574 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS
Example 1. The current flowing through a 1000-Qresistor is 50 mA. What is the power dissipated by the resistor? Convert mA to A or 0.05 A. Use Eq. (A.16):
P c (0.05)2 × 1000
c 2.5 W.
Example 2. The potential drop across a 600-Qresistor is 3 V. What power is being dissipated by the resistor? Use Eq. (A.15):
P c E2/ R c 32/ 600 c 0.015 W or 15 mW.
A.6 INTRODUCTION TO ALTERNATING CURRENT CIRCUITS
A.6.1 Magnetism and Magnetic Fields
The phenomenon of magnetism was known by the ancient Greeks when they discovered a type of iron ore called magnetite. Moorish navigators used a needle-shaped piece of this ore as the basis of a crude compass. It was found that when this needle-shaped piece of magnetite was suspended and allowed to move freely, it would always turn to the north. What we are dealing with here are permanent magnets. Remember, we used permanent magnets in the telephone subset to set up a magnetic field in its earpiece (Section 5.3.2). Figure A.9 shows an artist’s rendition of the magnetic field around a bar magnet, the most common form of permanent magnet.
In the early nineteenth century it was learned that this magnetic property can be artificially induced to a family of iron-related metals by means of an electric current. The magnetic field was maintained while current is flowing; it collapsed when the current stopped.
Magnets, as we know them today, are classed as either permanent magnets or electromagnets. A typical permanent magnet is a hard steel bar that has been magnetized. It can be magnetized by placing the bar in a magnetic field; the more intense the field, the more intense is the induced magnetism in the steel bar. The influence of a magnetic field can be detected in various ways, such as by a conventional compass or iron filings on paper where the magnet is placed below the paper.
Figure A.9 An artist’s rendition of the magnetic field around a bar magnet.
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575
With this latter approach with the iron filings we develop an arrangement of the filings that will appear almost identical with Figure A.9. The lines that develop are commonly known as lines of magnetic induction. All of the lines as a group are referred to as the flux, which is designated by the symbol f. The flux per unit area is known as the flux density and is designated by B. We arbitrarily call one end of the bar magnet the north pole, and the opposite end, the south pole. Many of us remember playing with magnets when we were children. It was fun to bring in a second bar magnet. It was noted that if its south pole was brought into the vicinity of the first magnet’s south pole, the magnets physically repelled each other. On the contrary, when the north pole of one magnet was brought into the vicinity of the south pole of the other, they were attracted.
Suppose the strength of the magnet in Figure A.9 is increased. The magnetic field will be strengthened in proportion, and is conventionally represented by a more congested arrangement of lines of magnetic induction. The force that will be exerted upon a pole of another magnet located at any point in the magnetic field will depend upon the intensity of the field at that point. This field intensity is represented by the notation H.
In the preceding paragraphs we stated that the flux density B is the number of lines of magnetic induction passing through a unit area. By definition, unit flux density is one line of magnetic induction per square centimeter in air. Thus, in air, field intensity H and the flux density B have the same numerical value.
A.6.2 Electromagnetism
Any conductor where electrical current flows sets up a magnetic field. When the current ceases, the magnetic field collapses and disappears. If we form that wire conductor into a loop and connect each end to an emf source, current will flow and a magnetic field is set up. Let’s say that the resulting magnetic field has an intensity H. Now take the loop and produce a second turn. The resulting magnetic field now has an intensity of 2H; with three turns, 3H, and so forth.
Figure A.10 shows a spool on which we can support the looped wire turns. Some will call this spool arrangement a solenoid. If a solenoid is very long when compared with its diameter, the field intensity in the air inside the solenoid is directly proportional to the product of the number of turns and the current and inversely proportional to the length of the solenoid. This can be expressed by the following relationship:
Figure A.10 A magnetic field surrounding a solenoid with an air-core.
576 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS
H c k(NI)/ l,
(A.17)
where l is the length of the solenoid, N is the number of turns, and I is the current. Parameter k adjusts the equation to the type of unit system used. When I is in amperes and l is in meters, then k c 1. H is defined as ampere-turns per meter. Of course, H is the field intensity.
A.7 INDUCTANCE AND CAPACITANCE
A.7.1 What Happens When We Close a Switch on an Inductive Circuit?
Figure A.11 shows a simple circuit consisting of an air-core inductance, a battery emf supply, and a switch. The resistance across the coil or inductance is 10 Q and the battery supply is 24 V. We close the switch, and calculate the current flowing in the circuit. Using Ohm’s law:
I c E/ R c 24/ 10 c 2.4 A.
