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Unformatted text preview: 22 AC Circuits 22—1 Impedances Most of our work in this course has been aimed at reaching the complete
equations of Maxwell. In the last two chapters We have been discussmg the con
sequences of these equations. We have found that the equations contain all the
static phenomena we had worked out earlier, as well as the phenomena of electro
magnetic waves and light that we had gone over in some detail in Volume I. The
Maxwell equations give both phenomena, depending upon whether one computes
the ﬁelds close to the currents and charges, or very far from them There is not
much interesting to say about the intermediate region; no special phenomena
appear there. There still remain, however, several subjects in electromagnetism that we
want to take up. We want to discuss the question of relativity and the Maxwell
equations—what happens when one looks at the Maxwell equations With respect
to moving coordinate systems. There is also the question of the conservation of
energy in electromagnetic systems. Then there is the broad subject of the electro
magnetic properties of materials; so far, except for the study of the properties
ofdielectrics, we have considered only the electromagnetic ﬁelds in free space And
although we covered the subject of light in some detail in Volume I, there are
still a few things we would like to do again from the point of View of the ﬁeld
equations. In particular, we want to take up again the subject of the index of re—
fraction, particularly for dense materials. Finally, there are the phenomena
associated with waves conﬁned in a limited region of space. We touched on this
kind of problem brieﬂy when we were studying sound waves. Maxwell’s equations
lead also to solutions which represent conﬁned waves of the electric and magnetic
ﬁelds. We will take up this subject, which has important technical applications,
in some of the followmg chapters. In order to lead up to that subject, we Will
begin by considering the properties of electrical Circuits at low frequenCies. We
Will then be able to make a comparison between those situations in which the
almost static approximations of Maxwell’s equations are applicable and those
situations in which highfrequency effects are dominant. So we descend from the great and esoteric heights of the last few chapters
and turn to the relatively lowlevel subject of electrical circuits. We will see, how
ever, that even such a mundane subject, when looked at in sufﬁc1ent detail, can
contain great complications We have already discussed some of the properties of electrical circuits in
Chapters 23 and 25 of Vol. I. Now we will cover some of the same material again,
but in greater detail. Again we are going to deal only With linear systems and With
voltages and currents which all vary sinus01dally; we can then represent all voltages
and currents by complex numbers, using the exponential notation described in
Chapter 22 of Vol. I. Thus a timevarying voltage V(t) will be written V(r) = Vem, (22.1) Where I7 represents a eomplex number that is independent of t. It is, of course,
understood that the actual timevarying voltage V0) is given by the real part of
the complex function on the righthand side of the equation. 22—1 2.2—1 Impedances
22—2 Generators 22—3 Networks of ideal elements;
Kirchhoff ’s rules 22—4 Equivalent circuits
22—5 Energy
22—6 A ladder network
22—7 Filters 22—8 Other circuit elements Rewew.’ Chapter 22, Vol. 1, Algebra Chapter 23, Vol. I, Resonance
Chapter 25, Vol 1, Linear
Systems and Review Hg.22—L Aninducfance. Similarly, all of our other timevarying quantities will be taken to vary
sinusoidally at the same frequency to. So we write I
8 I em (current), a em (emf), (22.2)
E = E e‘“‘ (electric ﬁeld), ll and so on.
Most of the time we will write our equations in terms of V, I, 8, . . . (instead of A In terms of V, i, 8, . . .), remembering, though, that the tlme variations are as
given in (22.2). In our earlier discussion of circuits we assumed that such things as inductances,
capac1tances, and resistances were familiar to you. We want now to look in a little
more detail at what is meant by these idealized circuit elements. We begin with
the inductance. An inductance is made by winding many turns of wire in the form of a coil
and bringing the two ends out to terminals at some distance from the coil, as shown
in Fig. 22—1. We want to assume that the magnetic ﬁeld produced by currents in
the coil does not spread out strongly all over space and interact with other parts of
the circuit. This is usually arranged by winding the coil in a doughnutshaped
form, or by conﬁning the magnetic ﬁeld by winding the coil on a suitable iron core,
or by placmg the coil in some suitable metal box, as indicated schematically in
Fig. 22—1. In any case, we assume that there is a negligible magnetic ﬁeld in the
external region near the terminals (1 and b. We are also going to assume that we
can neglect any electrical resistance in the wire of the coil. Finally. we will assume
that we can neglect the amount of electrical charge that appears on the surface of
a wire 1n building up the electric ﬁelds. With all these approximations we have what we call an “ideal” inductance.
(We Will come back later and discuss what happens in a real inductance.) For an
ideal inductance we say that the voltage across the terminals is equal to L(dI/dt).
Let’s see why that is so. When there is a current through the inductance, a magnetic
ﬁeld proportional to the current is bullt up inside the coil. If the current changes
with time, the magnetic ﬁeld also changes. In general, the curl of E is equal to
—dB/dt; or, put differently, the line integral of E all the way around any closed
path 15 equal to the negative ofthe rate of change of the ﬂux ofB through the loop
Now suppose we consrder the followmg path: Begin at termlnal a and go along
the c011 (staylng always inside the wire) to terminal b; then return from terminal b
to terminal a through the air in the space outside the inductance. The line integral
of E around this closed path can be written as the sum of two parts: /E~ds= [abEds+ Eds. (22.3) wt; outsrde
corl As we have seen before, there can be no electric ﬁelds 1nside a perfect conductor.
(The smallest ﬁelds would produce inﬁnite currents.) Therefore the integral from
a to b via the coil is zero. The whole contribution to the line integral of E comes
from the path outside the inductance from terminal b to terminal a. Since we have
assumed that there are no magnetic ﬁelds in the space outside of the “box,” this
part of the integral is independent of the path chosen and we can deﬁne the po
tentlals of the two terminals. The dlfference of these two potentials is what we
call the voltage difference, or simply the voltage V, so we have V: —/b“E~ds= —y§Eds. The complete line integral is what we have before called the electromotive
force 8 and is, of course, equal to the rate of change of the magnetic ﬂux in the
0011. We have seen earlier that this emf is equal to the negative rate of change of 22—2 the current. so we have :11 where L is the inductance of the coil. Since dl/dt = iwI, we have V = iwLI. (22.4) The way we have described the ideal inductance illustrates the general approach
to other ideal circuit elements—usually called “lumped” elements. The properties
of the element are described completely in terms of currents and voltages that
appear at the terminals. By making suitable approx1mations, it is pOSSible to
ignore the great complextties of the ﬁelds that appear inside the object. A separation
is made between what happens inside and what happens outside. For all the circuit elements we will ﬁnd a relation like the one in Eq. (22.4), in
which the voltage is proportional to the current with a proportionality constant
that is, in general, a complex number. This complex coefﬁcient of prOportionality
is called the impedance and is usually written as 2 (not to be confused with the
zcoordinate). It is, in general, a function of the frequency w. So for any lumped
element we write V i7
— 7 z . 22.5
I I z ( ) For an inductance, we have z(inductance) = 2,, = iwL. (22.6) Now let’s look at a capacitor from the same point of view.* A capacitor con—
Sists ofa pair of conducting plates from which two wires are brought out to suitable
terminals. The plates may be of any shape whatsoever, and are often separated
by some dielectric material. We illustrate such a situation schematically in Fig.
