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Transcript of Fundamentals of Electromagnetics for Teaching and Learning: A Two-Week Intensive Course for Faculty...
Fundamentals of ElectromagneticsFundamentals of Electromagneticsfor Teaching and Learning:for Teaching and Learning:
A Two-Week Intensive Course for Faculty inA Two-Week Intensive Course for Faculty inElectrical-, Electronics-, Communication-, and Electrical-, Electronics-, Communication-, and
Computer- Related Engineering Departments in Computer- Related Engineering Departments in Engineering Colleges in IndiaEngineering Colleges in India
byby
Nannapaneni Narayana RaoNannapaneni Narayana RaoEdward C. Jordan Professor EmeritusEdward C. Jordan Professor Emeritus
of Electrical and Computer Engineeringof Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign, USAUniversity of Illinois at Urbana-Champaign, USADistinguished Amrita Professor of EngineeringDistinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, IndiaAmrita Vishwa Vidyapeetham, India
Program for Hyderabad Area and Andhra Pradesh FacultySponsored by IEEE Hyderabad Section, IETE Hyderabad
Center, and Vasavi College of EngineeringIETE Conference Hall, Osmania University Campus
Hyderabad, Andhra PradeshJune 3 – June 11, 2009
Workshop for Master Trainer Faculty Sponsored byIUCEE (Indo-US Coalition for Engineering Education)
Infosys Campus, Mysore, KarnatakaJune 22 – July 3, 2009
6-3
Module 6Statics, Quasistatics, and
Transmission Lines6.1 Gradient and electric potential6.2 Poisson’s and Laplace’s equations6.3 Static fields and circuit elements6.4 Low-frequency behavior via quasistatics6.5 Condition for the validity of the quasistatic approximation6.6 The distributed circuit concept and the transmission-line
6-4
Instructional Objectives42. Understand the geometrical significance of the gradient operation43. Find the static electric potential due to a specified charge distribution by applying superposition in conjunction with the potential due to a point charge, and further find the electric field from the potential44. Obtain the solution for the potential between two conductors held at specified potentials, for one- dimensional cases (and the region between which is filled with a dielectric of uniform or nonuniform permittivity, or with multiple dielectrics) by using the Laplace’s equation in one dimension, and further find the capacitance per unit area (Cartesian) or per unit length (cylindrical) or capacitance (spherical) of the arrangement
6-5
Instructional Objectives (Continued)
45. Perform static field analysis of arrangements consisting of two parallel plane conductors for electrostatic, magnetostatic, and electromagnetostatic fields46. Perform quasistatic field analysis of arrangements consisting of two parallel plane conductors for electroquastatic and magnetoquasistatic fields47. Understand the condition for the validity of the quasistatic approximation and the input behavior of a physical structure for frequencies beyond the quasistatic approximation 48. Understand the development of the transmission-line (distributed equivalent circuit) from the field solutions for a given physical structure and obtain the transmission- line parameters for a line of arbitrary cross section by using the field mapping technique
6-7
Gradient and the Potential Functions
0
x y z
x y z
x y z
x y z
x y zA A A
x y z
x y z
x y zA A A
a a a
× A
× A × A × A × A
6-86-8
Since 0,B
B can be expressed as the curl of a vector.
Thus
A is known as the magnetic vector potential.
B = × A
Then
t
t
× E = × A
A×
6-9
t
AE +
is known as the electric scalar potential.
0t
A× E +
t
AE =
is the gradient of
x y z
x y z
x y z
x y z
a a + a
a a a
6-116-11
Basic definition of :
d d l
, ,P x y z
, ,Q x dx y dy z dz
x y z x y zd dx dy dzx y z
d
a a a a a a
l
For a constant surface, d = 0. Therefore is normal to the surface.
P
6-126-12
cos
cos
n
nd d
dl
d d
dl dn
a
a l
Thus, the magnitude of at any point P is the rate of increaseof normal to the surface, which is the maximum rate of increase at that point. Thus
n
d
dn
a
Useful for finding unit normal vector to the surface.
