Notes 18 - Multistage transformers
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Transcript Notes 18 - Multistage transformers
ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 18
Multistage Transformers
1
Single-stage Transformer
The transformer length is arbitrary in this analysis.
Step
= 1
Z0
Z1 line
0
S21
Z1
ZL
0
S22
S110
, Z in
Z in - Z 0
Z in Z 0
L
e- j1 Load
S120
Z L is real
e- j1
L
Z L - Z1
Z L Z1
From previous notes:
Step
Impedance
change
0
S110 - S 22
0
S 21
S120
Z1 - Z 0
Z1 Z 0
2 Z 0 Z1
Z0
1 S110
Z1
Z1 Z 0
2
Single-stage Transformer (cont.)
From the self-loop formula, we have (as derived in previous notes)
- j 21
0
0
S
S
e
S110 21 012 L - j 21
1 - S22 L e
For the numerator:
2
2 Z 0 Z1
4 Z 0 Z1
0 0
S21S12
Z1 Z 0 Z Z 2
1
0
Next, consider this calculation:
1 S110
2
2
2
Z1 Z 0
Z1 - Z 0
Z12 Z 02 2 Z 0 Z1 Z1 Z 0 Z1 Z 0 2 Z 0 Z1
1
1
1
2
2
2
Z
Z
Z1 Z 0
Z1 Z 0
Z1 Z 0
0
1
2
2
2
Z
2
1
Z 02 2 Z 0 Z1 Z12 Z 02 2 Z 0 Z1
2
Z1 Z 0
2
4 Z 0 Z1
Z1 Z 0
2
Hence
S S 1- S11
0
21
0
12
2
0
3
Single-stage Transformer (cont.)
We then have
S110
1 S
02
11
L
e- j 21
0
1 - S22
L e- j 21
Putting both terms over a common denominator, we have
S110 S110 2 L e- j 21 1 S110 2 L e- j 21
1 S110 L e- j 21
or
S110 L e- j 21
1 S110 L e- j 21
4
Single-stage Transformer (cont.)
S110 L e- j 21
1 S110 L e- j 21
Assuming small reflections
S
0
11
L 1
Note: It is also true that
0
S110 S21
S120 L e- j 21
But S210 S120 1- S110 2 1
S110 L e- j 21
e j1
0
Denote 0 S11
, 1 L
0 1 e- j 21
Z -Z
Z -Z
0 1 0 ; 1 L 1
Z1 Z 0
Z L Z1
0 S110
L
e j1
5
Multistage Transformer
i i
1
Z0
Z1
3
2
Z2
i
N -1
. . .
Z3
Z N -1
N
ZN
ZL
Assume 1 2 3 N
Assuming small reflections:
e j
0
e j
e j
1
e j
e j
2
e j
e j
3
N -2
e j
e j
N -1
N L
e j
6
Multistage Transformer (cont.)
Hence
0 1 e- j 2 2 e- j 4 3e- j 6 ..... N e- j 2 N
Z n1 - Z n
n
Z n1 Z n
Note that this is a polynomial in powers of z = exp(-j2).
i i
1
Z0
Z1
2
Z2
3
Z3
. . .
i
N -1
N
Z N -1
ZN
ZL
Assume 1 2 3 N
7
Multistage Transformer (cont.)
0 1 e- j 2 2 e- j 4 3e- j 6 ..... N e- j 2 N
If we assume symmetric reflections of the sections (not a symmetric layout
of line impedances), we have
0 N , 1 N -1 , 2 N -2 , . . .
e- jN 0 e jN e- jN 1 e j ( N -2) e- j ( N -2) . . .
N odd last term N -1 e e
j
- j
Last term
2
N even last term N
2
8
Multistage Transformer (cont.)
Hence, for symmetric reflections we then have
- jN
2e
2e- jN
1
cos
N
cos
N
2
...
cos
N
2
n
...
1
n
N ; N even
0
2 2
cos
N
cos
N
2
...
cos
N
2
n
...
cos
1
n
N -1
0
; N odd
2
Note that this is a finite Fourier cosine series.
9
Multistage Transformer (cont.)
Design philosophy:
If we choose a response for ( ) that is in the form of a polynomial (in
powers of z = exp(-j2 )) or a Fourier cosine series, we can obtain the
needed values of n and hence complete the design.
