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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
IMPEDANCE TRANSFORMERS
AND TAPERS
Lecturers: Lluís Pradell ([email protected])
Francesc Torres ([email protected])
March 2010
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
The quarter-Wave Transformer* (i)
Zin
Z1 ZLZ0
A quarter-wave transformer can be used to match a real impedance ZL to Z0
40
L
in Z
ZZ
21
tjZZ
tjZZZZ
L
Lin
1
11
tgtgt
If The matching condition at fo is 01 ZZZ L
At a different frequency and the input reflection coefficient is
00
0
0
0
20 ZZtjZZ
ZZ
ZZ
ZZ
LL
L
in
inZin
220
0
cos
41
1
ZZ
ZZ
L
Lin
The mismatch can be computed from:
0ZZ in
*Pozar 5.5
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
The quarter-Wave Transformer (ii)
If Return Loss is constrained to yield a maximum value , the
0
0
2
2
1cos
ZZ
ZZ
L
L
m
mm
frequency that reaches the bound can be computed from:
m
Where for a TEM transmission line
00
0
24
2
4 f
f
f
v
v
f
vp
pp
And the bound frequency is related to the design frequency as:
02 f
f mm
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
The quarter-Wave Transformer (iii)
Finally, the fractional bandwdith is given by
0,05m
02
fl
f
0/ 10LZ Z
0/ 4LZ Z
0/ 2LZ Z
18,1 %BW
4,5 %BW
0
0
2
1
0
0 2
1cos
42
2
ZZ
ZZ
f
fff
L
L
m
mm
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Multisection transformer* (i)
That is, in the case of small reflections the permanent reflection is dominated by the two first transient terms: transmission line discontinuity and load
The theory of small reflections
01
010 ZZ
ZZ
1
1
ZZ
ZZ
L
LL
In the case of small reflections, the reflection coefficient can be approximated taking into account the partial (transient) reflection coefficients:
jLe 2
0
L
L0
40
*Pozar 5.6
20
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
0
1Z2Z LZNZ0Z
1 2
N
Multisection transformer (ii)The theory of small reflections can be extended to a multisection transformer
2 4 2 10 1 2
1
... ; ( 0,1,..., )j j jN i iN i
i i
Z Ze e e i N
Z Z
It is assumed that the impedances ZN increase or decrease monotically
( 2) ( 2)0 1 ...jN jN jN j N j Ne e e e e
0 1 1 2 2, , ,...N N NSymmetric
The reflection coefficients can be grouped in pairs (ZN may not be symmetric)
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
0 1 / 2
12 cos cos( 2) ... cos( 2 ) ...
2jN
i Ne N N N i for N even
0 1 ( 1) / 22 cos cos( 2) ... cos( 2 ) ... cosjNi Ne N N N i
for N odd
Finite Fourier Series: periodic function (period: )
Multisection transformer (iii)
The reflection coefficient can be represented as a Fourier series
Any desired reflection coefficient behaviour over frequency can be synthesized by properly choosing the coefficients and using enough sections:
•Binomial (maximally flat) response
•Chebychev (equal ripple) response
i
L
02
F
0
0
ZZ
ZZ
L
LL
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Binomial multisection matching transformer (i)
2(1 ) 2 cosNj N N jNA e Ae
0
0
2 N L
L
Z ZA
Z Z
Binomial function
AN2)0(
The constant A is computed from the transformer response at f=0:
The transformer coefficients are computed from the response expansion:
n
N
n
jnNn eCA
0
2)( !!
