Quality Factor
Microwave Engineering
EE 172
Dr. Ray Kwok
Ray Kwok & Ji-Fuh Liang
Characterization of High-Q Resonators for Microwave-Filter Applications
IEEE Trans MTT vol.47, p111-114, 1999
& references therein.
Q factors - Dr. Ray Kwok
Quality Factor
• Often referred to as the Q-factor.
• It indicates how good a quality the “device” has.
“Good” here means low loss.
e.g. a capacitor is said to be high-Q when it’s low loss.
• Unloaded-Q (Qu), loaded-Q (QL) and external-Q (Qe)
of resonators are often quoted in literature.
• Q-factor is often difficult to calculate precisely.
Engineers measure it directly using either S12 or S11.
Q factors - Dr. Ray Kwok
Effect of Q in a bandpass filter
FilterResponse
Filter1Response
0
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50
Q=5000
Q=100
-60
-60
4
DB(|S[1,1]|)
Filter
DB(|S[2,1]|)
Filter
5
6
Frequency(GHz)
7
4
5
6
Frequency(GHz)
DB(|S[1,1]|)
Filter1
DB(|S[2,1]|)
Filter1
7
Q factors - Dr. Ray Kwok
Qu – dictates design types
Insertion L oss
0
Qu=5000
-1
-2
Qu=100
-3
-4
4.75
DB(|S[1,2]|)
Filte r
DB(|S[1,2]|)
Filte r1
5.25
5.75
Frequency (GH z)
6.25
Q factors - Dr. Ray Kwok
Unloaded Q
energy _ stored
Qu ≈
energy _ dissipated
1
1
1
=
+
+ ⋅⋅⋅
Qu Qc Qd
conduction
loss
adding series “resistance” !!!
dielectric
loss
for a rectangular waveguide. Rs is the surface resistance of the resonator.
See Pozar Ch.6.
Q factors - Dr. Ray Kwok
Loaded Q
Input / output coupling “de-Q” resonators.
The coupling is related to the external Q (Qe).
1
1
1
=
+
Q L Q u Qe
QL is measured by fo/∆f at the 3 dB (usually in S12).
Qu is the desired parameters for any passive design.
Qe can be calculated or evaluated for any given coupling structure.
Q factors - Dr. Ray Kwok
Transmission Measurement
output
S21
3 dB bandwidth
Q ≡ fo/∆ f
Quality Factor
fo
f
many resonants
Q factors - Dr. Ray Kwok
S21 requires weak coupling < 30 dB
weak coupling means 1/Qe ~ 0. QL ~ Qu
Q 50 0
-3 0
S21 (dB)
fo = 1.5002 GHz
∆f = 3.06 MHz
-4 0
Q = fo/∆f = 490
-5 0
-6 0
1.46
D B (|S [1 ,2 ]|)1 .4 8
S ch e m atic 1
1.5
F req u e n cy (G H z)
1 .52
1 .5 4
Q factors - Dr. Ray Kwok
Transmission vs. Reflection
|S11|2 + |S12|2 ≈ 1
Easier, no additional parts to
make.
Use existing coupling feature….
Often just need to change
coupling strength.
Precise measurement. True Qu
of that structure.
combline
filter
Q factors - Dr. Ray Kwok
Reflection < 3 dB?!
S11 (dB)
Q 500
0
-0.05
-0.1
∆ωx
-x dB
QL(x) ≡ ωo/∆ω
-0.15
-0.2
ρo
ωo
D B (|S[1 ,1]|)
Sc hematic 1
-0.25
1.48
1.49
1.5
Frequency (G H z)
1.51
1.52
Q factors - Dr. Ray Kwok
Equivalent Circuit - resonator
AA
L
K 01
Zo
r
rA
S 11
A ’A ’
K2
r = 01
Z
A
o
C
L
C
g
L
C
g
C
r
(a )
L
gA
S 11
rA
ρA
J01
Yo
gA
ρA
(b )
F ig . 1 (a ) E q u iv a le n t c irc u it o f a s e rie s re s o n a to r c o u p le d to a s o u rc e im p e d a n c e Z o .
(b ) E q u iv a le n t c irc u it o f a p a ra lle l re s o n a to r c o u p le d to a so u rc e a d m itta n c e Y o .
Q factors - Dr. Ray Kwok
L
One-port Reflection
rA
C
r
Γin
 ω ωo 
1 
j

 ≡ r + jωo LΩ
= r + jωL1 − 2  = r + jωo L
−
Zin = r + jωL −
ωC
 ω LC 
 ωo ω 
 rA  ωL
Ω
1 −  + j
(1 − β) + jQu Ω
Zin − rA (r − rA ) + jωLΩ 
r 
r
Γin =
=
=
≡
(
Zin + rA (r + rA ) + jωLΩ  rA  ωL
1 + β) + jQ u Ω
1
+
j
+
Ω


r
r


ω ωo
1
Ω≡
−
ωo =
ωo ω
LC
Qu ≡
β≡
ωL
r
rA
r
coupling parameter
Q factors - Dr. Ray Kwok
Around ωo
Ω≡
ω ωo ω − ωo (∆ω)x
1
−
≈
=
=
ωo ω
ωo
ωo
Q L ( x , β)
generalized QL
2
magnitude of Γin
ρx
2
 Qu 
2

