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```                            001 Rao Elements 2004 front matter
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01 Rao Elements 2004 ch1
02 Rao Elements 2004 ch2
03 Rao Elements 2004 ch3
04 Rao Elements 2004 ch4
05 Rao Elements 2004 ch5
06 Rao Elements 2004 ch6
07 Rao Elements 2004 ch7
08 Rao Elements 2004 ch8
09 Rao Elements 2004 ch9
10 Rao Elements 2004 ch10
11 Rao Elements 2004 ch11
12 Rao Elements 2004 appA
13 Rao Elements 2004 appB
14 Rao Elements 2004 appC
15 Rao Elements 2004 Suggested Reading
16 Rao Elements 2004 answers
17 Rao Elements 2004 index
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Page 2

Elements of Engineering
Electromagnetics

Sixth Edition

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Page 428

AAAVARQ0

6.5 Lines with Initial Conditions 403

them. Sketches of line voltage and current versus z for fixed values of t can be drawn from
these bounce diagrams in the usual manner. Sketches of line voltage and current versus t
for any fixed value of z also can be drawn from the bounce diagrams in the usual manner.
Of particular interest is the voltage across which illustrates how the line discharges
into the resistor. The time variation of this voltage is shown in Fig. 6.45.

It is also instructive to check the energy balance, that is, to verify that the energy dissipat-
ed in the resistor for is indeed equal to the energy stored in the line at
since the line is lossless. To do this, we note that, in general, energy is stored in both electric
and magnetic fields in the line, with energy densities and respectively.Thus, for
a line charged uniformly to voltage and current the total electric and magnetic stored
energies are given, respectively, by

(6.66a)

=
1
2

cV0
2

12lc T = 12 V02Z0 T We =

1
2 cV0

2 l = 12 cV0
2vp T

I0,V0

1
2 lI

2,12 cV
2

t = 0- ,t 7 0150-Æ

RL,

2 4 6

75

0
t, ms

[V]RL, V

37.5
18.75

9.375

FIGURE 6.45

Time variation of voltage across for in
Fig. 6.43(a) for

and T = 1 ms.RL = 150 Æ,
Z0 = 50 Æ,V0 = 100 V,

t 7 0RL

0 0

1

3

5

2

4

6

t, ms

� 1/2
� 1

100 V

75

37.5

18.75

�6.25

�6.25

�25

�25

�12.5

�12.5

100

50

25

12.5

z � 0 z � l

0 0

1

3

5

2

4

6

� �1/2 �
� �1

0 A

�0.5

–0.25

�0.125

�0.125

0.125

0.5

�0.5

�0.25

0.25

0

0

0

0

z � 0 z � l

FIGURE 6.44

Voltage-and current-bounce diagrams depicting the transient phenomenon for for the line
of Fig. 6.43 (a), for and T = 1 ms.V0 = 100 V, Z0 = 50 Æ, RL = 150 Æ,

t 7 0

Energy
balance

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Page 429

404 Chapter 6 Transmission-Line Essentials for Digital Electronics

and

(6.66b)

Since for the example under consideration, and
and Thus, the total initial stored energy in the line is Now,

denoting the power dissipated in the resistor to be we obtain the energy dissipated in
the resistor to be

which is exactly the same as the initial stored energy in the line, thereby satisfying the en-
ergy balance.

K6.5. Initial conditions;Arbitrary distribution; Uniform distribution; Bounce-diagram
technique.

D6.11. For the line of Fig. 6.39 with the initial voltage and current distributions as given
in the figure, find: (a) (b) (c) and
(d)
Ans. (a) 37.5 V; (b) 0.75 A; (c) 25 V; (d)

D6.12. In the system shown in Fig. 6.46, a line of characteristic impedance and
charged to 10 V is connected at to another line of characteristic imped-
ance and charged to 5 V. The one-way travel time T is equal to for
both lines. Find (a) the value of the voltage at the instant of time when both
lines are charged to the same voltage throughout their lengths; (b) the value of
the current to which the lines are charged at that instant of time; and (c) the en-
ergy stored in the system at any instant of time.
Ans. (a) 7 V; (b) 0.04 A; (c) 11/12 mJ.

1 ms50 Æ
t = 0

75 Æ
-0.5 A.

