bab 2 stts
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ElektronikaTRANSCRIPT
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CHAPTER 2
THE A-C STEADY STATE LINES WITH NO REFLECTION
2.1.The rotating vector.
The most convenient method of handling sinusoidally varying quantities mathematically is to
express them in terms of rotating vectors, and then to represent the vectors by means of complex
numbers. In transmission line theory it is desirable to use a somewhat more abstract notation than
is necessary for the simpler a circuits, and so we shall devote a small amount of time to examining
the mathematical basis for this method of representation.
In Fig.2.1 is shown a vector of length �� whose origin is at O and which rotates
counterclockwise with a uniform angular velocity of � rad/see. If the vector is horizontal at t = 0,
then, after a time t has gone by, the vector will make an angle Wt radians with the horizontal axis. It
is not hard to see from the geometry of the figure that the projection of the tip of the rotating
vector on any stationary straight line is a point that executes a simple harmonic motion with the
amplitude. ��From a geometrical point of view, it is easies to project on the vertical axis, and this is
commonly done in many treatments of a-c circuit theory. However, it will turn out to be more
convenient for our present purposes if we project instead on the horizontal axis. From the
trigonometry of the figure, it is evident that the projection on this axis is given by.
e = �� cos wt
The cosine function will go through one complete cycle for each revolution of the generating
vector. The period T is the amount of time during which �t changes by 2� radians; i.e., T= 2�/� sec.
The frequency, or number of cycles per second, is the reciprocal of this, or
f = ��
The quantity � is often called the angular frequency to distinguish it from f. Next, we examine
the properties of the complex number ��, where it is a real number and j =√�1. Euler’s formula
states that.
2
�� = cos � + j sin �
(2.2)
Therefore, �� is a complex number with the real part cos �
and the imaginary part sin �. Now refer to Fig. 2.2, which shows
a vector of unit length drawn on the complex plane so that it
makes an angle � with the real axis. Obviously the real and
imaginary components of this vector are cos � and sin �,
respectively, which are precisely the components of ��.
Therefore, we now have a geometrie representation for �� ; it
is a unit vector on the complex plane and makes an angle �
with the real axis. In the symbolic notation sometimes used,
�� = 1/�
If we now let the angle a increase uniformly with time, we shall haven unit rotating vector. If the
vector starts in the horizontal position at t = 0 and rotates with an angular velocity �(radians per
second), its angle at any moment will be� = � t radians. The unit rotating vector will be expressed
algebraically as.
��� = cos �� + j sin ��
(2.3)
The projection of the tip of this rotating vector on the horizontal axis (the axis of reals) is the
function cos wt, as can be seen from the diagram. An equivalent statement is that the real
component of ���� is cos wt, and, looking back at Eq. (2.3), we see that this is indeed true. The
shorthand notation that is frequently used is the following:
Cos �t = Re[���]
(2.4)
Where the symbols Re is read “the real part of.” The imaginary part of ���� is sin wt, and this
would do quite as well in our analysis, but, algebraically, it seems more natural to take the real part.
This is why we preferred at the beginning of this treatment to project our rotating vectors on the
horizontal axis.
Now, we can represent the rotating vector of Fig.2.1a algebraically in terms of complex numbers
as ������ , and its projection on the real axis [compare Eq. (2.1)] as
e = Re[�����]
(2.5)
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There are several reasons for following this curious procedure. First, it is much more convenient
to represent a sinusoidally varying quantity by a vector (or by its associated complex number) than
by the trigonometric expression. The advantage becomes more apparent as the number of
sinusoidal quantities (perhaps currents and voltage) in a given problem becomes greater. A second
reason is the mathematical simplification introduced in taking derivatives of the exponential does
not change its mathematical form when differentiated. For example, the derivative with respect to
time of Eq. (2.1) is
���� = -��� sin �t
(2.6)
On the other hand, the derivative of the exponential form (2.5) is
���� = Re[�������]
(2.7)
Which represents a vector of starts in the vertical position at t = 0. It is not hard to see that the
projection of this rotating vector on the horizontal axis is an negative sine function with an
amplitude ��� , which checks with Eq. (2,6).
To show algebraically that the two expressions are identical, we apply Euler’s formula to Eq (2.7)
and write.
���� = Re[�����cos �� ! � "#$ ��%]
= Re[������ cos �� � "#$ ��%]
= - ��� "#$ ��
It is common practice to omit the symbol re and to write simply.
e = �����
(2.8)
And
���� = j������
(2.9)
However, when this is done, one must not lose sight of the fact that the instantaneous variation
of e is given by the projection of the rotating vector on the axis of reals, a fact that is not indicated
explicitly in Eqs. (2,8) and (2,9).
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A sinusoidally varying quantity e, which leads the foregoing one by an angle a can be expressed
as.
�� = ��� cos (�� ! &%
(2.10)
or as
�� = Re[�������' (%] = Re[����(���]
(2.11)
or simply as
�� = ����(���
(2.12) The complex quantity ����( = ���/& represents the vector in its position at t =0.
To illustrate an elementary application of the foregoing method, consider a circuit containing an
inductance L and a resistance R in series. An emf e = �� cos �t is appied to the circuit and we wish
to find the steady state alternating current that results. The voltage drop across the inductance is L
di/dt and that across the resistance is Ri. The sum of the two is equal to the impressed voltage:
L �)�� + Ri = �� *+" ��
(2.13)
Let us write �� cos �t as Re (�����) and then omit the symbol Re. Similarly, weshall write the
current as I = Re (,����) where I m is a complex, or vector current whose length is equal to the
maximum value of the current i. Making these substitutions into Eq. (2.13) and taking the indicated
derivative, we obtain the relation.
