The 10 Percent Rule: This rule is a standard method for selecting R 1 and R 2 that takes into account
2.24 Inductors
In the preceding section we saw how a capacitor stored electrical energy in the form of an electric field. Another way to store electrical energy is in a magnetic field.
FIGURE 2.108
Circular radiating magnetic fields can be generated about a wire any time current passes through it. Increasing or decreasing current flow through the wire increases and decreases the magnetic field strength, respectively. During such changes in mag- netic field strength, we encounter a phenomenon known as inductance. Inductance is a property of circuits somewhat analogous to resistance and capacitance; however, it is not attributed to heat production or charge storage (electric field), but rather it is associated with magnetic fields—more specifically, how changing magnetic fields influence the free electrons (current) within a circuit. Theoretically, any device capa- ble of generating a magnetic field has inductance. Any device that has inductance is referred to as an inductor. To understand inductance requires a basic understanding of electromagnetic properties.
2.24.1 Electromagnetism
According to the laws of electromagnetism, the field of a charge at rest can be rep- resented by a uniform, radial distribution of electric field lines or lines of force (see Fig. 2.110a). For a charge moving at a constant velocity, the field lines are still radial and straight, but they are no longer uniformly distributed (see Fig. 2.110b). At the same time, the electron generates a circular magnetic field (see Fig. 2.110c). If the charge acceler- ates, things get a bit more complex, and a “kink” is created in the electromagnetic field, giving rise to an electromagnetic wave that radiates out (see Fig. 2.110d and e).
As depicted in Fig. 2.110c, the electric field (denoted E) of a moving electron—or any charge for that matter—is, in effect, partially transformed into a magnetic field (denoted B). Hence, it is apparent that the electric and magnetic fields are part of the same phenomenon. In fact, physics today groups electric and magnetic fields together into one fundamental field theory, referred to as electromagnetism. (The work of Maxwell and Einstein helped prove that the two phenomena are linked. Today, certain fields in physics paint a unique picture of field interactions using virtual photons being emitted and absorbed by charges to explain electromagnetic forces.
Fortunately, in electronics you don’t need to get that detailed.)
FIGURE 2.109 The three cornerstones of electronics are resistance, capacitance, and inductance. Inductance, unlike the other two, involves alternations in a circuit’s current and voltage characteristics as a result of forces acting upon free elec- trons resulting in the creation and collapse of magnetic fields, usually concentrated in a discrete inductor device. Like a capacitor, however, inductive effects occur only during times of change, when the applied voltage/current increases or decreases with time. Resistance doesn’t have a time dependency. Can you guess what will happen to the brightness of the lamp in each of the circuits in the figure when the switch is closed? What do you think will occur when the switch is later opened? We’ll discuss this a bit later.
The simplest way to generate a magnetic field is to pass a current through a con- ductor. Microscopically, each electron within a wire should be generating a magnetic field perpendicular to its motion. However, without any potential applied across the wire, the sheer randomness of the electrons due to thermal effects, collision, and so on, cause the individual magnetic fields of all electrons to be pointing in random directions. Averaged over the whole, the magnetic field about the conductor is zero (see Fig. 2.111a.1). Now, when a voltage is applied across the conductor, free electrons gain a drift component pointing from negative to positive—conventional current in the opposite direction. In terms of electron speed, this influence is very slight, but it’s enough to generate a net magnetic field (see Fig. 2.111a.2). The direction of this field is perpendicular to the direction of conventional current flow and curls in a direc- tion described by the right- hand rule: your right thumb points in the direction of the conventional current flow; your finger curls in the direction of the magnetic field. See Fig. 2.111b. (When following electron flow instead of conventional current flow, you’d use your left hand.)
The magnetic field created by sending current through a conductor is similar in nature to the magnetic field of a permanent magnet. (In reality, the magnetic field pattern of a permanent bar magnet, in Fig. 2.111c, is more accurately mimicked when the wire is coiled into a tight solenoid, as shown in Fig. 2.111e.) The fact that both a current- carrying wire and a permanent magnet produce magnetic fields is no coin- cidence. Permanent magnets made from ferromagnetic materials exhibit magnetic closed- loop fields, mainly as a result of the motion of unpaired electrons orbiting about the nucleus of an atom, as shown in Fig. 2.112, generating a dipole magnetic field. The lattice structure of the ferromagnetic material has an important role of fix- ing a large portion of the atomic magnetic dipoles in a fixed direction, so as to set up a net magnetic dipole pointing from north to south. This microscopic motion of
FIGURE 2.110 The electric and magnetic fields are part of the same phenomenon called electromagnetism. Magnetic fields appear whenever a charge is in motion. Interestingly, if you move along with a moving charge, the observable mag- netic field would disappear—thanks to Einstein’s relativity.
