ELEC-C4720 Advanced Circuit Theory - Lecture 10: Wideband ...

A!Aalto UniversitySchool of ElectricalEngineering

ELEC-C4720 Advanced Circuit Theory

Lecture 10: Wideband impedance matching

Anu Lehtovuori

|Sivu 1 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching

Copyright c© 2020 Anu Lehtovuori


Agenda today

A short review of different ways to tackle the wideband matchingproblem:

◮ Revision of impedance matching

◮ Theoretical background of wideband matching◮ Bode-Fano criteria◮ Optimization challenge◮ Real Frequency Technique

◮ Practical approaches and examples

“In practice, often the circuit designer throws circuit after circuit at

the problem and hopes for a lucky hit.”




Why do we need impedance matching?

Impedance matching

◮ provides reliable and predictable interconnections betweencomponents in a system

◮ improves the signal-to-noise ratio of the system in sensitivereceiver components (antenna, LNA, etc.)

◮ reduces amplitude and phase errors in a power distributionnetwork (such as antenna array feed network)

The maximum power is delivered when the generator is matched tothe load. Therefore, a crucial task in transmitter, amplifier,receiver, antenna and other RF applications is design of animpedance matching equalizer network.In practice, maximizing efficiency is most important!

S11 =Zin − ZG

Zin + ZG|S21|2 = 1 − |S11|2




Why do we need wideband matching?

◮ High data rates require wide frequency bands ⇒ trendtowards wider frequency bands

◮ Measuring the bandwidth as a relative frequency band: atcenter frequency 800 MHz, achieving 200 MHz band (0.25relative bandwidth) is much more challenging than realizingthe same band at center frequency 2 GHz (0.1 relativebandwidth) ⇒ trend towards new, higher frequencies incommunication




Impedance matching problem

+

−E

ZG

LC

I1

ZL

Zin

◮ Conjugate matching: in optimal case, the input impedance is thecomplex conjugate of ZG, i.e. Zin = Z∗

G = RG − jXG.

◮ Thus, the maximum-available generator power PA =|Eg|2

4RGis

delivered to the load ZL

◮ The power delivered to the load PL = ℜe {ZL} |I|2 =RL|Eg|2

|ZG+ZL|2

◮ The goal is to find a network that minimizes the mismatch, thusmaximizing the ratio of powers: the transducer power gain (TPG)

TPG =PL

PA

=4RLRG

(RL + RG)2 + (XL + XG)2.




Wideband matching problem

A complex load impedance ZL = RL + jXL should be matchedto a source impedance ZG = RG + jXG over a wide frequencyband.

+

−

E

ZG

LC ZL

ZinDifferent network termination arrangements:

1. Filter networks are terminated with only resistances RL and Rg.

2. The simpler wideband single match case involves a complex loadand a resistive source, i.e. XG=0 and often RG = 50.

3. The most general wideband double match case, where both loadand source impedances are complex.




Crucial tools for wideband matching

◮ Filter theory

◮ Darlington’s theorem

◮ Bode-Fano limits

◮ Smith chart

and also bandwidth estimators, optimization, experience, ...

Different approaches: for a real load, for a simple load, for aresonator load, ..., at one frequency, for a narrow band, over areally wide band, ...




The use of filter theory in the case of simple loads

◮ If both termination impedances are real ⇒ explicit elementvalues of the Chebyshev filter are known and found from thetables

◮ If the load includes only one reactive element, it can bethought to be a part of the filter if the value of the reactiveelement is not too large (loads with small Q). The filtertheory can still be utilized.

The following examples are from the book B. S. Yarman: "Design of ultra widebandpower transfer networks", Wiley, 2010.Observations for Chebyshev filters:

◮ As the ripple factor ε increases, the first component slightly increases as degreeincreases.

◮ For a fixed value of the first element x1, as the total number of elementsincreases, the ripple coefficient ε decreases and the second element x2 increases.