(j�- ! .%,���� = �� ���
(2.14)
Canceling ��� on both sides and solving for the complex current ,�, we obtain
,� = /01'��2
(2.15)
or
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,� = /0√13' �323 /- �4$56 �-/.
(2.16)
The quantity R + jwl is, of course, the complex impedance of the circuit.Equation (2.16) gives the
current vector in its position at t = 0. Usually, the algera stops at this point, for we have foundthe
essential information, namely, the magnitude of the current and its phase angle with respect to the
voltage. But wi should not lose sight of the fact that the instantaneous current in the circuit is not
given directly by Eq. (2.16). Instead, we must imagine the vector to torate at the angular speed w
and take its projection. Algebraically this is done by writing
i = Re[,����]
(2.17)
Which, after substituting from Eq. (2.16), expanding, and taking the real part, becomes
I = /0√13' �323 cos 7�� � �4$56 �21 8
(2.18)
In all the development w far, we have referred to the maximum value of the sinusoidal wave,
Em. For a sinusoid, the effective value E is related to the maximum value by Em = V2E. If both sides
of Eq. (2.10) are divided by v2, the result will be expressed in terms of the effective value of current
and voltage:
I = /01'��2 =
/√13' �323 /- �4$56 �-/.
(2.19)
This contains the same essential information as the previous expression, namely, the magnitude
of the current (now in rms amperes) and its phase angle with respect to the voltage. The equations
of circuit theory are usually written in this way because of the convenience of using rms values.
Again, however, a word of caution is in order. If we wish to obtain a true picture og the
instantaneous current in the circuit, it is now necessary to multiply the expression (2.19) by v2
before we rotate the vector and take its projection, or, algebraically speaking, before we multiply by
e wt and take the real part. For a further discussion of this subject, the reader should refer to the
literature1.
2.2. A.C Steady-state Solution for the Uniform Line.
1 See, for example, R. H. Frazier, “Elementary Electric Circuit Theory” chap. IV, McGraw Hill company, inc., New
York, 1945; E.A. Guillemin, “Communication Network”, Vol. I, chap. III, John Wiley & Sons, inc., New York, 1931;
W.C. Johnson, “Mathematical And Physical Principles of Engineering Analisys,” chap. VII, Mc Graw Hill Book
Company inc., New York 1944;, MIT Staff, “Electric Circuit,” chap. IV, John Willey & Sons. Inc., New York, 1940.
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In the preceding chapter we derived the partial differential equations which apply to the
uniform line under both transient and steady state conditions, and we used these equations that
apply under steady state sinusoidal conditions can be found in either of two ways: they can be
derived from the more general partial differential equations (1.2) and (1.3), or they can be found by
writing the steady state relations that must hod across an infinitesimal cection of line. We shall first
show how the steady state can be derived from the partial differential equations, and shall then
show that the second method yields the same result as the first.
from Eqa. (1.2) and (1.3) we have.
9�9: = -Ri - L9)9�
and (2.20)
9)9: = -Ge - C9�9�
As outlined in the preceding section, we shall express the steady state sinusoidal voltages and
currents on the line as the projection of rotating vectors, thus,
e = Re[�����]
and (2.21)
i = Re[,����]
Where �� and ,� are the complex amplitudes of the voltage and current, respectively, and ω is
2π times the frequency of the driving source. Observe that �� and ,� will vary along the line and,
therefore, will have derivatives with respect to χ. Substituting Eqs. (2.21) into the differential
equations (2.20), taking the indicated derivatives, and cancelling ����, we obtain
�/09: = - R,� - j�-,�
and (2.22)
�;09: = - G�� - j�<��
Total derivatives are now used because there is only one independent variable, x. the effective
values E and I are related to the amplitudes �� and ,� by �� = √2� and �� = √2,. Making this
substitution, we obtain
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�/9: = - (R + j�-)I
and (2.23)
�;9: = - (G + j�<)E
These relations could also have been deduced from a consideration of the a-c steady state
circuit properties of the small section of line as visualized in fig. 1.3. The change in voltage across a
section of length ∆χ, which is expressed as �>� >?⁄ % ∆x, is caused by the current I flowing through
the series impedance of the section, R ∆x + jωL ∆x. The minus sign is used because a positive value
of I causes E to decrease with increasing x. Similarly, the change in current between the two ends of
the section is caused by the voltage E acting on the shunt admittance G ∆x + jωC ∆x.
It is usual in transmission line theory to denote the a-c series impedance per unit length of line
the symbol Z and the shunt admittance by the letter Y:
Z = R + j�- ohms per unit length
and (2.24)
Y = G + j�< mhos per unit length
The differential equations (2.23) can then be written more compactly as
�/�: = -ZI
(2.25)
and
�;�: = -YE
(2.26)
We now have two equations with two unknowns, E and I. To eliminate I, we take the derivative
of the first equation with respect to x and obtain
�3/�:3 = �A �;�:
Into this we substitute the expression for >, >?⁄ from Eq. (2.26) and obtain
�3/�:3 = (YZ)E
8
(2.27)
The solution must be a function which, when differential twice, yields the original function
multiplied by the quantity YZ. One form of this solution is
E = B� √CD:E + B�√CD:
(2.28)
Where the A’s are constants with the dimensions of voltage. To find the corresponding
expression for I, substitute the above result into Eq. (2.25). Then we obtain
I = 6FD C⁄ 7B�5√CD: � B�√CD:8
(2.29)
The quantity FA G⁄ is a characteristic of the line and has the dimensions of an impedance :
HIJ�K LM)� N�MO�J⁄�JIK LM)� N�MO�J⁄ = ohms. This is the characteristic impedance of the line. We shall denote it by the
symbol AI, as in Chap. 1.