22—2. Again we make several Simplifying assumptions. We assume that the
plates and the Wires are perfect conductors. We also assume that the insulation
between the plates is perfect, so that no charges can ﬂow across the insulation
from one plate to the other. Next, we assume that the two conductors are close
to each other but far from all others, so that all ﬁeld lines which leave one plate
end up on the other. Then there are always equal and opposite charges on the two
plates and the charges on the plates are much larger than the charges on the sur—
faces of the leadin wires. Finally, we assume that there are no magnetic ﬁelds
close to the capacitor. Suppose now we consider the line integral of E around a closed loop which
starts at terminal a, goes along inside the wire to the top plate of the capaCitor,
jumps across the space between the plates, passes from the lower plate to terminal
b through the wire. and returns to terminal a in the space outside the capacitor.
Since there is no magnetic ﬁeld, the line integral of E around this closed path is
zero. The integral can be broken down into three parts: fEds=/ Eds+/ Eds+ Eds. along between
Wires plates (22.7) outstde The integral along the wires is zero, because there are no electric ﬁelds inside per—
fect conductors. The integral from b to a outside the capac1tor is equal to the nega—
tive of the potential difference between the terminals. Since we imagined that the
two plates are in some way isolated from the rest of the world, the total charge on * There are people who say we should call the objects by the names “inductor” and
“capaCitor” and call their properties “inductance” and “capacitance” (by analogy With
“reSistor” and “reSistance”). We would rather use the words you Will hear in the labora
tory. Most people still say “inductance” for both the phySical c011 and its inductance L.
The word “capaCitor” seems to have caught on—although you Will still hear “condenser”
fairly often—and most people still prefer the sound of “capac1ty” to “capaCitance.” 22—3 #ﬂ; 0
1 V
_> b
l________i I
Fig. 22—2. A capacitor (or con
denser). Fig. 22—3. A resistor. I IwL l—(JE Fig. 22—4. The ideal lumped circuit elements (passive). R the two plates must be zero; if there is a charge Q on the upper plate, there is an
equal. opposite charge — Q on the lower plate. We have seen earlier that if two
conductors have equal and opposite charges, plus and minus Q, the potential
difference between the plates is equal to Q/ C, where C is called the capacity of the
two conductors. From Eq. (22.7) the potential difference between the terminals
(1 and b is equal to the potential difference between the plates. We have, therefore,
that _ 2
V _ C
The electric current I entering the capacitor through terminal a (and leaving
through terminal b) is equal to dQ/dt, the rate of change of the electric charge on the plates. Writing dV/dt as sz, we can put the voltage current relationship for
a capacitor in the following way: 2 £9
or
I
V — Ew—C (22.8)
The impedance 2 of a capacitor, is then
2 (capacitor) = 20 = (22.9)
MC The third element we want to consider is a reSistor. However, since we have
not yet discussed the electrical properties of real materials, we are no: yet ready
to talk about what happens inside a real conductor. We will just have to accept
as fact that electric ﬁelds can eXist inside real materials, that these electric ﬁelds
give rise to a ﬂow of electric charge—that is, to a current—and that this current
is proportional to the integral of the electric ﬁeld from one end of the conductor
to the other. We then imagine an ideal resistor constructed as in the diagram of
Fig. 22—3. Two wires which we take to be perfect conductors go from the terminals
a and b to the two ends of a bar of resistive material. Followmg our usual line of
argument, the potential difference between the terminals a and b is equal to the
line integral of the external electric ﬁeld, which is also equal to the line integral of
the electric ﬁeld through the bar of resistive material. It then follows that the cur
rent I through the resistor is proportional to the terminal voltage V: where R is called the res1stance. We Will see later that the relation between the
current and the voltage for real conducting materials is only approximately linear.
We Will also see that this approximate proportionality is expected to be independent
of the frequency of variation of the current and voltage only if the frequency 18
not too high. For alternating currents then, the voltage across a resistor is in phase
With the current, which means that the impedance is a real number. 2 (reSistance) = 2,, = R. (22.10) Our results for the three lumped Circu1t elements—the inductor, the capaCitor,
and the reSistor—are summarized in Fig. 22—4. In this ﬁgure, as well as in the
preceding ones, we have indicated the voltage by an arrow that is directed from one
terminal to another. If the voltage is “pOSitive”——that is, if the terminal a 18 at a
higher potential than the terminal b—the arrow indicates the direction of a positive
“voltage drop.” Although we are talking about alternating currents, we can of course include
the special case of circuits with steady currents by taking the limit as the frequency
w goes to zero. For zero frequency—that is, for DC—the impedance of an induc
lance goes to zero; it becomes a short circuit. For DC, the impedance of a condenser 224 goes to inﬁnity; it becomes an open circuit. Since the impedance of a resistor is
independent of frequency, it IS the only element left when we analyze a circuit
for DC. In the Circu1t elements we have described so far, the current and voltage are
proportional to each other. If one is zero, so also is the other. We usually think in
terms like these: An applied voltage is “responsible” for the current, or a current
“gives rise to” a voltage across the terminals; so in a sense the elements “respond”
to the “applied” external conditions. For this reason these elements are called
passive elements. They can thus be contrasted with the active elements, such as
the generators we will consider in the next section, which are the sources of the
oscillating currents or voltages in a Circuit. 22—2 Generators Now we want to talk about an active circuit element—one that is a source of
the currents and voltages in a circuit—namely, a generator. Suppose that we have a cod like an inductance except that it has very few
turns, so that we may neglect the magnetic ﬁeld of its own current. This coil,
however, sits in a changing magnetic ﬁeld such as might be produced by a rotating
magnet, as sketched in Fig. 22—5. (We have seen earlier that such a rotating mag
netic ﬁeld can also be produced by a su1table set of COIiS with alternating currents.)
Again we must make several simplifying assumptions. The assumptions we Will
make are all the ones that we described for the case of the inductance. In particular,
we assume that the varying magnetic ﬁeld is restricted to a deﬁnite region in the
vicinity of the cm] and does not appear outside the generator in the space between
the terminals. Following closely the analysis we made for the inductance, we consider the
line integral of E around a complete loop that starts at terminal a, goes through the
coil to terminal b and returns to its starting point in the space between the two
terminals. Again we conclude that the potential difference between the terminals
is equal to the total line integral of E around the loop: V: —3§Eds. This line integral is equal to the emf in the circuit, so the potential difference V
across the terminals of the generator is also equal to the rate of change of the mag
netic ﬂux linking the coil: d
V = —a = it (ﬂux). (22.11) For an ideal generator we assume that the magnetic ﬂux linking the coil is deter
mined by external conditions—such as the angular velocity of a rotating magnetic
ﬁeld—and is not inﬂuenced in any way by the currents through the generator.