FIGURE 2.111 (a) Magnetic field generated free electrons moving in unison when voltage is applied. (b) Right- hand rule showing direction of magnetic field in relation to conventional current flow. (c) Permanent magnet. (d) Magnetic dipole radiation pattern created by current flowing through single loop of wire. (e) Energized solenoid with a magnetic field similar to permanent magnet. (f) Electromagnet, using ferromagnetic core for increased field strength.
FIGURE 2.112 Microscopically, the magnetic field of a permanent magnet is a result of unpaired valence electrons fixed in a common direction, generating a magnetic dipole. The fixing in orientation is a result of atomic bonding in the crystal- line lattice structure of the magnet.
the unpaired electron about the nucleus of the atom resembles the flow of current through a loop of wire, as depicted in Fig. 2.111d. (Electron spin is another source of magnetic fields, but is far weaker than that due to the electron orbital.)
By coiling a wire into a series of loops, a solenoid is formed, as shown in Fig. 2.111e.
Every loop of wire contributes constructively to make the interior field strong. In other words, the fields inside the solenoid add together to form a large field component that points to the right along the axis, as depicted in Fig. 2.111e. By placing a ferromagnetic material (one that isn’t initially magnetized) within a solenoid, as shown in Fig. 2.111f, a much stronger magnetic field than would be present with the solenoid alone is cre- ated. The reason for such an increase in field strength has to do with the solenoid’s field rotating a large portion of the core’s atomic magnetic dipoles in the direction of the field. Thus, the total magnetic field becomes the sum of the solenoid’s magnetic field and the core’s temporarily induced magnetic field. Depending on material and construction, a core can magnify the total field strength by a factor of 1000.
2.24.2 Magnetic Fields and Their Influence
Magnetic fields, unlike electric fields, only act upon charges that are moving in a direction that is perpendicular (or has a perpendicular component) to the direction of the applied field. A magnetic field has no influence on a stationary charge, unless the field itself is moving. Figure 2.113a shows the force exerted upon a moving charge placed within a magnetic field. When considering a positive charge, we use our right hand to determine the direction of force upon the moving charge—the back of the
FIGURE 2.113 Illustration showing the direction of force upon a moving charge in the presence of a fixed magnetic field.
hand points in the direction of the initial charge velocity, the fingers curl in the direc- tion of the external magnetic field, and the thumb points in the direction of the force that is exerted upon the moving charge. For a negative charge, like an electron, we can use the left hand, as shown in Fig. 2.113b. If a charge moves parallel to the applied field, it experiences no force due to the magnetic field—see Fig. 2.113c.
In terms of a group of moving charges, such as current through a wire, the net magnetic field of one wire will exert a force on the other wire and vice versa (pro- vided the current is fairly large), as shown in Fig. 2.114. (The force on the wire is possible, since the electrostatic forces at the surface of the lattice structure of the wire prevent electrons from escaping from the surface.)
Likewise, the magnetic field of a fixed magnet can exert a force on a current- carrying wire, as shown in Fig. 2.115.
Externally, a magnet is given a north- seeking (or “north,” for short) and south- seeking (or “south,” for short) pole. The north pole of one magnet attracts the south pole of another, while like poles repel—see Fig. 2.115b. You may ask how two station- ary magnets exert forces on each other. Isn’t the requirement that a charge or field must be moving for a force to be observed? We associate the macroscopic (observed) force with the forces on the moving charges that comprise the microscopic internal magnetic dipoles which are, at the heart, electrons in motion around atoms. These orbitals tend to be fixed in a general direction called domains—a result of the lattice binding forces.
FIGURE 2.114 Forces exerted between two current- carrying wires.
FIGURE 2.115 (a) The force a current- carrying wire experiences in the presence of a magnet’s field. (b) Illustration showing how bar magnets attract and repel.
