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1. Please write the results of following identities, in which A and B are vector
functions, and ¢ and If! are scalar functions. (20 %
V'(AxB)=
(d) V(¢IfI)=
(g) VX(Vx A) =
U) V(A.B)=
(a)
(b)
(e)
(h)
)
V(V.A) =
Vx(V¢}=
V.(VxA) =
(c)
(t)
(i)
VX(¢A)=
V.(¢A}=
Vx (A x B) =
2. Two coupled inductors of inductances L 1 and L2 and coupling coefficient k if they
C
N
are connected
(a) in a series configuration as shown in Fig.2a, (80/0) and
(b) in a parallel configuration as shown in Fig.2b (8 %
) .
Please find the equivalent inductance (Leq) in terms of L 1, L 2 and k as shown in
Fig.2c.
+ •
vI
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+
J...
+
v
v
•
ti
Iii
Lt.
2
+
~
v
k
...........--.
.
..... 1
•
Fig.2a
Fig.2b
3. (a) A symmetrical two-wire transmission line with permittivity
Fig.2c
&0 in
the air as shown
in Fig.3a. The charge per unit length of line is Q', and the radii of the wire
conductors,
G,
are much smaller than the distance between the conductor axes, d.
Please calculate the capacitance per unit length of the line. (8 %
)
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s: 'f ~k~ 100~if-li~~±Jjl{?3 !£:Af~s\~s\~
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(b) Based on previous result in (a), consider a transmission line that consists of a
wire conductor and a grounded conducting plane as depicted in Fig.3b. The medium
is air, the wire is parallel to the plane, the height of the wire axis with respect to the
grounded plane is h, and the wire radius is a (a « h). Please find the capacitance
per unit length of the line. (6%.)
E1
E2
N
Q'
h
M2
x
~
V
d-x
Fig.3a
Fig.3b
C
4. (a) Referring the impedance Smith chart given in Fig.4a and the point
ZL
marked in
this figure, if the possible equivalent circuit are shown in Fig.4b or Fig.4c, in which
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the possible value for R is 31.2Sn or 80n, the possible value for L is 1.S91SnH or
4.2441nH, and the possible value for C is 0.6632pF or 1.7684pF, please construct an
equivalent circuit representing ZL using the circuit in Fig.4b or Fig.4c and parameters
described above. (Assume that the characteristic impedance is son, and the
operation frequency is 3GHz). Explain your answer. (100/0)
(b) Following the condition in (a), if the load impedance
ZL
can be equivalent by a
section of transmission line with a pure resistive load, find the length of the
transmission line and the terminating resistance. (80/0)
R
Fig.4a
Fig.4b
R
Fig.4c
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±- J}1fg !£ :3lf~~~~~
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:
it~J:..~:f:~ T~Jl
5. A conducting sliding bar oscillates over two parallel conducting rails in a
sinusoidally varying magnetic field:
13 = ii)Ocos OJ t
(mT),
as shown in Fig.5. The position of the sliding bar is given by x = 0.5(1 - cosOJt) (m),
and the rails are terminated in a resistance R = 0.2 (n). Find i. (100/0)
Yi
i
0
0
0.2 (m)
o
N
°l~
~
0
0
0
0
R
8
0
-----. x
l.O(m)
C
·1
Fig.5
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6. A 3 (GHz) , y-polarized uniform plane wave propagates in the + x-direction in a
nonmagnetic medium having a dielectric constant 2.5 and a loss tangent 10-2 .
(a) Determine the distance over which the amplitude of the propagating wave will be
cut in half. (5 °/0)
(b) Determine the intrinsic impedance, the wavelength, the phase velocity, and the
group velocity of the wave in the medium. (12 % )
(c) Assuming
E = iiy10sin( 6Jl' x 109 t + Jl' 13)
expression for H for all t and x. (5°1o)
(Vim) at x
= 0, write the instantaneous