European Master of Research
on Information Technology
Design and Analysis of RF and Microwave Systems
IMPEDANCE
TRANSFORMERS
AND
TAPERS
Lecturers:
March 2010
Lluís Pradell (pradell@tsc.upc.edu)
Francesc Torres (xtorres@tsc.upc.edu)
Design and Analysis of RF and Microwave Systems
European Master of Research
on Information Technology
The quarter-Wave Transformer* (i)
A quarter-wave transformer can be used to match a real impedance ZL to Z0
Zin
Z0
If

Z1

0
4
Z12
Z in 
ZL
At a different frequency Z in  Z 0
in
Z0

ZL
Z in  Z1
Z L  jZ1t
Z1  jZ L t
The matching condition at fo is Z1  Z L Z 0
and the input reflection coefficient is
Z in  Z 0
Z L  Z0

Z in  Z 0 Z L  Z 0  j 2t Z L Z 0
The mismatch can be computed from:
1
in 
1
*Pozar 5.5
t  tg  tg
4Z L Z 0
Z L  Z 0 2 cos2 
European Master of Research
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Design and Analysis of RF and Microwave Systems
The quarter-Wave Transformer (ii)
If Return Loss is constrained to yield a maximum value m , the
frequency that reaches the bound can be computed from:
m
cos m 
2 Z LZ0
1  m
2
Z L  Z0
Where for a TEM transmission line
   
 0
vp 4

2f v p
 f

v p f0 4 2 f0
And the bound frequency is related to the design frequency as:
fm 
2 m f 0

European Master of Research
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Design and Analysis of RF and Microwave Systems
The quarter-Wave Transformer (iii)
Finally, the fractional bandwdith is given by

m
2 f 0  f m 
4
f 
 2  cos1 
f0

 1  m
2

2 Z LZ0 
Z L  Z0 

BW  4,5 %
Z L / Z0  10
Z L / Z0  4
Z L / Z0  2
BW  18,1 %
m  0,05
  l 
 f
2 f0
European Master of Research
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Design and Analysis of RF and Microwave Systems
Multisection transformer* (i)
The theory of small reflections

In the case of small reflections, the reflection
coefficient  can be approximated taking into account
the partial (transient) reflection coefficients:
L
0 
  0  L e
2 j
Z1  Z 0
Z1  Z 0
  
Z L  Z1
L 
Z L  Z1

0
4
0 

2
That is, in the case of small reflections the permanent reflection is
dominated by the two first transient terms: transmission line discontinuity 0
and load L
*Pozar 5.6
European Master of Research
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Design and Analysis of RF and Microwave Systems
Multisection transformer (ii)
The theory of small reflections can be extended to a multisection transformer



Z0
0
Z1
ZN
Z2
1
2
ZL
N
   
 0  1 e2 j  2 e4 j  ...   N e2 jN ; i 
Zi 1  Zi
Zi 1  Zi
(i  0,1,..., N )
It is assumed that the impedances ZN increase or decrease monotically
Symmetric  0  N , 1  N 1, 2  N 2 ,...
The reflection coefficients can be grouped in pairs (ZN may not be symmetric)


    e jN 0 e jN  e jN   1 e j ( N 2)  e j ( N 2)   ...
European Master of Research
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Design and Analysis of RF and Microwave Systems
Multisection transformer (iii)
The reflection coefficient can be represented as a Fourier series
1


    2e jN  0 cos N  1 cos( N  2)  ...  i cos( N  2i )  ...   N / 2 
2


for N even
    2e  jN   0 cos N  1 cos( N  2)  ...  i cos( N  2i )  ...   ( N 1) / 2 cos   for N odd
Finite Fourier Series: periodic function (period:   )
F  
Any desired reflection coefficient
behaviour over frequency can be
L
synthesized by properly choosing the
coefficients i and using enough
sections:
• Binomial (maximally flat) response
• Chebychev (equal ripple) response
L 
0