Thus a generator—at least the ideal generator we are cons1dering—is not an
impedance. The potential difference across its terminals is determined by the
arbitrarily assigned electromotive force 8(1). Such an ideal generator 15 represented
by the symbol shown in Fig. 22—6. The little arrow represents the direction of the
emf when it is positive. A pOSitive emfin the generator of Fig. 22—6 will produce
a voltage V = 8, with the terminal a at a higher potential than the terminal b.
There is another way to make a generator which is quite different on the
inside b1} which is indistinguishable from the one we have just described insofar
as what happens beyond its terminals. Suppose we have a coil of Wire which
is rotated in a ﬁxed magnetic ﬁeld, as indicated in Fig. 22—7. We show a bar
magnet to indicate the presence of a magnetic ﬁeld; it could, of course, be replaced
by any other source of a steady magnetic ﬁeld, such as an additional coil carrying
a steady current. As shown in the ﬁgure, connections from the rotating coil are
made to the outside world by means of sliding contacts or “slip rings.” Again,
we are interested in the potential difference that appears across the two terminals 22—5 Fig. 22—5.
0 ﬁxed coil and a rotating magnetic field. 0
O C". Fig. 22—6. erator. A generator consisting of \
/ $ymbo for an ideal gen Fig. 22—7. A generator consisting of
a coil rotating in CI ﬁxed magnetic ﬁeld. a and b, which is of course the integral of the electric ﬁeld from terminal a to ter
minal b along a path outside the generator. Now in the system of Fig. 22—7 there are no changing magnetic ﬁelds, so we
might at ﬁrst wonder how any voltage could appear at the generator terminals
In fact, there are no electric ﬁelds anywhere inside the generator. We are, as usual,
assuming for our ideal elements that the wires inside are made of a perfectly con—
ducting material, and as we have said many times, the electric ﬁeld inside a perfect
conductor is equal to zero. But that is not true. It is not true when a conductor
is moving in a magnetic ﬁeld. The true statement is that the total force on any
charge inside a perfect conductor must be zero. OtherWise there would be an
inﬁnite ﬂow ofthe free charges. So what is always true is that the sum of the electric
ﬁeld E and the cross product of the veloc1ty of the conductor and the magnetic
ﬁeld B—which is the total force on a unit charge—~must have the value zero
in51de the conductor: = E + v X B = 0 (in aperfect conductor), (22.12) where v represents the velocity of the conductor. Our earlier statement that there
is no electric ﬁeld ins1de a perfect conductor is all right if the veloc1ty v of the
conductor is zero; otherwise the correct statement is given by Eq. (22.12). Returning to our generator of Fig. 22—7, we now see that the line integral of
the electric ﬁeld E from terminal a to terminal b through the conducting path of
the generator must be equal to the line integral of v X B on the same path, b b
f Eds = —/ (v x B)ds. (22.13)
insiiie insildc
conductor conductor It is still true, h0wever, that the line integral of E around a complete loop, including
the return from b to a outside the generator, must be zero, because there are no
changing magnetic ﬁelds. So the ﬁrst integral in Eq. (22.13) is also equal to V,
the voltage between the two terminals. It turns out that the righthand integral
of Eq. (22 13) is just the rate of change of the ﬂux linkage through the c011 and is
therefore—by the ﬂux rule—equal to the emf in the coil. So we have again that
the potential dilTerence across the terminals is equal to the electromotive force in
the circuit, in agreement With Eq. (22.11). So whether we have a generator in which
a magnetic ﬁeld changes near a ﬁxed coil, or one in which a c011 moves in a ﬁxed
magnetic ﬁeld, the external properties of the generators are the same. There is a
voltage difference V across the terminals, which is independent of the current in
the Circuit but depends only on the arbitrarily assigned conditions inside the
generator. So long as we are trying to understand the operation of generators from the
point of view of Maxwell’s equations, we might also ask about the ordinary chemi
cal cell, like a ﬂashlight battery It is also a generator, Le, a voltage source, al
though it will of course only appear in DC Circuits. The simplest kind of cell to
understand is shown in Fig. 22—8. We imagine two metal plates immersed in some 226 chemical solution. We suppose that the solution contains positive and negative
ions. We suppose also that one kind of ion, say the negative, is much heavier than
the one of opposite polarity, so that its motion through the solution by the process
of diffusion is much slower. We suppose next that by some means or other it is
arranged that the concentration of the solution is made to vary from one part of
the liquid to the other, so that the number of ions of both polarities near, say, the
lower plate is much larger than the concentration of ions near the upper plate.
Because of their rapid mobility the positive ions will drift more readily into the
region of lower concentration, so that there will be a slight excess of positive charge
arriving at the upper plate. The upper plate wrll become positively charged and
the lower plate will have a net negative charge. As more and more charges diffuse to the upper plate. the potential of this plate
will rise until the resulting electric ﬁeld between the plates produces forces on the
ions which just compensate for their excess mobility, so the two plates of the cell
quickly reach a potential difference which is characteristic of the internal con
struction. Arguing just as we did for the ideal capacitor, we see that the potential differ
ence between the terminals a and b is just equal to the line integral of the electric
ﬁeld between the two plates when there is no longer any net diffusion of the ions.
There IS, of course, an essential difference between a capacitor and such a chemical
cell. If we shortcircuit the terminals of a condenser for a moment, the capacitor
is discharged and there is no longer any potential difference across the terminals.
In the case of the chemical cell a current can be drawn from the terminals con
tinuously without any change in the emf—until, of course, the chemicals inside
the cell have been used up. In a real cell it is found that the potential difference
across the terminals decreases as the current drawn from the cell increases. In
keeping with the abstractions we have been making, however, we may imagine an
ideal cell in which the voltage across the terminals is independent of the current.
