symmetrical. If the wave form isasymmetrical, as with voice and music, the peak-to-peak value will probablybe less or more than twice the peak reading, depending on whether we aremeasuring the peak value of the positive or negative half of the wave.
For sine wave signals, the peak amplitude can be found with the equation:
A subscript p = 0.5 A subscript (p-p)
where A subscript p is the peak amplitude and A subscript (p-p) is the peakto peak value.
The rms value or root-mean-square value of amplitude of the wave isdirectly related to power. It is therefore the value most often used inaudio measurements. Thermocouple instruments and those with carefullydesigned networks read rms directly irrespective of the waveform becausethey are reading the ability of the signal to produce heat. The effectiveor rms value is:
A subscript rms = square root of (sin 10 degree)squared + (sin 20degree)squared over 18
A subscript rms = square root of 2 A subscript p
A subscript rms = 0.707 A subscript p
A subscript peak = square root of 2 A subscript p
A subscript peak = 1.414 A subscript p
We most often measure amplifier power using the rms voltage and the ampimpedance, hence the term rms power. In truth, it is not rms power, butpower derived from rms voltage. The equation for power is:
P subscript avg = V subscript rms squared over Z
where P subscript avg is the power derived from the voltage and impedance,V subscript rms is the voltage, and Z is the impedance of the load.
Because (0.707)superscript 2/1 = 0.5, the peak power is exactly twice theaverage power rating when the sine wave signal is the source employed forthe measurement.
Most AC-measuring voltmeters or sound pressure meters, unless specificallystated that it measures true rms voltage, have DC movements with afull-wave rectifier attached, so they respond to the average, not theeffective or rms value. They assume the AC wave is a pure sine wave andmeasure A subscript avg, and scale the A subscript avg to indicate Asubscript rms. This is quite acceptable as long as the wave is a sine wave,but it can be disastrous on an extremely asymmetrical wave.
The average value of the sine wave is the value a DC meter indicates and is:
A subscript avg = square root of (sin 10 degree)squared + (sin 20degree)squared + … + (sin 180 degree)squared over 18
A subscript avg = 0.636 A subscript p
where A subscript avg is the rectified average value, and A subscript p iseither the positive or negative peak amplitude.
Voltage and current amplitudes can be characterized by their crest factoror their form factor:
crest factor = amplitude subscript peak over amplitude subscript rms = Asubscript p over A subscript rms
form factor = amplitude subscript rms over amplitude subscript peak = Asubscript rms over A subscript peak
You can see how the crest factor can become rather large for non-sinusoidalwaves, which constitute most of the waves we see in audio. Voice andmusical instruments, for instance, can have a crest factor of 10 or more.
For sine waves, the crest factor and the form factor values are easilycalculated:
crest factor = 1 / (0.707) = 1.41
where 1 is the peak amplitude, and 0.707 is the rms amplitude.
form factor = (0.707) / (0.636) = 1.11
where 0.707 is the rms amplitude, and 0.636 is the average amplitude.
The form factor is the value that is used to make a rectified typeAC-indicating meter indicate rms values. Other useful equations we can usewith sine waves are:
A subscript rms = A subscript peak / (square root of 2) = 0.707 A subscriptpeak
A subscript rms = p / (2 square root of 2) A subscript avg = 1.11 Asubscript avg
A subscript p = (p / 2) A subscript avg = 1.57 A subscript avg
As we can see, an AC sinusoidal signal’s amplitude may be measured aspeak-to-peak amplitude, peak amplitude (0.5 peak-to-peak), rectifiedaverage amplitude (0.636 peak), and root-mean-square (rms) amplitude (0.707peak).
As stated previously, rms is the most frequently used value because inlinear circuits, the dissipated power depends directly on the rms oreffective value. Figure 1 shows the three basic parameters of the simplesine wave. The first is period (p); the primitive period (p) of sin x is 2por 360 degree (i.e., one cycle). The second, amplitude, is measuredpeak-to-peak (A subscript p-p). The amplitude may be in volts (V), current(I), sound pressure level (SPL) and so on. The third is time (t), and thetime interval is N subscript p in seconds, where N is the number of periods.
The period (p) can most easily be observed on an oscilloscope. Most modernscopes have accurately calibrated time scales as well as amplitude scales,and the period can be directly read from the graticule. For example, if asine wave on the screen from one zero degree crossing to the next takes 10divisions on the graticule when the horizontal sweep is 0.1 ms perdivision, the period would be 10 divisions 0.1 ms or 1.0 ms. Thus, itsfrequency is:
f = t / p
where t is the time in seconds p is the period, and because t = 0.001 s or1.0 ms,
f = 1 / (0.001) = 1,000 Hz
This means the observed waveform is revolving (360 degree) 1,000 timesevery second (i.e. the rate of phase change is 360,000degree/s).
The simple sine wave can be used to measure many things includingdistortion, frequency response and power. If we have two or more sine waveswe can measure delay, polarity and phase shift. Periodic waves that are thesame frequency and have the same waveform are said to be in phase if theyreach corresponding amplitudes simultaneously.
Phase is the fraction of the whole period that has elapsed, measured from afixed datum. If two identical sine waves are started at the same, they willbe in-phase at all times. If one of the waves is started one-half waveafter the first wave, it would be 180 degree out of phase, or would belagging by 180 degree. This condition will remain as long as the wavescontinue.
If the two waves are different frequencies even though they are started atthe same time, the phase between them will constantly change. In fact,after a period of time, the phase difference could be many thousanddegrees. When we measure distance using a sine wave, we are starting acounter with the input wave and stopping the counter with the wave we pickup at the unknown distance. What we are really measuring is the wave weinserted into the input but many cycles later at the pickup. When weconvert this into time, we can do many things with it. For instance, oncewe know this time, we can use it to trigger circuits for measuringdistance, opening measuring windows or a myriad of other tests.
Care must always be taken to ensure that the data obtained is in referenceto the desired signal and that the system is being operated in its linearrange. Some quick tests you should use to ensure the proper test conditionsinclude being certain that there is zero output with zero input, makingsure that doubling the input signal doubles the output signal, examiningfor spurious outputs (i.e., instabilities that are transient in nature),and employing proper termination impedances.
Overall, the ubiquitous sine wave is one of the best test signals becauseof our wealth of mathematical knowledge about its behavior.