AI = HDC = H1'��2P'��Q
For lines with negligible loss, the characteristic impedance reduces to F- <⁄ , which is a pure
resistance independence of frequency (compare Sec. 1.5).
Observe that the characteristic impedance does not involve the length of the line is the
character of the terminating load, but is determined only by the characteristic of the line per unit
length. Observe also that it is not an impedance that the line itself possesses. It will be shown later
that a load impedance equal to FA G⁄ is the only one which will cause no reflection of a received
wave, and that, for this termination only, the steady state a-c sending end impedance of the line will
be equal to FA G⁄ regardless of the length of the line. This is the principal significance of the
characteristic impedance.
Reffering again to Eqs. (2.28) and (2.29), the quantity √GA is seen to govern the manner in
which E and I vary with x; i.e., it governs the way in which the waves are propagated. Therefore, it is
given the name propagation constant. It will be denoted by the symbol γ. Actually, γ is a function of
frequency and might better be called the propagation function. Explicitly,
γ = √GA = F�. ! ��-%�R ! ��<%
(2.31)
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The propagation constant will, in general, be a complex number. The real part is given the
symbol α and is found to determine the way in which the waves die out, or attenuate, as they travel.
Hence the real part α is called the attenuation constant. The imaginary part of γ is given symbol β
and is found to determine the variation in phase of E and I along the line. For this reason β is called
the phase constant.
The unit of α is called the neper per unit length, and the unit of β is the radian per unit length.
The word neper is a variation of the spelling of the name Napier.
Example. Consider a typical open wire telephone line which has R = 10 ohms/mile, L = 0.0037
henry/mile, C = 0.0083 x 105T farad/mile, and G = 0.4 x 105T mho/mile. At a frequency of 1,000 cps,
we have from Eq. (2.30) :
AI = H 1'��2P '��Q = H 6U '��V.��U.W'�X�.6% : 6UEY
= H �X.V TT.Z[⁄X�.6 : 6UEY Z\.T[⁄ = F48.5 ? 10W �22.8I⁄
= 697 �11.40I⁄ = 683 – j 138 ohms
From Eq. (2.31) the propagation constant at this frequency is
γ = F�. ! ��-%�R ! ��<% = F�25.3 66.8I⁄ %�52.1 ? 105T 89.6I⁄ %
= F13.2 ? 105W 156.4I⁄
= 0.0363 78.2I⁄ = 0.0074 + j0.0356 per mile
Therefore, the attenuation constant is
C = 0.0074 neper/mile
And the phase constant is
β = 0.0356 rad/mile
2.3. The Line with NO Reflection.
The simplest line from a mathematical viewpoint is one on which there is no reflected wave.
This condition would be obtained on a hypothetical smooth line of infinitive length, for no reflection
could ever return from the far end. We shall also show that a similar condition is obtained for a line
terminated in its characteristic impedance.
Consider the general solution (2.28) in connection with an infinitive line (see Fig. 2.3a). the
second term of the solution involves a positive exponent and would tend toward infinity as x
increased. This is physically impossible from an energy standpoint; hence, B� must be zero. We can
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evaluate the constant B6in terms of the sending end voltage by substituting x = 0 and E = �U into the
solution, thus obtaining (since B� = 0)
�I = B6
Using this, the solution (2.28) can be written as
E = �IdE √CD:
(2.32)
In the preceding section we defined the propagation constant γ = √GA and its real and
imaginary parts α and β. In terms of these, we can write
E = �K5(:5�e: = �K5(: /-f?
(2.33)
We obtain the corresponding solution for current from eq. (2.29) by using B6 g �K 4$> B� g 0,
which yields
I = /hDi 5(:5�e:
(2.34)
Upon comparison with eq. (2.33), we see that the current and voltage at every point on the
infinite line are simply by.
I = /Di
(2.35)
This expression applies, of course, to the sending end of the line as well as to all other points.
Hence, the sending end, or input, impedance of the infinite line is equal to its characteristic
impedance AU
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Now suppose that we cut the infinite line any point x = l, as indicated in fig. 2.3. the line to the
right of the cut is still infinite in length an has an input impedance equal to AU. So far the section to
the left of the cut is concerned, the infinite portion to the right can be replace by a lump receiving
end impedance equal to AUthe finite section will still behave as though it were infinite in length, and
the solution (2.33), (2.34) and (2.35) will still apply. The sending end impedance of the line will be
equal to AU regardless of the length l. for low loss lines this impedance is nearly a pure resistance
independent of frequency (F-/<; hence, moderate changes in frequency will not affect the load
voltage or power appreciably.
A line terminated in its characteristic impedance is sometimes called a correctly terminated or
nonresonant line. On a line so terminated, energy flows from the generator, travel down the line in
the form of a wave, and from a circuit standpoint, should all be absorbed by the load without
reflection. From the point of view of traveling electric and magnetic fields, it can be seen that a
lumped load impedance will not supply on continuation of the fields as did the infinite line.
However, the circuit theory is a quite accurate except when the conductor spacing is an appreciable
part of a quarter wave length beyond the load. The theory of this is discussed in chap. 6.
The solution (2.33) and (2.34) have been given in terms of the sending end voltage of the line,
although, as suggested in fig. 2.3, the open circuit voltage of the generator and its internal
impedance may be known instead. The sending end voltage can be found by the solving the
equivalent sending end circuit of fig. 2.4, in which the line is replaced by an equivalent lumped
impedance equal to AK. When �K has been computed, eqs. (2.33) and (2.34) can be used to find the
voltage and current at any point on the line.