!
nnN
NC N
n
The transformer impedances Zn are then computed, starting from n=0, as:
0
1 ln2lnZ
ZC
Z
Z LNn
N
n
n
Nnn AC
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Binomial multisection matching transformer (ii)
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Binomial multisection matching transformer (iii)
11/
1arccos arccos
2
NN
m mm
LA
Bandwidth of the binomial transformer
mNN
m A cos2
The maximum reflection at the band edge is given by:
02
fl
f
1
0/ 2LZ Z
71 %
( 3)
BW
N
m
05.0The fractional bandwitdh is then:
1/
0
0 0
2( ) 4 4 12 2 arccos
2
N
mm mf f f
f f A
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Chebyshev multisection matching transformer
1( ) cos( cos ), 1nT x n x for x 1( ) cosh( cosh ), 1nT x n x for x
0
0
1
1cos
L
LN
m
Z ZA
Z ZT
1
11 0
0
cosh1
cos11 1 coshcosh cosh
cos
L
m
L
mm L
m NZ Z
N Z Z
cos
cosjN
Nm
A e T
Chebyshev polynomial
02
fl
f
0,05m
0/ 2LZ Z
102 %
( 3)
BW
N
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Chebyshev transformer design
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Chebyshev transformer design
Application: Microstrip to rectangular wave-guide transition: both source and load impedances are real.
Rectangular guide
Ridge guide: five λ/4 sections: Chebychev design
Steped ridge guideMicrostrip line
Ridge guidesection
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
TRANSFORMER EXAMPLE (1):ADS SIMULATION
S_ParamSP1
Step=10 MHzStop=20 GHzStart=0 GHz
S-PARAMETERS
MSUBMSub1
Rough=0 umTanD=9e-4T=17.5 umHu=1.0e+036 umCond=5.8e+7Mur=1Er=2.17H=257 um
MSub
TermTerm2
Z=100 OhmNum=2
TermTerm1
Z=50 OhmNum=1
MLINTL5
L=5509.460000 umW=767.037000 umSubst="MSub1"
MLINTL3
L=5678.81 umW=283.802 umSubst="MSub1"
MSTEPStep3
W2=207.139 umW1=283.802 umSubst="MSub1"
MSTEPStep1
W2=429 umW1=616.935 umSubst="MSub1"
MSTEPStep2
W2=283 umW1=429.655 umSubst="MSub1"
MLINTL2
L=5611.44 umW=429.655 umSubst="MSub1"
MLINTL1
L=5548.47 umW=616.935 umSubst="MSub1"
MSTEPStep4
W2=616 umW1=767.037 umSubst="MSub1"
MLINTL4
L=5725.100000 umW=207.139000 umSubst="MSub1"
Chebyshev transformer, N = 3, |M|=0.05 (ltotal = 3/4)
87,14 70,71 100 57,37 50
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TRANSFORMER EXAMPLE (2):ADS SIMULATION
2 4 6 8 10 12 14 16 180 20
-40
-30
-20
-10
-50
0
freq, GHz
dB
(S(1
,1))
2 4 6 8 10 12 14 16 180 20
-0.6
-0.4
-0.2
-0.8
0.0
freq, GHz
dB
(S(1
,2))
CHEBYSCHEV N=3
2 4 6 8 10 12 14 16 180 20
0.1
0.2
0.3
0.0
0.4
freq, GHz
mag(S
(1,1
))
BW = 102 %0,05m
microstrip loss
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Tapered lines (i)
LZ
zL0
0Z Z z
zz z z
Z Z Z
In the limit, when z 0:
Taper: transmission line with smooth (progressive) varying impedance Z(z)
Z
Z
ZZZ
ZZZ
2
zZdZ
zd2
1
The transient ΔΓ for a piece Δz of transmission line is given by:
dz
zfd
zfdz
zfLd n
)(
1
dzdz
zfLdzfd
zfn
2
1
)(2
1
This expression can be developed taking into account the following property:
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Tapered lines (ii)
dz
dz
zZLdzd n
2
1
Taking into account the theory of small reflections, the input reflection coefficient is the sum of all differential contributions, each one with its associated delay:
Fourier Transform