Off hand, one would say that at the very moment the switch is closed, the 2.4 A are developed. That is not true. It takes a finite time to build from the zero value of current when the switch is open to the 2.4-A value when the switch is closed. This is a change of state, from off to on. For every electrical change of state there is a finite amount of time required to fully reach the changed-state condition. Imagine a steam-locomotive starting off. It takes considerable time to bring the railroad train to full speed.
When there is an inductance in a circuit, such as in Figure A.11, the buildup is slowed still further because a counter emf is being generated in the coil (L). This is due to the dynamics of the buildup. As current starts to flow in the circuit once the switch is closed, a magnetic field is developing in the coil windings. By definition, this field generates a second field just in the windings themselves. This second magnetic field generates a voltage in the opposite direction of the voltage resulting from the primary field due to the current flow in the circuit. This is called counter emf. This counter emf continues so long as there is a dynamic condition, so long as there is a change in current flow. Once equilibrium is reached and we have a current flow of exactly 2 A, the counter emf disappears.
Now open the switch in Figure A.11. At the very moment of opening the switch we see an electric arc across the switch contacts. When the switch is opened there is
Figure A.11 A simple circuit with an inductance or coil.
A.7 INDUCTANCE AND CAPACITANCE
577
no longer support (i.e., no current flows in the circuit) for the magnetic field in coil L. The energy stored in that field must dissipate somewhere. It dissipates back through the circuit again in the opposite direction of original current flow. Again we are faced with the fact that on opening the switch, the current does not drop to zero immediately, but takes a finite amount of time to drop to zero amperes. So a magnetic field of an electromagnet actually stores electricity.
Regarding Figure A.11, we are faced with two conditions: first the buildup of current to its steady-state value, and the decay of electric current to a zero value. It may be said that an electric circuit reacts to such current changes.
The magnitude of the induced emf, that reactive effect, is a function of two factors:
1. The first factor is the number of turns of wire in the coil; whether the coil has an iron core; and the properties of the iron core; and
2. The second factor is the rate of change of current in the coil.
These two properties only come into play when the circuit is dynamic—its electrical conditions are changing. There are two such properties that remain latent under steadystate conditions, but come into play when the current attempts to change its value. These are inductance and capacitance. For analogies, let’s consider that inductance is something like inertia in a mechanical device, and capacitance is like elasticity.
Inductance. The property of a circuit which we have called inductance is represented by the symbol L. Its unit of measure is the henry (H). The henry is defined as the inductance of a circuit that will cause an induced emf of 1 V to be set up in the circuit when the current is changing at the rate of 1 A per second. Now the following relationship can be written:
E1 c LI/ t,
(A.18)
where E1 is the induced emf; L is the inductance in henrys; I is the current in amperes; and t is time in seconds.
If we have coils in series, then to calculate the total inductance of the series combination, add the inductances such as we add resistances for resistors in series. In a similar manner, if we have inductances in parallel, use the same methodology as though they were resistances in parallel.
Now distinguish between self-inductanceand mutual inductance. Self-inductance is the property of a circuit which creates an emf from a change of current values when the reaction effects are wholly within the circuit itself. If the electromagnetic induction is between the coils or inductors of separate circuits, it is called mutual inductance.
Capacitance. Capacitance was introduced in Section 2.5.1.1, during a discussion of its buildup on wire pair as it was extended. An analogy of capacitance is a tank of compressed air. Air is elastic, and the quantity of air in the tank is a function of the air pressure and capacity of the tank.
Similar to Figure A.11, consider Figure A.12, which shows a capacitor connected across a battery emf supply. In its simplest form, a capacitor may consist of two parallel plates (conductors) separated by an insulator, in this case air. The circuit is equipped with an on–off switch. Close the switch. Unlike the inductance scenario in Figure A.11, there is a surge of current in the circuit. The current is charging the capacitor to a voltage equal to the emf of the battery. As the capacitor becomes charged, the value
578 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS
Figure A.12 A circuit model to illustrate capacitance.
of the current decreases until the capacitor is fully charged, when the current becomes zero.
A capacitor is defined as a device consisting of two conductors separated by an insulator (in its simplest configuration). The greater the area of the conductors, the greater the capacitance of the device. (Remember Section 2.5.2.2 where we dealt with a wire pair). Here we have two conductors separated by the insulation on each wire in the pair. Likewise, the longer the wire pair, the greater the capacitance it displays.