(1 − β) + 
 Q L ( x , β) 
=
2


(1 + β)2 +  Qu 
 Q L ( x , β) 
Define a mapping function
F( x , β) ≡
Qu
Q L ( x , β)
2
F( x , β) =
(1 + β) 2 ρ x − (1 − β) 2
1 − ρx
2
Note: F & QL are functions of x and β, but Qu is independent of both.
Q factors - Dr. Ray Kwok
Mapping Function F(x,β
β)
At fo
Ω=
ω ωo
−
=0
ωo ω
so
ρo =
1− β
1+ β
and
RL o = −20 log ρo
Note: ρo = 0
β=
1 − ρo
1 + ρo
if β > 1 (over-coupled)
β=
1 + ρo
1 − ρo
if β < 1 (under-coupled)
is the return loss at resonant
then
2
F( x , β) =
1 m ρo
if β = 1 (critically-coupled)
2
ρ x − ρo2
1 − ρx
2
for the over- / under-coupled cases
Q factors - Dr. Ray Kwok
Example (not ideal, RL too low)
S11 (dB)
Q500
0
-0.05
∆ωx
-0.1
-x ~ -0.075 dB
ρx = 10-x/20 = 0.991
QL(x) ∼ 1.5/0.004 ~ 375
-0.15
-0.2
F = 1.342
Qu ~ 503
ρo=10-0.21/20 = 0.976
DB (|S[1,1]|)
Sc hematic 1
ωo
-0.25
1.48
1.49
1.5
Frequency (GHz)
1.51
1.52
Q factors - Dr. Ray Kwok
For x = 3 dB
ρo = 0.70795
3
2.5
F(3,b)
2
1.5
1
over-coupled
under-coupled
0.5
0
0
5
10
15
20
25
Return Loss at resonant (dB)
30
35
40
Q factors - Dr. Ray Kwok
Critically-Coupled (β
β = 1)
Smith Chart
1.0
Swp Max
2.5GHz
2.
0
6
0.
0
0 .8
S[1,1]
Schematic 1
ReturnLoss
0.
4
0
3.
0
4.
-10
5. 0
0. 2
10.0
5.0
4.0
3.0
2.0
1.0
0.8
0.6
0.4
0
0.2
1 0. 0
-20
- 10 .0
2
-0 .
-4
.0
-5 .
0
1.502
sharp & high return loss
1.503
.0
-2
1.5
1.501
Frequency(GHz)
-1.0
1.499
- 0.8
1.498
.4
-0
-0
.6
-30
1.497
-3
.0
DB(|S[1,1]|)
Schematic1
Swp Min
0.5GHz
radius ~ 1 circle
Q factors - Dr. Ray Kwok
Over-Coupled (β
β > 1)
Smith Chart
1.0
Swp Max
2.5GHz
2.
0
6
0.
0
0 .8
S[1,1]
Schematic 1
ReturnLoss
0.
4
0
3.
0
4.
-10
5. 0
0. 2
10.0
5.0
4.0
3.0
2.0
1.0
0.8
0.6
0.4
0
0.2
1 0. 0
-20
- 10 .0
2
-0 .
-4
.0
-5 .
0
lower return loss
1.502
1.503
.0
-2
1.5
1.501
Frequency(GHz)
-1.0
1.499
- 0.8
1.498
.4
-0
-0
.6
-30
1.497
-3
.0
DB(|S[1,1]|)
Schematic1
radius > 1
Swp Min
0.5GHz
Q factors - Dr. Ray Kwok
Under-Coupled (β
β < 1)
Smith Chart
1.0
Swp Max
2.5GHz
2.
0
6
0.
0
0 .8
S[1,1]
Schematic 1
ReturnLoss
0.
4
0
3.
0
4.
-10
5. 0
0. 2
10.0
5.0
4.0
3.0
2.0
1.0
0.8
0.6
0.4
0
0.2
1 0. 0
-20
- 10 .0
2
-0 .
4
.0
-5 .
0
lower return loss
1.502
1.503
.0
-2
1.5
1.501
Frequency (GHz)
-1.0
1.499
- 0.8
1.498
.4
-0
-0
.6
-30
1.497
-3
.0
DB(|S[1,1]|)
Schematic 1
radius > 1
Swp Min
0.5GHz
Q factors - Dr. Ray Kwok
External Q – input coupling
Graph 1
180
Ang(S[1,1]) (Deg)
Schematic 1
90
0
RA∆f = ∆f/gog1
-90
-180
1.495
RA =
1.496
1.497
1.498
1.499
1.5
1.501
Frequency (GHz)
rA
∆f / f o
1.502
1.503
1.504
1.505
Use delay instead for dc shift
See paper for more info.