The 10 Percent Rule: This rule is a standard method for selecting R 1 and R 2 that takes into account
2.20 AC Circuits
A circuit is a complete conductive path through which electrons flow from source to load and back to source. As we’ve seen, if the source is dc, electrons will flow in only one direction, resulting in a direct current (dc). Another type of source that is fre- quently used in electronics is an alternating source that causes current to periodically change direction, resulting in an alternating current (ac). In an ac circuit, not only does the current change directions periodically, the voltage also periodically reverses.
Figure 2.77 shows a dc circuit and an ac circuit. The ac circuit is powered by a sinu- soidal source, which generates a repetitive sine wave that may vary in frequency from a few cycles per second to billions of cycles per second, depending on the application.
The positive and negative swings in voltage/current relative to a zero volt/amp reference line simply imply that the electromotive force of the source has switched
FIGURE 2.76
FIGURE 2.77
directions, causing the polarity of the voltage source to flip, and forcing current to change directions. The actual voltage across the source terminals at a given instant in time is the voltage measured from the 0- V reference line to the point on the sinusoidal waveform at the specified time.
2.20.1 Generating AC
The most common way to generate sinusoidal waveforms is by electromagnetic induction, by means of an ac generator (or alternator). For example, the simple ac generator in Fig. 2.78 consists of a loop of wire that is mechanically rotated about an axis while positioned between the north and south poles of a magnet. As the loop rotates in the magnetic field, the magnetic flux through it changes, and charges are forced through the wire, giving rise to an effective voltage or induced voltage.
According to Fig. 2.78, the magnetic flux through the loop is a function of the angle of the loop relative to the direction of the magnetic field. The resultant induced voltage is sinusoidal, with angular frequency ω (radians per second).
Real ac generators are, of course, more complex than this, but they operate under the same laws of induction, nevertheless. Other ways of generating ac include using a transducer (e.g., a microphone) or even using a dc- powered oscillator circuit that uses special inductive and capacitive effects to cause current to resonate back and forth between an inductor and a capacitor.
Why Is AC Important?
There are several reasons why sinusoidal waveforms are important in electronics. The first obvious reason has to do with the ease of converting circular mechanical motion into induced current via an ac generator. However, another very important reason for using sinusoidal waveforms is that if you differentiate or integrate a sinusoid, you get a sinusoid. Applying sinusoidal voltage to capacitors and inductors leads to sinusoi- dal current. It also avoids problems on systems, a subject that we’ll cover later. But one of the most important benefits of ac involves the ability to increase voltage or decrease voltage (at the expense of current) by using a transformer. In dc, a transformer is useless, and increasing or decreasing a voltage is a bit tricky, usually involving some
FIGURE 2.78 Simple ac generator.
resistive power losses. Transformers are very efficient, on the other hand, and little power is lost in the voltage conversion.
2.20.2 Water Analogy of AC
Figure 2.79 shows a water analogy of an ac source. The analogy uses an oscillating piston pump that moves up and down by means of a cam mechanism, driven by a hand crank.
In the water analogy, water particles, on average, appear to simply swish back and forth as the crank is turned. In an ac electrical circuit, a similar effect occurs, though things are a bit more complex. One way to envision what’s going on is that within a conductor, the drift velocity of the sea of electrons is being swished back and forth in a sinusoidal manner. The actual drift velocity and distance over which the average drift occurs are really quite small (fractions of millimeter- per- second range, depending on conductor and applied voltage). In theory, this means that there is no net change in position of an “average” electron over one complete cycle. (This is not to be confused with an individual electron’s thermal velocity, which is mostly ran- dom, and at high velocity.) Also, things get even more complex when you start apply- ing high frequencies, where the skin effect enters the picture—more on this later.
2.20.3 Pulsating DC
If current and voltage never change direction within a circuit, then from one per- spective, we have a dc current, even if the level of the dc constantly changes. For example, in Fig. 2.80, the current is always positive with respect to 0, though it varies periodically in amplitude. Whatever the shape of the variations, the current can be referred to as “pulsating dc.” If the current periodically reaches 0, it is referred to as
“intermittent dc.”
FIGURE 2.79
From another perspective, we may look at intermittent and pulsating dc as a com- bination of an ac and a dc current. Special circuits can separate the two currents into ac and dc components for separate analysis or use. There are also circuits that com- bine ac and dc currents and voltages.
2.20.4 Combining Sinusoidal Sources
Besides combining ac and dc voltages and currents, we can also combine separate ac voltages and currents. Such combinations will result in complex waveforms.
Figure 2.81 shows two ac waveforms fairly close in frequency, and their resultant combination. The figure also shows two ac waveforms dissimilar in both frequency and wavelength, along with the resultant combined waveform.
FIGURE 2.80
FIGURE 2.81 (Left) Two ac waveforms of similar magnitude and close in frequency form a composite wave. Note the points where the positive peaks of the two waves combine to create high composite peaks: this is the phenomenon of beats. The beat note frequency is f2 − f1 = 500 Hz. (Right) Two ac waveforms of widely different frequencies and amplitudes form a composite wave in which one wave appears to ride upon the other.
Later we will discover that by combining sinusoidal waveforms of the same frequency—even though their amplitudes and phases may be different—you always get a resultant sine wave. This fact becomes very important in ac circuit analysis.