The quantity of electricity stored by a capacitor is a function of its capacitance and the emf across its terminals. It is expressed by the relationship:
q c EC,
(A.19)
where q is the quantity of electricity in coulombs, E is the emf in volts, and C is the capacitance in farads.5 The capacitance value is a function of the dimensions of the capacitor conductor plates. In the practical world, the farad is too large a unit. As a result we usually measure capacitance in microfarads (mF) or 1 × 10−6 farads, or picofarads (pF) or 10−12 farads.
Restating the Eq. (A.19):
q c e0EA
(A.20)
where e0 is the permittivity constant, 8.85 × 10−12 farads/ meter or 8.85 pF/ m; A is the area of the plate; and E is the emf.
Substituting q from Eq. (A.19) in Eq. (A.20) and noting that the capacitance is inversely proportional to the distance, d, between the parallel plates, we have:
C c e0(A/ d)
(A.21)
This tells us that capacitance is a function of geometry; it is directly proportional to the area of the capacitor plates and inversely proportional to the distance between the plates. This assumes that the insulator between the plates is air.
5Coulomb (IEEE, Ref. 2) is the unit of electric charge in SI units. The coulomb is the quantity of electric charge that passes any cross section of a conductor in 1 second when the current is maintained at 1 A.
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579
Table A.2 Some Properties of Dielectrics
Dielectric
Dielectric
Strength
Material
Constant (k)a
(kV/ mm)
Air (1 atm)
1.00054
3
Polystyrene
2.6
24
Paper
3.5
16
Transformer oil
4.5
Pyrex
4.7
14
Ruby mica
5.4
Porcelain
6.5
Silicon
12
Germanium
16
Ethanol
25
Water (208C)
80.4
Water (258C)
78.5
Titania ceramic
130
Strontium titanate
310
8
For a vacuum, k c unity.
a Measured at room temperature, except for the water.
Source: Fundamentals of Physics—Extended, 4th ed., p. 751, Table 27-2. (Ref. 3, reprinted with permission.)
Suppose the insulator between the plates is some other material. Table A.2 lists some of the typical materials used as insulators and their dielectric constant. The capacitance increases by a numerical factor k, which is the dielectric constant. The dielectric constant of a vacuum is unity by definition. Because air is nearly “empty” space, its measured dielectric constant is only slightly greater than unity. The difference is insignificant.
Let L represent some geometric dimensions as a function of length. For a parallel plate capacitor with an air dielectric (insulator), L c A/ d (from Eq. (A.21)). Capacitance
(C) can be related to dielectric constant, the geometrical property L, and a constant by:
C c e0kL
(A.22)
where e0 is the permittivity constant, 8.85 pf/ m, and k is the dielectric constant from Table A.2.
Another effect of the introduction of a dielectric (insulator) including air is to limit the potential difference between the conductors to some maximum voltage value. If this value is substantially exceeded, the dielectric material will break down and form a conducting path between the plates. Every dielectric material has a characteristic dielectric strength, which is the maximum value of the electric field that it can tolerate without breakdown. Several dielectric strengths are listed in Table A.2.
Capacitors in Series and in Parallel. When there is a combination of capacitors in a circuit, sometimes we can replace the combination with a single capacitor with an equivalent capacitance value. Just as resistances and inductances can be in series and in parallel, we can have capacitors in series and in parallel. Figure A.13 shows several capacitors in parallel. As we see in the figure, each capacitor has the same potential difference across its plates (i.e., the battery emf). By direct inference from the figure we can see the equivalent capacitance of the three capacitors is the sum of each of the individual capacitances or:
580 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS
Figure A.13 Capacitors in parallel.
Ceq c C1 + C2 + C3.
(A.23)
Suppose the values of the capacitors were 2, 3, and 4 nF, respectively. What would the equivalent capacitance be? From Eq. (A.23), the value would be 9 nF.
Capacitors in series (see Fig. A.14) are handled in a similar manner as resistances in parallel. The total equivalent capacitance is:
1/ Ceq c 1/ C1 + 1/ C2 + 1/ C3.
(A.24)
Suppose, again, the values of the three capacitors were 2, 3, and 4 nF. What would the equivalent capacitance be?
1/ Ceq c 1/ 2 + 1/ 3 + 1/ 4
c 6/ 12 + 4/ 12 + 3/ 12 c 13/ 12, Ceq c 12/ 13 c 0.923 nF.
A.7.2 RC Circuits and the Time Constant
An RC circuit is illustrated in Figure A.15. The capacitor in the figure remains uncharged until the switch is closed. To charge it we throw the switch S to a closed position. The
Figure A.14 Capacitors in series.
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581
Figure A.15 An RC circuit.
circuit consists of a resistor (R) and capacitor (C) in series. When we close the switch, there is a surge of current whose intensity decreases with time as the charge of the capacitor builds up.