LnzjzjL
in dzdz
zZLdeezd
0
22
0 2
1
L Taper electrical length
zZ•Exponential taper
•Triangular taper
•Klopfenstein taper
dz
zZLd n
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Exponential Taper
0zZ z Z e
for 0 <z < L
0
1ln LZ
L Z
0ln / sin( )
2j LLZ Z L
L eL
(sinc function)
L
LZLZ
ZZ 00
dz
zZLd n
L zj
in dze0
2
2
1
Fourier Transform
Lmin2max
L
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Triangular taper
22
ln2 0
024 2
1 ln2 0
0
Zz LZL
Zz z LL ZL
Z e
Z e
Z z
20
20
4ln
0
4( ) ln
lnL
L
Lz Z
ZL
ZL z
ZL
Zd
Z
dz
0 / 2
/ 2
z L
L z L
0 / 2
/ 2
z L
L z L
2
0
1 sin( / 2)ln
2 / 2j L LZ L
L eZ L
(squared sinc function)
- lower side lobes- wider main lobe L
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Klopfenstein Taper
2 2cos
coshj L
L
L AL e
A
:passband L A
0
0 0
1ln
2L L
LL
Z Z Z
Z Z Z
coshL
m A
LShortest length for a specified |M|
Lowest |M| for a specified taper length
( )L A
ltaper =
0/ 2LZ Z
Based on Chebychev coefficients when n→∞. Equal ripple in passband
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Microstrip to rectangular wave-guide transition
Example of linear taper: ridged wave-guide
Microstrip line
Ridgedguide
Rectangularguide
SECTION A-A’
SECTION B-B’SECTION C-C’
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Rectangular wave-guide to finline to transition
Example of taper: finline wave guide
Finline mixer configuration
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
TAPER EXAMPLE (1):ADS SIMULATION
TermTerm2
Z=100 OhmNum=2
TermTerm1
Z=50 OhmNum=1
MTAPERTaper1
L=LtotW2=W11W1=W1Subst="MSub1"
MSUBMSub1
Rough=0 milTanD=9e-4 T=17.5 umHu=3.9e+34 milCond=5.8e7 Mur=1.0 Er=2.17 H=10.0 mil
MSub
ADS taper model
S_ParamSP1
Step=10 MHzStop=20 GHzStart=0 GHz
S-PARAMETERS
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
TAPER EXAMPLE (2):ADS SIMULATION
Aproximation to exponential taper using ADS : 10 sections of
MLINTL10
L=L11W=W11Subst="MSub1"
MLINTL9
L=L10W=W10Subst="MSub1"
MLINTL7
L=L8W=W8Subst="MSub1"
MLINTL8
L=L9W=W9Subst="MSub1"
MLINTL6
L=L7W=W7Subst="MSub1"
MSTEPStep4
W2=W5W1=W4Subst="MSub1"
MSTEPStep2
W2=W3W1=W2Subst="MSub1"
MLINTL15
L=L2W=W2Subst="MSub1"
MSTEPStep1
W2=W2W1=W1Subst="MSub1"
MLINTL14
L=L1W=W1Subst="MSub1"
MLINTL19
L=L6W=W6Subst="MSub1"
MLINTL18
L=L5W=W5Subst="MSub1"
TermTerm4
Z=100 OhmNum=4
MSTEPStep9
W2=W11W1=W10Subst="MSub1"
MSTEPStep8
W2=W10W1=W9Subst="MSub1"
MSTEPStep7
W2=W9W1=W8Subst="MSub1"
MSTEPStep6
W2=W8W1=W7Subst="MSub1"
MSTEPStep5
W2=W7W1=W6Subst="MSub1"
MSTEPStep11
W2=W6W1=W5Subst="MSub1"
MLINTL17
L=L4W=W4Subst="MSub1"
MLINTL16
L=L3W=W3Subst="MSub1"
TermTerm3
Z=50 OhmNum=3
MSTEPStep3
W2=W4W1=W3Subst="MSub1"
50 53,59 57,44 61,56 65,97 70,71
75,79 81,22 87,05 93,30 100
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
TAPER EXAMPLE (3):ADS SIMULATION
Aproximation to exponential taper using ADS : 10 sections of
50 53,59 57,44 61,56 65,97
70,71 75,79 81,22 87,05 93,30
100 VARVAR11Ltot=L1+L2+L3+L4+L5+L6+L7+L8+L9+L10
EqnVar
VARVAR23L11=2.290040 mm
EqnVar
VARVAR22W11=207.139000 um
EqnVar
VARVAR6L3=2.219510 mm
EqnVar
VARVAR5W3=615.883000 um
EqnVar
VARVAR4L2=2.211550 mm
EqnVar
VARVAR3W2=688.516000 um
EqnVar
VARVAR1W1=767.037000 um