The power transmitted down the line at any point can be computed from the relation P = |E| .
|I| cos �, where V is the phase angel the characteristic impedance, or by |,|� .U, when .U is the
resistive component AU. For the low loss lines, AU is very nearly a pure resistance, an which case cos � 1 and .U k AU.
Example. Consider an open wire telephone line which is 200 miles in length and terminate in its
characteristic impedance at the receiving end. At the sending end, it is driven by generator which
has a frequency of 1,000 cps, an open circuit emf of 10 volts rms, and an internal impedance of 500
ohms resistance. At this frequency the line has AU = 683 – j138 ohms
and l 0.0074 + j0.0356 per mile. Find the sending end voltage,
currnet,and power, and the receiving end voltage, current, and
power.
Using the equivalent sending circuit of fig. 2.4 wit �O = 10 volts, AO = 500 + j0 ohms, and AK = AU = 683 + j138 ohms, we can solve for
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the magnitude of the sending end current2:
|�K| g m /nDn' Dhm g 6U|XUU'TZV5�6VZ|
g 6U6,6\U g 8.40 ? 105Z amp rms
The magnitude of the sending end voltage is
|�K| g |,KAK| g 8.40 ? 105ZF�683%� ! �138%�
g 5.58 o+p�" qr"
The average power entering the sending end of the line is
sK g |,K|�.K g �8.40 ? 105Z%� ? 683
g 48.2 ? 105Z watt = 48.2 mw
For the receiving end voltage we use eq. (2.33) with x equal to the length of the line. Then,
arbitrarily taking �K to be real, we have
�1 g �K5(N5�eN g 5.855U.UUtW : �UU5�U.UVXT : �UU
g 1.335�t.6� g 1.33 /-7.12 radians = 1.33 /-408U volts
The magnitude of the receiving end voltage is 1.33 volts rms. The line is 7.12/2π = 1.134
wavelengths long, and the receiving end voltage lags that of the sending end by 1.134 cycles. In so
far as the phase position of the receiving end voltage is concerned, we can substract 2π radians or 360U without changing the result, thus giving
�1 g 1.33 /-0.84 radian g 1.33 /-48U o+p�"
The magnitude of the receiving end current is
|,1| g m�1A1m g 1.33697 g 1.91 ? 105V 4ru qr"
The average value of the power absorbed by the terminating impedance is
s1 g |,1|�.1 g �1.91 ? 105V%� ? 683
2 In this text, we have been using the italic symbols E, I, and Z to represent the complex, or vector, voltage, current,
and impedance. Theirs magnitudes, or absolute values, will be indicate by placing the symbols between vertical
bars. Thus: |E|, |I|, and |Z|.
13
g 2.49 ? 105V v4�� g 2.49 rv
2.4. The Travelling Wave and Its Characteristics.
Equations (2.33) and (2.34) show that, on a line terminated in is characteristic impedance, that
voltage and current decrease exponentially in amplitude by the factor �5w: as the distance from the
generator increases. This is caused by the line losses, which absorb energy from the wave as it
travels. A second important effect is the progressively increasing lag in phase as x increases, as
shown by the factor �5��: = rumus. This lag is caused by the finite time required for the wave to
travel the distance x.
The traveling wave on the line can be expressed most neatly by using the method of Sec. 2.1 for
obtaining instantaneous values: First express Eq. (2.33) in terms of the maximum value, then
multiply by the unit rotating vector ��LK�, and finally take the real part of the resulting expression. If
we take E, to be real, this process gives us
� g .� x√2 �K5(:5�e:5���y g √2 �K5(: .�x5����5 e:%y
or
� g √2 �K5(: cos��� � f?%
(2.36)
Now, we can follow the same reasoning regarding traveling waves that we used in Chap. 1.
Imagine that an observer is following the wave cos��� � f?% so that he stays with a point of
constant phase, for which the angle �� � f? g 4 *+$"�4$�. To find the velocity of the point, we
take the time derivative of this expression and obtain.
� � f >?>� g 0
But >?/>� is the velocity we desire (the phase velocity) and this is seen to be
o g �f
(2.37)
It should be observed that this is the velocity at which the steady state a-c wave and its
accompanying electric and magnetic fields are propagated, but it is not the velocity of the electrons
in the wire. The electrons may be visualized as executing an oscillatory motion as shown in Fig. 2.5
(this motion is superimposed on their usual random velocity). The phase of the oscillation lags in the
direction of motion of the wave, and this gives rise to a sinusoidal distribution of charge which
apparently travels along the wire, as suggested in the illustration. The portions marked A are regions
14
of deficiency of charge, and one marked B is a region of excess charge. The charged regions travel to
the right with the phase velocity. An analogous situation is the propagation of sound, which travels
at about 1,100 ft/sec in air at sea level. The actual motion of the air molecules, however, is only a
sinusoidal one, and the velocity of an individual molecule is not comparable with the velocity of
propagation of the wave.
For a line with negligible losses, Eq. (2.31) becomes merely l g ��√-< ; i.e., & g 0 and f g �√-<. The velocity �/f in this case is simply
o g 1√-<
(2.38) As mentioned in Sec. 1.5, the product LC perfect conductors immersed in a loseless dielectric is a
constant which depends only on the dielectric constant and the permeability of the dielectric. For air
the velocity turns
15
out to be approximately 3 ? 10Z r���q"/"�*. A solid dielectric will increase C and thus reduce that
velocity. This effect is frequently expressed in terms of a velocity constant:
o�p+*#�z *+$"�4$� g w{�LwN |JwK� }�NI{)�~}�NI{)�~ I� N)OJ� )M ���� K|w{� (2.39) A low-loss coaxial line with a solid dielectric may have a velocity constant of about 0.6 or 0.7. As
shown by Eq. (1.1), the wavelength for a given frequency is reduced by the same factor as the phase
velocity.