A real cell can then be looked at as an ideal cell in series with a resistor. 22—3 Networks of ideal elements; Kirchhoff ’5 rules As we have seen in the last section, the description of an ideal circuit element
in terms of what happens outside the element is quite simple. The current and
the voltage are linearly related. But what is actually happening insrde the element
is quite complicated, and it is quite difﬁcult to give a precise description in terms of
Maxwell’s equations. Imagine trying to give a precise description of the electric
and magnetic ﬁelds of the inside of a radio which contains hundreds of resistors,
capacitors, and inductors. It would be an impossible task to analyze such a thing
by using Maxwell’s equations. But by making the many approxrmations we have
described in Section 22—2 and summarizing the essential features of the real
circuit elements in terms of idealizations, it becomes possible to analyze an elec
trical circuit in a relatively straightforward way. We will now show how that
is done. Suppose we have a circuit consisting of a generator and several impedances
connected together, as shown in Fig. 22—9. According to our approximations there
is no magnetic ﬁeld in the region outside the individual circuit elements. Therefore
the line integral of E around any curve which does not pass through any of the
elements IS zero. Consider then the curve I‘ shown by the broken line which goes
all the way around the circuit in Fig. 22—9. The line integral of E around this curve
is made up of several pieces. Each piece is the line integral from one terminal of a
circuit element to the other. This line integral we have called the voltage drop
across the Circuit element. The complete line integral is then just the sum of the
voltage drops across all of the elements in the circuit: fads: EV”. Since the line integral is zero, we have that the sum of the potential differences
22—7 Fig. 22—8. A chemical cell. Fig. 22—9. The sum of the voltage
drops around any closed path is zero. around a complete loop of a circuit is equal to zero: 2 V,. = 0. (22.14) around
any loop
This result follows from one of Maxwell’s equations—that in a region where there
are no magnetic ﬁelds the line integral of E around any complete loop is zero.
Suppose we con51der now a circuit like that shown in Fig. 22—10. The hori
zontal line Joining the terminals a, b, c, and d is intended to show that these ter
minals are all connected, or that they are joined by wires of negligible resistance.
In any case, the drawing means that terminals a, b, c, and d are all at the same
potential and, similarly, that the terminals e,f, g, and h are also at one common
potential. Then the voltage drop V across each of the four elements is the same.
Now one of our idealizations has been that negligible electrical charges ac
cumulate on the terminals of the impedances. We now assume further that any
electrical charges on the w1reslom1ng terminals can also be neglected. Then the
conservation of charge requires that any charge which leaves one Circuit element
immediately enters some other circu1t element. Or, what is the same thing, we
require that the algebraic sum of the currents which enter any given Junction must
be zero. By a junction, of course, we mean any set of terminals such as a, b, c,
and d which are connected. Such a set of connected terminals is usually called a
“node.” The conservation ofcharge then requ1res that for the Circuit of Fig. 22~10, Fig. 22—10. The sum of the currents
into any node is zero. 11 ~ 12 — 13 — 14 = 0. (22.15) The sum of the currents entering the node which consists of the four terminals
e,f, g, and h must also be zero: —11 + 12 + 13 + [4 : 0. (22.16) This is, of course, the same as Eq. (22.15). The two equations are not independent.
The general rule IS that the sum of the currents into any node must be zero. 2 1,, = 0. (22.17) into
a node Our earlier conclusion that the sum of the voltage drops around a closed loop
15 zero must apply to any loop in a complicated Circuit. Also, our result that the
sum of the currents into a node is zero must be true for any node. These two equa
tions are known as Kirchhoﬂ’s rules. With these two rules it is possible to solve for
the currents and voltages in any network whatever. Suppose we consider the more complicated Circuit of Fig. 22—11. How shall
we ﬁnd the currents and voltages in this Circuit? We can ﬁnd them in the followmg
straightforward way. We con31der separately each of the four subsidiary closed
loops which appear in the Circuit. (For instance, one loop goes from terminal a to
terminal b to terminal e to terminal dand back to terminal a.) For each of the loops
we write the equation for the ﬁrst of Kirchhoﬁ’s rules——that the sum of the voltages
around each loop is equal to zero. We must remember to count the voltage drop Fig. 22—1]. Analyzing a circuit with as positive if we are going in the direction of the current and negative if we are
Kirchhoff's rules. going across an element in the direction opposite to the current; and we must
remember that the voltage drop across a generator is the negative of the emf in
that direction. Thus if we consider the small loop that starts and ends at terminal
a we have the equation 2111 + 2313 + Z414 ‘ 51 = 0 Applying the same rule to the remaining loops, we would get three more equations
of the same kind. Next, we must write the current equation for each of the nodes in the Circuit.
For example, summing the currents into the node at terminal b gives the equation [1 —‘ [3 — 12 = 0.
22—8 Similarly, for the node labeled e we would have the current equation
I3—I4+18—I5=0. For the circuit shown there are ﬁve such current equations. It turns out, however,
that any one of these equations can be derived from the other four; there are,
therefore, only four independent current equations. We thus have a total of eight
independent, linear equations: the four voltage equations and the four current
equations. With these eight equations we can solve for the eight unknown currents.
Once the currents are known the c1rcuit is solved. The voltage drop across any
element is given by the current through that element times its impedance (or, in
the case of the voltage sources, it is already known). We have seen that when we write the current equations, we get one equation
which is not independent of the others. Generally it is also possible to write down
too many voltage equations. For example, in the circuit of Fig. 22—11, although
we have considered only the four small loops, there are a large number of other
loops for which we could write the voltage equation. There IS, for example, the
loop along the path abcfeda. There is another loop which follows the path
abcfehgda. You can see that there are many loops. In analyzing complicated cir
cuits it is very easy to get too many equations. There are rules which tell us how to
proceed so that only the minimum number of equations is written down, but
usually with a little thought it is possible to see how to get the right number of
equations in the simplest form. Besides, writing an extra equation or two doesn’t
do any harm. They will not lead to any wrong answers, only perhaps a little
unnecessary algebra. In Chapter 25 of Vol. I we showed that if the two impedances 21 and 22 are
in series, they are equivalent to a single impedance 2, given by 23 = 21 + 22. (22.18) We also showed that if the two impedances are connected in parallel, they are
equivalent to the single impedance 2,, given by l 2122 _ (1/2,) + (1/22) = z, + 22 (22.19) 222 If you look back you will see that in deriving these results we were in effect making
use of Kirchhoff ’5 rules. It is often possible to analyze a complicated circuit by
repeated application of the formulas for series and parallel impedances. For in
stance, the circult of Fig. 22—12 can be analyzed that way. First, the impedances
24 and 25 can be replaced by their parallel equivalent, and so also can 20 and Z7.
Then the impedance 22 can be combined with the parallel equivalent of 26 and Z7
by the series rule. Proceeding in this way, the whole circuit can be reduced to a
generator in series with a single impedance Z. The current through the generator
is then just 8/2. Then by working backward one can solve for the currents in
each of the impedances. There are, however, quite simple circuits which cannot be analyzed by this
method, as for example the circuit of Fig. 22—13. To analyze this circuit we must 22—9 Fig. 22—13.
analyzed in terms of series and parallel
combinations. A circuit which can be Fig. 22—12.
analyzed in terms of series and parallel
combinations. A circuit that cannot be Fig. 22—14. A bridge circuit. (b) 6 Zeff. Fig. 22—l5. Any twoterminal net
work of passive elements is equivalent to
an effective impedance. write down the current and voltage equations from Kirchhoff ’5 rules. Let’s do it.
