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## IMPEDANCE TRANSFORMERS AND TAPERS

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**IMPEDANCE TRANSFORMERS AND TAPERS**Lecturers: Lluís Pradell (pradell@tsc.upc.edu) Francesc Torres (xtorres@tsc.upc.edu) March 2010**The quarter-Wave Transformer* (i)**A quarter-wave transformer can be used to match a real impedance ZL to Z0 Zin Z0 Z1 ZL 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 And the bound frequency is related to the design frequency as:**The quarter-Wave Transformer (iii)**Finally, the fractional bandwdith is given by**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**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)**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**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: The transformer impedances Zn are then computed, starting from n=0, as:**Binomial multisection matching transformer (iii)**1 Bandwidth of the binomial transformer The maximum reflection at the band edge is given by: The fractional bandwitdh is then:**Chebyshev multisection matching transformer**Chebyshev polynomial**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 Rectangular guide Ridge guide: five λ/4 sections: Chebychev design**TRANSFORMER EXAMPLE (1):ADS SIMULATION**Chebyshev transformer, N = 3, |GM|=0.05 (ltotal = 3l/4) 57,37 W 70,71 W 87,14 W 100 W 50 W**TRANSFORMER EXAMPLE (2):ADS SIMULATION**BW = 102 % microstrip loss**Tapered lines (i)**Taper:transmission line with smooth (progressive) varying impedance Z(z) The transient ΔΓfor a piece Δz of transmission line is given by: In the limit, when Dz 0: This expression can be developed taking into account the following property:**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**Exponential Taper**for 0 <z < L Fourier Transform bL (sinc function)**Triangular taper**(squared sinc function) - lower side lobes - wider main lobe bL**Klopfenstein Taper**Based on Chebychev coefficients when n→∞. Equal ripple in passband ltaper = l Shortest length for a specified |GM| bL Lowest |GM| for a specified taper length**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**Example of taper: finline wave guide**Rectangular wave-guide to finline to transition Finline mixer configuration**TAPER EXAMPLE (1):ADS SIMULATION**ADS taper model**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 100 W 93,30 W 87,05 W 81,22 W 75,79 W**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**TAPER EXAMPLE (4):ADS SIMULATION**− 10 section approx. − ADS model**TAPER EXAMPLE (5):ADS SIMULATION**ltaper = l @ 10 GHz − 10 section approximation − ADS model**TAPER EXAMPLE (6):ADS SIMULATION**(li=l/2) (li=l/10) − ADS model − 10 section approximation is periodic.**MATCHING NETWORKS**LEVY DESIGN Lecturers: Lluís Pradell (pradell@tsc.upc.edu) Francesc Torres (xtorres@tsc.upc.edu)**MATCHING NETWORKS**Z0 Pd1 PdL Matching Network (passive lossless) r (f) Vs r1 (f) Maximize Gt(w2) Minimize |r1 (f)|**CONVENTIONAL CHEBYSHEV FILTER (1)**LC low-pass filter Conversion from Low-Pass to Band- Pass filter Relative bandwidth Center frequency**CONVENTIONAL CHEBYSHEV FILTER (2)**Pass-band ripple Chebychev polynomials**CONVENTIONAL CHEBYSHEV FILTER (3)**Fix pass-band ripple and filter order “n” g0, g1,.., gn+1 are thelow-pass LC filter coefficients:**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**LEVY NETWORK (1)**SOLUTION: An additional parameter is introduced: Kn<1**LEVY NETWORK (2)**SOLUTION: Additional design equations Example: n = 2**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 parameterg1**LEVY NETWORK (4)**d) Choose x, compute y, OPTIMAL DESIGN: minimize Example: usual case n=2: Optimum x For n=2: Select Ls1 (or Cs1) and n. Compute g1. and x-y. Then determine x, y and Kn, en: The matched bandwith can be increased from ~5% to ~20% with n=2, with moderate Return Loss requirements (~20 dB) x y b a **LEVY NETWORK EXAMPLE (3):ADS SIMULATION**A transformer is necessary since g3≠1 (R3≠50 Ω). This transformed must be eliminated from the design**Norton Transformer equivalences**• STEPS:1) the capacitor C2 is pushed towards the load through the transformer • 2) The transformer is eliminated using Norton equivalences**SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES**IMPLEMENTED USING SHORT TRANSMISSION LINES L, C elements are then synthesized by means of short transmission lines:**SMALL SERIES INDUCTANCES AND PARALLEL CAPACITANCES**IMPLEMENTED USING SHORT TRANSMISSION LINES: EXAMPLE