Figure 2.6 shows a traveling wave of voltage on a lossy line at three successive instants of time,
as plotted from Eq. (2.36). The amplitude of the wave diminishes exponentially by the factor 5(:
as it travels. The accompanying current wave will be similar in form but will be out of phase with the
voltage by an amount equal to the angle of Zo, "#$*� , g �/AU. The transmitted power will
decrease down the line by the factor 5�(:
The instantaneous voltage at any point on the line will vary sinusoidally as the traveling wave
slides past, and will have an amplitude √2�K5(: and an rms value �K5(:. The phase lags
progressively down the line; for instance, a quarter wavelength from the generator the voltage lags
by a quarter of a cycle, or 90U. The phase lag at any point with respect to the input is f? radians.
The wavelength � is equal to the distance between successive crests of the wave at any
moment; i.e., it is equal to the change in x which makes the angle f? inerease by 2� radians.
Therefore, f� g 2� and
� g 2�f
(2.40)
Comparison with Eq.(2.37) shows that the phase velocity and wavelength are related by
� g 2�o� g o�
2.41)
16
This relation was first stated in Chap. 1, Eq. (1.1). As an exercise, the student should return to
the example given in the preceding section and show that the phase velocity was 176,200 miles/sec
and the wavelength 176.2 miles.
The variation of rms voltage and current along a line is sketched in Fig. 2.7. When the over all
losses are small, the factor 5(N Is nearly unity, and if the line is terminated in AU, the current and
voltage will be practically uniform in magnitude over the whole length. Such a line is said to be “flat”
Figure 2.8 is a polar representation of the vector � g �K5(:5�e:along the length of a lossy
line. The locus of F is a logarithmic spiral, of the magnitude decreases exponentially with increasing
angle. At this point the student should review in his mind the process by which any one of these
vectors can be translated into the corresponding instantaneous line-to-line voltage.
2.5. Note on Characteristic Impedance.
Figure 2.9 show a symmetrical four-terminal T network composed of lumped elements. Network
theory shows that there is one particular value of receiving-end impedance which will cause AK, to
be precisely equal to A1. This particular impedeance is called the characteristic impedance of the
network. It is not harl to show that for a symmetrical T network the characteristic impedance is
given by
AU g �A6A� ! A6�4
(4.42)
where A6, is the total series impedance of the section and A�, is the impedance of the shunt branch.
A simple numerical example which involves only resistances is shown in Fig. 2.10. Here we have A6 g 10 +�r", A� g 20 +�r", and so the characteristic impedance is
17
AU g �10 ? 20 ! 10�4 g 15 0�r"
Any number of identical sections can be connected in cascade to form a uniform ladder
network, as shown in Fig. 2.10b. If the last section is terminated in AU, its input impedance will be AU, as the student can show by combining resistances in series and parallel in Fig. 2.10. This will
terminate the next-to-last section in AU, and so on through the network. Consequently, the sending
–end impedance of the whole combination is equal to the characteristic impedance. For this reason
the more descriptive name iterative impedance is often used in the theory of four-terminal
networks.
Suppose that the T network of Fig. 2.9 is to represent a small length of transmission line, in
which case
A6 g �. ! ��-%∆? 4$> A� g 6�P'��Q%∆:
As ∆? mis made smaller to approximate a smooth line more closely, the product A6A� will remain
constant while the quantity A6�will vanish. In the limit, Eq. (2.42) will yield AU g F�. ! ��-%/�R ! ��<%, which is the expression previously derived for the smooth line.
2.6. The Decibel and the Neper.
We have shown previously that, on a line terminated in its characteristic impedance, the
magnitudes of both voltage and current decrease as 5(:, where & is the real part of the
propagation constant √GA . The unit of & is the neper per unit length. Consider two points 1 and 2
separated by a distance ∆? on a properly terminated line. The ratios of the magnitudes of E and I at
the two points are
m�6��m g m,6,�m g 5(∆:
(2.43)
The quantity & ∆? represents the total attenuation, measured in nepers, of the traveling wave
between the two points. Taking the logarithm to the base of both sides of Eq.(2.43), we obtain an
expression for the number of nepers of attenuation:
��r��q +� $�u�q" g p+�d m�6��m g p+�d m,6,�m (2.44)
This definition, expressed in terms of the natural logarithm, explains the association of Napiers
name with the unit
18
A logarithmic scale for the measurement of current or voltage is highly convenient in a
transmission system. Figure 2.11 shows two four-terminal networks (perhaps transmission lines)
which are connected in cascade. The attenuation in nepers through the separate networks will be
denoted by �6 and ��, where
�6 g p+�d m�6��m and
�� g p+�d m���Vm
The total attenuation through the combination is by adding the attenuations of the separate
networks:
�+�4p � g �6 ! �� g p+�d m�6��m ! p+�d m���Vm which, by the rule of adding logarithms, becomes
� g p+�d �/�/3� Thus, the addition of the logarithmic attenuations is equivalent to multiplying the voltage ratios.
The convenience of this method becomes greater as more and more networks are added in cascade.
Another reason for using a logarithmic scale is that, within wide limits, the human senses follow
a similar law. Thus, the minimum perceptible change in sound intensity that the ear can perceive is
roughly proportional to the amount of sound already present.