d l
0 =
QP
dn
0 d =
na
6-136-13
D5.1 Finding unit normal vectors to the surface
2 2 2, , 2 2x y z x y z
2 2 22 2 8x y z at several points:
2 2 2
2 2 2
2 2 2
2 2
2 2
2 2
4 4 2
x
y
z
x y z
x y zx
x y zy
x y zzx y z
a
a
a
a a a
6-146-146-14
( ) At 2, 2, 0 ,a
4 2 4 2
24 2 4 2
x y x yn
x y
a a a aa
a a
( ) At 1,1, 2 ,b
4 4 4
4 4 4 3x y z x y z
n
x y z
a a a a a aa
a a a
( ) At 1, 2, 2 ,c
4 4 2 2 2 2 2
74 4 2 2 2
x y z x y zn
x y z
a a a a a aa
a a a
6-166-16
Using , we obtaint
A =
22
2t A
A J
22
2t
Potential function equations
2
22
1t t
t t
AA J
AA A J
(2)
6-17
Laplacian of scalar
Laplacian of vector
In Cartesian coordinates,
2
2 A = A × × A
2 2 22
2 2 2x y z
2 2 2 2x x y y z zA A A A = a a a
6-186-18
B B
A A
B A
BA
A B
d d
d
E
E l l
also known as the potential difference between A andB, for the static case.
Voltage between and
B
A BAd V V
A B
E lBut,
For static fields, 0,t
6-19
Given the charge distribution, find V using superposition.Then find E using the above.
V VE
since
agrees with the previously known result.
For a point charge at the origin,
4
QV
r
2
4
4
r
r
r
VV
r
Q
r r
Q
r
E a
a
a
6-21
Considering the element of length dz at (0, 0, z), we have
Using
0
224
L dzdV
r z z
tan z z r 2 secd z r d
20
0
sec
4 sec
sec4
L
L
r ddV
r
d
6-226-22
for z a
1
2
1
2
0
0
0 2 2
1 1
2
0
2
sec 4
1n sec tan 4
sec tan 1n
4 sec tan
1n4
a
z a
L
L
L
2
L
2
V dV
d
r z a z a
r z a z a
6-24
Review Questions6.1. What is the divergence of the curl of a vector?6.2. What is the expansion for the gradient of a scalar in Cartesian coordinates? When can a vector be expressed as the gradient of a scalar?6.3. Discuss the basic definition of the gradient of a scalar.6.4. Discuss the application of the gradient concept for the determination of unit vector normal to a surface.6.5. Define electric potential. What is its relationship to the electric field intensity?6.6. Distinguish between voltage as applied to time-varying fields and potential difference.6.7. What is the electric potential due to a point charge? Discuss the determination of electric potential due to a charge distribution.
6-25
Review Questions (Continued)6.8. What is the Laplacian of a scalar? What is the expansion for the Laplacian of a scalar in Cartesian coordinates?6.9. What is the magnetic vector potential? How is it related to the magnetic flux density?
6-29
Problem S6.3. Finding the image charge(s) for a point charge in the presence of a conductor (Continued)
6-31Poisson’s Equation
For static electric field,
Then from
If is uniform,
D =
Poisson’s equation
0t
B
× E =
VE =
V
V
V
V
6-32If is nonuniform, then using
Thus
Assuming uniform , we have
For the one-dimensional case of V(x),
,
V V V
V V
V V
2 2 2
2 2 2
V V V
x y z
2
2
V
x
6-33
Anode, x = d V = V0
Cathode, x = 0 V = 0
Vacuum Diode
D5.7
(a) 8x dV
4 3 4 3
0 0 1 2
00 2 3
1 1
88
1
8 4
V V
VV
4 3
0
xV V
d
6-34
(b) 8 8
8
1 3
0
8
1 3
0
0
4
3
4 1
3 8
2
3
d d
xd
x
d
x
x
x x
x
x
V
V
x
V x
d d
V
d
V
d
E
a
a
a
a
6-35
(c)
8 0 8
2
0 2
8
2 3
00 2
8
2 3
00 2
2 300 2
0 02
4
9
4 1
9 8
48
916
9
d d
d
d
x x
x
x
V
V
x
V x
d d
V
d
V
dV
d
6-366-36Laplace’s Equation
Let us consider uniform first.