There 15 just one current equation: [I + 12 + 13 = 0’
so we know immediately that
13 = _(II + 12) We can save ourselves some algebra if we immediately make use of this result in
writing the voltage equations. For this Circuit there are two independent voltage
equations; they are
—81 + 1222 — [121 = 0
and
82 * (11 + I2)Za — 1222 = 0 There are two equations and two unknown currents. Solving these equations for
11 and 12, we get L282 — (22 + 2:081 I =
1 21(22 + Z3) + Z223 (22.20) and 2182 + 2381 1 _.
2 21(22 + Z3) + 2223 (22.21) The third current is obtained from the sum of these two. Another example of a circuit that cannot be analyzed by using the rules for
series and parallel impedance is shown in Fig. 22—14. Such a circuit is called a
“bridge.” It appears in many instruments used for measuring impedances. With
such a circuit one is usually interested in the question: How must the various
impedances be related if the current through the impedance 23 is to be zero? We
leave it for you to ﬁnd the conditions for which this is so. 22—4 Equivalent circuits Suppose we connect a generator 8 to a circuit containing some complicated
interconnection of impedances, as indicated schematically in Fig. 22—15(a). All
of the equations we get from Kirchhoff ’3 rules are linear, so when we solve them
for the current I through the generator, we will get that I is proportional to 8.
We can write [:83 Zeff where now 2,.“ is some complex number, an algebraic function of all the elements
in the circuit. (If the Circuit contains no generators other than the one shown, there
is no additional term independent of 8.) But this equation is Just what we would
write for the circuit of Fig. 22—15(b). So long as we are interested only in what
happens to the left of the two terminals a and b, the two Circu1ts of Fig. 22—15 are
equzvalent. We can, therefore, make the general statement that any twoterminal
network of passive elements can be replaced by a Single impedance 2,.“ without
changing the currents and voltages in the rest of the Circu1t. This statement is, of
course, just a remark about what comes out of Kirchhoff ’s rules—and ultimately
from the linearity of Maxwell’s equations. The idea can be generalized to a circuit that contains generators as well as
impedances. Suppose we look at such a Circuit “from the point of View” of one of
the impedances, which we will call Zn, as in Fig. 22~16(a). If we were to solve the
equation for the whole circuit, we would ﬁnd that the voltage V" between the two
terminals a and b is a linear function of I, which we can write where A and B depend on the generators and impedances in the circuit to the left
22—10 of the terminals. For instance, for the circuit of Fig. 22—13, we ﬁnd V] = 1121.
This can be written (by rearranging Eq. (22.20)] as Z223 V1=[(Eﬁ73)52 ‘ 8I] ‘ The complete solution is then obtained by combining this equation with the one for the impedance 21, namely, V1 = [121, or in the general case, by combining
Eq. (22.22) with (22.23) Vn = Inz" . 1f now we consider that 2,, is attached to a simple series circuit of a generatoi
and a current, as in Fig. 22—15(b), the equation corresponding to Eq. (22.22) is Vn = 80“ — Inzeffs which is identical to Eq. (22.22) provided we set 8L.“ 2 A and 2m 2 B. So if we
are interested only in what happens to the right of the terminals a and b. the arbi
trary circuit of Fig. 22—16 can always be replaced by an equwalent combination of
a generator in series with an impedance. 22—5 Energy We have seen that to build up the current I in an inductance, the energy = lL12 must be provided by the external circuit. When the current falls back to zero, this energy is delivered back to the external circuit. There is no energyloss mechanism in an ideal inductance. When there is an alternating current through an inductance, energy ﬂows back and forth between it and the rest of the ClI'CUlI, but the average rate at which energy is delivered to the circuit is zero. We say that an inductance is a nondissipative element; no electrical energy is dissipated—that is,
“lost”——in it. Similarly, the energy of a condenser, U : lCVz, is returned to the external
circuit when a condenser is discharged. When a condenser is in an AC circuit
energy ﬂows in and out of it, but the net energy ﬂow in each cycle is zero. An ideal
condenser is also a nondissipative element. We know that an emf is a source of energy. When a current I ﬂows in the
direction of the emf, energy is delivered to the external circuit at the rate dU/dt =
81. If current is driven against the emf—by other generators in the circuit—the
emf will absorb energy at the rate SI; since [15 negative, dU/dt will also be negative. If a generator is connected to a resistor R. the current through the resistor
is I : S/R. The energy being supplied by the generator at the rate 8115 being
absorbed by the resistor. This energy goes into heat in the reSistor and is lost
from the electrical energy of the cirCuit. We say that electrical energy is dissipated
in a reSistor. The rate at which energy is dissipated in a resistor is dU/dt 2 R12. In an AC circuit the average rate of energy lost to a master is the average of
R12 over one cycle. Since I = fem—by which we really mean that I varies as
cos cot—the average of 12 over one cycle is [IF/2, since the peak current is Il and
the average of cos2 cat is 1/2. What about the energy loss when a generator is connected to an arbitrary
impedance 2? (By “loss” we mean, of course, conversion of electrical energy into
thermal energy.) Any impedance 2 can be written as the sum of its real and im
ginary parts. That is, Z = R + ix, (2224) where R and Xare real numbers. From the point of view of equivalent Circuits we
can say that any impedance is equivalent to a resistance in series with a pure
imaginary impedance—called a reactance—as shown in Fig. 22—17. We have seen earlier that any circuit that contains only L’s and Cs has an
impedance that IS a pure imaginary number. Since there is no energy loss into any
of the L’s and Cs on the average, a pure reactance containing only L’s and C’s
Will have no energy loss. We can see that this must be true in general for a reactance. 22—11 In
a —>
Zeff
(M Z"
b Fig. 22—16. Any twoterminal net
work can be replaced by a generator in
series With an impedance. Fig. 22—17. Any impedance is equiv—
alent to a series combination of a pure
resistance and a pure reactance. 23 23 Zl+22 ﬂ '—
= = a
D.____l ‘2—
 _ l I _
‘2; — 72 + 2: ZS —Zl + 24
Fig. 22—18. The effective impedance of a ladder. Fig. 22—19.
22—12 If a generator with the emf 8 is connected to the impedance z of Fig. 22—17,
the emf must be related to the current I from the generator by a = I(R + iX). (22.25) To ﬁnd the average rate at which energy is delivered, we want the average of the
product 81. Now we must be careful. When dealing with such products, we must
deal with the real quantities 8(1) and I(t). (The real parts of the complex functions
Will represent the actual phySical quantities only when we have linear equations;
now we are concerned with products, which are certainly not linear.) Suppose we choose our origin of I so that the amplitude I is a real number,
let’s say I 0; then the actual time variation I is given by I = 10 cos wt.