10
Binomial (Butterworth*) Multistage Transformer
Consider:
A 1 e- j 2
N
Ae- jN e j e- j A e- jN 2 N cos N
N
A 2 N cos
N
Choose all lines to be a quarter wavelength at the center frequency so that
f f 0 i
2
0
2
(We have a perfect match at
the center frequency.)
dn
Also,
0 for n 1, 2, ..., N -1
d n
2
1st N - 1 derivatives are zero maximally flat
*The name comes from the British physicist/engineer Stephen Butterworth, who
described the design of filters using the binomial principle in 1930.
11
Binomial Multistage Transformer (cont.)
Use the binomial expansion so we can express the Butterworth response in terms of
a polynomial series:
N
1 z
N
CnN z n
where CnN
n 0
N!
N - n ! n !
A binomial type of response is obtained if we thus choose
A 1 e
- j 2
N
N
A CnN e- j 2 n
n 0
We want to use a multistage transformer to realize this type of response.
0 1 e- j 2 2 e- j 4 3 e- j 6 ...... N e- j 2 N
A 1 e
- j 2
N
N
A CnN e- j 2 n
n 0
Set equal
(Both are now in the form of polynomials.)
12
Binomial Multistage Transformer (cont.)
Note that as f 0 0
zero length transmisison lines
Z L - Z0
A 2N
Z L Z0
Hence
Z L - Z0
A2
Z
Z
0
L
-N
Note: A could be positive or negative.
Equating responses for each term in the polynomial series gives us:
n ACnN , n 1,2,......., N
Hence
Z -Z
Z n 1 - Z n
2- N L 0 CnN
Z n 1 Z n
Z L Z0
This gives us a
solution for the line
impedances.
Z0 Z0 , Z N 1 Z L
13
Binomial Multistage Transformer (cont.)
Note on reflection coefficients
n ACnN , n 1,2,......., N
CnN
N!
N - n ! n !
Note that
CNNn
N!
N!
CnN
N - ( N n) ! ( N n)! n! ( N n)!
Hence
n N n
Although we did not assume that the reflection coefficients were
symmetric in the design process, they actually come out that way.
14
Binomial Multistage Transformer (cont.)
Note: The table only shows data for ZL > Z0 since the design can be reversed
(Ioad and source switched) for ZL < Z0 .
15
Binomial Multistage Transformer (cont.)
Example showing a microstrip line
g1 / 4
50 line
Z1
g 2 / 4
Z2
g 3 / 4
100 line
Z3
ZL
Z0
A three-stage transformer is shown.
16
Binomial Multistage Transformer (cont.)
Note: Increasing the
number of lines
increases the
bandwidth.
Figure 5.15 (p. 250)
Reflection coefficient magnitude versus frequency for multisection binomial matching
transformers of Example 5.6. ZL = 50Ω and Z0 = 100Ω.
17
Binomial Multistage Transformer (cont.)
Use a series approximation for the ln function:
X -1 1
ln X ;
X 1 2
X 1
Recall
Z n 1 - Z n
- N Z L - Z0 N
2
Cn
Z n 1 Z n
Z L Z0
1 ln Z n1 2- N CnN 1 ln Z L
2
Zn
2
Z0
Hence
ve
recursi
ship
n
o
i
t
a
l
e
r
ZL
ln Z n 1 2 C ln ln Z n
Z0
-N
N
n
18
Binomial Multistage Transformer (cont.)
Bandwidth
m 2 N A cos N m
1
N
-1 1 m
m cos
2 A
Maximum acceptable reflection
m
m
The bandwidth is then:
f
f0
f -f
2 0 m
f0
f / 2
fm
/2
f / 2
f0
fm
m
4 m
4
1
2-2
2-2
2 2 - cos -1 m
f0
/2
2 A
Hence
1
N
f
4
1
2 - cos-1 m
2 A
f0
- m
2 f0 - fm
1
N
19
Binomial Multistage Transformer (cont.)
f
f0 2
Summary of Design Formulas
A 1 e
- j 2
N
N
A CnN e- j 2 n
Reflection coefficient response
n 0
CnN
Z L - Z0
A2
Z
Z
0
L
-N
A coefficient
ZL
ln Z n 1 2 C ln ln Z n
Z0
-N
N
n
1
N
f
4
1
2 - cos-1 m
2 A
f0
N!
N - n ! n !
