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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
A!Aalto UniversitySchool of ElectricalEngineering
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.”
|Sivu 2 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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
|Sivu 3 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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
|Sivu 4 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 5 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 6 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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, ...
|Sivu 7 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 8 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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!
|Sivu 9 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 10 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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!
|Sivu 11 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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
|Sivu 12 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 13 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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
|Sivu 14 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 15 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
Bode-Fano limits (from Pozar)
|Sivu 16 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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
|Sivu 17 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 18 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.)
|Sivu 19 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
* 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.)
|Sivu 20 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
* 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.
|Sivu 21 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
* 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.
|Sivu 22 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 23 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
Example of bandwidth saturation for a simple load
from doctoral thesis by J. Ollikainen, 2002.
|Sivu 24 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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!
|Sivu 25 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 26 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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)
|Sivu 27 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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
|Sivu 28 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 29 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori
A!Aalto UniversitySchool of ElectricalEngineering
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.
|Sivu 30 / 30 | ELEC-C4720 Lecture 10: Wideband impedance matching
Copyright c© 2020 Anu Lehtovuori