Utilizing the values of the Chebyshev filter (by Yarman)

The first normalized reactive element x1

x1 =2 sin( π

2n)

sinh[( 1

n) sinh−1( 1

ε)]

The idea: the reactive element C = x1 = 3 in the load is taken to be a

part of the filter!




For more complex (still simple) load (by Yarman)

Two possible circuit configurations for the load R||C + L withR = 1, C = 3, L = 0.82. (the load is on the left in these figures)

x2 = x′(2) + L

For n = 7, x2 ≈ 0.88, x′(2) ≈ 0.06 based on the table at theprevious slide. Note that for n = 3, the value of x2 is too smallto realize a solution.




Darlington’s theorem

Darlington theorem: Any PR function can be realized with alossless two-port loaded with a resistor.

+

−

E1

RG

LC LC RL

Zin

◮ If Zin is a PR function, also Z∗in is a PR function. (Complex

conjugation changes only the imaginary part.)

◮ Darlington says that Z∗in can be realized with an LC network

terminated to a real value (= Zg).

Any complex impedance can be matched to a real impedance atsingle frequency using a L section.Note that this does not hold for complex impedances: Complex impedance cannot be

matched to all complex impedances!




Conjugate matching with a L section

L1

B2

X1L2

X2

B1

The load ZL = RL + jXL is assumed to be on the right.

X1 = ±√

RL(Z0 − RL) − XL

B2 = ±

√

(Z0 − RL)/RL

Z0

X2 =1

B+

XLZ0

RL

−Z0

BRL

B1 =XL ±

√

RL/Z0

√

R2L + X2

L − Z0RL

R2L + Z2

L




Smith chart illustrates the matching

Possible L-sections to achieve conjugate matching in different areasof the Smith chart

The Smith chart is a transformation of all impedances in the right-halfplane (RHP) onto a unit circle.




Example of L-section matching

A simple RLC load matched with four different L-sections. For asimple load, the difference in bandwidth is quite small.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Frequency (GHz)

|S11

| (dB

)

sCpCsCpLpCsLpCsC




Analytic broadband matching theory

Conjugate matching is not physically possible over a finitefrequency band. For a given load, the ratio of bandwidth andreflection coefficient is fixed. The bandwidth can only be increasedat the expense of the reflection coefficient.The analytical theory answers the following questions:

◮ What is the trade-off between the maximum allowablereflection in the passband and the bandwidth?

◮ How complex must the matching network be to reach thegiven specification?

The main rule: a good match over a narrow band or a poorermatch over a wide band.




Bode-Fano limits (from Pozar)




Bode-Fano in practice

Increasing the matching level requirement decreases the availablebandwidth

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−12

−10

−8

−6

−4

−2

0

Frequency (GHz)

|S11

| (dB

)

−4 dB−5 dB−6 dB−7 dB




Analytical gain-bandwidth theory

Analytical theory

◮ studies the poles and zeros

◮ requires precise load models

◮ requires determining N Cauchy integral constraints for Nelements ⇒ Applying gain-bandwidth limits to simple lumpedterminations

◮ can solve only simple RC or RLC single-match problems

◮ provides a benchmark against which a practical design can becompared.




Numerical optimization in wideband matching

Numerical optimization is the basis of many design techniques inwideband matching.Optimization is the technique for minimizing a nonlinear scalar function of many

variables. In simple notation, the objective is to minimize a function f(x) subject to

constraints c(x) ≤ 0, where x is a vector of scalar variables and c(x) is a vector of

scalar constraint functions.

◮ Two major weaknessess inherent in numerical optimization:1. determining the initial values for optimization variables2. uncertainty of the search finding a global (deepest) minimum

◮ Different choices of optimization variables:◮ matching desired coefficients of rational polynomials at

complex frequency p = σ + jω,◮ varying polynomial pole and zero locations in the complex

plane,◮ varying the L and C values in candidate matching networks.

(This is less ill-conditioned by many orders of magnitude andthus more accurate.)