The emf of Eq. (22.25) is the real part of IOeWR + iX)
OF a = 10R cos wt — IoXsin wt. (22.26) The two terms in Eq. (22.26) represent the voltage drops across R and X
in Fig. 22—17. We see that the voltage drop across the resistance is In phase with
the current, while the voltage drop across the purely reactive part 15 out of phase
with the current. The average rate of energy loss, (PM, from the generator is the integral of
the product 81 over one cycle divided by the period T; in other words, T T T
(P)uv = 1/ 81611 = 1/ 13R cos2 wzdl — Ichos wt sin wt dz.
0 T 0 T 0
The ﬁrst integral is %I§R, and the second integral is zero. So the average
energy loss in an impedance z = R + iX depends only on the real part of z,
and is 1312/2, which is in agreement with our earlier result for the energy loss in a
resistor. There IS no energy loss in the reactive part. 22—6 A ladder network We would like now to conSider an interesting circuit which can be analyzed
in terms of series and parallel combinations. Suppose we start with the circuit of
Fig. 22—18(a). We can see right away that the impedance from terminal a to ter
minal b is simply 21 + 22. Now let’s take a little harder Circuit, the one shown in
Fig. 22—18(b). We could analyze this c1rcu1t using Kirchhoﬁ‘s rules, but it is
also easy to handle with series and parallel combinations. We can replace the
two impedances on the righthand end by a single impedance Z3 2 21 —+— 22, as
in part (c) of the ﬁgure. Then the two impedances 22 and 23 can be replaced by
their equivalent parallel impedance Z4, as shown in part (d) of the ﬁgure. Finally,
21 and Z4 are equivalent to a Single impedance Z5, as shown in part (e). Now we may ask an amusing question: What would happen if in the network
of Fig. 22—18(b) we kept on adding more sections forever—as we indicate by the
dashed lines in Fig. 22—l9(a)? Can we solve such an inﬁnite network? Well, that’s Cl C O M The effective impedance of an inﬁnite ladder. not so hard. First, we notice that such an inﬁnite network is unchanged if we add
one more section at the “front” end. Surely, if we add one more section to an
inﬁnite network it is still the same inﬁnite network. Suppose we call the impedance
between the two terminals 0 and b of the inﬁnite network 20; then the impedance of
all the stuﬁ" to the right of the two terminals 6 and dis also 20. Therefore, so far as
the front end is concerned, we can represent the network as shown in Fig. 22—19(b).
Combining the parallel combinations 2220 and adding the result in series with 21,
we can immediately write down the impedance of this combination:
1 2220
z = 21 + gm or z = —«—  (1/22) + (1/20) But this impedance is also equal to 20, so we have the equation Z220
Z2+Zo 20 = 321 + Veg/4) +7122. 20 = 21 +
We can solve for zo to get
(22.27) So we have found the solution for the impedance of an inﬁnite ladder of repeated
series and parallel impedances. The impedance 20 is called the characteristic
impedance of such an inﬁnite network. Let’s now consider a speCiﬁc example in which the series element is an in
ductance L and the shunt element is a capacitance C, as sh0wn in Fig. 22—20(a).
In this case we ﬁnd the impedance of the inﬁnite network by setting 21 = icoL
and 22 = l/le. Notice that the ﬁrst term, 21/2, in Eq. (22.27) is just onehalf
the impedance of the ﬁrst element. It would therefore seem more natural, or at
least somewhat simpler, if we were to draw our inﬁnite network as shown in Fig.
22—20(b). Looking at the inﬁnite network from the terminal a’ we would see the
characteristic impedance 2'0 = V(L/C)  (MU/4) Now there are two interesting cases, depending on the frequency w. If (.02 is less
than 4/LC, the second term in the radical will be smaller than the ﬁrst, and the
impedance 20 Will be a real number. On the other hand, if (02 is greater than
4/LC the impedance 20 Will be a pure imaginary number which we can write as (22.28) 20 = i\/(w2L2/4) — (L/C) We have said earlier that a circuit which contains only imaginary impedances,
such as inductances and capac1tances, will have an impedance which is purely
imaginary. How can it be then that for the circuit we are now studying—which has
only Us and C ’s—the impedance is a pure resistance for frequencies below \/ 4/LC?
For higher frequencies the impedance is purely imaginary, in agreement with our
earlier statement. For lower frequencies the impedance is a pure resistance and
will therefore absorb energy. But how can the circuit continuously absorb energy,
as a resistance does, if it is made only of inductances and capacitances? Answer:
Because there is an inﬁnite number of inductances and capacitances, so that when
a source is connected to the circuit, it supplies energy to the ﬁrst inductance and
capacitance, then to the second, to the third, and so on. In a circuit of this kind,
energy is continually absorbed from the generator at a constant rate and ﬂows
constantly out into the network, supplying energy which is stored in the induc
tances and capacitances down the line. This idea suggests an interesting point about what is happening in the circuit.
We would expect that if we connect a source to the front end, the eﬁects of this
source Will be propagated through the network toward the inﬁnite end. The
propagation of the waves down the line is much like the radiation from an antenna
which absorbs energy from its driving source; that is, we expect such a propagation
to occur when the impedance is real, which occurs if to is less than x/4/LC. But
when the impedance is purely imaginary, which happens for to greater than V4/LC,
we would not expect to see any such propagation. 22—1 3 n. . T . I . I _
Fig. 22—20. An LC ladder drawn
in two equivalent ways. 22—7 Filters We saw in the last section that the inﬁnite ladder network of Fig. 22—20 absorbs
energy continuously if it is driven at a frequency below a certain critical frequency
\/4/LC, which we will call the cutoff frequency wo. We suggested that this effect
could be understood in terms of a continuous transport of energy down the line.
On the other hand, at high frequencies, for w > wo, there is no continuous ab
sorption of energy; we should then expect that perhaps the currents don’t “pene
trate” very far down the line. Let’s see whether these ideas are right. Suppose we have the front end of the ladder connected to some AC generator
and we ask what the voltage looks like at, say, the 754th section of the ladder.