I1l>4, 1 ms2.
V1l>4, 1 ms2;I1l>2, 0.25 ms2;V1l>2, 0.25 ms2;

=
2 * 10-6

150
* 752 a1 + 1

4
+

1
16

+ Á b = 10-4 J

= L
2 * 10-6

0

752

150
dt + L

4 * 10-6

2 * 10-6

37.52

150
dt + L

6 * 10-6

4 * 10-6

18.752

150
dt + Á

Wd = L
q

t = 0
Pd dt

Pd,
10-4 J.Wm = 0.We = 10-4 J

T = 1 ms,V0 = 100 V, I0 = 0,

=
1
2

lI0
2

12lc T = 12 I02Z0 T

Wm =
1
2

lI0
2 l =

1
2

lI0
2vp T

Z0 � 75 �
T � 1 ms

10 V

Z0 � 50 �
T � 1 ms

5 V

t � 0

FIGURE 6.46

For Problem D6.12.

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830 Index

Uniform plane wave propagation. See
Wave propagation.

Uniform plane waves
bouncing obliquely of, 539
normal incidence of, 263
superposition of, 537

Unit conductance circle, 479
Unit impulse response

frequency response from, 390
of transmission-line system, 387–89

Unit pattern, 696
Unit vector, 4, 6

along line between two points, 14
Unit vector normal to a surface

from cross product, 16
from gradient, 285

Unit vectors
cross products of, 8
dot products of, 7, 25
in Cartesian coordinates, 13
in cylindrical coordinates, 21
in spherical coordinates, 23
left-handed system of, 4
right-handed system of, 4

Units
International system of, 32, 795
MKSA rationalized, 795
table of, 796–98

V

V. See Electric potential; Voltage.
Vector

circulation of, 81
curl of. See Curl.
definition of, 3
divergence of. See Divergence.
division by a scalar, 6
graphical representation of, 4
joining two points, 13–14
Laplacian of. See Laplacian of

a vector.
magnitude of, 6
multiplication by a scalar, 6
position, 13, 26
unit, 4, 6

Vector algebra, summary of rules of, 64
Vector fields

graphical description of, 29–31
sinusoidally time-varying, 178–84

Vector potential. See Magnetic vector
potential.

Vector product. See Cross product of
vectors.

Vectors
addition of, 5, 6
conversions between coordinate

systems, 24–25
cross product of, 8
dot product of, 7
examples of, 4–5
scalar triple product of, 10
subtraction of, 6
triple cross product of, 9
unit. See Unit vectors.
versus scalars, 3–4

Velocity
drift, 209
group. See Group velocity.
phase. See Phase velocity.

Velocity of light, in free space, 166
Velocity of propagation, 166, 367. See also

Phase velocity.
VHF TV channels, 176
Volt, definition of, 34, 796
Voltage, 81

around closed path, 81
compared to potential difference, 291

Voltage reflection coefficient, 375, 420
for some special cases, 377
generalized, 452

Voltage transmission coefficient, 387
Volume, differential. See Differential

volume.
Volume charge density, 41

units of, 41
Volume current density, 53

units of, 53
Volume integral, evaluation of, 107

W

Watt, definition of, 796
Wave

traveling. See Traveling wave.
uniform plane. See Uniform plane wave.

Wave equation
for material medium, 240
one-dimensional, 163
solution of, 163–64, 240–41

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Index 831

Wave plates, 657, 663
full-wave, 657
half-wave, 657
quarter-wave, 657

Wave propagation
in anisotropic dielectric, 655–57
in free space, 160–178
in good conductor, 251–54
in imperfect dielectric, 251
in material medium, 239–49
in perfect dielectric, 250
in terms of voltage and current, 363

Waveguide
compared to transmission line, 527
cylindrical. See Cylindrical waveguide.
dielectric slab. See Dielectric slab

waveguide.
graded-index. See Graded-index guide.
metallic. See Metallic waveguide.
optical. See Optical fiber.
parallel-plate. See Parallel-plate

waveguide.

rectangular. See Rectangular
waveguide.

Wavelength, 174
guide, 541, 588, 621
in good conductor, 252
in imperfect dielectric, 250–51
in material medium, 246
in perfect dielectric, 250
times frequency, 175

Waveguide dispersion, 644
Waveguides, optical. See Optical

waveguides.
Waves

classification of, 175
electromagnetic. See Electromagnetic

waves.
sinusoidal. See Sinusoidal waves.
standing. See Standing waves.
TE, 536
TEM, 366
TM, 543

Work, in moving a test charge, 77–78

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