Design of line impedances
Bandwidth
20
Example
Example: three-stage binomial transformer
Given:
Z L 50 []
100
N 3
Z1
Z2
Z3
50
50 -100
A 2-3
-0.0417
50 100
1
3
f
4
-1 1 0.05
BW
2 - cos
2 0.0417
f0
0.713
Z 0 100 []
m 0.05
dB
m -26.0 [dB]
BW 71.3%
21
Example (cont.)
ZL
ln Z n1 ln Z n 2 C ln
Z0
-N
Z1 :
N
n
50
ln Z1 ln Z 0 2-3 C03 ln
4.519
100
Z1 91.7 []
Z2 :
50
ln Z 2 ln Z1 2-3 C13 ln
4.259
100
CnN
N!
N - n ! n !
C 30 = 1
C13 = 3
C 32 = 3
C 33 = 1
Z 2 70.7 []
Z3 :
50
ln Z 3 ln Z 2 2-3 C23 ln
3.999
100
Z 3 54.5[]
22
Example (cont.)
Using the table in Pozar we have:
ZL / Z0 2 :
Z1, Z2 , Z3 / Z0 1.0907, 1.4142,1.8337
(The above normalized load impedance is the reciprocal of what we actually have.)
Hence, switching the load and the source ends, we have
Z1, Z2 , Z3 / Z0 1.8337, 1.4142, 1.0907;
Z0 50[]
Therefore
Z1 91.685 []
Z 2 70.711 []
Z 3 54.585 []
23
Example (cont.)
g1 / 4
50 line
Z1
g 2 / 4
Z2
g 3 / 4
Z3
100 line
ZL
Z0
S11 dB 20 log10 f
-26
3.29 GHz
f0 5.0GHz
6.74 GHz
BW 69.0%
Response from Ansoft Designer
24
Chebyshev Multistage Matching Transformer
Chebyshev polynomials of the first kind:
cos n cos 1 x , x 1
Tn x
1
cosh
n
cosh
x, x 1
T1 x x
T2 x 2 x 2 1
For -1 x 1: Tn x 1
T3 x 4 x 3 - 3 x
For x 1: Tn x 1
Tn x 2 xTn -1 x - Tn -2 x
We choose the response to be in the form of a Chebyshev polynomial.
25
Chebyshev Multistage Transformer (cont.)
Figure 5.16 (p. 251)
The first four Chebyshev polynomials Tn(x).
26
Chebyshev Multistage Transformer (cont.)
A Chebyshev response will have equal ripple within the bandwidth.
Ae- jN TN secm cos
This can be put into a form
involving the terms cos (n ) (i.e.,
a finite Fourier cosine series).
Tn x
m
1
B
A
A
B
n=
2
m A
n=
1
2
n
ni
n=
2
n=
3
g
sin
a
e
cr
x
-1
1
f0 -
f
2
m
f0
/2
f0
f
2
f
- m
Note: As frequency decreases, x increases.
27
Chebyshev Multistage Transformer (cont.)
We have that, after some algebra,
T1 sec m cos sec m cos
T2 sec m cos sec 2 m 1 cos 2 -1
T3 sec m cos sec3 m cos 3 3cos - 3sec m cos
Tn sec m cos 2 sec m cos Tn-1 sec m cos - Tn-2 sec m cos
T1 x x
T2 x 2 x 2 1
Hence, the term TN (sec, cos)
can be cast into a finite cosine
Fourier series expansion.
T3 x 4 x 3 - 3 x
Tn x 2 xTn -1 x - Tn -2 x
28
Chebyshev Multistage Transformer (cont.)
Transformer design
Ae- jN TN sec m cos
2 e- jN 0 cos N 1 cos N - 2 .... n cos N - 2n .....
From the above formula we can extract the coefficients n (no general formula is given here).
As f 0 0
0 ATN sec m
Z L - Z0
Z L Z0
Z -Z
1
A L 0
Z L Z 0 TN sec m
29
Chebyshev Multistage Transformer (cont.)
At m
m m A TN sec m cos m A TN 1 A
A m
At 0 :
0 ATN sec m
TN sec m 0
Z L - Z0
Z L Z0
sec m 1
A has the same sign as Z L - Z 0
Hence
A sgn ZL - Z0 m
30
Chebyshev Multistage Transformer (cont.)
