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ENE 428
Microwave
Engineering
Lecture 11 Excitation of Waveguides
and Microwave Resonator
1
Excitation of WGs-Aperture coupling
coupling aperture
wg1
feed wg
cavity
wg2
(a)
coupling aperture
(b)
microstrip1
er
er
Ground
plane
wg
stripline
er
microstrip2
(c)
(d)
 WGs can be coupled through small apertures such as for
directional couplers and power dividers
2
A small aperture can be represented as an
infinitesimal electric and/or magnetic dipole.
Fig 4.30
 Both fields can be represented by their respective
polarization currents.
 The term ‘small’ implies small relative to an electrical
wavelength.
3
Electric and magnetic polarization
ˆ n ( x  x0 ) ( y  y0 ) ( z  z0 ),
Pe  e 0e nE
Pm  m H t ( x  x0 ) ( y  y0 ) ( z  z0 ).
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.
Aperture shape
e
m
Round hole
2r03
3
 ld 2
16
4r03
3
 ld 2
16
Rectangular slot
(H across slot)
4
Electric and magnetic polarization can be
related to electric and magnetic current
sources, respectively
From Maxwell’s equations, we have
  E   j B  M   j0 H  j0 P m  M
  H  j D  J  je 0 E  j P e  J
j0 Pm
Thus sinceM andJ has the same role as
we can define equivalent currents as
j P e
and
,
J  j P e
and
M  j0 Pm
5
Coupling through an aperture in the broad
wall of a wg (1)
y
y
2b
4
3
b
1
2
z
0
a/2
a
x
 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.
6
Coupling through an aperture in the
broad wall of a wg (2)
 The incident fields can be written as
 x  j z
E y  A sin
e ,
a
A
 x  j z
Hx 
sin
e .
Z10
a
 The excitation field a the center of the aperture at x =
a/2, y = b, z = 0 can be calculated.
E y  A,
A
Hx 
.
Z10
7
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
Pe  e 0 e nEn ( x  x0 ) ( y  y0 ) ( z  z0 ),
Pm  m H t ( x  x0 ) ( y  y0 ) ( z  z0 ).
Note that we have excited both an electric and a magnetic
dipole.
a
J y  je 0 e A ( x  ) ( y  b) ( z ),
2
j0 m A
a
Mx 
 ( x  ) ( y  b) ( z ).
Z10
2
8
Coupling through an aperture in the
broad wall of a wg (4)
 Let the fields in the upper guide be expressed as
 x j z
E y  A sin
e , for z  0,
a

A
 x j z
H x 
sin
e , for z  0,
Z10
a

y

E  A sin
x
a
e j z ,


A
 x  j z

Hx 
sin
e ,
Z10
a
for
for
z  0,
z  0,
where A+, A- are the unknown amplitudes of the forward
and backward traveling waves in the upper guide,
respectively.
9
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
0 m
1
 j A



An 
(e 0e  2 ),
V ( E y J y  H x M x )dv 
P10
P10
Z10
and for the backward waves as
0 m
1
 j A



(e 0e  2 ),
V ( E y J y  H x M x )dv 
P10
P10
Z10
ab
P

.
where 10
Z10
An
10
Microwave Resonator
 A resonator is a device or system that exhibits resonance
or resonant behavior, that is, it naturally นoscillates 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.
11
Basic characteristics of series RLC
resonant circuits (1)
L
R
Zin
C
AC
I
 The input impedance is
1
Zin  R  j L  j
.
C
 The complex power delivered to the resonator is
1
1 2
1
Pin  VI   I ( R  j L  j
).
2
2
C
12
Basic characteristics of series RLC
resonant circuits (2)
 The power dissipated by the resistor, R, is
1 2
Ploss  I R.
2
 The average magnetic energy stored in the inductor, L, is
1 2
Wm  I L.
4
 The average electric energy stored in the capacitor, C, is
1 2
1 2 1
We  Vc C  I
.
2
4
4
C
 Resonance occurs when the average stored magnetic and
electric energies are equal, thus
Zin 
Ploss
 R.
1 2
I
2
13
The quality factor, Q, is a measure of
the loss of a resonant circuit.
 At resonance,
0 
1
LC
 Lower loss implies a higher Q
 the behavior of the input impedance near its resonant
frequency can be shown as
Z in
R j
2 RQ
0
.
14
A series resonator with loss can be
modeled as a lossless resonator
 0 is replaced with a complex effective resonant
frequency.