EqnVar
VARVAR2L1=2.203780 mm
EqnVar
VARVAR8W4=548.755000 um
EqnVar
VARVAR7L4=2.227670 mm
EqnVar
VARVAR10W6=429.647000um
EqnVar
VARVAR9L6=2.244580 mm
EqnVar
VARVAR13W7=377.052000 um
EqnVar
VARVAR12L7=2.253330 mm
EqnVar
VARVAR15W5=486.783000 um
EqnVar
VARVAR14L5=2.236020 mm
EqnVar
VARVAR17W8=328.727000 um
EqnVar
VARVAR16L8=2.262270 mm
EqnVar
VARVAR19W9=284.432000 um
EqnVar
VARVAR18L9=2.271390 mm
EqnVar
VARVAR21W10=243.966000 um
EqnVar
VARVAR20L10=2.280660 mm
EqnVar
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
TAPER EXAMPLE (4):ADS SIMULATION
m3freq=m3=-32.452
9.980GHzm1freq=m1=-22.116
7.170GHz
2 4 6 8 10 12 14 16 180 20
-60
-40
-20
-80
0
freq, GHz
dB(S
(1,1
))
9.880G-32.36
m3
dB(S
(3,3
))
7.170G-22.12
m1
EXPONENTIAL / ADS TAPER
− 10 section approx.− ADS model
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
TAPER EXAMPLE (5):ADS SIMULATION
2 4 6 8 10 12 14 16 180 20
0.1
0.2
0.3
0.0
0.4
freq, GHz
mag
(S(1
,1))
mag
(S(3
,3))
Exponential taper
0,05m
ltaper = @ 10 GHz
− 10 section approximation− ADS model
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
TAPER EXAMPLE (6):ADS SIMULATION
m1freq=m1=-9.824
49.41GHz
m2freq=m2=-65.843
9.980GHz
20 40 60 800 100
-60
-40
-20
-80
0
freq, GHz
dB(S
(1,1
))dB
(S(3
,3))
48.85G-9.995
m1
9.980G-65.84
m2
10 section taper: periodicity in frequency
(li=/2)
(li=/10)
− ADS model − 10 section approximation is periodic.
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
MATCHING NETWORKS
LEVY DESIGN
Lecturers: Lluís Pradell ([email protected])
Francesc Torres ([email protected])
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
MatchingNetwork
(passive lossless)
Z0
fVs
22 21
11
11 1dL d
tavS avS
P PG
P P M
f
Minimize |1 (f)| Maximize Gt(2)
Pd1 PdL
MATCHING NETWORKS
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CONVENTIONAL CHEBYSHEV FILTER (1)
20
0
0
0 0
1si si pi pi
isi
ipi
L C L C
g ZL
w
gC
w Z
20 0 0
020 0
1,
1,
sisi i
pipi i
wC
L g Z
wZL
C g
' 0
0
2 1 2 1
0 0
20 2 1
1
w
f fw
f
0Z
1SL 1SC
2PL 2PC
3SL 3SC
PNL PNC
1 1 0.N NR g Z
Conversion from Low-Pass to Band-
Pass filter
1g 3g
1 1n ng R 0g
2g ng
LC low-pass filter
Center frequency
Relative bandwidth
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
CONVENTIONAL CHEBYSHEV FILTER (2)
( 0)
( 1)
22 2
2
2
1'
1 '
1
1
1
10log 10log 1
MAX Tn
MIN Tn
MAX
MIN
tn n
t
tn
tn
t
GT
G
G
Gr dB
G
1
' 2( )Gt
1'
2( )Gt
2
1
1 n
1
1 0 2
20 1. 2
1
1
cos cos , 1
cosh cosh , 1n
n x xT x
n x x
Pass-band ripple
Chebychev polynomials
![Page 33: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/33.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
CONVENTIONAL CHEBYSHEV FILTER (3) 0
1
12 2
1
1
2 • sin( ) 12 , sinh( ) ,sinh( )
2 1 2 14 • sin( ) • sin( )
2 2•sin ( )
( 1,2,...., 1)
2 • sin( )2
n
i i
n n
g
ng x ax na
i in ng g
ix
n
i n
ng gx
Fix pass-band ripple and filter order “n”
g0, g1,.., gn+1 are the low-pass LC filter coefficients: 1'1 w
![Page 34: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/34.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
APPLICATION TO A MATCHING NETWORK
1 1 1
1
1 1
1
' '
''
'
,
,
s s e s e
e ss s e
e s
L L L L L
C CC C C
C C
1
1 1
e1 e
0 0
20
2.sin .R.R 2. . .
