The quarter Wave TransformerIMPEDANCE TRANSFORMERS
The quarter-Wave Transformer* (i)
Zin
Z1
ZL
Z0
A quarter-wave transformer can be used to match a real impedance ZL to 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
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
*
The quarter-Wave Transformer (iii)
*
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
In the case of small reflections, the reflection coefficient can be approximated taking into account the partial (transient) reflection coefficients:
*Pozar 5.6
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
*
for N even
for N odd
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
Binomial multisection matching transformer (i)
Binomial function
The constant A is computed from the transformer response at f=0:
The transformer coefficients are computed from the response expansion:
*
Binomial multisection matching transformer (ii)
*
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:
1
Chebyshev multisection matching transformer
Chebyshev transformer design
Chebyshev transformer design
Rectangular guide
Steped ridge guide
TRANSFORMER EXAMPLE (1):
87,14 W
70,71 W
100 W
57,37 W
50 W
TRANSFORMER EXAMPLE (2):
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:
*
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:
Fourier Transform
Exponential Taper
Triangular taper
Klopfenstein Taper
Lowest |GM| for a specified taper length
ltaper = l
*
Microstrip to rectangular wave-guide transition
Example of linear taper: ridged wave-guide
Microstrip
line
Ridged
guide
Rectangular
guide
Rectangular wave-guide to finline to transition
Example of taper: finline wave guide
Finline mixer configuration
TAPER EXAMPLE (1):
TAPER EXAMPLE (2):
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
TAPER EXAMPLE (3):
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
TAPER EXAMPLE (4):
TAPER EXAMPLE (5):
TAPER EXAMPLE (6):
*
MATCHING NETWORKS
LEVY DESIGN
Matching
Network
CONVENTIONAL CHEBYSHEV FILTER (1)
LC low-pass filter
CONVENTIONAL CHEBYSHEV FILTER (2)
CONVENTIONAL CHEBYSHEV FILTER (3)
g0, g1,.., gn+1 are the low-pass LC filter coefficients:
Design and Analysis of RF and Microwave Systems
APPLICATION TO A MATCHING NETWORK
Solution (?): increase en (n constant) a, x decrease
or increase n (en 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
Transistor Model
LEVY NETWORK (1)
1
LEVY NETWORK (2)
Example: n = 2
LEVY NETWORK (3)
Design procedure
a) 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
Design and Analysis of RF and Microwave Systems
LEVY NETWORK (4)
OPTIMAL DESIGN: minimize
For n=2:
Select Ls1 (or Cs1) and n. Compute g1. and x-y. Then determine x, y and Kn, en:
x y b
Optimum x
The matched bandwith can be increased from ~5% to ~20% with n=2, with moderate Return Loss requirements (~20 dB)
Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE (1)
LEVY NETWORK EXAMPLE (2)
LEVY NETWORK EXAMPLE (3):
ADS SIMULATION
A transformer is necessary since g3≠1 (R3≠50 Ω). This transformed must be eliminated from the design
Design and Analysis of RF and Microwave Systems
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
LEVY NETWORK EXAMPLE (4):
SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES IMPLEMENTED USING SHORT TRANSMISSION LINES
L, C elements are then synthesized by means of short transmission lines:
Z0h
Z0l
SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES IMPLEMENTED USING SHORT TRANSMISSION LINES: EXAMPLE
Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE ADS SIMULATION (5):
Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE: ADS SIMULATION (6):
Design and Analysis of RF and Microwave Systems
LEVY NETWORK EXAMPLE (7):
LEVY NETWORK EXAMPLE (8):
LEVY NETWORK EXAMPLE (9):
LEVY NETWORK EXAMPLE (10):
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