A second logarithmic unit which is frequently used is the decibel, abbreviated db. The decibel is
one-tenth the size of the bel, which was named in honor of Alexander Graham Bell. The decibel is
defined fundamentally in terms of a power ratio:
��r��q +� >�*#��p" g 10 p+�6U s6s�
(2.45)
19
If the powers s6 and s6 are associated with equal impedance, as would be the case on a
transmission line terminated in its characteristic impedance, the power ratio can be expressed as
the square of either the voltage or the current ratio. Under these conditions,
��r��q +� >�*#��p" g 20 p+�6U m s6s�m g 20 p+�6U m ,6,�m (2.46)
The decibel is a unit of convenient size. A change in level of 3 db corresponds very nearly to a
power ratio of two; and 10 db correspond to a power ratio of 10. A change in sound power of 1 db is
roughly the minimum change that is perceptible by the human ear.
The advantage of a logarithmic scale and the convenient size of the decibel have given rise to a
tendency to express voltage ratios in terms of decibels even when the impedances are different at
the two points. This usage is most common in connection with cascaded voltage amplifiers in which
the voltage ratio is of primary interest, and amounts to replacing the definitions of eq. (1.45) by a
new one, namely, 20 p+�6U |��/�6|. The two definitions are not equivalent except when the
impedance level is the same at both points, and confusion may arise if the usage is not made clear.
The student should show that, in a system having the same impedance level at two points in
question, an attenuation of 1 neper is equivalent to 8.686 db.
2.7. Variation of AU, &, and f with Frequency.
The characteristic impedance of a uniform line is given by eq. (2.30) as
AU g �. ! ��-R ! ��<
(2.47)
At zero frequency, i.e., with steady state direct current, the characteristic impedance reduces to F./R . This may be a somewhat variable quantity on an open wire line, for G will depend to a large
extent on the amount of moisture present on the insulators.
As the frequency increases, G becomes negligible compared with �<, and R becomes negligible
compare wit �-, giving a high frequency characteristic impedance very nearly equal to F-/< .
20
On lines with good insulation the ratio G/C is much smaller than R/L, and so G becomes negligible at
a much lower frequency than R. if R �� �<, eq. (2.47) can be written more simply as
AU g �-< �1 � � .�-
(2.48) The resistive and reactive componenets of this are plotted in fig. 2.12 as functions of �-/.. The
resistive component approaches the high frequency value F-/< rather rapidly, while the reactive
component tends toward zero more slowly. The dashed portions of the curve indicate roughly the
effect of the neglegted conductance at the lowest frequencies.
A typical open wire telephone line, composed of two No. 10 AWG copper wire spaced 12 in.
apart, has the following constants: . g 10.2 +�r"/r#p�, - g 0.00367 ��$qz/r#p�, and < g 0.00821 ? 105T�4q4>"/r#p�. The leakage conductance of an open wire line is greatly affected by
the amount of moisture present. We shall use R g 0.3 ? 105T r�+/r#p�, which is an approximate
dry weather value for aline in good condition. On this line the reactance �- is equal to the
resistance R at an angular frequency � g ./- g 2,780 q4>/"�*, or � g 442 *u". This
frequency, therefore, will correspond to �-/. g 1 in fig. 2.12. The approximate formula (2.48)
fails at the lower frequencies where G becomes appreciable compared with �<. For this line, G is
equal to ωC at a frequency of about 6 cps.
The propagation constant for the uniform line is, from Eq. (2.31),
21
l g & ! �f g F�. ! ��-%�R ! ��<%
(2.49)
The zero-frequency limit of l is simply √.R . This, being real, is an attenuation constant; the
phase constant is zero.
As the frequency increases, we finally obtain �- � . and �< � R. Under these condition we
can expand Eq. (2.49) by the binomial theorem and retain only the fire two terms, i.e., we use �4 ! �%6/� k 46/� ! 456/��/2 for 4 � �. Then we ebtain
l g �. ! ��-%6/� �R ! ��<%6/�
g ����-%6/� ! .2 ���-%6/�� ����<%6/� ! R2 ���<%6/��
Upon expansion and neglect of the small term involving the product RG, the high-frequency
propagation constant is found to be
l g 1� HQ2 ! P� H2Q ! ��√-< (2.50)
Since AU k √-< for high frequencies, we now have
& g .2AU ! RAU2
(2.51)
and, as for the lossless line,
f k � √-<
(2.52)
In Eq. (2.51), the term ./2AU is the attenuation caused by energy losses in the conductors,
while RAU/2 is the attenuation caused by energy losses in the insulation.
General expressions for & and f can be found by the following process: Square both sides of Eq.
(2.49), separate the result into two equations by equating reals to reals and imaginaries to
imaginaries, and then solve for & and f. This process yields the rather cumbersome expressions
&� g 6� �F�.� ! ��-�%�R� ! ��<�% ! �.R � ��-<�
22
and (2.53)
f� g 12 �F�.� ! ��-�%�R� ! ��<�% � �.R � ��-<�
These equations can be used to study in detail the variation of & and f with frequency. Figures
2.13 and 2.14 show graphs of & and f vs angular frequency as calculated for the typical open-wire
telephone line mentioned
previously. Although only the comparatively low-frequency region is shown (0 to 478 cps), both &
and f are already seen to be approaching their high-frequency asymptotes.