E6.1. Parallel-plate capacitor
If Poisson’s equation becomes
x = d, V= V0
x = 0, V = 0
0 for uniform V
0 for nonuniform V V
6-38Boundary conditions
Particular solution
0
00
0 at 0
at
0 0 0
0
V x
V V x d
B B
VV Ad A
d
0
x
x
VV
xV
d
E a
a
0VV x
d
6-39
x = d
x = 0
xa
xx a
0
0
0
0
0
for 0
for
for 0
for
for 0
for
x xS n
x x d
x x
x x
x
x d
Vx
dV
x dd
Vx
dV
x dd
a Ea D
a E
a a
a a
6-42
21 xd
V A
x
0 for 0V x
0 for V V x d
2 1n 12
xV Ad B
d
0 2 1n 1 0Ad B B
00 3
2
32 1n
2 2 1n
VV Ad A
d
0 1n 11n 1.5 2
V xV
d
6-43
0
2
1
2 1n 1.5 1
x
xxd
VV
xV
d
Ε a
a
0 0
0 00
0 0
0
0
0
2 1n 1.5
2 1n 1.5
2 1n 1.5
2 1n 1.5
2 1n 1.5
x
S Sx x d
S
V
d
V
d
V AQ A
d
AQC
V d
C
A d
D Ε = a
6-44
Review Questions6.10. State Poisson’s equation for the electric potential. How is it derived?6.11. Outline the solution of the Poisson’s equation for the potential in a region of known charge density varying in one dimension.6.12. State Laplace’s equation for the electric potential. In what regions is it valid?6.13. Outline the solution of Laplace’s equation in one dimension by considering a parallel-plate arrangement.6.14. Outline the steps in the determination of the capacitance of a parallel-plate capacitor.
6-45
Problem S6.4. Solution of Poisson’s equation for a space charge distribution in Cartesian coordinates
6-46
Problem S6.5. Finding the capacitance of a spherical capacitor with a dielectric of nonuniform permittivity
6-486-48
Classification of Fields
Static Fields ( No time variation; ) Static electric, or electrostatic fields
Static magnetic, or magnetostatic fieldsElectromagnetostatic fields
Dynamic Fields (Time-varying)
Quasistatic Fields (Dynamic fields that can be analyzed as though the fields are static) Electroquasistatic fields Magnetoquasistatic fields
0t
6-496-496-49
Static Fields
For static fields, , and the equations reduce to
D =
B 0
J 0
E d l0C
HdlC JdS
SDdS dv
VSB dS 0
SJdS 0
S
0t
0 ×E
× H J
6-506-50
Solution for charge distribution
Solution for point charge
Electric field dueto point charge
Solution for Potential and Field
1 ( )( )
4 V
V dv
r
rr r
( )( )
4
QV
rr
r r
E(r) Q( r )(r r )
4 r r 3
6-516-51
Laplace’s Equation and One-Dimensional Solution
2
20
d V
dx
Laplace’s equation
For Poission’s equation reduces to
Ax + B
2 0V
6-526-52
Example of Parallel-Plate Arrangement:Capacitance
0( ) ( )V
V x d xd
S
S
(b)
y
z
x
(a)
x
y z
w
x= 0
x = d
l
z – l
d
z z = 0
V0
z = – l z = 0z
––
– – – – – –
E, DV0
x 0, V =V0
x d, V = 0
6-536-53
D V0
dax
00( )
V wlQ wl V
d d
C QV0
wld
0x
VV
d E a
We 12
Ex2
(wld) 1
2wld
V0
2 12
CV02
Capacitance of the arrangement, F
Electrostatic Analysis of Parallel-Plate Arrangement
C w ld
6-546-54
Magnetostatic Fields
Poisson’s equation formagnetic vector potential
2A –J
B 0
Hdl JdSSC
BdS 0S
× H J
6-556-55
Solution for current distribution
Solution for current element
Magnetic field dueto current element
( )( )
4 V
dv
J r
A rr r
( )( )
4
I d
l r
A rr r
2A = 0 For current-free region
Solution for Vector Potential and Field
3
4
I d
l r × r rB r
r r
6-566-56
Example of Parallel-Plate Arrangement:Inductance
(b)x
y zx = 0
x d
z – l z = 0z
H, BI0
JS
(a)
y
z
xx = 0
x = d
z – l z = 0
l
w d
z
I0
6-576-57
0 0 0
x y z
x y z
xH H H
a a a Bxx
0
H I0w
ay
Magnetostatic Analysis of Parallel-Plate Arrangement
JS (I0 w)az on the plate x 0
(I0 w)ax on the plate z 0
(I0 w)az on the plate x d
6-58
B I0w
ay
I0w
(dl) dl
w
I0
L I0
dlw
Wm 12
H2
(wld) 1
2dlw
I0
2 12
LI02
Inductance of the arrangement, H
Magnetostatic Analysis of Parallel-Plate Arrangement (Continued)
L dlw
6-59
Electromagnetostatic Fields
D 0
B 0
(J Jc E)
Edl0C
HdlC JcdS
S EdSS
DdS 0S
BdS 0S
x E 0
x H Jc E
6-606-60
Example of Parallel-Plate Arrangement
(b)
(a)
y
z
x
x
y z
w
x = 0
x = d
l
z – l
d
z z = 0
V0
z = – l z = 0z
x d, V 0
––
– – – – – –
H, BDE,V0
Ic
x 0, V V0
Jc ,
JS
JS
S
SS
