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Pure beauty: The Sine Wave
a mostly non-mathematical look at a fundamental of nature

Sine waves are the measured or plotted results of many natural waveforms and oscillatory phenomena. An easy example is the trace made by the end of a swinging pendulum through time, when the time progression is constant. A pendulum is a natural harmonic oscillator, so sine waves are the resulting output of various harmonic oscillators. Another classic example is the vertical motion of a weight suspended from a spring—when pulled down and released, the spring pulls the weight up and then gravity pulls the weight back down, and a harmonic oscillation occurs—the tracing of the motion of the weight through time yields a sine wave.

Sine waves are more familiar in electronics, since they are the easily seen waveform produced by many electronic oscillators and generators. Below is a photo of the trace of the output of a sine wave oscillator on an oscilloscope screen. The 'scope plots amplitude (in this case voltage) on the vertical axis against time (here, in microseconds) on the horizontal axis:

1kHz sine wave

The 1kHz sine wave's amplitude, above, can be measured by multiplying the volts per division, 5, shown on-screen, by the number of vertical divisions from lowest peak to highest peak, 5.6, yielding 28; and its period by multiplying the time per division, 200usec, shown on-screen, by the number of horizontal divisions for one complete cycle, 5, obtaining 1000usec = 1msec.

Like any other kind of oscillator, the output of a sine wave oscillator is defined by several attributes, among them period, frequency, amplitude, and purity.

Period — In the photo above, the period of the wave is the length of time measured on the horizontal axis between any two identical points on the waveform. So from one positive peak or maximum voltage to the next, the time period is 1 one-thousandth of a second, or 1 millisecond, shortened to 1msec.

Frequency — Because frequency and period are reciprocals of each other, the frequency of this sine wave is 1/1msec, or 1 thousand Hertz (complete cycles per second), or 1 kiloHertz, shortened to 1kHz. This frequency of a sine wave is known as the fundamental frequency or first harmonic.

Amplitude — Using the oscilloscope's vertical axis, the voltage can be measured from the positive peak value to the negative peak value (shortened to p-p) and from this, an average value or the more complex root-mean-square value can be calculated—this is made possible because the relationships of a sine wave's p-p value to its average and RMS values is well understood. Other ways to measure the amplitude involve various types of meters, either analog or digital, to yield the result in the form wanted.

Sine waves (and their orthogonal siblings, cosine waves) have a unique property among waveforms—if perfectly pure (which is a property we will look at shortly) they only have output at a single frequency—all other waveforms, no matter how "pure" they are, have outputs that are harmonically rich; that is, they have outputs that simultaneously contain many other frequencies in addition to their fundamental frequency. This single-frequency quality of sine (or cosine) waves makes them crucial in the design and testing of electronic circuits—when passing through linear circuits, like filters, their waveform doesn't change, and so accurate basic voltage measurements are easily accomplished.

Purity — This property is a little more difficult to describe. Practically, it is the extent to which a sine wave only has one frequency component—the fundamental. Any additional frequencies present in the sine wave, whether harmonically related to the fundamental or not, are known as distortions. For some uses, the purity of a sine wave is vital, because the extent to which the circuit the sine wave passes through adds these distortions is what defines its quality or its affect on the signal.

An audio amplifier, for example, should add as little distortion as possible, because distortion can make one musical instrument sound like another. You might do this purposely for some reason but, if not intended, it can be unpleasant. Other circuits purposely add distortion, and knowing how much distortion and what kind, can be very useful to a circuit designer. Consider the electronic music synthesizer—since it is the difference in harmonic content that makes one musical instrument sound different from another when playing the same note, knowing how a circuit will change the harmonic content of a pure sine wave can make it easier to get the desired result.

In some circuits, preserving signal purity is vital—this is true, for example, for analog-to-digital converters (ADCs) and digital-to-analog converters (DACs). Using pure sine waves as sources helps the designer of such converters achieve the highest conversion accuracy, which affects everything from music transmission and reproduction to the quality of measurement instrumentation.

Why is it called a sine wave?
We need trigonometry to understand this. The sine (a scalar quantity) of an angle formed in a right triangle by the hypotenuse and the side opposite the angle, is computed by the ratio of the length of the opposite side to the length of the hypotenuse. This can be seen in the diagram below, which shows a right triangle with an angle of 45°:

The length of the hypotenuse is exactly 1, and the length of the opposite side is 0.707107... , so the sine of 45° = 0.707107... If the instantaneous value of the amplitude of a sine wave is measured with reference to its zero point, and the progress of a complete wave is considered not in time but in degrees of rotation, with one complete cycle being 360 degrees, then the instantaneous amplitude at each point of rotation will be the value of the sine of that angle.