Note: The table only shows data for ZL > Z0 since the design can be reversed
(Ioad and source switched) for ZL < Z0 .
31
Chebyshev Multistage Transformer (cont.)
Bandwidth
BW
4
f
2- m
f0
At f 0 : ATN sec m
Z L - Z0
1 Z - Z 1 Z L - Z0
TN sec m L 0
Z L Z0
A Z L Z0 m Z L Z0
cosh N cosh -1 sec m
1
1 Z L - Z0
-1
sec m cosh cosh
m Z L Z0
N
Hence
X -1 1
ln X ;
X 1 2
X 1
1
1
Z L
-1
sec m cosh cosh
ln
N
2
m Z 0
32
Chebyshev Multistage Transformer (cont.)
f
f0 2
Summary of Design Formulas
Ae- jN TN secm cos
Reflection coefficient response
1
1
Z L
-1
secm cosh cosh
ln
N
2
m Z0
A sgn ZL - Z0 m
A coefficient
No formula given for the line impedances. Use the Table
from Pozar or generate (“by hand”) the solution by
expanding ( ) into a polynomial with terms cos (n ).
f
4
2 - m
f0
m term
Design of line impedances
Bandwidth
33
Example
Example: three-stage Chebyshev transformer
Given
Z L 100[]
50[]
Z1
Z2
Z3
100[]
Z0 50[]
m 0.05
Assumed symmetry : Γ3 = Γ0 , Γ 2 = Γ1
A sgn Z L - Z 0 m A 0.05
N 3 A e- j 3 T3 sec m cos
Ae- j 3 sec3 m cos 3 3cos - 3sec m cos
2 e- j 3 0 cos 3 1 cos
(finite Fourier cosine series form)
Equate
34
Example: 3-Section Chebyshev Transformer
Equating coefficients from the previous equation on the last slide, we have
2 0 A sec3 m
2 1 3 A sec3 m - 3 A sec m
1
1
100
ln
2
0.05
50
Also, sec m cosh cosh -1
3
1.408
m 44.7 o 0.780[rad] BW 1.007 100.7%
BW
0
1
1
3
0.051.408
2
f
4
2 - m
f0
0 3 0.0698
1
3
3 0.05 1.408 - 3 0.05 1.408
2
1 2 0.1037
35
Example: 3-Section Chebyshev Transformer
Next, use
Z n 1 - Z n
n
Z n 1 Z n
Z n 1 Z n
1 n
1- n
1 0.0698 57.5
1- 0.0698
1 0.1037 70.8
Z 2 57.5
1- 0.1037
1 0.1037 87.2
Z 3 70.8
10.1037
Z1 50
Z1 57.5
Z 2 70.8
Z 3 87.2
Checking consistency :
Z 4 Z L 87.2
1 0.0698 100.3
1- 0.0698
36
Example: 3-Section Chebyshev Transformer
Alternative method:
Z n 1 - Z n 1 Z n 1
n
ln
ln Z n 1 ln Z n 2 n
Z n 1 Z n 2 Z n
ln Z1 ln Z 0 2 0
ln 50 2 0.0698
4.051
Z1 57.49
ln Z 2 ln Z1 21
4.259
Z 2 70.74
ln Z 3 ln Z 2 2 2
4.466
Z 3 87.05
37
Example: 3-Section Chebyshev Transformer
From Table m 0.05, N 3, Z L / Z 0 2, Z 0 50
Z1 1.1475 50 57.4
Z 2 1.4142 50 70.7
Z 3 1.7429 50 87.1
38
Example: 3-Section Chebyshev Transformer
g1 / 4
50 line
Z1
g 2 / 4
Z2
g 3 / 4
Z3
100 line
ZL
Z0
S11 dB 20 log10 f
-26
2.51 GHz
7.5 GHz
f0 5.0GHz
BW 99.8%
Response from Ansoft Designer
39
Example: 3-Section Chebyshev Transformer
Comparison of Binomial (Butterworth) and Chebyshev
The Chebyshev design has a higher bandwidth (100% vs. 69%).
The increased bandwidth comes with a price: ripple in the passband.
Note: It can be shown that the Chebyshev design gives the highest possible
bandwidth for a given N and m.
40
Tapered Transformer
The Pozar book also talks about using continuously tapered lines to match between an
input line Z0 and an output load ZL. (pp. 255-261). Please read this.
41