S
6-616-61
E V0d
ax
Jc V0
dax
Ic V0d
(wl) wl
dV0
Electromagnetostatic Analysis of Parallel-Plate Arrangement
6-62
G IcV0
wld
R V0Ic
dwl
Pd (E2 )(wld) wld
V0
2
GV02
V02
R
Conductance, S
Resistance, ohms
Electromagnetostatic Analysis of Parallel-Plate Arrangement (Continued)
6-636-63
0yH V
z d
Internal Inductance
Electromagnetostatic Analysis of Parallel-Plate Arrangement (Continued)
Li 1Ic
N d z l
0
H V0
dz ay
13
dlw
H Hy(z)ay
Hyd(dz 1Ic
– z l
z –l
0
6-64
Electromagnetostatic Analysis of Parallel-Plate Arrangement (Continued)
V0
R d wl
Li 13
dlw
C wld
Equivalent Circuit
Alternatively, from energy considerations,
Li 1Ic
2 (dw ) Hy2
dzzl
0
13
dlw
6-65
Review Questions6.15. Discuss the classification of fields as static, dynamic, and quasistatic fields.6.16. State Maxwell’s equations for static fields in (a) integral form, and (b) differential form.6.17. Outline the steps involved in the electrostatic field analysis of a parallel-plate structure and the determination of its capacitance.6.18. Outline the steps involved in the magnetostatic field analysis of a parallel-plate structure and the determination of its inductance.6.19. Outline the steps involved in the electromagnetostatic field analysis of a parallel-plate structure and the determination of its circuit equivalent.6.20. Explain the term, “internal inductance.”
6-66
Problem S6.6. Finding the internal inductance per unit length of a cylindrical conductor arrangement
6-68
Quasistatic Fields
For quasistatic fields, certain features can be analyzed as though the fields were static. In terms of behavior in the frequency domain, they are low-frequency extensions of static fields present in a physical structure, when the frequency of the source driving the structure is zero, or low-frequency approximations of time-varyingfields in the structure that are complete solutions to Maxwell’s equations. Here, we use the approach of low-frequency extensions of static fields. Thus, for a given structure, we begin with a time-varying field having the same spatial characteristics as that of the static field solution for the structure and obtain field solutions containing terms up to and including the first power (which is thelowest power) in for their amplitudes.
6-69
Electroquasistatic Fields
Vg t V0 cos t
x
y z
z = –l z = 0z–
– – –
Ig (t)
+–
x 0
x d
– – – –
H1
E0
JS
S
6-706-70
1 0 0 siny xH D V
tz t d
Electroquasistatic Analysis of Parallel-Plate Arrangement
E0 V0
dcos t ax
H1 V0z
dsin t ay
6-71
whereI g j CV g C wld
Electroquasistatic Analysis of Parallel-Plate Arrangement (Continued)
C w ld
Ig (t)w Hy1 z l
wld
V0 sin t
CdVg (t)
dt
6-72
Electroquasistatic Analysis of Parallel-Plate Arrangement (Continued)
Pin wd Ex0 Hy1 z0
wld
V0
2sin t cos t
ddt
12
CVg2
6-73
Magnetoquasistatic Fields
x
y zx = 0
x d
z – l z = 0z
– – – – –
–Vg (t)
E1
H0
JS
Ig t I0 cos t
S
6-746-74
01 0 sinyxBE I
tz t w
Magnetoquasistatic Analysis of Parallel-Plate Arrangement
H0 I0w
cos t ay
E1 I0z
wsin t ax
6-75
V g jL I g where
Magnetoquasistatic Analysis of Parallel-Plate Arrangement (Continued)
L dlw
L dlw
Vg (t) d Ex1 z l
dlw
I0 sin t
LdIg(t)
dt
6-76
Magnetoquasistatic Analysis of Parallel-Plate Arrangement (Continued)
Pin wd Ex1Hy0 z l
dlw
I0
2sin t cos t
ddt
12
LIg2
6-77
Quasistatic Fields in a Conductor
(b)
(a)
(c)
x
y z
z = –l z z = 0
Ex1
z = –l z z = 0
Hy 0
Ex 0
z = –l z = 0z
x 0
x d
+–
IgH0E0, J c0
Hyd1 Hyc1
Vg t V0 cos t
6-786-78
01 0 sinyxBE V z
tz t d
2 201 sin
2x
VE z l t
d
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor
Jc0 E0 V0
dcos t ax
E0 V0d
cos t ax
H0 V0z
dcos t ay
6-796-79
2 3 20 0
1
3sin sin
6y
V z zl V zH t t
d d
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued)
1 01
22 20 0sin sin
2
y xx
H EE
z t
V Vz l t t
d d
6-80
E x V gd
j 2d
z2 l2 V g
H y zd
V g j zd
V g j 2 z3 3zl2