Think of the hypotenuse as the radius of a "unit circle" centered on a grid that has a horizontal axis and a vertical axis, and that the radius line is rotating counterclockwise through 360°, starting at the right-side 0° position on the horizontal axis. At each point on the circle, the vertical line that drops (or rises) from the end of the radius line to the horizontal axis has a value, and the ratio of that value to the unity value 1 of the hypotenuse is the sine of the angle at the origin between the hypotenuse and the horizontal axis:
Sine circle

At 0°, the vertical value is 0, and the ratio is 0/1, so the sine = 0. At +90°, the radius line is vertical and its vertical value is the same as its length, 1. The ratio is 1/1, so the sine = 1. Rotating on around the circle, the value of the sine varies from 0 to 1 to 0 to -1 and back to 0 for a complete cycle. So when plotted in these two dimensions of amplitude and degrees of rotation, the sine wave looks like a circle.

But sine waves live on other axes as well—obviously it takes time for the unit radius to rotate through the 360° of the plot, and it may well keep on moving. As seen in the scope picture, when the sine signal is plotted as amplitude vs. time, it looks like a two-dimensional figure that is probably familiar—maybe you've looked at the threads of a screw or bolt from the side; or maybe you've played with a toy called a "Slinky" and viewed, from the side, the coils hanging down; or you've looked at the springy coil cord of a telephone handset—these are real sine waves, because sine waves are always at least three-dimensional. Sine waves are actually cylindrical spirals called helixes. The animation below, from the Wikipedia entry on helixes, demonstrates the progression through time described above and plots the sine and cosine waves produced by the rotating end of the radius line:

Click on this link to see propagating sine wave and cosine wave functions—use the browser back button to return to this page

In nature, waves can have additional properties to the ones already mentioned, including propagation velocity and wavelength. If you drop a stone into a large pool of water, the cross-section of the concentric ripples that form are sine waves, and they obviously have a period, frequency, and amplitude; but the wave crests also move outward through the water in time and space. If we measure the distance that an individual wave crest moves for some known period of time, we get the additional attributes of propagation velocity and wavelength, two attributes that are linked with the others. If we measure the wavelength and the frequency, we can calculate the velocity—the equation is: wavelength, Λ, = velocity, v, divided by frequency, f. Similarly, knowing any two can give you the third.

For example, the velocity of sound in air is approximately 1130 feet per second. A sine wave of frequency 1kHz being reproduced by a loudspeaker will have a wavelength of 1000 / 1130 = 0.885 feet.

When we look at the ripples of the stone dropped in the water, we are viewing the sine waves in four dimensions, and some sine waves may also exist in even larger numbers of dimensions, which is actually spooky, since it is impossible for me to visualize.

When a sine wave has variations in any one or more attributes over some period of time, these variations generate changes in the values of the sine function, and these changes lead to the production of harmonics, as well as intermodulations of the fundamental and the harmonics, which all produce noise and spurious frequencies. Measuring these distortions requires a variety of electronic instruments that can examine signals in the amplitude vs.time domain, like the scope; in the amplitude vs. frequency domain, like a wave analyzer or spectrum analyzer; in the amplitude vs. phase domain, like a phase monitor; and many others.

1kHz square wave

Above is the scope display of a "square" wave, so-called because of its square corners and its periods of on and off or high and low being approximately equal, giving the display a square shape. A square wave is rich in harmonics, primarily of odd orders, that is 3rd, 5th, 7th , 9th, etc. Note that the rise and fall of the vertical portions of the wave are too fast to register on the scope screen.

Square spectrum

As you can see from the spectrum analyzer display above, the fundamental frequency of 1kHz on the left side is followed by a train of large spikes that decrease in amplitude and occur at multiples of 1kHz—3kHz, 5kHz, 7kHz, etc. The shorter spikes are the even order harmonics and are much lower. The fact that these signals are exact multiples of 1kHz indicates that they are harmonics. But these signals also interact with the fundamental and each other, and with noiser or other signals intruding into the circuit, generating low-amplitude signals that are modulation products of each other around the base of each harmonic—these frequencies are multiples of the sums and differences of the various signals interacting with, or modulating each other.