Another aspect of magnetic fields is their ability to force electrons within conduc- tors to move in a certain direction, thus inducing current flow. The induced force is an electromotive force (EMF) being set up within the circuit. However, unlike, say, a battery’s EMF, an induced EMF depends on time and also on geometry. According to Faraday’s law, the EMF induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit:
EMF d , , EMF
dt B dA Nd
dt
M
M M
∫
= − Φ
Φ = ⋅ =− Φ
(for a coiled wire of N loops) (2.49) where ΦM is the magnetic flux threading through a closed- loop circuit (which is equal to the magnetic field B dotted with the direction surface area—both of which are vec- tors, and summed over the entire surface area—as the integral indicates). According to the law, the EMF can be induced in the circuit in several ways: (1) the magnitude of B can vary with time; (2) the area of the circuit can change with time; (3) the angle between B and the normal of A can change with time; (4) any combination of these can occur. See Fig. 2.116.
The simple ac generator in Fig. 2.117 shows Faraday’s law in action. A simple rotating loop of wire in a constant magnetic field generates an EMF that can be used to power a circuit. As the loop rotates, the magnetic flux through it changes with time, inducing an EMF and a current in an external circuit. The ends of the loop are connected to slip rings that rotate with the loop, while the external circuit is linked to the generator by stationary brushes in contact with the slip rings.
A simple dc generator is essentially the same as the ac generator, except that the contacts to the rotating loop are made using a split ring or commutator. The result is a pulsating direct current, resembling the absolute value of a sine wave—there are no polarity reversals.
A motor is essentially a generator operating in reverse. Instead of generating a cur- rent by rotating a loop, a current is supplied to the loop by a battery, and the torque
FIGURE 2.116 Illustration of Faraday’s law of induction.
acting on the current- carrying loop causes it to rotate. Real ac generators and motors are much more complex than the simple ones demonstrated here. However, they still operate under the same fundamental principles of electromagnetic induction.
The circuit in Fig. 2.118 shows how it is possible to induce current within a sec- ondary coil of wire by suddenly changing the current flow through a primary coil of wire. As the magnetic field of the primary expands, an increasing magnetic flux per- meates the secondary. This induces an EMF that causes current to flow in the second- ary circuit. This is the basic principle behind how transformers work; however, a real transformer’s primary and secondary coils are typically wound around a common ferromagnetic core to increase magnetic coupling.
2.24.3 Self- Inductance
In the previous section, we saw how an EMF could be induced in a closed- loop circuit whenever the magnetic flux through the circuit changed with time. This phenome- non of electromagnetic induction is used in a number of mechanisms, such as motors, generators, and transformers, as was pointed out. However, in each of these cases, the induced EMF was a result of an external magnetic field, such as the primary coil
FIGURE 2.117 Basic ac generator.
FIGURE 2.118 An induced voltage, or EMF, is generated in the secondary circuit whenever there is a sudden change in current in the primary.
in relation to a secondary coil. Now, however, we will discuss a phenomenon called self- induction. As the name suggests, self- induction typically involves a looped wire inflicting itself with an induced EMF that is generated by the varying current that passes through it. According to Faraday’s law of induction, the only time that our loop can self- inflict is when the magnetic field grows or shrinks in strength (as a result of an increase or decrease in current). Self- induction is the basis for the induc- tor, an important device used to store energy and release energy as current levels fluctuate in time- dependent circuits.
Consider an isolated circuit consisting of a switch, a resistor, and a voltage source, as shown in Fig. 2.119. If you close the switch, you might predict that the current flow through the circuit would jump immediately from zero to V/R, according to Ohm’s law. However, according to Faraday’s law of electromagnetic induction, this isn’t entirely accurate. Instead, when the switch is initially closed, the current increases rapidly. As the current increases with time, the magnetic flux through the loop rises rapidly. This increasing magnetic flux then induces an EMF in the circuit that opposes the current flow, giving rise to an exponentially delayed rise in current. We call the induced EMF a self- induced EMF.
Self- induction within a simple circuit like that shown in Fig. 2.119 is usually so small that the induced voltage has no measurable effect. However, when we start incorpo- rating special devices that concentrate magnetic fields—namely, discrete inductors—
time- varying signals can generate significant induced EMFs. For the most part, unless otherwise noted, we shall assume the self- inductance of a circuit is negligible com- pared with that of a discrete inductor.
