TRANSMISSION LINE
RESONATORS
Series and Parallel Resonator Circuits
T
Zin
C
R
V
Z in  R  jL  j
L
1
C
1 * 1
Pin  VI  Z in I
2
2
Ploss 
2
1
V
 Z in
2
Z in
2
1 2
1 
 I  R  jL  j

2 
C 
1 2
I R
2
ENEE 482 Spring 2001
2
1 2
1
1 2 1
2
Wm  I L , We  Vc C  I
4
4
4
 2C
Pin  Ploss  2 j (Wm  We )
Z in 
2 Pin
I
2

Ploss  2 j (Wm  We )
2
I /2
Resonance occurs when Wm  We .
Z in 
Ploss
2
I /2
R
, 0 
1
LC
Wm  We
2Wm  0 L
1
Q  0
 0


: Quality factor
Ploss
Ploss
R
 0 RC


  2   02
1
  R  j L 
Z in  R  jL1 
 2
  LC 



ENEE 482 Spring 2001



3
 2   02  (   0 )(   0 )   (2   )
 2 for small value of 
2 RQ 
Z in  R  j 2 L  R  j
0
A resonator with loss can be treated as a lossless resonator
whose resonant frequency  0 has been replaced with a complex
effective resonant frequency :

j 

 0   0 1 
 2Q 
The half - power fractional bandwidth : 2

0
 BW .
1
BW 
Q
ENEE 482 Spring 2001
4
Parallel Resonant Circuit
1
1

1
Z in   
 jC 
 R j L

1 * 1
1
2
Pin  VI  Z in I  V
2
2
2
2
1
Z in*
1 2 1
j

 V  
 j C 
2
 R L

2
1V
1 2
1
1 2 1
2
Ploss 
, We  V C , Wm  I L L  V
2 R
4
4
4
 2L
Pin  Ploss  2 j (Wm  We )
ENEE 482 Spring 2001
5
Z in 
2 Pin
I
2

Ploss  2 j (Wm  We )
2
I /2
2Wm
R
, Q  0

  0 RC
Ploss  0 L
Let    0   , where  is small 
 1 1   /  0

Z in   
 j 0 C  jC 
j 0 L
R

1
1
j
1

1

   2  jC     2 jC 
 R  0L

R

R
R


1  2 jRC 1  2 jQ /  0
1

j 
, The half - power bandwidth occur at frequncies
 0   0 1 
 2Q 
R2
1
2
such that Z in 
,
BW 
2
Q
ENEE 482 Spring 2001
6
TRANSMISSION LINE RESONATORS
• LENGTHS OF T.L TERMINATED IN SHORT CIRCUITS
l
n g 0
2
Z
Zin
T
C
T
R
Zin
L
0
tanh   j tan 
1  j tan  tanh 
tanh    ,   1
  0  
 


vp
vp
vp
Z in  Z 0
   / 2  v p /  0 for    0
ENEE 482 Spring 2001
7


tan   tan   
0


 
  tan

0
0


  j ( /  0 )


Z in  Z 0
 Z 0   j
1  j ( /  0 )
0




1
R  Z 0l    Z 0  go
2
X  0L  1
 0C ,
Z 0
L
2 0
  0 

Z in  R  jX 

 0  
QX




R 2 2
ENEE 482 Spring 2001
8
Open Circuited line
l
n g 0
Y0
Zin
T
T
2
L
G
C
T
1  j tan  tanh 
Z in  Z 0 coth(  j )  Z 0
tanh   j tan 
   / 2 at    0
   
ENEE 482 Spring 2001

 
, tan   tan

0

0
9
Z0
Z0 
  j ( /  0 )
n
G  Y0l   Y0  go
2
B   0C  1
0L
, C

2 0 Z 0
  0 

Yin  G  jB

 0  
QB
ENEE 482 Spring 2001


  0 RC 

G
2 2
10
WAVEGUIDE RESONATORS
• RECTANGULAR WAVEGUIDE RESONATORS
RESONANT FREQUENCIES OF TEl,m,n OR Tml,m,n
m 2 n 2
 mn  k  (
) ( )
a
b
 mn d   ,   1,2,3,...
2
Y
2
2
2


