Post on 06-Jul-2021
1
ENE 428
Microwave
Engineering
Lecture 11 Excitation of Waveguides and Microwave Resonator
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)
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.
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
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
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
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 z
y
j z
x
xE A e
a
A xH e
Z a
10
,
.
y
x
E A
AH
Z
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 neP nE x x y y z ze
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 z
y
j z
x
j z
y
j z
x
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
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 for
the 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 e
j AA E J H M dv
P P Z
e
00 2
10 10 10
1( ) ( ),m
Vn y y x x e
j AA E J H M dv
P P Z
e
10
10
.ab
PZ
))2cos(1(2
1sin 2
11
(note: details of integration calculation from the previous slide….
dxdya
x
Za
xdszheP )sin(
1)sin(2ˆ2
10
101010
aa b
dxa
x
Z
bdydx
a
x
Z 010
0 0
2
10
)]2
cos(1[2
12)(sin
2
10Z
ab
Vxxyyn dvMHJE
PA )(
1
10
dvzbya
xAjea
xAdvJE eo
zj
yy )()()2
()sin( e
dzzedybydxa
xa
xAAj zj
eo )()()2
()sin(
e
Ajea
aAj eo
j
eo e
e 01)2
sin(
12
dvzbya
xAZ
je
a
x
Z
AdvMH mozj
xx )()()2
()sin(1010
dzzedybydxa
xa
xj
Z
A zj
mo )()()2
()sin(2
10
mo
j
mo jZ
Ae
a
aj
Z
A
2
10
0
2
10
1)2
sin(
Vxxyyn dvMHJE
PA )(
1
10
][2
10
10moeo j
Z
AAj
ab
Ze
)(1
2
1010 ZAj
P
moeo
e
13
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.
14
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 j
C
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 I
C
VjCdt
dVCI )(
)( jC
IV
C
I
jC
IV
)(
16
2.
1
2
lossin
PZ R
I
Resonance occurs when the average stored magnetic and electricenergies are equal, thus
)(2 emlossin WWjPP
2
)(2222
I
WWjP
I
PZ emlossin
in
LCo
1At resonance:
em WW and so
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())((22
ooo
o
in
RQjRLjRZ
22
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
(2Q
jLjZ ooin
)(2 oo LjQ
L
LjR 2
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
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
0
1
LC
o
in
RQjRLjRZ
22
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
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 2in
jP VI V j C
R L
Anti-resonance
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
.2
loss
VP
R
2 2
2
1 1 1.
4 4m LW I L V
L
21.
4eW V C
IjLdt
dILV )(
L
VI
24
2.
1
2
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 emlossin
in
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!
25
At “near” resonance, letting , so
1))()(
11(
Cj
LjRZ o
o
in
Zin can be simplified using , where ...11
1
x
x
1))(
)1(
11(
Cj
LjR
o
o
o
o
x
1))1(1
(
CjCjLjR
Z o
o
oin
o
1
2)
11(
CjCj
LjLjRo
oo
But CjLj
o
o
1
and at resonance,LC
o
1
26
so 1
2)
11(
CjCj
LjLjRZ o
oo
in
RCj
RCj
R
21)2
1( 1
1
2)
11(
CjCj
L
j
LjRo
oo
1
2)
)1(
1(
Cj
LLC
j
R
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
28
Let’s replace with , soo )
21(
Q
jo
)2
1(2
1
)(2
1
Q
jjC
jCZ
oo
in
)2
(2
1
Q
jjC o
o
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
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
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
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.
RL
Resonant
circuit Q
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
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)
34
Transmission line resonators: Short-circuited /2 line
(1b)
At , the transmission line length (l): and 2
l
2where 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
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
)/(
o
o
o
ooin jlZ
lj
jlZZ
1)/( ol
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
o
2
1
l
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
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: )
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 .in
ZZ R
l
0 .4 2
Q RCl
ooZC
4
CL
o
2
1
2l
40