Learning how to combine inductors in series and parallel
Seeing how inductors are used in electronic circuits
I have a lot of books about electronics on my bookshelf, covering a wide range of topics. Some of them are new; some are a few decades old. My favorite of them all is a called A Course in Electrical Engineering, by Chester L. Dawes. It was written in 1920, when electronics was in its infancy. Radio was brand new. Vacuum tubes were just catching on. The transistor wouldn’t be invented for another quarter of a century.
You’d think that an electrical engineering book written in this era would be completely obsolete. But it’s amazing how much of this book is still accurate. For example, Ohm’s law hasn’t changed since 1920. Dawes’s explanation of Ohm’s law is as good as any I’ve ever read.
What fascinates me most about this old book is the way it starts. Chapter 1 is titled “Magnetism and Magnets,” and it begins like this:
Magnets and magnetism are involved in the operation of practically all electrical apparatus. Therefore an understanding of their underlying principles is essential to a clear conception of the operation of all such apparatus.
This makes perfect sense when you consider the state of electrical technology in those days. We’ve moved beyond simple magnetism in our ability to exploit the amazing properties of electricity. Yet the basic relationship that exists between electric current and magnetism is still at the heart of many different types of essential circuits.
For example, a power adapter that converts 120 V AC from a wall outlet to a safer level, say 9 V, uses a transformer that relies on magnetism to step down the voltage. An analog multimeter (the kind with a needle that moves with voltage, current, or resistance) relies on magnetism to deflect the needle. And electric motors convert electric current into magnetism, which is then converted into motion.
In this chapter, I turn your attention to a special class of components called inductors that exploit the nature of magnetism and its relationship to electric current. Inductors are sort of like cousins to capacitors, in that they can be used to do similar things and they play by similar rules. For example, an important characteristic of a capacitor is its ability to oppose changes in voltage. Inductors have a similar ability to oppose changes, not in voltage but rather in current.
This chapter is a bit of a departure from the other chapters in this minibook, in that it contains no construction projects. The practical applications for inductors are in circuits that you might not be ready to build yet, such as radio tuners or power supplies. You learn the important concepts of how inductors work in this chapter so that later, when it’s time to use an inductor in an actual circuit, you’ll know what it does. You can always return to this chapter at that time to refresh your memory on how inductors work. In fact, I won’t hold it against you if you just skim through this chapter on your first reading through this book, or even skip it altogether for now. You can always come back to this chapter when the need arises.
What Is Magnetism?
When Albert Einstein was 5 years old and sick in bed, his father gave him a compass to play with. Young Albert saw that no matter how he spun the compass around the needle always swung back to point north, and he was amazed. Years later, he wrote that this compass was what launched his lifelong interest in physics. He realized then that there was “something behind things, something deeply hidden.”
In the case of the compass, that deeply hidden thing that Einstein marveled at was magnetism. We’re all familiar with magnetism, though exactly what it is remains mysterious. A magnetic field extends through space and either attracts or repels certain materials. Materials that are strongly affected by magnetic fields are called magnetic. Materials that create magnetic fields are called magnets.
The north and south of magnetism
Magnetic fields and electric fields are distinct things, but they’re closely related and have much in common. For example, magnetic fields are polarized in much the same way that electric fields are polarized. Electric fields exist between electric charges of opposite polarity (negative and positive). Magnetic fields exist between opposite magnetic poles, called north and south.
Just as opposite electric charges attract and like charges repel, opposite magnetic poles attract and like magnetic poles repel. That’s why if you take two strong magnets and try to push the two north poles together or the two south poles together, the magnet fights back, and you won’t be able to get the poles together. But if you turn one of the magnets around and try to hook up the north pole with the south pole, the magnets attract each other, and you’ll have trouble keeping them apart.
A magnetic field has a distinct shape that you can visualize by using a simple experiment that’s commonly done in grade school. All you need is a bar magnet, some iron filings, and a piece of paper. Put the paper over the magnet and sprinkle the filings on the paper. The filings magically line up to reveal the shape of the magnetic field, as shown in Figure 4-1. As you can see, the filings align themselves in lines that are aligned from one pole of the magnet to the other.