2

Z L  Z0
Z L  Z0

European Master of Research
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Design and Analysis of RF and Microwave Systems
Binomial multisection matching transformer (i)
Binomial function
    A(1  e
2 j
)  2 Ae
N
N
 jN
 cos 
N
The constant A is computed from the transformer response at f=0:
A  2 N
(0)  2 N A
The transformer coefficients n
N
( )  A C e
Z L  Z0
Z L  Z0
are computed from the response expansion:
N!
C 
N  n !n!
N
n
n  ACnN
The transformer impedances Zn are then
computed, starting from n=0, as:
 Z n 1   N N  Z L 
  2 Cn ln 
ln
 Zn 
 Z0 
n 0
N
n
 2 jn
European Master of Research
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Design and Analysis of RF and Microwave Systems
Binomial multisection matching transformer (ii)
European Master of Research
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Design and Analysis of RF and Microwave Systems
Binomial multisection matching transformer (iii)
Bandwidth of the binomial transformer
The maximum reflection at the
band edge is given by:
m  2N A cosN m
1   
 m  arccos   m 
 2  A 
1/ N

 
  arccos  m 

 L 
m
Z L / Z0  2
1
N
BW  71 %
( N  3)
The fractional bandwitdh is then:
0.05
 1   1/ N 
f 2( f 0  f m )
4m
4

 2
 2  arccos   m  
f0
f0


 2  A  
1
  l 
 f
2 f0
European Master of Research
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Design and Analysis of RF and Microwave Systems
Chebyshev multisection matching transformer
Chebyshev polynomial
 cos  
    A e jN TN 

cos

m 

Z L / Z0  2
Tn ( x)  cos(n cos1 x), for x  1
Tn ( x)  cosh(n cosh1 x), for x  1
BW  102 %
( N  3)
A
Z L  Z0
Z L  Z0
m  0,05
1
 1 
TN 

cos

m


  
cosh  L 
 m 
N
 1 
cosh 1 

 cos m 
1
1
cos  m 
1
 1 Z L  Z0
cosh  cosh 1 
 N
 m Z L  Z 0




  l 
 f
2 f0
European Master of Research
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Design and Analysis of RF and Microwave Systems
Chebyshev transformer design
European Master of Research
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Design and Analysis of RF and Microwave Systems
Chebyshev transformer design
Application: Microstrip to rectangular wave-guide transition: both source
and load impedances are real.
Ridge guide
section
Microstrip line
Steped ridge guide
Ridge guide: five λ/4 sections: Chebychev design
Rectangular guide
European Master of Research
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TRANSFORMER EXAMPLE (1):
ADS SIMULATION
Chebyshev transformer, N = 3, |M|=0.05 (ltotal = 3/4)
50 W
57,37 W
70,71 W
87,14 W
100 W
Design and Analysis of RF and Microwave Systems
European Master of Research
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TRANSFORMER EXAMPLE (2):
ADS SIMULATION
CHEBYSCHEV N=3
mag(S(1,1))
0.4
0.3
0.2
BW = 102 %
0.1
m  0,05
0.0
0
2
4
6
8
10
12
14
16
18
20
microstrip loss
freq, GHz
0
0.0
-0.2
dB(S(1,2))
dB(S(1,1))
-10
-20
-30
-0.4
-0.6
-40
-0.8
-50
0
2
4
6
8
10
12
freq, GHz
14
16
18
20
0
2
4
6
8
10
12
freq, GHz
14
16
18
20
Design and Analysis of RF and Microwave Systems
European Master of Research
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Tapered lines (i)
Taper: transmission line with smooth (progressive) varying impedance Z(z)
The transient ΔΓ for a piece Δz of transmission line is given by:
 