1
s
s s
g nLw x w
L C
eR
'1SL '
1SC
2PL2PC
eLeC
PNL PNC
1 1.N N eR g R
TransistorM odel
1 1 ?n s e
rgiven a x g L L
n
Solution (?): increase n (n constant) a, x decrease
or increase n (n constant) a, x decrease
Transistor modeled with a dominant RLC behaviour in the pass-band to be matched
The final design may be out of specifications: n too high (too many sections) or r too large
![Page 35: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/35.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK (1)
( 0)
( 1) 2
2
1
10log 10log 1
MAX Tn
MIN Tn
MAX
MIN
t n
nt
n
tn
t
G K
KG
Gr dB
G
1
1
cos cos , 1
cosh cosh , 1n
n x xT x
n x x
2
2 2' ( 1)
1 'n
t nn n
KG K
T
' 2( )Gt
1'
2( )Gt
21n
n
K
1 0 2
20 1. 2
21n
n
K
nK
nK
SOLUTION: An additional parameter is introduced: Kn<1
![Page 36: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/36.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK (2)
0
1
12 2 2
1
1
2.sin2
2 1 2 14.sin .sin
2 2 , ( 1,2...., 1)sin 2. . .cos
2.sin2
i i
n n
g
ngx y
i in ng g i n
i ix y x y
n n
ng gx y
2
2
sinh ( )
sinh
1
sinh
sinh1 , ( 1)
sinhn n
n
x a
n
y b new freedom degree
nbK K
a
na
0
1
2 2 21
32
1
2.sin4
1 2·
1
2.sin1 4·
g
gx y
gg x y
gg x y
Example: n = 2
2
2
2 22
sinh ( )
sinh
1
sinh 2
sinh 21 , ( 1)
sinh 2
y b new freedom degree
bK
a
a
K
x
a
SOLUTION: Additional design equations
![Page 37: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/37.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK (3)
1 1 1 1
1 1 1 1
2 20
2 20
1 1
1 1
s s e e s e s ee
s s e e s e s ee
If L C L C take C C L L
If L C L C take L L C C
Design procedurea) Choose Cs1 or Ls1 taking into account the load to be matched
c) Compute x-y from the parameter g1
b) Choose network order (n) and compute g1
1
2.sin2nx y
g
1
1
01 1
e 0 e
. .
