UNIT 9 Applications of Thermodynamics -10 ENE ENE ENE ENE ...
ENE 428 Microwave Engineering
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1
ENE 428Microwave Engineering
Lecture 11 Excitation of Waveguides and Microwave Resonator
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2
Excitation of WGs-Aperture coupling
WGs can be coupled through small apertures such as for directional couplers and power dividers
(a)
wg1
wg2
coupling aperture
feed wg cavity
(b)
coupling aperture microstrip1
microstrip2
Ground planeer
er er
wg stripline
(c) (d)
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3
A small aperture can be represented as an infinitesimal electric and/or magnetic dipole.
Both fields can be represented by their respective polarization currents.
The term ‘small’ implies small relative to an electrical wavelength.
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4
Electric and magnetic polarization
Aperture shape
e m
Round hole
Rectangular slot
(H across slot)
0 0 0 0ˆ ( ) ( ) ( ),e e nP nE x x y y z ze ))))))))))))))
0 0 0( ) ( ) ( ).m tmP H x x y y z z ))))))))))))))))))))))))))))
e is the electric polarizability of the aperture.m is the magnetic polarizability of the aperture.(x0, y0, z0) are the coordinates of the center of the aperture.
302
3
r 304
3
r
2
16
ld 2
16
ld
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5
From Maxwell’s equations, we have
Thus since and has the same role as and , we can define equivalent currents as
and
Electric and magnetic polarization can be related to electric and magnetic current sources, respectively
0 0
0
m
e
E j B M j H j P M
H j D J j E j P J
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
e
M))))))))))))))
J))))))))))))))
0 mj P))))))))))))))
ej P))))))))))))))
eJ j P))))))))))))))))))))))))))))
0 mM j P))))))))))))))))))))))))))))
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6
Coupling through an aperture in the broad wall of a wg (1)
Assume that the TE10 mode is incident from z < 0 in the lower guide and the fields coupled to the upper guide will be computed.
2b
1 2
34
z
y y
xaa/20
b
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7
Coupling through an aperture in the broad wall of a wg (2) The incident fields can be written as
The excitation field a the center of the aperture at x = a/2, y = b, z = 0 can be calculated.
10
sin ,
sin .
j zy
j zx
xE A e
aA x
H eZ a
10
,
.
y
x
E A
AH
Z
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8
Coupling through an aperture in the broad wall of a wg (3) The equivalent electric and magnetic dipoles for
coupling to the fields in the upper guide are
Note that we have excited both an electric and a magnetic dipole.
0
0
10
( ) ( ) ( ),2
( ) ( ) ( ).2
y e
mx
aJ j A x y b z
j A aM x y b z
Z
e
0 0 0( ) ( ) ( ).m tmP H x x y y z z ))))))))))))))))))))))))))))
0 0 0 0( ) ( ) ( ),e ne
P nE x x y y z ze ))))))))))))))
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9
Coupling through an aperture in the broad wall of a wg (4) Let the fields in the upper guide be expressed as
where A+, A- are the unknown amplitudes of the forward and backward traveling waves in the upper guide, respectively.
10
10
sin , 0,
sin , 0,
sin , 0,
sin , 0,
j zy
j zx
j zy
j zx
xE A e for z
a
A xH e for z
Z a
xE A e for z
a
A xH e for z
Z a
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10
Coupling through an aperture in the broad wall of a wg (5) By superposition, the total fields in the upper guide due
to the electric and magnetic currents can be found forthe forward waves as
and for the backward waves as
where Note that the electric dipole excites the same fields in both directions but the magnetic dipole excites oppositely polarized fields in forward and backward directions.
00 2
10 10 10
1( ) ( ),m
Vn y y x x ej A
A E J H M dvP P Z
e
00 2
10 10 10
1( ) ( ),m
Vn y y x x ej A
A E J H M dvP P Z
e
1010
.ab
PZ
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))2cos(1(2
1sin 2
11
(note: details of integration calculation from the previous slide….
dxdya
x
Za
xdszheP )sin(
1)sin(2ˆ2
10101010
aa b
dxa
x
Z
bdydx
a
x
Z 010
0 0
2
10
)]2
cos(1[2
12)(sin
2
10Z
ab
V xxyyn dvMHJE
PA )(
1
10
dvzbya
xAjea
xAdvJE eo
zjyy )()()
2()sin( e
dzzedybydxa
xa
xAAj zj
eo )()()2
()sin( e
Ajea
aAj eo
jeo ee 01)
2sin(
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12
dvzbya
xAZ
je
a
x
Z
AdvMH mozjxx )()()
2()sin(
1010
dzzedybydxa
xa
xj
Z
A zjmo )()()
2()sin(
210
moj
mo jZ
Ae
a
aj
Z
A 2
10
02
10
1)2
sin(
V xxyyn dvMHJE
PA )(
1
10
][2
10
10moeo j
Z
AAj
ab
Z e
)(1
21010 Z
AjP
moeo
e
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Microwave Resonator A resonator is a device or system that exhibits
resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequency, with greater amplitude than at others.
Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal.
The operation of microwave resonators is very similar to that of the lumped-element resonators of circuit theory.
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Basic characteristics of series RLC resonant circuits (1)
The input impedance is
The complex power delivered to the resonator is
1.inZ R j L j
C
21 1 1( ).
2 2inP VI I R j L jC
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15
Basic characteristics of series RLC resonant circuits (2) The power dissipated by the resistor, R, is
The average magnetic energy stored in the inductor, L, is
The average electric energy stored in the capacitor, C, is
Because
21.
2lossP I R
21.
4mW I L
2 2
2
1 1 1.
4 4e cW V C IC
VjCdt
dVCI )(
)( jC
IV C
I
jC
IV
)(
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2.
12
lossin
PZ R
I
Resonance occurs when the average stored magnetic and electric energies are equal, thus
)(2 emlossin WWjPP
2
)(2222
I
WWjP
I
PZ emlossinin
LCo
1At resonance:
em WW and so
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17
Q is “quality factor” which is a measure of the loss of a resonance circuit. High Q means low loss.
Therefore, at resonance,
so that Q decreases as R increases.
At “near” resonance: where is small,
RCR
L
P
WQ
o
o
loss
mo
12
o
)()1()1
1(2
22
2
2
2
oo
in LjRLjRLC
LjRZ
2)2())((22ooo
oin
RQjRLjRZ
2
2
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Substitute with , so :
18
A resonator with loss can be treated as a “lossless” resonator with
replaced by . For example, Zin with no loss (R = 0) is
o
)2
1(Q
jo
))((
)()1
(1 2
oooin jLjL
LCjL
CjLjZ
)(22
ojLjL
)2
1(Q
jo o )
2(2
QjLjZ o
oin
)(2 oo LjQ
L
LjR 2
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19
Half-power fractional bandwidth of resonator
When frequency is such that , the average (real) power
delivered to the circuit is that delivered at resonance.
222RZ in
2
1
o
BW
2
222)( RBWjRQR and
QBW
1so
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20
The quality factor, Q, is a measure of the loss of a resonant circuit.
At resonance,
Lower loss implies a higher Q
the behavior of the input impedance near its resonant frequency can be shown as
01
LC
oin
RQjRLjRZ
2
2
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21
A series resonator with loss can be modeled as a lossless resonator 0 is replaced with a complex effective resonant
frequency.
Then Zin can be shown as
This useful procedure is applied for low loss resonators by adding the loss effect to the lossless input impedance.
0 0 1 .2
j
Q
02 ( ).inZ j L
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22
Basic characteristics of parallel RLC resonant circuits (1)
The input impedance is
The complex power delivered to the resonator is
11 1
.inZ j CR j L
21 1 1( ).
2 2inj
P VI V j CR L
Anti-resonance
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23
Basic characteristics of parallel RLC resonant circuits (2) The power dissipated by the resistor, R, is
The average magnetic energy stored in the inductor, L, is
Because , so
The average electric energy stored in the capacitor, C, is
21
.2loss
VP
R
2 2
2
1 1 1.
4 4m LW I L VL
21.
4eW V C
IjLdt
dILV )(
L
VI
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24
2.
12
lossin
PZ R
I
Resonance occurs when the average stored magnetic and electric energies are equal, thus
)(2 emlossin WWjPP
2
)(2222
I
WWjP
I
PZ emlossinin
LCo
1At resonance:
em WW and so
Parallel resonance circuit:
RCL
R
P
WQ o
oloss
mo
2The quality factor:
so Q increases as R increases!