Other waveforms have different harmonic and modulation structures—the triangle wave, for example, which looks like a line of symmetrical sawteeth, primarily has even order harmonics—2nd, 4th, 6th, etc.

The value of purity
If we want to know how faithfully some circuit component, or design, or amplifier reproduces, transmits, or amplifies signals, a basic tool is to feed it an extremely pure sine wave and then measure the resulting distortion at the output. A very convenient way to do this is to look at the resulting output signal with a spectrum analyzer, which sweeps through a defined frequency band and then plots the amplitudes and frequencies of all the distortion components that are present in the resulting signal.

Low distortion sine wave straight into ADC

This spectrum analyzer plot shows the output of a very pure sine wave oscillator, one whose actual summed distortion and noise products are less than 0.0002% (<2 parts per million) of the amplitude of the fundamental. This sine wave is fed into the input of an analog-to-digital converter system, and then computer software uses the digitized data to do the analysis of the distortion products.

However, as you can see, the distortion reads much more than 0.0002%, at 0.0024%. Why? Because the measuring tool, an audio spectrum analyzer, has distortion of its own that shows up in the plot. The instrument used to measure the oscillator's distortion has more distortion than the oscillator! What to do? Either get a better ADC system, or take advantage of the fact that lower fundamental signal levels at the input of the ADC system will give it better performance.

1kHz sine wave through notch filter

Here is the same oscillator being fed into a deep, very sharp active notch filter circuit known as an Active Twin-T Filter, which lowers the amplitude of the 1kHz fundamental to 1/100th of its true value—40dB lower, while leaving the other signal components essentially untouched. This means that the ADC circuits can operate in a more linear, lower distortion part of their ranges, allowing us to see the oscillator signal with greater resolution. Now the distortion products are all at -120dB or less, ≤1 ppm, compared to the fundamental; and this still is not a true picture of the actual performance of this oscillator.

It's not easy
Obviously, making a totally pure sine wave signal is very difficult; and actually is impossible, because there is always something in nature perturbing things a little (or a lot). Noise is inevitable in everything. This has led to a quest over time to make better and better equipment to both generate and measure sine waves and their distortion products.

It is important to define what purity means—one general specification is for Total Harmonic Distortion plus Noise, or THD+N. This definition or specification largely ignores modulation products because they generally are far lower in sine wave oscillators and analyzers than the harmonic products, and so generally constitute noise.

In the past, electronic sine wave oscillators having high purity used notch filter circuits in the feedback loops of amplifiers, or combinations of amplifiers in so-called state-variable circuits to generate the sine waves, and negative feedback circuits called automatic level controls or automatic gain controls, to detect the output amplitude and use that to control a non-linear element like a lamp, or transistor, or diode, or thermistor, to change the overall circuit gain in response to a change in output amplitude of the oscillator amplifier, in order to stabilize the amplitude. Today, high-precision digital-to-analog converters are often used to turn a precise digital table of sine values into an analog signal having low noise and distortion—performance can approach THD+N of 0.0001% or better. But the lowest distortion oscillators are still analog designs.

The ability to generate pure sine waves has always led the ability to measure their distortion, and that is still true today. In the 1950s, test equipment could resolve about 100 ppm THD+N; in the 1970s, the ability improved to about 10 ppm; today that ability is around 1 ppm, while the ability to view individual harmonic products is up to two orders of magnitude better—these are measurements of purity in parts per billion, good work by any measure.

In development are oscillators which use unique solid-state elements called Josephson Junctions in circuits that can produce sine waves having amplitude stabilities and wave shape accuracies better than 0.1 ppm, with frequency stabilities of very high order. While very expensive, these circuits are used in precision calibration equipment that is used to define and maintain primary standards for voltage, both alternating current (AC), like sine waves, and direct current (DC), like the output of a battery, for calibrating all of the measuring apparatus found in the modern world.

For some reason, the pure sine wave has a shape that is very appealing to me in its symmetry, regularity, and smooth, even delicate curves. For me, it has a natural beauty that is a reflection of its purity. When listened to over a loudspeaker or through a headset or earphones, this quality of the sine wave is immediately apparent, especially when compared to any other waveform, all of which to one degree or another sound raspy, or tinny, or buzzy, while the pure sine wave sounds... well... pure.

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