AC Waveform Composition

Part 1:  AC Voltages and Current: AC Waveform Composition

The sine wave we have been discussing is a very common waveform but it, of course, is not the only waveform shape we are likely to see.  It does however have special significance.  As it turns out, electrical components that are sensitive to frequency, which we will cover in a later chapter, are most easily understood in the context of a sinusoidal AC waveform.  For that reason, this section explores how sine waves relate to other waveform shapes.

The following figure depicts both a sine wave and a square wave.  They have characteristics in common.  Their frequency and peak amplitude are the same, but they are clearly different.  How, other than describing the shape, are they different?

Figure 1.  A sine wave and a square wave.

Before we investigate the differences between a square wave and a sine wave, let’s step back for a moment and consider something else.  What if it were possible to decompose a waveform into parts?  The following figure illustrates how this might be done.

Figure 2.  An offset sine wave and its parts.

In Figure 2, the waveform shown on the far left is, strictly speaking, not an AC waveform.  This is because it never changes polarity, it is always a positive voltage.  It fits the definition of a time variant voltage.  We can disassemble, so to speak, this waveform into two other waveforms.  They are a DC voltage and an AC sine wave.  The time variant waveform shown on the left of Figure 2 is obtained by adding together the DC voltage and sine wave shown to the right of the figure.  You can envision adding two waveforms together by thinking of them as being made up of a number of small points, or voltage measurements, spread out in time.  To add two waveforms together, you add together the value of a point on one waveform with the like point (the one at the same time) on the other.

You can now see why the time variant voltage, although descriptive, is not really a unique thing.  It is actually an AC voltage that has a DC voltage added to it.  The fact that the offset waveform in Figure 2, is composed of a DC voltage (which serves as the fixed offset) and a sine wave, is relatively easy to see.  It is however, hard to imagine what to add together to get a square wave.  The answer, surprisingly, is a number of sine waves at different frequencies and amplitudes.

As improbable as it may seem, any periodic waveform shape can be produced by adding together sine waves.  The sine waves that are added together need to be of different frequencies and amplitudes to accomplish this.  As you will see in following chapters, this concept, that all periodic waveforms are composed of sine waves of different frequencies and amplitudes added together, can be very useful in understanding how circuits that use certain types of components work.  In the case of a square wave, the following figure illustrates the concept.

Figure 3.  Adding different sine waves.

If you wish to explore how this works you can use LTspice to run a simulation.  In the same way that placing two or more DC voltage sources in series results in the voltages being summed, AC voltage sources can also be placed in series to sum their voltages.  The following figure depicts an LTspice schematic (in white) annotated (in green) with a description of the voltage present at each of the labeled nodes.  Each voltage is shown in the form used in Equation 5 in the previous section.  It is not important to fully grasp the mathematical description shown in the figure.  What is important to understand is that the voltages present on “B” through “D” are a sum of the sine wave voltages that are in series between the common return and the labeled node.

Figure 4.  LTspice schematic summing six sine waves. 

The following figure depicts the waveforms present at each of the summation points (nodes “A” through “F”) obtained from the simulation.

Figure 5. Adding sine waves in LTspice. 

The sine wave shown for node “A” is called the fundamental or first harmonic.  In this case it is a 100Hz sine wave.  The second voltage source, node “B”, is the third harmonic (which means 3 times the fundamental), which is 300Hz.  Each voltage source in sequence is the next odd harmonic of the fundamental with frequencies of 500Hz, 700Hz, 900Hz, and 1,100Hz. Each harmonic’s amplitude is 1/n times the fundamental amplitude, where n is the harmonic number.

As you can see by looking at Figure 5, the waveform begins to resemble a square wave fairly quickly.  By the time the third or fifth harmonic is added (waveform C or D), a reasonably usable square wave is obtained.  To obtain a perfect square wave would require and infinite sequence of square waves to be added together.  To get a square wave that is nearly indistinguishable from perfect requires a dozen or so sine waves.

The technique of adding together different harmonics with different amplitudes can produce any desired waveform shape.  When combined with an additional DC term to provide the offset, you can describe all possible periodic waveforms.

This means that any waveform, other than a sine wave, has as part of its content, frequencies that are greater than the obvious fundamental frequency you see by measuring its period.

Why is this information useful?  In the following chapters we will begin to work with a variety of different electrical components.  All of them are sensitive, in one way or another, to frequency.  It is therefore very useful to keep in mind that a complex waveform, such as a square wave, is composed of sine waves of different frequencies.  There are in fact, from an analytical standpoint, two different types of voltages.  An AC sine wave and a DC voltage.  For instance, a circuit that may operate well when it has a sine wave of 1,000Hz used as an input but fail to work properly when the input is a 1,000Hz square wave do to the complex nature of the square wave.