2.24.4 Inductors
Inductors are discrete devices especially designed to take full advantage of the effects of electromagnetic induction. They are capable of generating large concentrations of magnetic flux, and they are likewise capable of experiencing a large amount of self- induction during times of great change in current. (Note that self- induction also
FIGURE 2.119 (a) Circuit is open and thus no current or magnetic field is generated. (b) The moment the circuit is closed, current begins to flow, but at the same time, an increasing magnetic flux is generated through the circuit loop. This increas- ing flux induces a back EMF that opposes the applied or external EMF. After some time, the current levels off, the magnetic flux reaches a constant value, and the induced EMF disappears. (d) If the switch is suddenly opened, the current attempts to go to zero; however, during this transition, as the current goes to zero, the flux decreases through the loop, thus generat- ing a forward induced voltage of the same polarity as the applied or external EMF. As we’ll see later, when we incorporate large solenoid and toroidal inductors within a circuit, opening a switch such as this can yield a spark—current attempting to keep going due to a very large forward EMF.
exists within straight wire, but it is usually so small that it is ignored, except in special cases, e.g., VHF and above, where inductive reactance can become significant.)
The common characteristic of inductors is a looplike geometry, such as a sole- noid, toroid, or even a spiral shape, as shown in Fig. 2.120. A solenoid is easily con- structed by wrapping a wire around a hollow plastic form a number of times in a tight- wound fashion.
The basic schematic symbol of an air core inductor is given by . Magnetic core inductors (core is either iron, iron powder, or a ferrite- type ceramic), adjustable core inductors, and a ferrite bead, along with their respective schematic symbols are shown in Fig. 2.121.
Magnetic core inductors are capable of generating much higher magnetic field den- sities than air core inductors as a result of the internal magnetization that occurs at the atomic level within the core material due to the surrounding wire coil’s magnetic field. As a result, these inductors experience much greater levels of self- induction when compared to air core inductors. Likewise, it is possible to use fewer turns when
FIGURE 2.120 Various coil configurations of an inductor—solenoid, toroid, and spiral.
FIGURE 2.121
using a magnetic core to achieve a desired inductance. The magnetic core material is often iron, iron powder, or a metallic oxide material (also called a ferrite, which is ceramic in nature). The choice of core material is a complex process that we will cover in a moment.
Air core inductors range from a single loop in length of wire (used at ultrahigh fre- quencies), through spirals in copper coating of an etched circuit board (used at very high frequencies), to large coils of insulated wire wound onto a nonmagnetic former.
For radio use, inductors often have air cores to avoid losses caused by magnetic hys- teresis and eddy current that occur within magnetic core–type inductors.
Adjustable inductors can be made by physically altering the effective coil length—
say, by using a slider contact along the uncoated coil of wire, or more commonly by using a ferrite, powdered iron, or brass slug screwed into the center of the coil. The idea behind the slug- tuned inductor is that the inductance depends on the perme- ability of the material within the coil. Most materials have a relative permeability close to 1 (close to that of vacuum), while ferrite materials have a large relative per- meability. Since the inductance depends on the average permeability of the volume inside the coil, the inductance will change as the slug is turned. Sometimes the slug of an adjustable inductor is made of a conducting material such as brass, which has a relative permeability near 1, in which case eddy currents flow on the outside of the slug and eliminate magnetic flux from the center of the coil, reducing its effective area.
A ferrite bead, also known as a ferrite choke, is a device akin to an inverted fer- rite core inductor. Unlike a typical core inductor, the bead requires no coiling of wire (though it is possible to coil a wire around it for increased inductance, but then you’ve created a standard ferrite core inductor). Instead, a wire (or set of wires) is placed through the hollow bead. This effectively increases the inductance of the wire (or wires). However, unlike a standard inductor that can achieve practically any inductance based on the total number of coil turns, ferrite beads have a limited range over which they can influence inductance. Their range is typically limited to RF (radiofrequency). Ferrite beads are often slipped over cables that are known to be notorious radiators of RF (e.g., computers, dimmers, fluorescent lights, and motors).
With the bead in place, the RF is no longer radiated but absorbed by the bead and converted into heat within the bead. (RF radiation can interfere with TV, radio, and audio equipment.) Ferrite beads can also be placed on cables entering receiving equipment so as to prevent external RF from entering and contaminating signals in the cable runs.
Inductor Basics
An inductor acts like a time- varying current- sensitive resistance. It only “resists”
during changes in current; otherwise (under steady- state dc conditions), it passes current as if it were a wire. When the applied voltage increases, it acts like a time- dependent resistor whose resistance is greatest during times of rapid increase in cur- rent. On the other hand, when the applied voltage decreases, the inductor acts like a time- dependent voltage source (or negative resistance) attempting to keep current flowing. Maximal sourcing is greatest during times of rapid decreases in current.
In Fig. 2.122a, when an increasing voltage is applied across an inductor, resulting in an increasing current flow, the inductor acts to resist this increase by generating a