Z  Z   Z
Z  Z   Z
In the limit, when z
dz  

Z
2Z
0:
1
d Z z 
2Z
This expression can be developed taking
into account the following property:
d Ln f z 
1 d  f z 

dz
f ( z ) dz
ZL
Z  z
Z0
0
L

z
Z  Z
Z
z
z  z
1
1 d Ln f z 
d  f  z  
dz
2 f ( z)
2
dz
z
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Tapered lines (ii)
1 d Ln Z  z 
d  z  
dz
2
dz
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:
in     dz e
L
 2 j z
0
  L
Z z 
Taper electrical length
•Exponential taper
•Triangular taper
•Klopfenstein taper
1 L  2 jz d Ln Z z 
  e
dz
2 0
dz
Fourier Transform
d Ln Z  z 
dz
Design and Analysis of RF and Microwave Systems
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Exponential Taper
Z  z   Z0e z
for 0 <z < L
Z 0  Z 0 

Z L   Z L 

1  ZL 
ln 

L  Z0 
d Ln Z  z 

dz
1 L  2 j z
in      e
dz
0
2
Fourier Transform
L
(  L ) 
ln Z L / Z 0  j L sin  L
e
2
L
(sinc function)
 min L  
L
max
2
European Master of Research
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Triangular taper


Z  z    Z0  4 z 2 z2   Z L 
 
1 ln 

 L L2
  Z0 
 Z0e

 2 z2   Z 

 ln  L 
 L2   Z0 
e 
 ZL 
d  ln   4 z  Z L 
ln
 Z 0    L2  Z0 
4
 ZL 
dz
 L2 ( L z )ln Z0 
0  z  L/2
L/2  z  L
0  z  L/2
L/2  z  L
 Z   sin(  L / 2) 
1
   L   e  j L ln  L  

2
 Z0    L / 2 
2
(squared sinc function)
- lower side lobes
- wider main lobe
L
European Master of Research
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Design and Analysis of RF and Microwave Systems
Klopfenstein Taper
   L  L e
 j L
cos
  L
2
 A2
cosh A

(  L  A)
Based on Chebychev coefficients
when n→∞. Equal ripple in passband
passband :  L  A
L 
Z L  Z0
1 Z 
 ln  L 
Z L  Z0
2  Z0 
L
m 
cosh A
Shortest length for a specified |M|
Lowest |M| for a specified taper length
Z L / Z0  2
ltaper = 
L
European Master of Research
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Example of linear taper: ridged wave-guide
Microstrip to rectangular wave-guide transition
SECTION C-C’
SECTION B-B’
SECTION A-A’
Rectangular
guide
Ridged
guide
Microstrip
line
European Master of Research
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Design and Analysis of RF and Microwave Systems
Example of taper: finline wave guide
Rectangular wave-guide to finline to transition
Finline mixer configuration
European Master of Research
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Design and Analysis of RF and Microwave Systems
TAPER EXAMPLE (1):
ADS SIMULATION
ADS taper model
Design and Analysis of RF and Microwave Systems
European Master of Research
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TAPER EXAMPLE (2):
ADS SIMULATION
Aproximation to exponential taper using ADS : 10 sections of /10
50 W
57,44 W
53,59 W
100 W
93,30 W
61,56 W
87,05 W
65,97 W
81,22 W
70,71 W
75,79 W
European Master of Research
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Design and Analysis of RF and Microwave Systems
TAPER EXAMPLE (3):
ADS SIMULATION
Aproximation to exponential taper using ADS : 10 sections of /10
50 W
53,59 W
57,44 W
61,56 W
70,71 W
75,79 W
81,22 W
87,05 W
100 W
65,97 W
93,30 W
European Master of Research
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TAPER EXAMPLE (4):
ADS SIMULATION
EXPONENTIAL / ADS TAPER
0
m1
m3
freq=7.170GHz freq=9.980GHz
m1=-22.116
m3=-32.452
m1
dB(S(3,3))
dB(S(1,1))
-20
m3
-40
-60
-80
0
2
− 10 section approx.
− ADS model
4
6
8
10
12
freq, GHz
14
16
18
20
European Master of Research
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Design and Analysis of RF and Microwave Systems
TAPER EXAMPLE (5):
ADS SIMULATION
Exponential taper
mag(S(3,3))
mag(S(1,1))
0.4
0.3
0.2
ltaper =  @ 10 GHz
0.1
m  0,05
0.0
0
2
4
− 10 section approximation
− ADS model
6
8
10
12
freq, GHz
14
16
18
20
European Master of Research
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Design and Analysis of RF and Microwave Systems
TAPER EXAMPLE (6):
ADS SIMULATION
10 section taper: periodicity in frequency
0
m1
m1
freq=49.41GHz
m1=-9.824
(li=/2)
dB(S(3,3))
dB(S(1,1))
-20
-40
m2
-60
-80
0
m2
freq=9.980GHz
m2=-65.843
(li=/10)
20
40
60
− ADS model
freq, GHz
− 10 section approximation is periodic.
80
100
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MATCHING NETWORKS
LEVY DESIGN
Lecturers:
Lluís Pradell (pradell@tsc.upc.edu)
Francesc Torres (xtorres@tsc.upc.edu)
European Master of Research
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Design and Analysis of RF and Microwave Systems
MATCHING NETWORKS
Z0
Vs
Pd1
PdL
Matching
Network
(passive lossless)
r1  f 
2
PdL
Pd 1
1
2
Gt   