FIGURE 4-1: The shape of a magnetic field revealed by iron filings.
Pondering permanent magnets
A permanent magnet is a material that creates its own magnetic field. Some naturally occurring materials, such as loadstone, are inherently magnetic and produce magnetic fields all on their own. But most permanent magnets are made from materials that aren’t inherently magnetic but become magnetic when they’re exposed to a powerful magnetic field.
You probably have several permanent magnets around your home. In fact, you probably have a few on your refrigerator, holding up your kid’s kindergarten picture or your shopping list.
Examining Electromagnets
Permanent magnets create a magnetic field all by themselves. An electromagnet relies on the key relationship between electricity and magnetism to create a magnetic field. Specifically, whenever electric current flows, a magnetic field is created. This magnetic field is created by the moving charges. In short, an electron in motion becomes a magnet.
As I say several times in this book, electrons are always in motion, so aren’t there little magnets everywhere? The answer is yes, in the same sense that electric current is everywhere. But when the motion of electrons within a material is random, the magnetic fields created by the electrons are oriented randomly, and so they end up just canceling each other out. But when you give the electrons a nudge with voltage, they all start moving in the same direction. This strengthens and organizes the magnetic fields so they can combine to form one large magnetic field.
The magnetic field created by current flowing through a single wire is measurable but small. However, if you coil the wire tightly as shown in Figure 4-2, the magnetic fields are strengthened because of their proximity to one another. For example, you can create a simple electromagnet by wrapping several feet of small, insulated wire around a pencil, a ballpoint pen, or any other rigid tube or cylinder.
The strength of the magnetic field of such a coil depends on several factors, the most important being these:
The number of turns in the coil.
The amount of current flowing through the electromagnet. Increasing the current increases the strength of the electromagnet exponentially. For example, if you double the current, the electromagnet is four times as strong.
The material used for the core that the electromagnet is wrapped around. Any coil wrapped around an iron core will be about ten times as strong as the same coil wrapped around an inert core such as air, plastic, or wood.
The polarity of an electromagnet is easy to determine: The negative side of the coil is the magnet’s north pole, while the positive side is the south pole.
Inducing Current
Electromagnets are possible because a moving current creates a magnetic field. Interestingly enough, the reverse is also true: A moving magnetic field creates an electric current. In other words, if you wave a magnet over a wire, you create a current in the wire. This effect is called electromagnetic induction, or sometimes just magnetic induction.
Remember our old friend Michael Faraday? In Chapter 3 of this minibook, I mention that even though he didn’t invent capacitors, his name is the basis of the farad, which is the unit of measure we use to measure capacitors. In 1831, Faraday discovered electromagnetic induction. It’s one of the most important discoveries in the history of electrical science, as it’s at the heart of nearly all forms of electrical power generation. Coal-burning, hydroelectric, and even nuclear power plants all use moving water to spin turbines that are connected to generators. Those generators use the principle of electromagnetic induction to turn the spinning motion of magnetic fields into electric current.
You can increase the strength of the current induced in a wire by coiling the wire into turns so that you can expose a greater length of wire to the magnetic field. Also, it doesn’t matter whether it’s the magnet or the coil that moves. Either way, a current is induced in the coil if the coil is moving relative to the magnet, provided of course that the coil passes through the magnet’s magnetic field.
Note that whenever current is flowing, there must be voltage. Thus, in addition to causing current to flow through the coil, electromagnetic induction creates a voltage across the coil.
Inductance and the art of resisting change
An inductor is a coil that’s designed for use in electronic circuits. Inductors take advantage of an important characteristic of coils called self inductance, also called just inductance. Inductors are simple devices, consisting of nothing more than a coil of wire, often wrapped around an iron core. But their ability to exploit the idea of self-inductance is a stroke of genius.