. .s
s
L w wg or g
R C R
![Page 38: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/38.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK (4)
2 2
cosh cosh
tgh a tgh b
a b
OPTIMAL DESIGN: minimize
2
2
2
1
cosh1 1
1 coshMAX
t
ntMIN
n
G
nbKG
na
0MAX
b
sinh sinha b x y ct
cosh cosh
tgh na tgh nb
a b
22 2 2
2
C Cx
For n=2: 2C x y
Select Ls1 (or Cs1) and n. Compute g1. and x-y. Then determine x, y and Kn, n:
x y b
a nnK
d) Choose x, compute y, max
Example: usual case n=2: Optimum x
The matched bandwith can be increased from ~5% to ~20% with n=2, with moderate Return Loss requirements (~20 dB)
![Page 39: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/39.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (1)
M atch ingN etwork
LeC e
R e R
15,2
0,528,86
0,62
50
e
ee
e
R
L nHf GHz
C pF
R
10
2
5,56,4226
7,5
f GHzf GHz
f GHz
![Page 40: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/40.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (2)
dBRL 75.17min
dBRL 98.23min
0
20
10
21
22 2
2
3
0,62 ( )
10,99
0,8195
2sin( )4 1,7257
21,978 1,434
20,2535 0,2509
0,996 0,015
0,114 0,071
0,4903 2,57 , 0,239
1
S e e
SS
S e
n tMAX
n tMIN
p p
C C pF f f
L nHC
wg
C R
C x yg
C Cx a
y b
K G dB
G dB
g C pF L nH
g
3 3, 2928 19,65eR g R
1
2
0
5,5
7,5
6,4226
20,3114
6,4226
f Ghz
f Ghz
f GHz
w
dBRL 75.17min
![Page 41: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/41.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (3):ADS SIMULATION
5 6 7 84 9
-20
-15
-10
-5
-25
0
freq, GHz
dB
(S(2
,1))
dB
(S(2
,2))
5 6 7 84 9
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
-1.0
0.0
freq, GHz
dB
(S(2
,1))
LEVY NETWORK (LUMPED COMPONENTS)
TFTF1T=1.5951
S_ParamSP1
Step=100 MHzStop=9.0 GHzStart=4 GHz
S-PARAMETERS
LLs
R=L=0.99 nH
TermTerm2
Z=50 OhmNum=2
TermTerm1
Z=15.2 OhmNum=1
CC2C=2.57 pF
LLp
R=L=0.239 nH
CCeC=0.62 pF
A transformer is necessary since g3≠1 (R3≠50 Ω). This transformed must be eliminated from the design
![Page 42: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/42.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
Norton Transformer equivalences
STEPS:1) the capacitor C2 is pushed towards the load through the transformer2) The transformer is eliminated using Norton equivalences
![Page 43: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/43.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (4):ADS SIMULATION
5 6 7 84 9
-20
-15
-10
-5
-25
0
freq, GHz
dB
(S(2
,1))
dB
(S(2
,2))
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.54.5 9.0
-0.8
-0.6
-0.4
-0.2
-0.0
-1.0
0.2
freq, GHz
dB
(S(2
,1))
S_ParamSP1
Step=100 MHzStop=9.0 GHzStart=4 GHz
S-PARAMETERS
LLe
R=L=0.52 nH
LL2
R=L=0.23 nH
CC2C=1.02 pF
LLp
R=L=0.38 nH
LLs
R=L=0.33 nH
TermTerm2
Z=50 OhmNum=2
TermTerm1
Z=15.2 OhmNum=1
CCeC=0.62 pF
![Page 44: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/44.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES IMPLEMENTED USING SHORT TRANSMISSION LINES
L l
Z0h 0 0 0 00
2 h h