Since the network is inﬁnite, whatever happens to the voltage from one section to
the next 15 always the same; so let’s just look at what happens when we go from
some section, say the nth to the next. We will deﬁne the currents In and voltages
V” as shown in Fig. 22—21(a). (b) Fig. 22—2]. Finding the propagation factor of a ladder. We can get the voltage VH1 from V” by remembering that we can always
replace the rest of the ladder after the nth section by its characteristic impedance 20;
then we need only analyze the circuit of Fig. 22—21(b). First, we notice that any
V”, since it is across 20, must equal Inzo. Also, the difference between V,, and VH+1
is just Inzlz 21
VII _ Vn+1 = 111.21 = V11 So we get the ratio =1_ 3 , 51:3 Vn Z (I Z 0 We can call this ratio the propagation factor for one section of the ladder; we’ll
call it a. It IS, of course, the same for all sections: 01 = mﬂ—u (22.29) The voltage after the nth section 15 then
V,, = MS. (22.30) You can now ﬁnd the voltage after 754 sections: it is Just a to the 754th p0wer
times 8. Suppose we see what a 18 like for the LC ladder of Fig. 22—20(11) Using 2‘,
from Eq. (22.27), and 2‘ : iwL, we get
t«/rL’/‘c"3“:“(717i43 — «wt/2)
a : AiAﬁ—ii 7 _77‘ o , WL/C) — (WU/4) + 1(wL/2)
If the driving frequency is below the cutoff frequency w“ = V1711“, the radical is a real number, and the magnitudes of the complex numbers in the numerator
and denominator are equal. Therefore, the magnitude of a is one; we can write (22.31) a = 6'5, which means that the magnitude of the voltage is the same at every section. only
22—14 its phase changes. The phase change 6 is, in fact, a negative number and represents
the “delay” of the voltage as it passes along the network. For frequencies above the cutoff frequency w“ it is better to factor out an 1
from the numerator and denominator of Eq. (22 3]) and rewrite it as a * yawn/4) — (L/C) — (wt/2y
«(TEE/4) — (L75) + (wL/z) (22 32) The propagation factor a is now a real number, and a number less than one. That
means that the voltage at any section is always less than the voltage at the pre
ceding sectioii by the factor a. For any frequency above can, the voltage dies
away rapidly as we go along the network. A plot of the absolute value of a as a
function of frequency looks like the graph in Fig. 22—22. We see that the behavior of a, both above and below coo, agrees with our
interpretation that the network propagates energy for w < w“ and blocks it for
w > to”. We say that the network “passes” low frequenCies and “reJects” oi
“ﬁlters out” the hiin frequencies. Any network deSigned to have its characteristics
vary in a prescribed way with frequency is called a “ﬁlter.” We have been analyzmg
a “lowpass ﬁlter.” You may be wondering why all this discussion of an inﬁnite network which
obviously cannot actually occur. The pomt is that the same characteristics are
found in a ﬁnite network if we ﬁnish it off at the end with an impedence equal to
the characteristic impedence 20. Now in practice it is not poss1ble to exactly
reproduce the characteristic impedance With a few Simple elements—like R’s.
L’s, and C’s. But it is often possible to do so With a fair approximation for a certain
range of frequenCIes. In this way one can make a ﬁnite ﬁlter network Whose
properties are very nearly the same as those for the infinite case For instance, the
LC ladder behaves much as we have described it if it is terminated in the pure
resistance R = VTfC. If in our LC ladder we interchange the positions of the L’s and C’s, to make
the ladder shown in Fig. 22—23(21), we can have a ﬁlter that propagates lug/1 fre
quencies and I‘EJECIS low frequencies. It is easy to see what happens with this net
work by usmg the results we already have. You Will notice that Whenever we change
an L to a C and Vice versa, we also change every rm to l/iw. So Whatever happened
at to before Will now happen at l/w. In particular, we can see how a Will vary With
frequency by us1ng Fig. 22—22 and changing the label on the was to 1/0), as we
have done in Fig. 22—23(b). The lowpass and highpass ﬁlters we have described have various technical
applications. An L—C lowpass ﬁlter is often used as a “smoothing” ﬁlter in a Dc
power supply. If we want to manufacture DC power from an AC source, we begin
with a rectiﬁer which permits current to ﬂow only in one direction. From the
rectiﬁer we get a series of pulses that look like the function V(l) shown in
Fig 22—24, which is lousy DC, because it wobbles up and down. Suppose we would
like a nice pure DC, such as a battery provides. We can come close to that by
putting a lowpass ﬁlter between the rectiﬁer and the load. We know from Chapter 50 of Vol. I that the time function in Fig. 22—24 can be
represented as a superposmon of a constant voltage plus a sine wave, plus a higher
frequency sine wave, plus a still higherfrequency sine wave, etc—by a Fourier
series. If our ﬁlter is linear (if, as we have been assuming, the L’s and C’s don't
vary With the currents or voltages) then What comes out of the ﬁlter is the super
pos1tion of the outputs for each component at the input. If we arrange that the
cutoff frequency w” of our ﬁlter is well below the lowest frequency in the function
V(I), the DC (for which to = 0) goes through ﬁne, but the amplitude of the ﬁrst
harmonic Will be cut down a lot. And amplitudes of the higher harmonics Will be
cut down even more. So we can get the output as smooth as we Wish, depending
only on how many ﬁlter sections we are Willing to buy. A high—pass ﬁlter is used if one wants to reJect certain low frequenCies. For
instance, in a phonograph ampliﬁer 21 highpass ﬁlter may be used to let the mUSlC 22—15 lal
l
l

l
I
(no cu Fig 22—22. The propagation factor
of o sectiOn of an LC lodder C C C C
L L L L
(0)
WI
l
0 Hate l/w
(b)
Fig. 22—23. (a) A highpass ﬁlter; (b) its propagation factor as 0 function
of] w. UH)
OW
Fig. 22—24. The output voltage of o fullwove rectiﬁer. W) ~———_____,_ Fig. 22—25. (a) A bandpass ﬁlter.
(b) A simple resonant ﬁlter. I, 12
Ll L2
(b)
Fig. 22—26. Equivalent crrcurt of a
mutual inductance. through, while keeping out the lowpitched rumbling from the motor of the
turntable. It is also possrble to make “bandpass” ﬁlters that reject frequencies below
some frequency ml and above another frequency (4)2 (greater than wl), but pass the
frequencies between col and (02. This can be done simply by putting together a
highpass and a lowpass ﬁlter, but it is more usually done by making a ladder in
Wthh the impedances 21 and 22 are more complicated—being each a combination
of L’s and Cs. Such a bandpass ﬁlter might have a propagation constant like
that shown in Fig. 22—25(a). It might be used, for example, in separating signals
that occupy only an interval of frequencies, such as each of the many vorce channels
in a highfrequency telephone cable, or the modulated carrier of a radio trans
missron. We have seen in Chapter 25 of Vol. I that such ﬁltering can also be done using
the selectivity of an ordinary resonance curve, which we have drawn for comparison
in Fig. 22—25(b). But the resonant ﬁlter is not as good for some purposes as the
bandpass ﬁlter. You will remember (Chapter 48, Vol. I) that when a carrier of
frequency w. is modulated with a “signal” frequency cos, the total srgnal contains
not only the carrier frequency but also the two sideband frequencies to. + w.