6dV g
Hy V0 z
dcos t
V0 zd
sin t 2V0 z3 3zl2
6dsin t
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued)
Ex V0
dcos t
V0
2dz
2 l2 sin t
6-81
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued)
Y in I gV g
j wld
wld
1 j l2
3
j wld
1d
wl 1 j l2
3
I g w H y z l
wld
j wld
j 2wl
3
3d
V g
6-826-82
Equivalent Circuit
Y in j wld
1d
wl j dl3w
j C 1R jLi
V0 C wld
R d wl
Li 13
dlw
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued)
6-83
Review Questions6.21. What is meant by the quasistatic extension of the static field in a physical structure?6.22. Outline the steps involved in the electroquasistatic field analysis of a parallel-plate structure and the determination of its input behavior. Compare the input behavior with the electrostatic case.6.23. Outline the steps involved in the magnetoquasistatic field analysis of a parallel-plate structure and the determination of its input behavior. Compare the input behavior with the magnetostatic case.6.24. Outline the steps involved in the quasistatic field analysis of a parallel-plate structure with a conducting slab between the plates and the determination of its input behavior. Compare the input behavior with the electromagnetostatic case.
6-85
Problem S6.7. Frequency behavior of a capacitor beyond the quasistatic approximation (Continued)
6-86
6.5 Condition for the validity ofthe quasistatic approximation (EEE, Sec. 5.5; FEME, Secs. 6.5, 7.1)
6-87
We have seen that quasistatic field analysis of a physical structure provides information concerning the low-frequency input behaviorof the structure. As the frequency is increased beyond that for which the quasistatic approximation is valid, terms in the infinite series solutions for the fields beyond the first-order terms need to be included. While one can obtain equivalent circuits for frequencies beyond the range of validity of the quasistatic approximation by evaluating the higher order terms, we shall here obtain the exact solution by resorting to simultaneous solution of Maxwell’s equations to find the condition for the validity of thequasistatic approximation, and further investigate the behavior for frequencies beyond the quasistatic approximation. We shall do this by considering the parallel-plate structure, and obtaining the wavesolutions, which will then lead us to the distributed circuit concept and the transmission-line.
6-886-88
Wave Equation
One-dimensional wave equation
For the one-dimensional case of
E Ex z, t ax and H = Hy z, t ay ,
t t
B H
×Et t
D E
× H
yxHE
z t
y x
H E
z t
2 2
2 2x xE E
z t
6-896-89
Solution to the One-Dimensional Wave Equation
Traveling wave propagating in the +z direction
–1
1 t 0
0z
1 2 f
1 f
t 4 t
2
( , ) cos cosxE z t A t z B t z
cos t z
6-906-90
Solution to the One-Dimensional Wave Equation
Traveling wave propagating in the –z direction
t 2
t 01
–1
0–z
1 2 f
1 f
1( , ) cos cosyH z t A t z B t z
t 4
cos t z
6-916-91
E x A e j z B e j z
H y 1 A e jz B e j z
Phase constant
Phase velocity
Intrinsic impedance
pv
General Solution in Phasor Form
1vp
A Ae j
, B Be j –
,
,
,
6-926-92
s
Vg t V0 cos t
Example of Parallel-Plate StructureOpen-Circuited at the Far End
E x V g
d cos lcos z H y
jV gd cos l
sin z
H y 0 at z = 0
E x V gd
at z =l
B.C.
x
y z
z = –l z = 0z
–
– – –
Ig (t)
+–
x 0
x d
– – – –
JS
E
HS
6-94
Complete Standing Waves
Complete standing waves are characterized by pure half-sinusoidal variations for the amplitudes of the fields. For values of z at which the electric field amplitude is a maximum, the magnetic field amplitude is zero, and for values of z at which the electric field amplitude is zero, the magnetic field amplitude is a maximum. The fields are also out of phase in time, such that at any value of z, the magnetic field and the electric field differ in phase by t = / 2.