2.20.5 AC Waveforms
Alternating current can take on many other useful wave shapes besides sinusoidal.
Figure 2.82 shows a few common waveforms used in electronics. The squarewave is vital to digital electronics, where states are either true (on) or false (off). Triangular and ramp waveforms—sometimes called sawtooth waves—are especially useful in timing circuits. As we’ll see later in the book, using Fourier analysis, you can create any desired shape of periodic waveform by adding a collection of sine waves together.
An ideal sinusoidal voltage source will maintain its voltage across its terminals regardless of load—it will supply as much current as necessary to keep the voltage the same. An ideal sinusoidal current source, on the other hand, will maintain its output current, regardless of the load resistance. It will supply as much voltage as necessary to keep the current the same. You can also create ideal sources of other waveforms.
Figure 2.83 shows schematic symbols for an ac voltage source, an ac current source, and a clock source used to generate squarewaves.
In the laboratory, a function generator is a handy device that can be used to gener- ate a wide variety of waveforms with varying amplitudes and frequencies.
2.20.6 Describing an AC Waveform
A complete description of an ac voltage or current involves reference to three proper- ties: amplitude (or magnitude), frequency, and phase.
FIGURE 2.82
FIGURE 2.83
Amplitude
Figure 2.84 shows the curve of a sinusoidal waveform, or sine wave. It demon- strates the relationship of the voltage (or current) to relative positions of a circular rotation through one complete revolution of 360°. The magnitude of the voltage (or current) varies with the sine of the angle made by the circular movement with respect to the zero point. The sine of 90° is 1, which is the point of maximum cur- rent (or voltage); the sine of 270° is −1, which is the point of maximum reverse current (or voltage); the sine of 45° is 0.707, and the value of current (or voltage) at the 45° point of rotation is 0.707 times the maximum current (or voltage).
2.20.7 Frequency and Period
A sinusoidal waveform generated by a continuously rotating generator will gener- ate alternating current (or voltage) that will pass through many cycles over time.
You can choose an arbitrary point on any one cycle and use it as a marker—say, for example, the positive peak. The number of times per second that the current (or voltage) reaches this positive peak in any one second is called the frequency of the ac. In other words, frequency expresses the rate at which current (or voltage) cycles occur. The unit of frequency is cycles per second, or hertz—abbreviated Hz (after Heinrich Hertz).
The length of any cycle in units of time is the period of the cycle, as measured from two equivalent points on succeeding cycles. Mathematically, the period is sim- ply the inverse of the frequency:
f Frequency in hertz 1
Period in secondsor 1
= = T (2.22)
and
Period in seconds 1 f
Frequency in hertz or T 1
= = (2.23)
FIGURE 2.84
Example: What is the period of a 60- Hz ac current?
Answer:
1
60 Hz 0.0167 s
T= =
Example: What is the frequency of an ac voltage that has period of 2 ns?
Answer:
f 1
2 10 s9 5.0 10 Hz 500 MHz8
= × = × =
−
The frequency of alternating current (or voltage) in electronics varies over a wide range, from a few cycles per second to billions of cycles per second. To make life easier, prefixes are used to express large frequencies and small periods. For example:
1000 Hz = 1 kHz (kilohertz), 1 million hertz = 1 MHz (megahertz), 1 billion hertz = 1 GHz (gigahertz), 1 trillion hertz = 1 THz (terahertz). For units smaller than 1, as in the measurements of period, the basic unit of a second can become millisecond (1 thousandth of a second, or ms), microsecond (1 millionth of a second, or µs), nanosecond (1 billionth of a second, or ns), and picosecond (1 trillionth of a second, or ps).
2.20.8 Phase
When graphing a sine wave of voltage or current, the horizontal axis represents time.
Events to the right on the graph take place later, while events to the left occur earlier.
Although time can be measured in seconds, it actually becomes more convenient to treat each cycle of a waveform as a complete time unit, divisible by 360°. A conven- tional starting point for counting in degrees is 0°—the zero point as the voltage or current begins a positive half cycle. See Fig. 2.85a.
By measuring the ac cycle this way, it is possible to do calculations and record measurements in a way that is independent of frequency. The positive peak voltage or current occurs at 90° during a cycle. In other words, 90° represents the phase of the ac peak relative to the 0° starting point.
Phase relationships are also used to compare two ac voltage or current waveforms at the same frequency, as shown in Fig. 2.85b. Since waveform B crosses the zero point in the positive direction after A has already done so, there is a phase difference between the two waveforms. In this case, B lags A by 45°; alternatively, we can say that A leads B by 45°. If A and B occur in the same circuit, they add together, produc- ing a composite sinusoidal waveform at an intermediate phase angle relative to the individual waveforms. Interestingly, adding any number of sine waves of the same frequency will always produce a sine wave of the same frequency—though the mag- nitude and phase may be unique.
In Fig. 2.85c we have a special case where B lags A by 90°. B’s cycle begins exactly one- quarter cycle after A’s. As one waveform passes through zero, the other just reaches its maximum value.
Another special case occurs in Fig. 2.85d, where waveforms A and B are 180° out of phase. Here it doesn’t matter which waveform is considered the leading or lagging waveform. Waveform B is always positive when waveform A is negative, and vice versa. If you combine these two equal voltage or current waveforms together within the same circuit, they completely cancel each other out.