Figure A.16 shows the change in the value of the current with time after the switch is closed. There is a corresponding change in the voltage drop across the capacitor C. This is illustrated with curve E. The instantaneous value of current, I, is determined solely by the value of resistor, R. The total voltage drop is then across R and the drop across C is zero. As the capacitor begins to charge, however, the voltage drop across C gradually builds up, the current decreases, and the voltage drop across R decreases correspondingly. When the capacitor reaches full charge and the current has fallen to a negligible value, the total voltage drop is now across the capacitor. Remember that at all times the sum of the voltage drops across R and C must be equal and opposite to the impressed voltage.
Where does time come into the picture (i.e., time constant)? It takes time for an RC circuit (or an RL circuit) to reach a steady-state value. This time, we find, is a function of the product of R and C. Actually, if we multiply the value of R in ohms by the value of C in farads, it is equal to time in seconds. In a series RC circuit as illustrated in Figure A.15, the product RC is known as the time constant of the circuit. By definition, it is the time required to charge the capacitor to 63% of its final voltage. If we plot the curve for voltage (E) in Figure A.16, we have an exponential function of time. This is expressed mathematically as:
Figure A.16 Current and voltage values as a function of time for an RC circuit.
582 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS
Ei c Emax[1 − e−t/ RC],
(A.25)
where Ei is the instantaneous voltage across the capacitor at any time t, Emax is the final voltage, and e is the natural number. This is the base of the natural logarithm or the Naperian logarithm.6 Its value is 2.718+. When t equals RC, the term in parentheses equals:
1 − e−1 c 1 − 1/ e c 1 − 2.718 c 0.63.
It should be noted that a similar relationship can be written for an RL circuit where the exponential buildup curve is current rather than voltage. However, here the exponent is L/ R rather than RC. Again, it is the time required for the current to build up to 63% of its final value.
RC circuits have wide application in the telecommunications field. Because of the precision with which resistors, capacitors, and inductors may be built, they enable the circuit designer to readily control the timing of current pulses to better than 1-ms accuracy. For such practical applications, the designer considers that currents and voltages in RC and RL circuits reach their final value in a time equal to five times their time constant [i.e., 5 × RC or 5 × (L/ R)]. It can be shown that at 5 × time constant the current or voltage has reached 99.33+% of its final value.
A.8 ALTERNATING CURRENTS
Alternating current (ac) is a current where the source emf is alternating, having a simple and convenient waveform, namely, the sine wave. Such a sine wave is illustrated in Figure A.17. Any sine wave can be characterized by its frequency, phase, and amplitude. These were introduced in Section 2.3.3, and much of that information is repeated here for direct continuity.
Note in the figure that half a cycle of the sine wave is designated by p and a full cycle by 2p. Remember that there are 2p radians in a circle or 360 degrees, and 1 p
Figure A.17 A typical sine wave showing amplitude and wavelength.
6In this text, unless otherwise specified, logarithms are to the base ten. These are called common logarithms. Under certain circumstances we will also use logarithms to the base 2 (binary system of notation). In the calculus, nearly all logarithms used are to the natural or Naperian base. This is named in honor of a Scottsman named Napier. However, Napier did not use the base e but used a base approximately equal to e−1.
A.8 ALTERNATING CURRENTS
583
Figure A.18a An ac circuit that is predominantly resistive.
radian is 180 degrees. Therefore 1 radian is 1808/p or 57.29582791 . . .8. That is an unwieldy number and is not exact. That is why we like to stick to using p, and the value is exact.
In Figure A.17 the maximum amplitude is +A and −A; the common notation for wavelength is l. To convert wavelength to frequency and vice versa we use the formula
given in Section 2.3.3, or
Fl c 3 × 108 m/ s (meters/ second).
(A.26)
We recognize the constant on the right-hand side of the equation as an accepted estimate of the velocity of light (or a radio wave). The period of a sine wave is the length of time it takes to execute 1 cycle (1 Hz). We then relate frequency, F, to period, T, with:
F c 1/ T,
(A.27)
where F is in Hz and T is in seconds (s).
Voltage and current may be out of phase one with the other. The voltage may lead the current or lag the current depending on circuit characteristics, as we will see.
When an AC circuit is resistive—there is negligible capacitance and inductance—the voltage and current in the circuit will be in-phase. Figure A.18a is a typical ac resistive circuit. The phase relations between voltage (VR) and current (IR) is illustrated in Figure A.18b.
Figure A.18b Phase relationship between voltage and current in an ac circuit that is resistive. Here the voltage and current remain in phase.