lf L Z l f L Z
C
l
Z0l 0 0 0 00
2 l l
lf C Y l f C Y
L, C elements are then synthesized by means of short transmission lines:
![Page 45: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/45.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES IMPLEMENTED USING SHORT
TRANSMISSION LINES: EXAMPLE
10
0
10,33 106
50S h
lL nH Z for
22 0
0
10,23 73,85
50h
lL nH Z for
32 0
0
11,02 15,26
10l
lC pF Z for
![Page 46: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/46.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE ADS SIMULATION (5):
5 6 7 84 9
-25
-20
-15
-10
-5
-30
0
freq, GHz
dB(S
(1,1
))dB
(S(2
,1))
5.5 6.0 6.5 7.0 7.55.0 8.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
-1.2
0.2
freq, GHz
dB(S
(2,1
))
MSUBMSub3
Rough=0 milTanD=9e-4 T=0.689 milHu=3.9e+34 milCond=5.8e7 Mur=1.0 Er=2.17 H=10.0 mil
MSub
MLINTL8
L=715.898000 umW=178.873000 umSubst="MSub3"
MLSCTL4
L=948.7005301751 umW=262 umSubst="MSub3"
MLINTL10
L=3278.680000 umW=3586.240000 umSubst="MSub3"
TermTerm2
Z=50 OhmNum=2
MLINTL9
L=701.096000 umW=396.160000 umSubst="MSub3"
S_ParamSP1
Step=10 MHzStop=9 GHzStart=4 GHz
S-PARAMETERS
LLe
R=L=0.52 nH
CCeC=0.62 pF
TermTerm1
Z=15.2 OhmNum=1
![Page 47: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/47.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE: ADS SIMULATION (6):
5 6 7 84 9
-10
-5
-15
0
freq, GHz
dB
(S(1
,1))
dB
(S(1
,2))
5.5 6.0 6.5 7.0 7.55.0 8.0
-2.0
-1.5
-1.0
-0.5
-2.5
0.0
freq, GHz
dB
(S(2
,1))
MTEETee1
W3=262.0 umW2=0.39616 mmW1=0.178873 mmSubst="MSub3"
MSTEPStep1
W2=3586.24 umW1=396.16 umSubst="MSub3"
MSUBMSub3
Rough=0 milTanD=9e-4 T=0.689 milHu=3.9e+34 milCond=5.8e7 Mur=1.0 Er=2.17 H=10.0 mil
MSub
MLINTL8
L=715.898000 umW=178.873000 umSubst="MSub3"
MLSCTL4
L=948.7005301751 umW=262 umSubst="MSub3"
MLINTL10
L=3278.680000 umW=3586.240000 umSubst="MSub3"
TermTerm2
Z=50 OhmNum=2
MLINTL9
L=701.096000 umW=396.160000 umSubst="MSub3"
S_ParamSP1
Step=10 MHzStop=9 GHzStart=4 GHz
S-PARAMETERS
LLe
R=L=0.52 nH
CCeC=0.62 pF
TermTerm1
Z=15.2 OhmNum=1
![Page 48: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/48.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (7):
ADS SIMULATION: optimization
VARVAR4L4=3213.73 um opt{ 2000 um to 4000 um }
EqnVar
VARVAR3L3=1074.12 um opt{ 500 um to 1400 um }
EqnVar
VARVAR2L2=368.083 um opt{ 300 um to 1200 um }
EqnVar
VARVAR1L1=553.743 um opt{ 300 um to 1200 um }
EqnVar
OptimOptim1
SaveCurrentEF=noUseAllGoals=yes
UseAllOptVars=yesSaveAllIterations=noSaveNominal=yesUpdateDataset=yesSaveOptimVars=noSaveGoals=yesSaveSolns=noSeed= SetBestValues=yesNormalizeGoals=noFinalAnalysis="SP1"StatusLevel=4DesiredError=0.0MaxIters=25OptimType=Random
OPTIM
MLINTL10
L=L4W=3586.240000 umSubst="MSub3"
MLSCTL4
L=L3W=262 umSubst="MSub3"
MLINTL9
L=L2W=396.160000 umSubst="MSub3"
MLINTL8
L=L1W=178.873000 umSubst="MSub3"
GoalOptimGoal2
RangeMax[1]=7.5 GHzRangeMin[1]=5.5 GHzRangeVar[1]="freq"Weight=Max=Min=-0.5SimInstanceName="SP1"Expr="insertion_loss"
GOAL