Self-inductance is similar to electromagnetic induction as described in the previous section, but with a subtle twist. Whereas electromagnetic induction refers to the ability of a coil to generate a current when it moves across a magnetic field, self inductance refers to the ability of a coil to create the very magnetic field that then induces the voltage. In other words, with self-inductance, the coil feeds back upon itself. A voltage applied across the coil causes current to flow, which creates a magnetic field, which in turn creates more voltage.
Inductance happens only when the current running through the coil changes. That’s because only a moving magnetic field induces voltage in a coil. Whenever the current changes in a coil, the magnetic field created by the current grows or shrinks, depending on whether the current increases or decreases. When the magnetic field grows or shrinks, it’s effectively moving, so a voltage is inducted in the coil as a result of this movement. When the current stays steady, no inductance occurs.
Self-inductance is a tricky concept to get your mind around, so don’t panic if this doesn’t make sense to you the first time through. It took me awhile to get this concept when I first learned about it. Let’s go over the idea in more detail, point by point:
When voltage is applied across a coil, the voltage causes current to flow through the coil. Remember, current always requires voltage, and voltage always results in a current when applied across a conductor.
The current flowing through the coil creates a magnetic field around the coil. Keep in mind that the coil that creates the magnetic field is itself within the field and can therefore be influenced by it.
If the current flowing through the coil changes, the magnetic field created by the current also changes. The magnetic field grows or shrinks, depending on whether the current increases or decreases. Either way, the changing magnetic field is in effect moving.
Because the magnetic field is moving, voltage is induced in the coil. This is an additional voltage, on top of the voltage that’s driving the main current through the coil.
The amount of voltage induced by the changing magnetic field depends on the speed in which the current changes. The faster the current changes, the more the magnetic field moves and therefore the more voltage is induced.
The polarity of the induced voltage depends on whether the current is increasing or decreasing. This is because the direction of movement in the magnetic field depends on whether the field is growing or shrinking, and the voltage induced by a moving magnetic field depends on which direction the field is moving, according to the following rules:
When the current increases, the polarity of the induced voltage is opposite to the polarity of the voltage driving the coil. This inducted voltage is often called back voltage because it has the opposite polarity as the supply voltage.
When the current decreases, the resulting self-induced voltage has the same polarity as the supply voltage.
The induced voltage creates a current in the coil that flows either with or against the main coil current, depending on whether the coil current is decreasing or increasing, according to the following rules:
When the coil current is increasing, the additional current flows against the main coil current. This has the effect of pushing back against the increasing main current, which effectively slows down the rate at which the current can change.
When the coil current is decreasing, the additional current flows with the main coil current, thus counteracting the decrease in coil current.
When the coil current stops changing, self-inductance stops. Thus, when current is steady, an inductor is simply a straight conductor. (It is also an electromagnet, as the current traveling through it produces a magnetic field.)
Because of self-inductance, an inductor is said to oppose changes in current. If the current increases, an opposite voltage is induced across the coil, which slows the rate at which the current can increase. If the current decreases, a forward voltage is induced across the coil, which slows the rate at which the current decreases. An inductor applies equal opposition to both increases and decreases in current.
It turns out that this ability to oppose changes in current is quite useful in electronic circuits, as I explain later in this chapter, in the section “Putting Inductors to Work.”
Here are some additional important details concerning inductors:
Inductors can’t stop changes in current; they can only slow them down. Measuring how much an inductor can slow down a change in current is the topic of the next section.
The magnetic field from one inductor can spill over into a nearby inductor and induce voltage in it too. To prevent this from happening, many inductors have special shielding around them to keep them magnetically isolated. If you use unshielded inductors in your circuits, be sure to space them as far apart as possible — unless, of course, you want them to interact.
Regarding henry
Inductance is only a momentary thing. Exactly how much of a momentary thing depends on the amount of inductance an inductor has. Inductance is measured in units called henrys (H).
The definition of one henry is simple: One henry is the amount of inductance necessary to induce one volt when the current in coil changes at a rate of one ampere per second.
As you might guess, one henry is a fairly large inductor. Inductors in the single-digit henry range are often used when dealing with household current (120 V AC at 60 Hz). But for most electronics work, you’ll use inductances measured in thousandths of a henry (millihenrys, abbreviated mH) or in millionths of a henry (microhenrys, abbreviated ).
