Part 1:  AC Voltages and Current: Time, Frequency, and Modulation Domains

This section is optional.  It expands on the previous section.  Nothing that follows in this chapter, or the next, relies on the information presented here.  That said, the concepts presented are useful.  The purpose of this section is to present different ways that an AC waveform can be used to carry information.

When making measurements in a circuit, the most common tool used is a voltmeter.  Voltmeters (usually a Digital Voltmeter or DVM) allow you to quickly check the voltage levels (among a number of other things) in your circuit.  However, a DVM is only useful for measurements that don’t change, or change very slowly.  An oscilloscope allows you to see the waveform and therefore make measurements that are impossible to make otherwise.  A DVM is used to measure a single value while an oscilloscope allows you to measure voltages over time.  AC waveforms,  particularly complex non sinusoidal waveforms that change over time (like an FM modulated waveform), contain information that can be hard to discern in a display of voltage versus time.  Given that an AC voltage is characterized by its amplitude and frequency, it makes sense that being able to view a waveform as voltage versus frequency plot, as opposed to a voltage versus time plot, would be useful.

A plot of amplitude versus frequency is called a spectrum plot.  An example of an idealized spectrum plot of a complex waveform, such as a square wave, is shown in Figure 1.  The vertical axis is the amplitude and the horizontal axis is the frequency.  Each bar in the figure represents the amplitude of one of the sine waves that are added to make up the waveform.  If you look back at the SPICE schematic in Figure 4 of the section titled AC Waveform Composition, you see that the (almost) square wave being generated is composed of six individual sine waves added together.  The spectrum shown below in Figure 1 displays the amplitude of each of sine waves as the height of the bar while the frequency of the individual sine waves shown on the horizontal axis. Figure 1.  Spectrum, or frequency domain, plot of a  square wave.

When viewing a waveform on an oscilloscope where the amplitude is plotted versus time, you are looking at it in the time domain.  When viewing a waveform’s spectrum where the amplitude is plotted versus frequency, you are viewing it in the frequency domain.

Other than viewing the content of a complex signal, are there any advantages in working, and more importantly thinking, in the frequency domain?  The answer is a definite yes, if the way that the circuit works calls for it.  Although we haven’t covered circuits, or components, that are sensitive to frequency yet, most are.  The frequency content of a complex signal can be used to carry, or communicate, the information in a circuit.  Even if the circuit was not intentionally designed to operate on the frequency content of the waveforms in it, given that many components are sensitive to frequency, they will effect the way the circuit works.  It is well worth your while to think in the frequency domain as you work your way though a circuit.

We have looked at signals in the time domain and frequency domain.  Of the three measures available to plot (time, amplitude, and frequency), we have discussed two of the three possible combinations.  The last combination, frequency versus time, is the modulation domain.

Each of the three quantities (time, amplitude, and frequency) can be thought of as a dimension or axis in a two dimensional plot. Figure 3 represents a three dimensional space where the three axes are amplitude, time, and frequency.  The corner of the box represents 0 amplitude, 0 frequency, and 0 time.

There are three planes (green at the back, blue on the left, and beige on the bottom) shown.  The green plane at the back represents the time domain (like you see on an oscilloscope) because its two axes are amplitude and time.  The blue plane on the left represents the frequency domain because its axes are amplitude and frequency.  This is shown above in Figure 1.  The  beige plane on the bottom displays the modulation domain because its axes are time and frequency.  Measurements in the modulation domain are frequently accomplished using an oscilloscope with some external circuitry (an adapter) that does the signal conversion or through the use of a meter that can measure a single value in time.  The plots shown in the three planes represent the same waveform. Figure 2.  Three domains.

Take some time to look at Figure 2 and try to understand what is being shown.  The three planes are plotting the same information from different viewpoints.  The green time domain plot is easy to interpret (we normally think in the time domain), and mimics the way most of the waveforms shown in this book.  As time goes forward, we see a higher frequency sine wave followed by a lower amplitude, lower frequency sine wave, which is followed again by a higher frequency, higher amplitude sine wave.

The frequency domain plot shown in the blue plane has two bars, one for each of the two frequencies shown in the time domain plot in green.  The bars are different lengths because the amplitude of the sine waves shown in the time domain plot changes.  If the equipment you are using displays information in real time (some do, others show an average over time), the tall bar for the high frequency sine wave would occur first.  This would be followed by the smaller bar indicating a lower amplitude, lower frequency sine wave.  This would then be followed by the taller bar at a higher frequency which represents that last of the time based waveform.  Normally a frequency domain plot (a spectrum) is used to see the content of a complex waveform rather than a group of sequential sine waves.

Modulation domain plots, where frequency is plotted against time, appear on the beige plane.  In this case the plot starts off higher in frequency (further away from the origin) until the time that the frequency decrease and the plot line moves closer to the origin.  The plot line continues at the lower frequency until the time at which the frequency increases again and the plot line moves to a higher frequency further away from the origin.  Modulation domain plots are typically used on sinusoidal waveforms of the type shown here.  This is because complex waveforms, such as a sine wave, contain more than one frequency at a time.

Working in the modulation domain is also common place.  One example of signals that can be thought of in the modulation domain, is telephone dialing. When you press a number key on your phone, two tones, or frequencies, are produced (it may sound like one, but there really are two frequencies produced for each button push). The amplitude of these signals is not terribly important  but the tone pairs in sequence, or their order over time, are what tell the telephone system what number you are calling.

A more sophisticated example of signals that are best considered in the modulation domain are Fax machines or modems.  If you can recall what they sound like (a rapid sequence of multiple tones) you can envision why looking at these signals, or thinking about them, as a representation of frequency versus time is important to understanding what is going on.

Key Concepts

• The most common way of thinking about an AC waveform is in the time domain.  The time domain refers to the way a voltage changes over time.  This is what an oscilloscope displays and the way most of the plots in this book are shown.

• In the frequency domain, a waveforms frequency versus amplitude is of interest.  This is particularly helpful when working with complex waveforms which are composed of a number of sine waves added together making it difficult to see what is going on in the time domain.  Plots in the frequency domain are referred to as spectrum plots.

• In the modulation domain, a waveforms frequency versus time is of interest.  Data plots of this type are rare.

• Although AC voltages are typically dealt with in the time domain, it is very useful to be aware of the frequency content of the waveform by thinking of it in the time or modulation domain when working with circuits given most circuits operate over a relatively narrow range of frequencies.