 1  r1    1
PavS PavS M1
2
Minimize |r1 (f)|
Maximize Gt(2)
r f
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CONVENTIONAL CHEBYSHEV FILTER (1)
g3
g1
LC low-pass filter
g n 1  Rn 1
g0
Conversion from
Low-Pass to BandPass filter
LS1
CS 1
LS 3
1 
0 


w  0
 
  1
f  f1
Relative bandwidth
w 2
 2
0
f0
' 
gn
g2
02  21
Center frequency
CS 3
LsiCsi 
Z0
LP 2
RN 1  g N 1.Z 0
CP 2
LPN CPN
1
02
 LpiC pi
Lsi 
gi Z 0
w0
, Csi 
1
w

02 Lsi 0 gi Z0
C pi 
gi
, Lpi 
1
wZ0

02C pi 0 gi
w0 Z 0
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CONVENTIONAL CHEBYSHEV FILTER (2)
Gt ( ' 2 )
1
Gt  '  
1   n2Tn2  ' 
2
1
Pass-band ripple
GtMAX ( T 0 )  1
n
GtMIN ( T 1) 
n
1
1   n2
'
1
r  10log
GtMAX
GtMIN
 10log 1   n2 
 dB 
Chebychev polynomials
cos  n cos 1  x   , x  1

Tn  x   
1
cosh  n cosh  x   , x  1
Gt ( 2 )
1
1
1   n2
 02  1. 2
1
0
2

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CONVENTIONAL CHEBYSHEV FILTER (3)
g0  1
g1 
2 • sin(

2n
Fix pass-band ripple and filter order “n”
)
,
x
g i • g i 1 
4 • sin(
x  sinh( a ) ,
2i  1
2i  1
 ) • sin(
)
2n
2n
i
x 2  sin 2 (
)
n
(i  1, 2,...., n  1)
g n g n 1 
2 • sin(

2n
)
x
g0, g1,.., gn+1 are the low-pass LC filter coefficients: w1'  1
n 
1
sinh( na )
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APPLICATION TO A MATCHING NETWORK
Ce
L'S 1
Le
CS' 1
Transistor modeled with a dominant RLC
behaviour in the pass-band to be matched
RN 1  g N 1.Re
Re
Transistor
Model
LP 2 CP 2
 
2.sin   . R e
g .R
 2n 
Ls1  1 e 
w.0
x.w.0
Ls1  Ls1  Le  Ls1 , Le
'
Cs1 
CeCs1
'
'
Ce  Cs1
given
 Cs1 , Ce
'
'
LPN CPN
Ls1Cs1 
1
r
   n  a  x  g1  Ls1  Le ?
n
02
increase n (n constant)
a, x decrease
or increase n (n constant)
a, x decrease
Solution (?):
The final design may be out of specifications: n too high (too many sections) or r too large
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LEVY NETWORK (1)
SOLUTION: An additional parameter is introduced: Kn<1
Gt  '2  
Gt ( ' 2 )
Kn
1   n2Tn2  ' 
( Kn  1)
Kn
Kn
1   n2
GtMAX ( T 0 )  K n
n
GtMIN ( T 1) 
n
r  10log
Kn
1   n2
GtMAX
GtMIN
 10log 1   n2 
cos  n cos 1  x   , x  1