Here are a few additional things to know about inductors and henrys:
The plural of henry is henrys, not henries.
The letter L is often used to represent inductance in formulas. Inductors in schematic diagrams are usually referenced by the letter L. For example, if a circuit calls for three inductors, they will be identified L1, L2, and L3.
Speaking of schematics, the symbol used to represent inductors in a schematic diagram is shown in the margin.
When I first learned about self-inductance and henrys, I hoped that since the henry is a measure of a coil’s ability to resist change in current, the henry should be named after someone who was famous for resisting change, such as Professor Henry Higgins from My Fair Lady.
Imagine my disappointment to discover that the henry is named after Joseph Henry. All he ever did was discover the self-inductance and invent the inductor. Well, that plus he was the first Secretary of the Smithsonian Institution. He must have known someone really important to get the henry named after him.
Calculating RL Time Constants
In the preceding chapter, you learn how to calculate the RC time constant for a resistor-capacitor circuit. A similar calculation can be done for inductors, except that instead of calling it the RC time constant, we call it the RL time constant. (Remember, L is the symbol used to represent inductance.)
The RL time constant indicates the amount of time that it takes to conduct 63.2 percent of the current that results from a voltage applied across an inductor. (Hmm. Where have we seen 63.2 percent before? Right! It’s the same percentage used to calculate time constants in resistor-capacitor networks. The value 63.2 percent derives from the calculus equations used to determine the exact time constants for both resistor-capacitor and resistor-inductor networks.)
Here’s the formula for calculating an RL time constant:
In other words, the RL time constant in seconds is equal to the inductance in henrys divided by the resistance of the circuit in ohms.
For example, suppose the resistance is , and the inductance is 100 mH. Before you do the multiplication, you first convert the 100 mH to henrys. Because one millihenry (mH) is one one-thousandth of a henry, you can convert millihenrys to henrys by dividing the millihenrys by 1,000. Therefore, 100 mH is equivalent to 0.1 H. Dividing 0.1 H by gives you a time constant of 0.001 second (s), or one millisecond (ms).
The following table gives you a helpful approximation of the percentage of current that an inductor passes after the first five time constants. For all practical purposes, you can consider the current fully flowing after five time constants have elapsed.
RL Time Constant Interval
Percentage of Total Current Passed
1
62.3%
2
86.5%
3
95.0%
4
98.2%
5
99.3%
Thus, in my example circuit in which the resistance is and the inductance is 0.1 H, you can expect that current will be flowing at full capacity within 5 ms of when the voltage is applied.
Five milliseconds is a very short amount of time. But electronic circuits are often designed to respond within very short time intervals. For example, the sine wave of standard household alternating current swings from its peak positive voltage to its peak negative voltage in about 8 ms. Sound waves at the upper end of the human ear’s ability to hear cycle in about (microseconds), and the time interval for radio waves can be in small fractions of microseconds. Thus, very small RL time constants can be very useful in certain types of electronic circuits.
Calculating Inductive Reactance
Although inductors oppose changes in current, they don’t oppose all changes equally. All inductors present more opposition to fast current changes than they do to slower changes, or put another way, inductors oppose current changes in higher-frequency signals more than they do in lower-frequency signals.
The degree to which an inductor opposes current change at a particular frequency is called the inductor’s reactance. Inductive reactance is measured in ohms, just like resistance, and can be calculated with the following formula:
Here, the symbol XL represents the inductive reactance in ohms, f represents the frequency of the signal in hertz (cycles per second), and L equals the inductance in henrys. Oh, and is that magic mathematical constant you learned about in high school, whose value is approximately 3.14, except here in Fresno where we round it down to 3 because it makes the math so much easier.
For example, suppose you want to know the reactance of a 1 mH inductor to a 60 Hz sine wave (not coincidentally, the frequency of household power). The math looks like this:
Thus, a 1 mH inductor has a reactance of about a third of an ohm at 60 Hz.