6-95
Input Admittance
For l << 1,
Y in I gV g
j wd
tan l
Y in jwd
(l)
j wld
wld
I g w H y z l
jwV g d
tan l
Y in j w
dl (l)
3
3 2(l)
5
15
6-96
Condition for the Validity of the Quasistatic ApproximationThe condition l << 1 dictates the range of validity for the quasistatic approximation for the input behavior of the structure. In terms of the frequency f of the source, this condition means that f << vp/2l, or in terms of the period
T = 1/f, it means that T >> 2(l/vp). Thus, quasistatic fields are low-frequency
approximations of time-varying fields that are complete solutions to Maxwell’s equations, which represent wave propagation phenomena and can be approximated to the quasistatic character only when the times of interest are much greater than the propagation time, l/vp, corresponding to the length of the
structure. In terms of space variations of the fields at a fixed time, the wavelength ( = 2 ), which is the distance between two consecutive points along the direction of propagation between which the phase difference is 2, must be such that l << /2 ; thus, the physical length of the structure must be a small fraction of the wavelength.
6-976-97
For frequencies slightly beyond the approximation l <<1,
Y in j wd
l (l)3
3
j wld
1 wl
d
dl
3w
1
13
1( ) 3
Zwl wl dl
jd d w
dlj
j wl d w
in
wld
13
dlw
6-99
E x jI g
w cos lsin z H y
I gw cos l
cos z
s
Ig t I0 cos t
Example of Parallel-Plate StructureShort-Circuited at the Far End
x
y zx = 0
x d
z – l z = 0z
– – – – –
–Vg (t)
JS
E
H
E x 0 at z 0
H y I gw
at z l
B.C.
6-101
Input Impedance
For l << 1,
Z in V gI g
jdw
tan l
Z in dw
(l)
j dlw
dlw
Z in j
dw
l ( l)3
3 2( l)
5
15
V g d E x z l
j dI g
wtan l
6-102
For frequencies slightly beyond the approximation l <<1,
Z in jdw
l ( l)3
3
j dlw
1 dl
w
wl
3d
Y in 1
j dlw
1 dl
w
wl
3d
1j (dl / w)
j wl3d
13
wld
dlw
6-104
Review Questions6.25. Outline the steps in the solution for the electromagnetic field in a parallel-plate structure open-circuited at the far end. 6.26. What are complete standing waves? Discuss their characteristics.6.27. What is the input admittance of a a parallel-plate structure open-circuited at the far end? Discuss its variation with frequency.6.28. State and discuss the condition for the validity of the quasistatic approximation.6.29. Outline the steps in the solution for the electromagnetic field in a parallel-plate structure short-circuited at the far end. 6.30. What is the input impedance of a a parallel-plate structure short-circuited at the far end? Discuss its variation with frequency.
6-105
Problem S6.8. Frequency behavior of a parallel-plate structure from input impedance considerations
6-106
Problem S6.8. Frequency behavior of a parallel-plate structure from input impedance considerations (Continued)
6.6 The Distributed Circuit Concept and the Transmission Line
(EEE, Secs. 6.1, 11.5; FEME, Secs. 6.5, 6.6)
6-134
Review Questions6.31. Discuss the phenomenon taking place in a parallel-plate structure at any arbitrary frequency.6.32. How is the voltage between the two conductors in a given cross-sectional plane of a parallel-plate transmission line related to the electric field in that plane?6.33. How is the current flowing on the plates across a given cross-sectional plane of a parallel-plate transmission line related to the magnetic field in that plane?6.35. Discuss transverse electromagnetic waves.6.36. What are transmission-line equations? How are they derived from Maxwell’s equations?
6-135
Review Questions6.37. Discuss the concept of the distributed equivalent circuit. How is it obtained from the transmission-line equations?6.38. Discuss the solutions for the transmission-line equations for the voltage and current along a line.6.39. Explain the “characteristic impedance” of a transmission line.6.40. Discuss the relationship between the transmission-line parameters.6.41. What are the transmission-line parameters for a parallel- plate line?6.42. Describe the curvilinear squares technique of finding the line parameters for a line with an arbitrary cross section.
6-137
Problem S6.10. Application of the curvilinear squares technique for an eccentric coaxial cable