Tn  x   
1
cosh  n cosh  x   , x  1
'
1
 dB 
Gt ( 2 )
Kn
Kn
1   n2
02  1. 2
1
0
2

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LEVY NETWORK (2)
SOLUTION: Additional design equations
g0  1
Example: n = 2
 
2.sin  
 2n 
g1 
x y
2i  1 
 2i  1 
4.sin 
  .sin 

2n 
2n 


gi gi 1 
 i 
 i
x 2  y 2  sin 2    2. x. y.cos 
n
n
 
2.sin  
 2n 
g n g n 1 
x y
x  sinh  a 
y  sinh  b  (new freedom degree)
1
n 
sinh  na 
sinh 2  nb 
Kn  1 
, ( K n  1)
sinh 2  na 
g0  1
 
2.sin  
4
g1 
x y



, (i  1, 2...., n  1)
g2 
1
2
· 2
g1 x  y 2  1
 
2.sin  
1
4
g3  ·
g2 x  y
x  sinh  a 
y  sinh  b  (new freedom degree)
2 
1
sinh  2a 
sinh 2  2b 
K2  1 
, ( K 2  1)
sinh 2  2a 
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LEVY NETWORK (3)
Design procedure
a) Choose Cs1 or Ls1 taking into account the load to be matched
If
Ls1Cs1 
If
Ls1Cs1 
1
02
1
02
 LeCe 
 LeCe 
1
e2
1
e2
take
Cs1  Ce
 Ls1  Le
take
Ls1  Le
 Cs1  Ce
b) Choose network order (n) and compute g1
g1 
Ls1 .w.0
Re
or
g1 
w
0 .Cs1 .Re
c) Compute x-y from the parameter g1
 
2.sin  
 2n 
x y 
g1
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LEVY NETWORK (4)
d) Choose x, compute y,
OPTIMAL DESIGN: minimize r
r  1  Gt
tgh  na  tgh  nb

cosh  a  cosh  b
2
r
2
MAX
 1  GtMIN  1 
cosh  nb 
Kn

1   n2 cosh  na 
Example: usual case n=2:
tgh  2a  tgh  2b 

For n=2:
cosh  a  cosh  b 
max
 r MAX
0
b
sinh  a   sinh b  x  y  ct
Optimum x
C2  C22  2
x
2
C2  x  y 
Select Ls1 (or Cs1) and n. Compute g1. and x-y. Then determine x, y and Kn, n:
xyb
a
Kn
n
The matched bandwith can be increased from
~5% to ~20% with n=2, with moderate Return
Loss requirements (~20 dB)
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Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE (1)
Ce
Le
Re
Matching
Network
Re  15, 2 W
 Le  0,52 nH 

  f e  8,86 GHz
C

0,62
pF
 e

R  50 W
 f1  5,5 GHz
 f 0  6, 4226 GHz

f

7,5
GHz
 2
R
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LEVY NETWORK EXAMPLE (2)
 f1  5,5 Ghz