2.8. The Distortionless Line.
The complex signals that are encountered in communications practice can be resolved by the
Fourier analysis into sinusoidal components3. In linear systems, such as those we are considering,
each of these components can be treated separately by the steady-state sinusoidal theory. When a
complex signal is impressed on a transmission system, the received wave will be of the same form as
the transmitted one only if all components are attenuated equally and if they all travel at the same
velocity. The first requirement for zero distortion is, therefore, an attenuation constant that is
independent of frequency. Second, to make the phase velocities �/f equal at all frequencies, the
phase constant must be linearly proportional to frequency. In general, these conditions are not
3
23
Satisfied, for & and f are, respectively, the real and imaginary parts of a rather complicated function
of frequency: F�. ! ��-%�R ! ��<%.
A distortionless line is obtained, however, if the constants of the line have the following relation:
.- g R<
(2.54)
Under these conditions, the propagation constant reduces simply to
l g √.R ! ��√-< (2.55)
The line is distortionless, for & is independent of frequency and B varies linearly with frequency.
The phase velocity is simply o g 1/√-< For all frequencies. The lossless line is a special case of this,
for the condition R = G = 0 satisfies Eq.(2.54).
The student should show that a distortionless line has (1) a characteristic impedance equal to √-< (2) equal electric and magnetic energies associated with a traveling wave, and (3) equal energy
lesses in the resistance and leakage conductance of the line. The last two conditions hold, of course,
only for a single traveling wave and not for the superposition of two oppositely traveling waves.
A condition of low distortion is particularly important for lines in which the ration of the
highest to the lowest frequency is rather large, as it is in telephone lines. On most transmission lines
the energy loss in the resistance of the conductors is greater than the loss in the insulator; that is,
24
./- is larger than R/<. (Exceptions to this occur when solid dielectrics are used in the ultra-high-
frequency region, where the dielectric loss may be quite high) The representative open-wire
telephone line mentioned in the preceding section had ./- g 2,780 and R < g 36.5. Even more
extreme is the telephone cable, in which the clo-e proximity of the conductors causes the
capacitance to be quite high and the inductance low. Typical constants for a cable are . g 60 +�r" r#p�, - g 0.001 ��$q�/r#p�, < g 0.060 ? 105T �4q4>/r#p�, and R g 2 ? 105T r�+/r#p�, givinf ./- g 60.000 and R/< g 33.
It is not ordinarily desirable to increase G artificially, as this will greatly increase the attenuation.
On the other hand, a reduction of R to satisfy Eq. (2.54) will require wires of unusually large
diameter. To a certain extent, L can be increased and C reduced by a larger spacing between
conductors, but there are practical limits to this. The inductance can also be increased by the
process known as inductive loading, which is described in the next section. In practice, a certain
amount of distortion can be tolerated, and a true distortionless line is rarely attempted.
2.9. Inductive Loading.
On some types of transmission lines, the inductance is increased over its normal value either by
wrapping the conductors with a high-permeability metal tape or by inserting inductance coils at
uniform intervals. This is called inductive loading. Since G/C is usually smaller than R/L, an increased
inductance will make the line approach more nearly a distortionless condition, with the result that
the characteristic impedance of frequency. Usually, however, the most desired effect of inductive
loading is the reduction in attenuation that it produces. This will now be investigated, assuming for
the moment that the added inductance is uniformly distributed so that the smooth-line formulas
will apply.
In Sec. 2.7 it was shown that the high-frequency limit of x is
& g 1� HQ2 ! P� H2Q (2.56)
When ./- � R/<, the first term of Eq. (2.56) will be larger than the second, indicating that
eonductor losses contribute the major portion of the attenuation. An incrense in L will reduce the
first term and increase the second by the same factor, but up to the point where the two terms are
equal, the net effect will be a reduction in the high-frequency limit of x. Physically, this means that
an increased L will raise the high-frequency characteristic impedance √-<, and thus a smaller
current will be required for a given transmitted power, resulting in lower I R losses. A higher voltage
will be required and this will increase the insulation losses, but, so long as these remain the minor
portion, the net effect will be a gain.
As an example, consider the line for which Fig. 2.13 was plotted. With out loading, the two
terms of Eq. (2.56) are, respectively, 7.63 ? 105V and 0,10 ? 105V neper /mile, indicating that the
copper loss is 76 times as great as the loss in the insulators. If L were increased by a factor of 4 with
no change in the other constants of 0.20 ? 105V, giving a total attenuation constant of 4.02 ? 105V
25
neper/mile, which is smaller than before. Referring again to Fiq. 2.13, the greatest possible
improvement would be obtained by reducing the high-frequency asymptote to the same value as
the zero-frequency intercept √-<. The line would then be distortionless. If L is inereased beyond
this point, the second term of Eq.(2.56) dominates and the high-frequency asymptote of x increases.
Inductive loading is much more necessary for telephone cables than for open-wire lines because
of the inherently low inductance and high capacitance of the cable. The limitations of space within
the cable sheath make desirable a smaller wire than is commonly used on open-wire lines, thus
causing a higher resistance. The normal inductance is low because of the proximity of the
conductors, and the presence of solid insulation contributes to the high capacitance. As a result, the
conductor loss dominates by a large factor. The attenuation and phase velocity vary widely within
the audio range, and the high-frequency asymptote of attenuation is quite high. Inductive loading is,
therefore, in general use on cable lines.
An audio-frequency transmission line can be smoothly loaded by wrapping the conductors with
a high-permeability metal tape. However, this method of construction is quite expensive and is used
principally on submarine cables where other methods are impractical. Cable lines on land are “lump-
loaded” with inductance coils spaced at uniform intervals. A disadvantage of this method is that the
lumped inductances in the line give rise to a low-pass filter effect which distorts the higher
frequencies. This effect is re4duced as the coils are placed conser together in a better approximation
to a smooth line. From physical reasoning, it can be seen that there must be at least several coils per
half wavelength at the highest desired frequency if the line is to behave with a reasonable
approximation to smoothness.