cl
am
abn
2
f ab  34.82

 2 
b
c 
 a
a, b, c ARE IN INCHES
f IS IN GHz
Z
ENEE 482 Spring 2001
b
X
c
a
11
MODE CHART OF RECTANGULAR RESONATOR
WITH A/B=2
f^2a^2 [GHz In.]^2
500
450
TE101
400
TM110
350
TE011
300
250
TE111,TM111
200
TM210
150
TM112,TE112
100
TE211,TM211
50
TE212,TM212
TE012
0
0
0.5
1
1.5
2
2.5
3
3.5
4
a^2/c^2
ENEE 482 Spring 2001
12
CYLINDRICAL RESONATORS
• CYLINDRICAL WAVEGUIDE RESONATORS
RESONANT FREQUENCIES OF TEl,m,n OR Tml,m,n

z

2

2
 xl ,m 
2 2
nD
f D  139.3
 


2
L



WHERE:
xl ,m  m th ROOT OF J l'  x   0 FOR TE - MODES
L
r
q
xl ,m  m th ROOT OF J l  x   0 FOR TM - MODES
f is in GHz
D
D AND L ARE IN INCHES
ENEE 482 Spring 2001
13
MEASUREMENTS OF CAVITY COUPLING
SYSTEM PARAMETERS
Zo
r
C
+
e
-
L
Zin
CAVITY EQUIVALENT CIRCUIT
NEAR ONE OF THE RESONANCES
ENEE 482 Spring 2001
14
RESONATOR’S Q-FACTORS
2 ENERGY STORED
Q =
ENERGY DISSIPATED PER CYCLE
UNLOADED Q: Qu = 2 fo (L I2/2)/(r I2/2) = o L/r
LOADED Q : QL = o L/(r + Zo) = Qu/(1+ Zo/r)
COUPLING PARAMETER : Zo/r ; Qu = (1+ QL
EXTERNAL Q : QE = Qu/  ;QL = Qu + QE
LOADED Q: INCLUDES ALL DISSIPATION SOURCES
UNLOADED Q: INCLUDES ONLY INTERIOR DISSIPATION
SOURCES TO CAVITY COUPLING SYSTEM
ENEE 482 Spring 2001
15
CIRCUIT PARAMETERS AND DEFINITIONS
1 

Z in  r  j  L 

C 

  o 
ˆ
Z in  r  jZ o 
 
 o  
where :
L
C
1
o 
LC
Zˆ o 
ENEE 482 Spring 2001
16
RESONATOR’S INPUT REFLECTION COEFFICIENT
  o   r  Zo 
ˆ

 
r  Z o  jZ o 

ˆ 


Z
Z in  Z o
 o
 o 
 in 

Z in  Z o
  o   r  Zo 
ˆ

r  Z o  jZ o 
  


  o    Zˆ o 
ENEE 482 Spring 2001
  o 

j 

 o  
  o 
j 
 
 o  
17
DEFINITIONS AND RELATIONSHIPS
AMONG THE RESONATOR’S Q’S
Qu 
o L
r

1 L
L 1 Zˆ o
 
 
C r
r
LC r
Zˆ o
1
L
L
1
QL 





r  Zo
C r  Zo r  Zo
LC r  Z o
o L
1
L
L 1 Zˆ o
QE 





Zo
C Zo Zo
LC Z o
o L
1
1
1


QL QE Qu
ENEE 482 Spring 2001
18
AMPLITUDE MEASUREMENTS
The reflection coefficient is:
 1
1    o 

  j

 
Qu QE    o  

 in 
 1
1    o 

  j

 
 Qu QE    o  
Magnitude of the reflection coefficient is:
2
 in
2
2
 1
1    o 

  

 
Qu QE    o  


2
2
 1
1    o 

  

 
 Qu QE    o  
ENEE 482 Spring 2001
19
Reflection Coefficient At Resonance :
 1
1 



Qu QE 

o 
 1
1 



 Qu QE 
At Angular Frequency L Where:
2
  L o 
1  1
1 

  2  



QL  Qu QE 
 o  L 
The Reflection Coefficient is Given By:
2
L
2
2
2
2
 1
 1
1   1
1 
1 













Qu QE   Qu QE 
1  Qu QE 
1 1
2


 
  o
2
2
2
2
2 2
 1
 1
1   1
1 
1 

  




2

 Qu QE   Qu QE 
 Qu QE 
ENEE 482 Spring 2001
20
• MEASURE REFLECTION COEFFICIENT 0 AT RESONANCE
• DETERMINE L FROM:
1 1 2
2
L 
2