How much reactance does the same inductor have at 20 kHz? Much more:
Increase the frequency to 100 MHz and see how much resistance the inductor has:
At low frequencies, inductors are much more likely to let current pass than at high frequencies. As the next section explains, this characteristic can be exploited to create circuits that block frequencies above or below certain values.
Combining Inductors
Just like resistors or capacitors, you can combine inductors in series or parallel and use simple equations to calculate the total inductance of the circuit. Note, however, that for the calculations to be valid, the inductors must be shielded. If the inductors aren’t shielded, they’ll not only be affected by their own magnetic fields but also by the magnetic fields of other inductors around them. In that case, all bets are off.
You calculate inductor combinations just like resistor combinations, using exactly the same formulas except substituting henrys for ohms. Here are the formulas:
Series inductors: Just add up the value of each individual inductor.
Two or more identical parallel inductors: Add them up and divide by the number of inductors.
Two parallel and unequal inductors: Use this formula:
Three or more parallel and unequal inductors: Use this formula:
Here’s an example in which three inductors valued at 20 mH, 100 mH, and 50 mH are connected in parallel:
In this example, the total inductance of the circuit is 12.5 mH.
Putting Inductors to Work
Now that you know all about how inductors work and you understand inductive reactance, time constants, and all that other good stuff, you may be wondering what inductors are actually used for in real-life circuits. Here are a few of the more common uses for inductors:
Smoothing voltage in a power supply: The final stage of a typical power supply circuit that converts 120 V AC household current to a useable direct current is often a filter circuit that removes any residual irregularities in the voltage due to the fact that it was derived from a 60 Hz AC input.
Filter: Selects frequencies that should be allowed to pass, or it selects frequencies that should be blocked. You’re probably familiar with the tone settings on a stereo system, which let you bump up the bass, tone down the midrange, and maybe ease up the upper range just for brightness. There are three different types of filters: high-pass filters which allow only frequencies above a certain value to pass, low-pass filters which allow only frequencies below a certain value to pass, and band-pass filters which allow only frequencies that fall between an upper and a lower value to pass.
Radio tuning circuits: Coils can be used to help a radio tuning circuit tune to a particular frequency signal and hold it.
Transformers: One of the most common uses of inductors is in transformers. The simplest transformer consists of a pair of inductors placed next to each other. The transformer serves two functions. First, it can increase or decrease voltage, and second, it can electrically isolate one part of a circuit from another. You learn more about transformers in Book 4, Chapter 1.
Examining the mystery of induction
This chapter is a bit of a departure from the other chapters in this minibook, in that it contains no construction projects. The practical applications for inductors are in circuits that you might not be ready to build yet, such as radio tuners or power supplies. You learn the important concepts of how inductors work in this chapter so that later, when it’s time to use an inductor in an actual circuit, you’ll know what it does. You can always return to this chapter at that time to refresh your memory on how inductors work. In fact, I won’t hold it against you if you just skim through this chapter on your first reading through this book, or even skip it altogether for now. You can always come back to this chapter when the need arises.
Note that whenever current is flowing, there must be voltage. Thus, in addition to causing current to flow through the coil, electromagnetic induction creates a voltage across the coil.
).
Speaking of schematics, the symbol used to represent inductors in a schematic diagram is shown in the margin.
, and the inductance is 100 mH. Before you do the multiplication, you first convert the 100 mH to henrys. Because one millihenry (mH) is one one-thousandth of a henry, you can convert millihenrys to henrys by dividing the millihenrys by 1,000. Therefore, 100 mH is equivalent to 0.1 H. Dividing 0.1 H by 
gives you a time constant of 0.001 second (s), or one millisecond (ms).
and the inductance is 0.1 H, you can expect that current will be flowing at full capacity within 5 ms of when the voltage is applied.
(microseconds), and the time interval for radio waves can be in small fractions of microseconds. Thus, very small RL time constants can be very useful in certain types of electronic circuits.
is that magic mathematical constant you learned about in high school, whose value is approximately 3.14, except here in Fresno where we round it down to 3 because it makes the math so much easier.