GoalOptimGoal1
RangeMax[1]=7.5 GHzRangeMin[1]=5.5 GHzRangeVar[1]="freq"Weight=Max=-10Min=SimInstanceName="SP1"Expr="matching"
GOALMeasEqnMeas1
insertion_loss=dB(S(2,1))matching=dB(S(1,1))
EqnMeas
MTEETee1
W3=262.0 umW2=0.39616 mmW1=0.178873 mmSubst="MSub3"
MSTEPStep1
W2=3586.24 umW1=396.16 umSubst="MSub3"
MSUBMSub3
Rough=0 milTanD=9e-4 T=0.689 milHu=3.9e+34 milCond=5.8e7 Mur=1.0 Er=2.17 H=10.0 mil
MSub
TermTerm2
Z=50 OhmNum=2
S_ParamSP1
Step=10 MHzStop=9 GHzStart=4 GHz
S-PARAMETERS
LLe
R=L=0.52 nH
CCeC=0.62 pF
TermTerm1
Z=15.2 OhmNum=1
![Page 49: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/49.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (8):ADS SIMULATION: optimization
5 6 7 84 9
-20
-15
-10
-5
-25
0
freq, GHz
dB
(S(1
,1))
dB
(S(2
,1))
5.0 5.5 6.0 6.5 7.0 7.54.5 8.0
-1.0
-0.5
0.0
0.5
1.0
-1.5
1.5
freq, GHz
dB
(S(2
,1))
![Page 50: IMPEDANCE TRANSFORMERS AND TAPERS](https://reader036.vdocuments.net/reader036/viewer/2022062409/5681474b550346895db48d80/html5/thumbnails/50.jpg)
Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (9):ADS SIMULATION: optimization
VARVAR2L2=554.494 um opt{ 200 um to 1200 um }
EqnVar
VARVAR4L4=2193.75 um opt{ 2000 um to 4000 um }
EqnVar
VARVAR3L3=1332.32 um opt{ 500 um to 1400 um }
EqnVar
VARVAR1L1=485.326 um opt{ 300 um to 1200 um }
EqnVarGoal
OptimGoal1
RangeMax[1]=7.5 GHzRangeMin[1]=5.5 GHzRangeVar[1]="freq"Weight=Max=-18Min=SimInstanceName="SP1"Expr="matching"
GOAL
GoalOptimGoal2
RangeMax[1]=7.5 GHzRangeMin[1]=5.5 GHzRangeVar[1]="freq"Weight=Max=Min=-0.2SimInstanceName="SP1"Expr="insertion_loss"
GOAL
OptimOptim1
SaveCurrentEF=noUseAllGoals=yes
UseAllOptVars=yesSaveAllIterations=noSaveNominal=yesUpdateDataset=yesSaveOptimVars=noSaveGoals=yesSaveSolns=noSeed= SetBestValues=yesNormalizeGoals=noFinalAnalysis="SP1"StatusLevel=4DesiredError=0.0MaxIters=25OptimType=Random
OPTIM
MLINTL10
L=L4W=3586.240000 umSubst="MSub3"
MLSCTL4
L=L3W=262 umSubst="MSub3"
MLINTL9
L=L2W=396.160000 umSubst="MSub3"
MLINTL8
L=L1W=178.873000 umSubst="MSub3"
MeasEqnMeas1
insertion_loss=dB(S(2,1))matching=dB(S(1,1))
EqnMeas
MTEETee1
W3=262.0 umW2=0.39616 mmW1=0.178873 mmSubst="MSub3"
MSTEPStep1
W2=3586.24 umW1=396.16 umSubst="MSub3"
MSUBMSub3
Rough=0 milTanD=9e-4 T=0.689 milHu=3.9e+34 milCond=5.8e7 Mur=1.0 Er=2.17 H=10.0 mil
MSub
TermTerm2
Z=50 OhmNum=2
S_ParamSP1
Step=10 MHzStop=9 GHzStart=4 GHz
S-PARAMETERS
LLe
R=L=0.52 nH
CCeC=0.62 pF
TermTerm1
Z=15.2 OhmNum=1
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Design and Analysis of RF and Microwave SystemsEuropean Master of Researchon Information TechnologyEuropean Master of Researchon Information Technology
LEVY NETWORK EXAMPLE (10):ADS SIMULATION: optimization
5 6 7 84 9
-20
-15
-10
-5
-25
0
freq, GHz
dB(S
(1,1
))dB
(S(2
,1))
dB(le
vy3_
amb_
T_o
ptim
..S(1
,1))
dB(le
vy3_
amb_
T_o
ptim
..S(2
,1))
5.0 5.5 6.0 6.5 7.0 7.5 8.04.5 8.5
-0.8
-0.6
-0.4
-0.2
0.0
-1.0
0.2
freq, GHzdB
(S(2
,1))
dB(le
vy3_
amb_
T_o
ptim
..S(2
,1))