 f 2  7,5 Ghz
f 0  6, 4226 GHz
2
w
 0,3114
6, 4226
CS  Ce  0, 62 pF
LS 
g1 
( fe  f0 )
1
 0,99 nH
02CS
w
CS 0 Re
 0,8195
2sin( )
4  1, 7257
C2  x  y 
g1
RLmin  17.75 dB
C2  C22  2
x
 1,978  a  1, 434
2
y  0, 2535  b  0, 2509
RLmin  17.75 dB
K n  0,996  GtMAX  0, 015dB
RLmin  23.98 dB
  0,114  G  0, 071dB
n
tMIN
g 2  0, 4903  C p  2,57 pF , L p  0, 239nH
g3  1, 2928  R3  g3 Re  19, 65 W
Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE (3):
ADS SIMULATION
A transformer is necessary since g3≠1
(R3≠50 Ω). This transformed must be
eliminated from the design
LEVY NETWORK (LUMPED COMPONENTS)
0
0.0
-0.1
-5
-0.2
dB(S(2,1))
dB(S(2,2))
dB(S(2,1))
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-10
-15
-20
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-25
-1.0
4
5
6
7
freq, GHz
8
9
4
5
6
7
freq, GHz
8
9
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Norton Transformer equivalences
STEPS:1) the capacitor C2 is pushed towards the load through the transformer
2) The transformer is eliminated using Norton equivalences
Design and Analysis of RF and Microwave Systems
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0
0.2
-5
-0.0
dB(S(2,1))
dB(S(2,2))
dB(S(2,1))
LEVY NETWORK EXAMPLE (4):
ADS SIMULATION
-10
-15
-0.2
-0.4
-0.6
-20
-0.8
-25
-1.0
4
5
6
freq, GHz
7
8
9
4.5
5.0
5.5
6.0
6.5
7.0
freq, GHz
7.5
8.0
8.5
9.0
Design and Analysis of RF and Microwave Systems
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SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES
IMPLEMENTED USING SHORT TRANSMISSION LINES
L, C elements are then synthesized by means of short transmission lines:
l
L
Z0h
2 f 0 L  Z 0 h  l  f 0 L  Z 0 h
l
0
l
C
Z0l
2 f 0 C  Y0l  l  f 0 C  Y0l
l
0
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SMALL SERIES INDUCTANCES AND PARALLEL
CAPACITANCES IMPLEMENTED USING SHORT
TRANSMISSION LINES: EXAMPLE
LS  0,33 nH  Z 0h  106 W
L2  0,23 nH  Z0h  73,85 W
C2  1,02 pF  Z 0l  15,26 W
for
l1
0
l2
for
for

0
l3
1
50

1
50
1

0 10
Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE ADS SIMULATION (5):
0
0.2
-5
0.0
dB(S(2,1))
dB(S(2,1))
dB(S(1,1))
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-10
-15
-20
-25
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-30
4
5
6
7
freq, GHz
8
9
5.0
5.5
6.0
6.5
freq, GHz
7.0
7.5
8.0
Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE: ADS SIMULATION (6):
0
0.0
-0.5
-5
dB(S(2,1))
dB(S(1,2))
dB(S(1,1))
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-10
-1.0
-1.5
-2.0
-15
-2.5
4
5
6
7
freq, GHz
8
9
5.0
5.5
6.0
6.5
freq, GHz
7.0
7.5
8.0
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Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE (7):
ADS SIMULATION: optimization
Design and Analysis of RF and Microwave Systems
European Master of Research
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LEVY NETWORK EXAMPLE (8):
ADS SIMULATION: optimization
0
1.5
1.0
dB(S(2,1))
dB(S(2,1))
dB(S(1,1))
-5
-10
-15
0.5
0.0
-0.5
-20
-1.0
-25
4
5
6
7
freq, GHz
8
9
-1.5
4.5
5.0
5.5
6.0
6.5
freq, GHz
7.0
7.5
8.0
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Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE (9):
ADS SIMULATION: optimization
Design and Analysis of RF and Microwave Systems
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0
-5
-10
-15
-20
-25
4
5
6
7
8
9
dB(levy3_amb_T_optim..S(2,1))
dB(S(2,1))
dB(levy3_amb_T_optim..S(2,1))
dB(levy3_amb_T_optim..S(1,1))
dB(S(2,1))
dB(S(1,1))
LEVY NETWORK EXAMPLE (10):
ADS SIMULATION: optimization
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
4.5
freq, GHz
5.0
5.5
6.0
6.5
freq, GHz
7.0
7.5
8.0
8.5