* Real Frequency Techniques I

The real frequency technique (RFT) does not require a loadmodel, because it is based on load characterization by samples inreal frequency only on the s = jω axis.

Different RFT approaches, which are based on optimization:

◮ Carlin’s RFT (line segment technique):

1. The matching network is described in terms of a driving-pointfunction over the entire jω axis

2. The real part is represented as a linear combination of straightlines with unknown break points Rk. The imaginary part isobtained through Hilbert transform. (The desired impedance isrepresented in a few points.)




* Real Frequency Techniques II

.3. With optimization, a nonnegative, even rational function of

R2(ω) is obtained.4. The second optimization varies both numerator and

denominator coefficients.5. Bode-Gewertz procedure is used to convert the rational R2(ω)

resistance function to a rational Z2(ω) LC impedancefunction.

6. Z2 is realized with Darlington synthesis to get the elementvalues.




* Real Frequency Techniques III

◮ Yarman’s RFT, (direct computational technique)

1. The Z2(ω) function directly represents a physical low-pass orband-pass double-match circuit

2. The parametric representation of Z2(ω) is a form of partialfraction expansion with numerator residues that are functionsof the (LHP) pole frequencies.

3. The transducer power gain is maximized by varying thepositive-real and imaginary parts of the N complex polefrequencies.

4. The laborious Gewertz procedure is not required and numericalstability is improved.

5. An intelligent guess of the initial variables is required.




H-infinity approach

◮ Gives the best possible performance limit by optimizing over allphysical broadband-matching circuits

◮ But: does not give the matching circuit.

The benefit obtained by increasing the number of matchingelements saturates quite rapidly.




Example of bandwidth saturation for a simple load

from doctoral thesis by J. Ollikainen, 2002.




Increasing the number of matching elements

The benefit obtained with additional elements decreases fast, so itis reasonable to use three (four) elements and study differenttopologies instead of just increasing the number of elements.

Π T

F1 F2

Γ1 Γ2

Note: T and Π topologies include all L-sections.In total 40 different alternative matching circuits already with three elements!




Example: matching with two, three, or four elements

The bandwidth increases when the number of matching elements isincreased - but you have to use the right topology!

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Frequency (GHz)

|S11

| (dB

)

critical Loptimal LΓ (3 elements)T+ (4 elements)

The dual-resonance behavior achieved with three and four elements.




One approach for designing Π (and T) circuits

A Π matching network constructed from two L-sections:

For the Z between the blocks: ℜe {Z} < Rmin = min(Rg, RL)(Orfanidis 2015)




A glance at research: designing mobile antennas

◮ Modern handset antennas consist of a radiating part and amatching circuit. To achieve an optimal performance,co-design of the radiating structure and matching circuit isneeded.

◮ Evaluating the available bandwidth with a matching circuit isan important aspect in antenna design in order to find thebest antenna structure.

◮ Bandwidth estimators describe the largest availablesymmetrical bandwidth over the center frequencies at acertain circuit topology. This facilitates1) evaluating the different antenna structures2) choosing the right matching topology




Example: Bandwidth estimator analysis by Valkonen, Lehtovuori

0.5p 20

1p 10n

0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

Frequency (GHz)

Ach

ieva

ble

rela

tive

band

wid

th

L1|L2ΠTLL2

R. Valkonen and A. Lehtovuori, "Determining Bandwidth Estimators and Matching

Circuits for Evaluation of Chassis Antennas", IEEE Transactions on Antennas and

Propagation, 2015.




Summary on wideband matching

◮ A general method does not exist.

◮ For simple loads, methods based on the quality factor Q exist.

◮ Complexity of the problem increases rapidly when increasingthe number of matching elements or the complexity of theload.

◮ The choice of topology is important.

◮ Number of matching elements is reasonable to keep small.

◮ Conjugate matching at a single frequency is not the beststarting point for optimization with wideband matching.



ELEC-C4720 Advanced Circuit Theory - Lecture 10: Wideband ...

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