IMPEDANCE TRANSFORMERS AND TAPERS

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IMPEDANCE TRANSFORMERS AND TAPERS
Lecturers Lluís Pradell (pradell_at_tsc.upc.edu)
Francesc Torres (xtorres_at_tsc.upc.edu)
March 2010
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The quarter-Wave Transformer (i)
A quarter-wave transformer can be used to match a
real impedance ZL to Z0
Zin
Z1
ZL
Z0
If
The matching condition at fo is
At a different frequency and
the input reflection coefficient is
The mismatch can be computed from
Pozar 5.5
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The quarter-Wave Transformer (ii)
If Return Loss is constrained to yield a maximum
value , the
frequency that reaches the bound can be computed
from
Where for a TEM transmission line
And the bound frequency is related to the design
frequency as
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The quarter-Wave Transformer (iii)
Finally, the fractional bandwdith is given by
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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
That is, in the case of small reflections the
permanent reflection is dominated by the two
first transient terms transmission line
discontinuity and load
Pozar 5.6
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Multisection transformer (ii)
The theory of small reflections can be extended
to a multisection transformer
It is assumed that the impedances ZN increase or
decrease monotically
The reflection coefficients can be grouped in
pairs (ZN may not be symmetric)
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Multisection transformer (iii)
The reflection coefficient can be represented as
a Fourier series
for N even
for N odd
Finite Fourier Series periodic function (period
q p)

• 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

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Binomial multisection matching transformer (i)
Binomial function
The constant A is computed from the transformer
response at f0
The transformer coefficients are
computed from the response expansion
The transformer impedances Zn are then computed,
starting from n0, as
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Binomial multisection matching transformer (ii)
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Binomial multisection matching transformer (iii)
Bandwidth of the binomial transformer
The maximum reflection at the band edge is given
by
The fractional bandwitdh is then
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Chebyshev multisection matching transformer
Chebyshev polynomial
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Chebyshev transformer design
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Chebyshev transformer design
Application Microstrip to rectangular wave-guide
transition both source and load impedances are
real.
Ridge guide section
Rectangular guide
Steped ridge guide
Microstrip line
Ridge guide five ?/4 sections Chebychev design
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TRANSFORMER EXAMPLE (1)ADS SIMULATION
Chebyshev transformer, N 3, GM0.05 (ltotal
3l/4)
87,14 W
70,71 W
100 W
57,37 W
50 W
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TRANSFORMER EXAMPLE (2)ADS SIMULATION
BW 102
microstrip loss
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Tapered lines (i)
Taper transmission line with smooth
(progressive) varying impedance Z(z)
The transient ?G for a piece ?z of transmission
line is given by
In the limit, when Dz 0
This expression can be developed taking into
account the following property
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Tapered lines (ii)
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
Taper electrical length
Fourier Transform

• Exponential taper
• Triangular taper
• Klopfenstein taper

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Exponential Taper
for 0 ltz lt L
Fourier Transform
bL
(sinc function)
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Triangular taper
(squared sinc function)
- lower side lobes - wider main lobe
bL
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Klopfenstein Taper
Based on Chebychev coefficients when n?8. Equal
ripple in passband
ltaper l
Shortest length for a specified GM
bL
Lowest GM for a specified taper length
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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
Microstrip line
Ridged guide
Rectangular guide
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Example of taper finline wave guide
Rectangular wave-guide to finline to transition
Finline mixer configuration
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TAPER EXAMPLE (1)ADS SIMULATION
ADS taper model
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TAPER EXAMPLE (2)ADS SIMULATION
Aproximation to exponential taper using ADS 10
sections of l/10
57,44 W
50 W
53,59 W
61,56 W
65,97 W
70,71 W
87,05 W
93,30 W
100 W
75,79 W
81,22 W
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TAPER EXAMPLE (3)ADS SIMULATION
Aproximation to exponential taper using ADS 10
sections of l/10
50 W
53,59 W
57,44 W
61,56 W
65,97 W
70,71 W
75,79 W
81,22 W
87,05 W
93,30 W
100 W
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TAPER EXAMPLE (4)ADS SIMULATION
- 10 section approx. - ADS model
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TAPER EXAMPLE (5)ADS SIMULATION
ltaper l _at_ 10 GHz
- 10 section approximation - ADS model
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TAPER EXAMPLE (6)ADS SIMULATION
(lil/2)
(lil/10)
- ADS model - 10 section approximation is
periodic.
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MATCHING NETWORKS

• LEVY DESIGN

Lecturers Lluís Pradell (pradell_at_tsc.upc.edu)
Francesc Torres (xtorres_at_tsc.upc.edu)
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MATCHING NETWORKS
Z0
Pd1
PdL
Matching Network (passive lossless)
r (f)
Vs
r1 (f)
Maximize Gt(w2)
Minimize r1 (f)
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CONVENTIONAL CHEBYSHEV FILTER (1)
LC low-pass filter
Conversion from Low-Pass to Band- Pass filter
Relative bandwidth
Center frequency
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CONVENTIONAL CHEBYSHEV FILTER (2)
Pass-band ripple
Chebychev polynomials
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CONVENTIONAL CHEBYSHEV FILTER (3)
Fix pass-band ripple and filter order n
g0, g1,.., gn1 are the low-pass LC filter
coefficients
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APPLICATION TO A MATCHING NETWORK
Transistor modeled with a dominant RLC behaviour
in the pass-band to be matched
Solution (?) increase en (n constant) a, x
decrease or increase n (en constant)
a, x decrease
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
Knlt1
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LEVY NETWORK (2)
SOLUTION Additional design equations
Example n 2
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LEVY NETWORK (3)
Design procedure
a) Choose Cs1 or Ls1 taking into account the load
to be matched
b) Choose network order (n) and compute g1
c) Compute x-y from the parameter g1
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LEVY NETWORK (4)
OPTIMAL DESIGN minimize
d) Choose x, compute y,
Example usual case n2
Optimum x
For n2
Select Ls1 (or Cs1) and n. Compute g1. and x-y.
Then determine x, y and Kn, en
x ? y ? b ?
The matched bandwith can be increased from 5 to
20 with n2, with moderate Return Loss
requirements (20 dB)
? a ?
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LEVY NETWORK EXAMPLE (1)
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LEVY NETWORK EXAMPLE (2)
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LEVY NETWORK EXAMPLE (3)ADS SIMULATION
A transformer is necessary since g3?1 (R3?50 O).
This transformed must be eliminated from the
design
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Norton Transformer equivalences
STEPS1) the capacitor C2 is pushed towards the
load through the transformer 2) The transformer
is eliminated using Norton equivalences
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LEVY NETWORK EXAMPLE (4)ADS SIMULATION
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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
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SMALL SERIES INDUCTANCES AND PARALLEL
CAPACITANCES IMPLEMENTED USING SHORT TRANSMISSION
LINES EXAMPLE
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LEVY NETWORK EXAMPLE ADS SIMULATION (5)
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LEVY NETWORK EXAMPLE ADS SIMULATION (6)
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LEVY NETWORK EXAMPLE (7)ADS SIMULATION
optimization
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LEVY NETWORK EXAMPLE (8)ADS SIMULATION
optimization
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LEVY NETWORK EXAMPLE (9)ADS SIMULATION
optimization
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LEVY NETWORK EXAMPLE (10)ADS SIMULATION
optimization