and w. — 0),. With a resonant ﬁlter, these sidebands are always attentuated some—
what, and the attenuatlon is more, the higher the signal frequency, as you can see
from the ﬁgure. So there is a poor “frequency response.” The higher musical
tones don’t get through. But if the ﬁltering is done with a bandpass ﬁlter designed
so that the width «:2 — wl is at least twice the highest signal frequency, the fre
quency response will be “ﬂat” for the signals wanted. We want to make one more point about the ladder ﬁlter: the LC ladder of
Fig. 22—20 is also an approximate representation of a transmissron line. If we
have a long conductor that runs parallel to another conductor—such as a wire in a
coaxral cable, or a wire suspended above the earth—there will be some capacitance
between the two conductors and also some inductance due to the magnetic ﬁeld
between them. If we imagine the line as broken up into small lengths M, each
length will look like one section of the LC ladder with a series inductance AL and
a shunt capacitance AC. We can then use our results for the ladder ﬁlter. If we
take the limit as A6 goes to zero, we have a good description of the transmission
line. Notice that as A6 is made smaller and smaller. both AL and ACdecrease, but
in the same proportlon, so that the who AL/AC remains constant. So if we take
the limit of Eq. (22.28) as AL and AC go to zero, we ﬁnd that the characteristic
impedance 20 is a pure resrstance whose magnitude is x/XL/TC—I We can also
write the ratlo AL/AC as LO/CO, where L0 and C0 are the inductance and capaci
tance of a unit length of the line; then we have _ Z6 20 ——»  _ 22.
CO ( 33) You will also notice that as AL and AC go to zero, the cutoﬁ" frequency
(.00 = x/4/LC goes to inﬁnity. There is no cutoff frequency for an ideal
transmission line. 22—8 Other circuit elements We have so far deﬁned only the ideal Circuit impedances—the inductance,
the capacnance, and the resrstance—as well as the ideal voltage generator. We want
now to show that other elements, such as mutual inductances or transistors or
vacuum tubes, can be described by using only the same basrc elements. Suppose
that we have two coils and that on purpose, or otherwrse, some ﬂux from one of
the coils links the other, as shown in Fig. 22—26(a). Then the two coils Will have a
mutual inductance M such that when the current varies in one of the c0115, there
will be a voltage generated in the other. Can we take into account such an eﬁect
in our equivalent circuits? We can in the followmg way. We have seen that the 22—16 induced emf ’s in each of two interacting coils can be written as the sum of two parts: 81 :2 —'L1 g}; i 5%:
(22.34) _ dlg dll ' ‘L2 dt * 71‘ The ﬁrst term comes from the selfinductance of the coil, and the second term
comes from its mutual inductance With the other cm]. The sign of the second term
can be plus or minus, depending on the way the ﬂux from one coil links the other.
Making the same approximations we used in describing an ideal inductance, we
would say that the potential difference across the terminals of each coil is equal to
the electromotive force in the coil. Then the two equations of (22.34) are the same
as the ones we would get from the Circuit of Fig. 22—26(b), prOVided the electro
motive force in each of the two circuits shown depends on the current in the
opposite circuit according to the relations 81 = iinlz, 82 = iinII. (22.35)
So what we can do is represent the effect of the self—inductance in a normal way but
replace the effect of the mutual inductance by an auxiliary ideal voltage generator.
We must in addition, of course, have the equation that relates this emf to the
current in some other part of the circuit: but so long as this equation is linear, we
have just added more linear equations to our circuit equations, and all of our
earlier conclusions about equivalent circuits and so forth are still correct. In addition to mutual inductances there may also be mutual capacitances.
So far, when we have talked about condensers we have always imagined that there
were only two electrodes, but in many situations, for example in a vacuum tube.
there may be many electrodes close to each other. If we put an electric charge on
any one of the electrodes. its electric ﬁeld will induce charges on each of the other
electrodes and affect its potential. As an example, consider the arrangement of
four plates shown in Fig. 22—27(a). Suppose these four plates are connected to
external circuits by means of the Wires A, B, C, and D. So long as we are only
worried about electrostatic effects, the equivalent Circuit of such an arrangement
of electrodes is as shown in part (b) of the ﬁgure. The electrostatic interaction of
any electrode with each of the others is equivalent to a capaCity between the
two electrodes. Finally, let’s consider how we should represent such complicated devices as
transistors and radio tubes in an AC Circuit. We should point out at the start that
such devices are often operated in such a way that the relationship between the
currents and voltages is not at all linear. In such cases, those statements we have
made which depend on the linearity of equations are, of course, no longer correct.
On the other hand, in many applications the operating characteristics are sufﬁCiently
linear that we may consider the transistors and tubes to be linear deVices. By this
we mean that the alternating currents in, say, the plate of a vacuum tube are linearly
proportional to the voltages that appear on the other electrodes, say the grid
voltage and the plate voltage. When we have such linear relationships, we can
incorporate the device into our equivalent circuit representation. As in the case of the mutual inductance, our representation will have to include
aux1liary voltage generators which describe the inﬂuence of the voltages or currents
in one part of the device on the currents or voltages in another part. For example,
the plate Circu1t of a triode can usually be represented by a resistance in series With
an ideal voltage generator Whose source strength is proportional to the grid voltage.
We get the equivalent Circuit shown in Fig. 22—28.* Similarly, the collector circuit * The equivalent Circu1t shown is correct only for low frequenCies. For high frequencies
the equivalent Circu1t gets much more complicated and Will include various socalled
“paraSitic” capaCitances and inductances. 22—17 Fig. 22—27.
mutual capacitance. Equivalent circuit of PLATE P GRID \xo THODE C
C 5* 7.qu Fig. 22—28. A lowfrequency equiv
alent circuit of a vacuum triode. Fig. 22—29. A lowfrequency equiv A alent circuit of a transistor. EMITTER BASE of a tranSIStor IS conveniently represented as a resistor in series with an ideal
voltage generator whose source strength is proportional to the current from the
emitter to the base of the tran51stor. The equivalent Circuit is then like that in Fig.
22—29. So long as the equations which describe the operation are llnear, we can
use such representations for tubes or transistors. Then, when they are incorporated
in a complicated network, our general conclusions about the equivalent representa
tion of any arbitrary connection of elements is still valid. There is one remarkable thing about transistor and radio tube circuits which
is different from circuits containing only impedances: the real part of the elTective
impedance 2..“ can become negative. We have seen that the real part of 2 represents
the loss of energy. But it is the important characteristic of transistors and tubes
that they supply energy to the Circuit. (Of course they don’t just “make” energy;
they take energy from the DC circuits of the power supplies and convert it into
AC energy.) So it 1s possible to have a circult with a negative resistance. Such a
circuit has the property that if you connect It to an impedance With a positive real
part, i.e., a positive resistance, and arrange matters so that the sum of the two
real parts is exactly zero, then there is no dissipation in the combined circuit. If
there is no loss of energy, any alternating voltage once started will remain forever.
This is the basic idea behind the operation of an oscrllator or signal generator which
can be used as a source of alternating voltage at any desired frequency. 22—18 ...
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 Spring '09
 LeeKinohara
 Physics, Resistance, Alternating Current, Electric charge

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