1

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)

3

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).

4

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

5

,� = /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.

6

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

7

�/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)

9

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

10

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

11

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

12

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.