j 
0  0 1 
.
 2Q 
Then Zin can be shown as
Zin  j 2L(  0 ).
 This useful procedure is applied for low loss resonators by
adding the loss effect to the lossless input impedance.
15
Basic characteristics of parallel RLC
resonant circuits (1)
I
Zin
AC
L
C
R
 The input impedance is
1
1

1
Zin   
 jC  .
 R j L

 The complex power delivered to the resonator is
1
1 2 1
j
Pin  VI   V ( 
 jC ).
2
2
R L
16
Basic characteristics of parallel RLC
resonant circuits (2)
 The power dissipated by the resistor, R, is
2
V
1
Ploss 
.
2 R
 The average magnetic energy stored in the inductor, L, is
1 2
1 2 1
Wm  I L L  V
.
2
4
4
 L
 The average electric energy stored in the capacitor, C, is
1 2
We  V C.
4
 Resonance occurs when the average stored magnetic and
electric energies are equal, thus
Ploss
Zin 
 R.
1 2
I
2
17
The quality factor, Q, of the parallel
resonant circuit
 At resonance,
1
0 
LC
Q increases as R increases
 the behavior of the input impedance near its resonant
frequency can be shown as
Zin
R
R

.
1  2 j  RC 1  2 jQ / 0
18
A parallel resonator with loss can be
modeled as a lossless resonator.
 0 is replaced with a complex effective resonant
frequency.

j 
0  0 1 
.
 2Q 
Then Zin can be shown as
Zin 
1
j 2C (  0 )
.
19
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.
Resonant
circuit Q
RL
 The effective resistance is the combination of R and the
load resistor RL.
20
The external quality factor, Qe, is
defined.
 0 L
for series circuits
 R
Qe  
 RL
for parallel circuits
 0 L
 Then the loaded Q can be expressed as
1
1 1

 .
QL Qe Q
21
Transmission line resonators: Shortcircuited /2 line (1)
Z0,,
Zin
l
 The input impedance is
tanh  l  j tan  l
Zin  Z0 tanh(  j  )l  Z0
.
1  j tan  l tanh  l
22
Transmission line resonators: Shortcircuited /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,
  l l
l  l  0 
.
vp vp
vp
 For  = 0, we have
Zin
or
 l  j ( / 0 )
Z0
1  j ( / 0 ) l

Z0 ( l  j
)
0
Zin  R  j 2L.
23
Transmission line resonators: Shortcircuited /2 line (3)
 This resonator resonates for  = 0 (l = /2) and its
input impedance is
Zin  R  Z0 l.
 Resonance occurs for l = n/2, n = 1, 2, 3, …
 The Q of this resonator can be found as
Q
0 L
R




.
2 l 2
24
Transmission line resonators: Shortcircuited /4 line (1)
 The input impedance is
1  j tanh  l cot  l
Zin  Z0
.
tanh  l  j cot  l
 Assume tanhl  l for small loss, it gives
Zin
 l  j l  / 20 )
Z0
 l  j / 20 )
Z0

( l  j
)
20
.
 This result is of the same form as the impedance of a
parallel RLC circuit
1
Z in
.
1
 2 j C
R
25
Transmission line resonators: Shortcircuited /4 line (2)
 This resonator resonates for  = 0 (l = /4) and its
input impedance is
Z0
Zin  R  .
l
 The Q of this resonator can be found as


Q  0 RC 

.
4 l 2
26