Microwave Engineering
8
thhWeek: Series Resonant Circuit
Definition of Microwave Resonator
: Microwave junction in which EM energy can be excited for one or more specific frequencies.
Transmission Line: EM wave is propagating.
Resonator: EM energy is stored.
Types of Microwave Resonators - Rectangular/Circular cavities
- Finite length of microstrip transmission line - Microstrip ring resonator
- Dielectric resonator (DR) - Fabry-perot open resonator
- Surface acoustic wave (SAW) resonator - Film bulk acoustic resonator (FBAR)
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Important Characteristics:
Ȧ
o, Q
L, Q
u, Q
e, Z
o, Insertion loss, coupling coefficient, tunability and temperature stability.
Analysis of Microwave Resonator
- Lumped element circuit theory using series and parallel RLC circuits: Microstrip ring resonator
- Field analysis: Cavity, Fabry-Perot open resonator Excitation of resonators
- Apertures, loop/probe couplings
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Microwave resonators are used in a variety of applications such as:
- Antennas - Sensors - Filters
- Oscillators
- Frequency meters - Tuned Amplifier
The series and parallel RLC circuit representation is
useful for analyzing microwave resonator near resonance.
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Near the resonance frequency, a microwave resonator can be modeled as a series or parallel RLC lumped-element equivalent circuit.
1 Zin R j L j
Z C Z
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AC
R L
C Zin
I
Power delivered to the resonator is
1 1 2 1
( ).
2 2
Pin VI I R j L j
C
Z
Z
Power dissipated by the resistor, R, is
1 2
2 .
Ploss I R
Average magnetic energy stored in the inductor, L, is 1 2 4 .
Wm I L
2 2
2
1 1 1
4 4 .
e c
W V C I
Z C
Average electric energy stored in the capacitor, C, is
V
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in
in I Z
P 2
2 1
2
2 I Zin Pin
1 2
2 .
Ploss I R 1 2
4 .
Wm I L
We 1 2 12 4 I .
Z C
2 loss
2 loss
2 m
2 m
ee
2 2 2
2
m eloss j W W
P 2
Z
¸¹
¨ ·
©
§
j C L
j R
I
Z Z
12
1 2
I C L j
j I R
I
Z
Z
12 2
2
1 2 2 2
C I
j L
I j
Ploss 2 2 12
4 2 1
4 2 1
Z Z
Z
¸¹
¨ ·
©
§
j I L I C
Ploss 2 2 12
4 1 4
2 1
Z Z
j C L
j R
Zin
Z Z
1in
2
^
Ploss j Wm We`
I
2
Z
2
2
2
m
m ee
2 2
Resonance occurs when the average stored magnetic (Wm) and electric energies (We) are equal and Zin is purely real.
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The frequency in which is called the resonant frequencyo 1 .
Z LC
1 2
4 .
Wm I L
2 2
2
1 1 1
4 4 .
e c
W V C I
Z C
1
o LC
Z
L Co 2
1
Z
^
loss m e`
in P j W W
I
Z 2 2
Z
2 2
2 R Ploss I
2 2
1 I
Zin Ploss R
Q is a measure of the loss of a resonant circuit.
Lower loss Î Higher Q
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loss e m
o P
W Q Z W
Quality Factor Q
less]
[dimension Loss
Power
Stored Energy
Average Z
Q
1 2
2 .
Ploss I R 1 2
4 .
Wm I L
We 1 2 12 4 I .
Z C
2 loss
2 loss
2 m
2 m
ee
2 2 2
2
Q increases as R decreases.
loss e o P
2W
° Z
°°
¯
°°
°
®
loss m o P
2W
Z R
oL Z R
I
L I
o 2
2
2 14 2 1 Z
oRC Z
1 R
I I C
o
o 2
2
2 1
1 4
2 1 Z Z
In order to see the input impedance versus a frequency, let’s see behavior of the input impedance near its resonant frequency.
Near resonance,
Z Z
o'Z V
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j C L
j R
Zin
Z Z
1¸
¹
¨ ·
©§
j L LC
R 21
1
Z
Z
¸¹
¨ ·
©§
1 2
1
Z Z
L LC jR
¸¸
¹
·
¨¨
©
§
1 022
Z Z
LZ
j
R ¸¸
¹
·
¨¨
©
§ 2 2 02
Z Z Z
LZ
j
R o 1
Z LC
Z Z Z Z Z Z
Z
Z
2 o2 o o # 2 ' Z' | 2Z
Z0
Z #
¸¹
¨ ·
©
§ '
# 2 2
Z Z Z
LZ
j R
Zin
R j 2 L ' Z
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o in
j RQ R
Z Z
Z '
| 2
R . Q ZoL
This form is useful for identifying equivalent circuits with distributed This form is useful for
element resonators.
element resonators.
Example: Microstrip resonators
Q factor of the series resonant circuit is
o
L RQ
Z
Z '
# R j L
Zin 2 Z
Z ¸¸¹'
¨¨ ·
© §
0
2 RQ j
R
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Lossy resonator
A resonator with loss can be modeled as a lossless resonator whose resonant frequency has been replaced with a complex effective resonant frequency.
Lossless resonator
oin
L j
L j
L j R Z
Z Z
Z
Z '
'
| 2 2
2
oin
L j R
L j R Z
Z Z
Z
'
|
2 2
¸¸
¹
·
¨¨
©
§ ¸¸
¹
¨¨ ·
©
§
| Q
L j j L
j
Zin o o
1 2 2
2 Z Z Z Z
¸¸¹
¨¨ ·
©
§
m Q
j
o
o Z 1 2
Z
o
o RQ
R L Q L
Z Z
Identical
back to gap coupled MSL resonator
oL j
R 2 Z Z
Zo
¹
¸¸¸¸·
¹¸¸
·¸¸
¸¸¸¸¹
¸¸·
¨¨¨¨ ¸¸
©¨¨
§¨¨
Q j
o 1 2
Z
Q L j
j L
j o o
2 2
2 Z Z Z
oo j L
Q
LZ Z Z
2
L j L j2
Q j Q Q 2QQ 2
oo o
L Q j
RQ Z Z Z
Z 2
Proof:
o
L RQ Z
Z L LZ
Q Z
o
RQ Z
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Q Q
RQ R
R R
Z R R
Z
o
in in
1 1 2
2 2
satified.
be must ,
707 2 . have 0
order to
In 2 2
'
'
'
Z Z Z
Z Z
Z
Z Z
2
2 2
1 2 ¸¸
¹
¨¨ ·
©
§ '
¸¸
¹
¨¨ ·
©
§ '
|
o in
o in
R RQ Q Z
j R
Z Z
Z Z Z
Z Z
·2
§
§ ·
Magnitude of input impedance versus a frequency
Fractional Bandwidth is defined as:
o
o o
o o
o
BW BW
Z Z
Z Z Z
Z Z
Z Z
Z Z Z
'
' '
'
2
1 2
2
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Z Z
Z2 o ' Z
Z
Z1 o '
• Bandwidth increases as R increases.
• Narrower bandwidth can be achieved at higher quality factor (Smaller R).
BW Q
o Q
, 1 1 Since 2'
Z Z
Magnitude of input impedance versus a frequency