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25
At “near” resonance, letting , so
1))()(
11(
Cj
LjRZ o
oin
Zin can be simplified using , where ...11
1
x
x
1))()1(
11(
Cj
LjR o
oo
o
x
1))1(1
(
CjCjLjR
Z oo
oin
o
12
)11
(
CjCjLjLjR o
oo
But CjLj o
o
1
and at resonance,LC
o
1
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26
so 12
)11
(
CjCjLjLjR
Z ooo
in
RCj
RCj
R
21)2
1( 1
12
)11
(
CjCjL
j
LjR ooo
1
2)
)1(
1(
Cj
LLC
j
R
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27
The effect of the loss can be accounted for by replacing with .o )2
1(Q
jo
Since , so for lossless Zin (at R = 0), LC
o
1
12
1 )1
()1
()(
Lj
LCCj
LjlosslessZ in
))(()(1
1
2
22
2
2
22
oo
o
o
o
o
LjLjLj
LC
Lj
oo
o
LCj
LLj
22
2
)(2
1
2
1
ojCjC
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28
Let’s replace with , soo )2
1(Q
jo
)
21(2
1
)(2
1
Qj
jCjC
Z
oo
in
)2
(2
1
Qj
jC oo
o
o
oo Qj
CQ
jCC
/21
)/(
)(2
1
)caselossy (/21 in
o
ZjQ
R
Half-power bandwidth edges occur at frequency such that
Therefore,
2
22 R
Z in
QBW
1
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29
The quality factor, Q, of the parallel resonant circuit At resonance,
Q increases as R increases
the behavior of the input impedance near its resonant frequency can be shown as
RCj
RCj
RZ in
21)2
1( 1
LCo
1
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30
A parallel resonator with loss can be modeled as a lossless resonator. 0 is replaced with a complex effective resonant
frequency.
Then Zin can be shown as
0 0 1 .2
j
Q
0
1.
2 ( )inZ j C
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31
Loaded and unloaded Q
An unloaded Q is a characteristic of the resonant circuit itself.
A loaded quality factor QL is a characteristic of the resonant circuit coupled with other circuitry.
The effective resistance is the combination of R and the load resistor RL.
RLResonant circuit Q
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32
The external quality factor, Qe, is defined.
Then the loaded Q can be expressed as
0
0
eL
Lfor series circuits
RQ
Rfor parallel circuits
L
1 1 1.
L eQ Q Q
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33
Transmission line resonators: Short-circuited /2 line (1)
Assuming the line is “lossy” so the input impedance is
0 0
tanh tantanh( ) .
1 tan tanhin
l j lZ Z j l Z
j l l
Note:)tanh()tanh(1
)tanh()tanh()tanh(
BA
BABA
, and )tan( ljZZ oin if = 0 (lossless)
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34
Transmission line resonators: Short-circuited /2 line (1b)At , the transmission line length (l): and
2
l
2
where and vp is the phase velocity of transmission line
o
At near resonance, , so o
pv
p
o
p v
l
v
ll
)(
opp
o
pp
o
vvv
l
v
l
2
2
22
ooo
l
)tan()tan()tan(and so
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35
Transmission line resonators: Short-circuited /2 line (2) For a small loss TL, we can assume l << 1 so
tan(l) l. Now let = 0+ , where is small. Then, assume a TEM line,
For = 0, we have
(because )
which can be written in the form
which is the Zin of series RLC
2 .inZ R j L
pp
o
p
o
p v
l
v
l
v
l
v
ll
)(
)()/(1
)/(
oo
o
ooin jlZ
lj
jlZZ
1)/( ol
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36
Transmission line resonators: Short-circuited /2 line (3) This resonator resonates for = 0 (l = /2) and its
input impedance is
where and
Resonance occurs for l = n/2, n = 1, 2, 3, …
The Q of this resonator can be found as
since at 1st resonance
0 .inZ R Z l
0 .2 2
LQ
R l
o
oZL
2
LC
o2
1
l
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37
Transmission line resonators: Short-circuited /4 line (1) The input impedance is
0
1 tanh cot.
tanh cotin
j l lZ Z
l j l
)cot(
)cot(
)tan()tanh(1
)tan()tanh(
lj
lj
ll
ljlZZ oin
At , the transmission line length (l): and 4
l
At near resonance, , so o
o pv
p
o
p v
l
v
ll
)(
opp
o
pp
o
vvv
l
v
l
24
2
44
ooo
l
2)
2tan()
22cot()cot(
and so
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38
Assume tanh(l) l for small loss, it gives
(because )
This result is of the same form as the impedance of a parallel RLC circuit
CjR
Z in
21
1
)2
()2/(
)2/1(
o
o
o
ooin j
l
Z
jl
ljZZ
1)2/( ol
)cot()cot(
1)cot()cot()cot(
BA
BABA
(Note: )
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39
Transmission line resonators: Short-circuited /4 line (2) This resonator resonates for = 0 (l = /4) and its
input impedance is
where and
The Q of this resonator can be found as
because at resonance.
0 .inZ
Z Rl
0 .4 2
Q RCl
ooZC
4
CL
o2
1
2
l
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40