Lecture 4

Microwave Applications

Resonators

Amr M. Ezzat Safwat, Ph.D.

Professor
Ain Shams University
Cairo Egypt

Course Content
 Review and introduction
 Planar transmission lines
 Network theory
 Resonators
 Filter design
 Microwave components
 Matching networks
 Amplifier design
 RF measurement

Microwave resonators Slide 2

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 Lecture outline

•Series and Parallel Resonant Circuits

•Transmission Line Resonators
•Rectangular Waveguide Cavity Resonators

Microwave resonators Slide 3

Series Resonant Circuit

1
Z in  R  jL  j
C
1 1 1
Pin  VI *  | I |2 ( R  jL  j )
2 2 C
1 2 1
Ploss  |I| R Wm  L | I |2
2 4

1 1 1
We  | Vc |2 C  | I |2 2
4 4 C

Pin  Ploss  2 j (Wm  We )

Microwave resonators Slide 4

2
 Series Resonant Circuit
1 2 1 1
Resonance occurs when W e=Wm | I | 2  | I |2 L
4 0 C 4
1
0 
LC

Quality Factor
Q is 2 times the ratio of the total energy stored
divided by the energy lost in a single cycle or Energy sto red W  We
equivalently the ratio of the stored energy to the
Q   m
Power loss Ploss
energy dissipated over one radian of the
oscillation

20Wm 20We 0 L 1
At resonance Q0    
Ploss Ploss R 0 RC

Microwave resonators Slide 5

Series Resonant Circuit

0 L 1 1 1  2  02
Q0   Z in  R  jL  j  R  jL(1  2 )  R  jL( )
R 0 RC C  LC 2

Near to resonance (=0+)

(  0 )(  0 )  (20   ) 20 
Z in  R  jL( )  R  jL( )  R  jL )
 2
 2

20  2 RQ
Z in  R  jL ) R j
 
Half-power fractional bandwidth ( |Zin|2=2R2)
2 2 RQ
Set 0 and BW   Z in  R  j  R  jRQBW
0 0

2 1
| Zin |2 | R  jRQBW |2  ( R 2  ( RQBW ) 2 )  2R 2 BW  
0 Q

Microwave resonators Slide 6

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 Parallel Resonant Circuit

Assignment: Find Q and BW for the parallel resonant circuit.

Microwave resonators Slide 7

Loaded and Unloaded Q

•The unloaded Q, Q0, is a characteristic of
the resonator itself, in the absence of any
loading effects caused by external
circuitry.

•In practice a resonator is invariably

coupled to other circuitry, which will have
the effect of lowering the overall, or
loaded Q, QL , of the circuit.

•If the resonator is a series RLC circuit,

the load resistor RL adds in series with R,
so the effective resistance is R + RL.

•If the resonator is a parallel RLC circuit,

the load resistor RL combines in parallel
with R, so the effective resistance in
•is RRL/(R + RL ).
Microwave resonators Slide 8

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Microwave resonators Slide 9

Transmission Line resonators

Ideal lumped circuit elements are often unattainable at
microwave frequencies, so distributed elements are frequently
used.

1. Half wavelength resonator

2. Quarter wavelength resonator

Microwave resonators Slide 10

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Half-wavelength Short Circuit resonators
Zin  Z 0 tanh(  j )l

tanh(l )  j tan(l )
Z in  Z 0
1  j tan(l ) tanh(l )

Near to resonance (=0+)

l (0   )l l 
l      
v v v 0
 
tan l  tan(  )
0 0
Please note that  is small as well
tanh(l )  j tan(l ) l  j ( / 0 ) 
Z in  Z 0  Z0  Z 0 (l  j )
1  j tan(l ) tanh(l ) 1  j ( / 0 )l 0

Microwave resonators Slide 11

Half wavelength Short Circuit resonators

tanh(l )  j tan(l ) l  j ( / 0 ) 
Z in  Z 0  Z0  Z 0 (l  j )
1  j tan(l ) tanh(l ) 1  j ( / 0 )l 0

Compared to the case of series resonance: Zin  R  j 2L

Z 0 1
R  Z 0l L C
20 02 L

0 L  
Q0   
R 2l 2

Microwave resonators Slide 12

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 Transmission Line resonators

•Open circuited half-wavelength

resonators behave as parallel
resonant circuit.
•Short-circuited quarter-
wavelength resonators also
behave as parallel resonant
circuit.

Assignment: Find the equivalent circuit model of these resonators.

Microwave resonators Slide 13

Waveguide Resonator
 
Et ( x, y, z )  e ( x, y)( A e j mnz  Ae j mnz )

m 2 n 2
 mn  k 2  ( ) ( )
a a

At z=0 and d, Et (the tangential

component) is zero.
A  A  0 A   A
A e jmnd  A e j mnd  2 j sin(  mn d )  0
 mn d  l l=1,2,3, …
l 2 m 2 n 2
)  k2 (
( ) ( )
d a b
2 m 2 n 2 l 2 c m 2 n 2 l 2
k  2 (
2
)  ( )  ( ) f mnl  ( ) ( ) ( )
v a b d 2  r  r a b d
Microwave resonators Slide 14

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 Waveguide Resonator TE10l
x x lz x lz
E y  A sin (e  j10z  e  j10z )  2 jA  sin sin  E0 sin sin
a a d a d
 jE0 x lz jE0 x lz
Hx  sin cos Hz  cos sin
ZTE a d ka a d

 abd 1 2
4
Wm  H H x* H z H z*dv  E02 (  2 2 2)
ZTE k  a
x 2
V
16
abd 2
( / a) abd 2 1 abd 2
2
 E02 (  2 2 ) E0 2  E0
16 k
2 2
k 16  16

 abd
4
We  E E dv  y
*
y E02
V
16

Microwave resonators Slide 15

Waveguide Resonator TE10l

b a d b
Rs 2 R
Pc   | H t | ds  s [2   | H x ( z  0) |2 dxdy  2   | H z ( x  0) |2 dydz 
2 Walls 2 y 0 x 0 z 0 y 0

Rs E022 l 2 ab bd l 2 a d
d a
2   (| H x ( y  0) |  | H z ( y  0) | )dxdz ] 
2 2
( 2  2  )
z 0 x 0
8 2 d a 2d 2a

 ' ' abd ' ' | E0 |2

b
1
Pd 
2V J .E *dv 
2 y 0 | E |2 dv 
2

20We
2 W (kad )3 b 1 Qd 
Qc  0 e  Pd
Pc 2 Rs l ab bd l 2 a d
2 2
( 2  2  ) ' 1
d a 2d 2a  
 ' ' tan 
1 1
Q0  (  ) 1
Qc Qd
Microwave resonators Slide 16

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 Excitation of Resonators

Microwave resonators Slide 17

Excitation of Resonators

Microwave resonators Slide 18

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 Next Time
Microwave Filters

Microwave resonators Slide 19

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