2
o
• OR USE CURVE OF L IN dB VS. o IN dB TO FIND L
• MEASURE THE FREQUENCIES  L FOR WHICH THE
REFLECTION COEFFICIENT IS EQUAL TO L
• CALCULAT QL FROM :
o L
QL  2
 L  o2
• CALCULATE QE FROM:
QE 
2QL
1  O
• THE SIGN TO USE IS DETERMINED FROM THE PHASE OF
0 USE +VE SIGN FOR r < Z0 AND -VE SIGN FOR r < Z0
ENEE 482 Spring 2001
21
LOCUS OF CAVITY IMPEDANCE ON SMITH CHART
NEAR RESONANCE
r < ZO
ENEE 482 Spring 2001
r = ZO
r > ZO
22
R.L. at fLVs. R.L. at fo
Return Loss at fl [dB]
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.0
5.0
10.0
15.0
20.0
25.0
30.0
Return Loss at fo [dB]
ENEE 482 Spring 2001
23
(Magnitude of Roh at
fL)^2
Reflection Coefficient for Amplitude
Measurements
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.00
0.20
0.40
0.60
0.80
1.00
(Magnitude of Roh at fo)^2
ENEE 482 Spring 2001
24
1.00
0.90
0.80
|roh|^2
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.985
0.990
0.995
1.000
1.005
1.010
1.015
Normalized Freq.
Fig. 2 Magnitude of Ref. Coeff. Squared Vs. Freq.
ENEE 482 Spring 2001
25
Phase (Degrees)
-60.00
-110.00
-160.00
-210.00
-260.00
-310.00
-360.00
0.985
0.990
0.995
1.000
1.005
1.010
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
1.015
Amplitude of Refl. Coeff
Squared
-10.00
Normalized Frequency
Fig. 4 Reflection Coeff. Magnitude & Phase for
Qu>QE
ENEE 482 Spring 2001
26
20.00
10.00
0.00
-10.00
-20.00
-30.00
0.985
0.990
0.995
1.000
1.005
1.010
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
1.015
Amoplitude of Refl. Coeff
Squared
Phase (Degrees)
30.00
Normalized Frequency
Fig. 3 Reflection Coeff. Magnitude & Phase for
QE>Qu
ENEE 482 Spring 2001
27
PHASE MEASUREMENTS
• MORE SUITABLE FOR LOW Q ( TIGHTLY COUPLED )
SYSTEMS
• AT FREQUENCY SHIFT u = fo / (2 Qu ) , THE IMPEDANCE IS:
Zu = r + j r
• INTERSECTION OF THE LOCUS OF Zu WITH THE LOCUS
OF THE CAVITY IMPEDANCE DETERMINES A POINT Pu
• MEASUREMENT OF u AND THE RESONANT FREQUENCY
fo YIELDS THE VALUE OF Qu = fo /( 2 u )
• AT FREQUENCY SHIFT L = fo / (2 QL ) , THE IMPEDANCE IS:
ZL = r + j(Zo + r )
• INTERSECTION OF THE LOCUS OF ZL WITH THE LOCUS
OF THE CAVITY IMPEDANCE DETERMINES A POINT PL
• MEASUREMENT OF L AND THE RESONANT FREQUENCY
fo YIELDS THE VALUE OF QL = fo /( 2 L )
ENEE 482 Spring 2001
28
PHASE MEASUREMENTS (ctd.)
• LOCUS OF Zu ON THE SMITH CHART CAN BE SHOWN TO
HAVE THE EQUATION:
X2 + ( Y + 1 ) 2 = 2
WHERE X = Re Y = Im LOCUS OF Zu IS A CIRCLE
OF CENTER (0,-1) AND RADIUS (2)1/2
• LOCUS OF ZL ON THE SMITH CHART CAN BE SHOWN TO
HAVE THE EQUATION:
X + Y =1
WHICH IS A STRAIGHT LINE OF SLOPE -1, PASSING
THROUGH THE POINTS (1,0) AND (0,1)
ENEE 482 Spring 2001
29
Phase Measurements
PL
Pu
r=0
r=
8
Locus of ZU
Locus of ZL
Zo
Locus of Zin
  o 
Z in  r  jZˆ o 
 

 o  
ENEE 482 Spring 2001
30