In the past it was common practice to load open-wire telephone lines to reduce their
attenuation. However, the loading had a variable effect which depended on the leakage
conductance as determined by moisture on the insulators. Also, the increasing requirements on
high-frequency response would have made necessary a rather small spacing and vacuum-tube
amplifiers are inserted at intervals frequent enough to keep the signals from being attenuated into
the noise level from which they could not be recovered.
2.10. Phase and Group Velocities.
In Sec. 24 we discussed the phase velocity, which is the velocity of a point of constant phase on
a sinusoidal traveling wave. As was shown in Eq. (2.37), this is given by the relation
ou g �u
(2.57)
where we now use the subscript p to distinguish this velocity from another that we shall discuss
in a moment.
26
When a complex signal is impressed on a transmission system which has a phase function B that
is not linearly proportional to frequency, the various sinusoidal components of the signal will have
different phase velocities and the wave will change shape as it travels along. Under these conditions,
the signal cannot be said to travel at any one velocity.
A particularly simple situation with regard to velocities is obtained in the important case of a
signal which has all its sinusoidal components grouped closely together in frequency. An example of
this kind of wave is an amplitude-modulated signal, in which the amplitude of a sinusoidal “carrier”
is varied in proportion to the instantaneous value of a low-frequency signal. One of the components
of such a wave has a frequency equal to that of the carrier itself. Each of the frequencies in the
original signal produces two “side-band” components, one below and the other above the carrier by
the amount of the modulating frequency4. If this set of voltages is impressed on a transmission
system, it is found that the envelope of the wave, which has the same shape as the original signal,
travels at one velocity, while the actual wave which is enclosed within the envelope travels at
another velocity.
Consider, for simplicity, an amplitude-modulated wave which has only two side bands. The
amplitude of the carrier will be denoted by Ev, its angular frequency by w, and the phase constant of
the transmission line at this frequency by B. The amplitude of each side band will be denoted by E2,
their frequencies by � � ∆� and � ! ∆�, respectively, and the eorresponding phase constants by f � ∆f and f � ∆f. The traveling wave consisting of these there components can be written as
� g �� cos ��� � ∆�%� � �f � ∆f%� ! �� cos��� � f?% ! �� cos ��� ! ∆�%� � �f ! ∆f%� (2.58)
If we expand the expressions for the side-band voltages by the trigonometric identity cos ( a +
b) = cos a cos b – sin a sin b and then collect terms, we obtain
� g ��6 ! 2�� cos�∆� � � ∆f ?%� cos��� � f?%
(2.59)
4 See F.E Terman, “Radio Enginnering,” 3d ed., secs. 1.5 and 9.1, McGraw Hill Book Company, Inc., New York, 1947;
M.I.T. Staff, “Applied Electronics,” pp. 632-638, John Wiley $ Sons, Inc., New York, 1943, or any standard textbook
on communication theory
27
We can regard the bracketed expression as the amplitude of a traveling wave cos��� � f?)which moves with the phase velocity �/>. The amplitude itself varies sinusoidally between the
extreme limits �6 ! 2�� and �6 � 2��, and, furthermore, it moves along the line with a velocity
∆�/∆f. For components grouped closely together, this velocity approaches the derivative >�/>f,
which is the re4ciprocal of the slope of the phase function.
This is the so-called group velocity :
o� g>�
>f
(2.60)
which, in general, is different from the phase velocity w/B. Note, however, that for the distortionless
case, where f g �√-<, the phase and gropup velocities are both equal to 1/√-<
The picture that we now get of the traveling amplitude-modulated wave is illustrated in Fig.
2.15. The envelope, which is the imaginary curve joining the peaks of the actual wave, moves along
the line at the group velocity, while the actual wave, moves along the line at the group velocity,
while the actual wave “slips through” the envelope at the phase velocity5.
In fig. 2.14 was shown a calculated curve of f o" � for a representative open wire telephone
line. The corresponding phase and group velocities are plotted in fig. 2.16.
5 See H.H. Skilling, “Fundamentals of Electric Waves,” 2d ed., pp. 228-231, John Wiley $ Sons, Inc., New York, 1948.
28
The concept of group velocity losses its usefulness when the components of a wave are spread
so widely in frequency that ∆�/∆f is not approximately equal to the derivative >�/>f.
It will be observed that, as in the example plotted in fig, 2.16, the group velocity may exceed the
speed of light. However, we are dealing here with a steady state wave which, in theory, has no
beginning and no end. Therefore, we cannot “tag” any of the energy as it enters the transmission
system and then identify it as it leaves, and so we cannot draw any final conclusion regarding the
velocity with which the actual energy is propagated. For this reason a group velocity greater than
that of light does not violate the relativity postulate. We can identify the position of the energy in a
signal of short duration, but the Fourier analysis of such a wave shows that the components are
widely spaced in frequency. The concept of group velocity then loses its usefulness. Further analysis
shows that the wave front of a suddenly applied signal cannot travel faster than light, and the
relativity postulate is not violated6.
6 For a more complete treatment of group velocity, see E.A. Gullemin, “Communication Networks,” Vol. II, Chap. III,
sec. 6, John Wiley & Sons, inc., New York, 1935. For a discussion of group and signal velocities, see J.A. Stratton,
“Electromagnetic Theory,” secs. 5.17 and 5.18, McGraw-Hill Book Company, Inc., New York, 1941, and R.I
Sarbacher and W.A. Edson, “Hyper and Ultra High Frequency Enginnering,” sec. 5.8, John Wiley $ Sons, Inc., New
York, 1943.