Generalized Coupled-line All-Pass Phasers

0.96 MB·

by  Shulabh Gupta

IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 1
Generalized Coupled-line All-Pass Phasers
Shulabh Gupta, Member, IEEE, Qingfeng Zhang, Member, IEEE, Lianfeng Zou, Student Member, IEEE, Li
Jun Jiang, Senior Member, IEEE, and Christophe Caloz, Fellow, IEEE
Abstract—Generalizedcoupled-lineall-passphasers,basedon processing [11], [12] and chip less tags for radio-frequency
transversally-cascaded (TC), longitudinally-cascaded (LC) and identification (RFID) systems [13].
hybrid-cascaded (HC) coupled transmission line sections, are
Phaserscanbeeitherofreflectivetypeortransmissiontype.
4 presented and demonstrated using analytical, full-wave and
1 experimental results. It is shown that for N commensurate Reflective-type phasers are single port structures which are
0 coupled-linesections,LCandTCphasersexhibitN groupdelay convertedintotwo-portstructuresusingabroadbandcirculator
2 peaks per coupled-line section harmonic frequency band, in or a hybrid coupler. They are mostly based on the principle
contrast to the TC configuration, which exhibits only one peak of Bragg reflections, and include microstrip chirped delay
n
within this band. It is also shown that for a given maximum
a lines[14],artificialdielectricsubstratebasedphasers[15]and
achievable coupling-coefficient, the HC configuration provides
J reflection-type waveguide phasers [16]. While reflection-type
the largest group delay swing. A wave-interference analysis is
3 finally applied to the various coupled-line phasers, explaining phasers impose less constraints on the design parameters of
theiruniquegroupdelaycharacteristicsbasedonphysicalwave- the phaser compared to transmission-type phasers [17], their
]
s propagation mechanisms. requirement of an external one-port to two-port conversion
c Index Terms—Dispersion engineering, group delay engineer- component, incurring additional loss along with undesired
i
t ing, phasers, C-sections, D-Sections, all-pass networks, radio- phase distortions in the overall delay response, is a major
p
analog signal processing (R-ASP). drawback. Transmission-type phasers, on the other hand,
o
. are inherently two-port components. Surface acoustic wave
s
c I. INTRODUCTION (SAW) devices [18], and magneto-static devices [19] are
si Radio-Analog Signal Processing (R-ASP) has recently some classical representatives of these phasers, but they are
y emerged as a new paradigm for monitoring, manipulating suitable only for very low frequencies and narrow-bandwidth
h and processing radio signals in real time [1]–[3]. Compared applications.While some recentlyproposedtransmission-type
p to conventional Digital Signal Processing (DSP) techniques, phasers, based on coupled-matrix analysis, offer great syn-
[
R-ASP operates on signals directly in their pristine analog thesis flexibility for advanced group delay engineering, they

1 form to execute specific operations enabling microwave or are usually restricted to narrow-band designs [20], [21]. For
v millimeter-wave and terahertz applications. It thus provides a broader-band applications, coupled-line all-pass phasers are
6
potential solution, specially at high frequencies, to overcome more suitable, offering also greater design simplicity and
7
6 the drawbacks of DSP techniques, which include high-cost benefiting from efficient synthesis procedures [22]–[25].
0 A/DandD/Aconversion,highpowerconsumption,low-speed Thecoupled-lineall-passphasersreportedtodatearebased
1. and high complexity. on transversal cascaded (TC) C-sections and/or D-sections
0 The heart of an R-ASP system is a phaser, which is a synthesizing prescribed group delay responses [25], [26]. On
4 componentexhibitinga specified frequency-dependentgroup- the other hand, a longitudinalcascade (LC) of commensurate
1 delay response within a given frequency range [1]. When a coupled transmission-lines was first demonstrated in [22]. In
v: broadband signal propagates through a phaser, its spectral this work, a third type of cascading configurationis proposed
i components progressively separate from one another in the based on a combination of a TC and an LC configuration,
X
time domain due to their different group velocities, a well- hereby termed as Hybrid cascade (TC). The corresponding
ar known phenomenon commonly referred to as chirping. This coupled-linephaseriscalledaHybrid-cascaded(HC)coupled-
spectral discrimination in the time domain allows manipu- line phaser. The HC coupled-line phaser was first introduced
lating spectral bands individually and directly in the time in[27]andlaterusedingroupdelayengineeringin[28].This
domain, thereby enabling several high-speed and broad-band set of three cascading configurations, LC, TC and HC, rep-
processing operations. Some recently demonstrated R-ASP resent the fundamental cascading schemes upon which more
applicationsincludesignalreceiversandtransmittersforcom- complexphaserscanbeconstructed,leadingtovastvarietyof
munications [4], [5], frequency meters and discriminators for coupled-line all-pass phasers with rich and exotic dispersion
cognitive networks [6], [7], spectrum analyzers for instru- characteristics.Further,thegroupdelaycharacteristicsofthese
mentation [8]–[10], signal manipulation for efficient signal three cascaded configurations are investigated and compared
in details and their unique properties are explained using a
S.GuptaandL.J.JiangarewiththeElectricalandElectronicEngineering rigorous wave-interference analysis.
DepartmentofTheUniversityofHongKong,China.Email:[email protected].
The paper is organized as follows. Section II introduces
Q.ZhangiswiththeDepartmentofElectricalandElectronicsEngineering,
South University of Science and Technology of China (SUSTC), Shenzhen, the TC, LC and HC coupled-line phaser topologies, with a
China. comparisonof their groupdelay characteristicsbased on their
L. Zou, and C. Caloz are with the Department of Electrical Engineering,
Poly-Grames Research Center, E´cole Polytechnique de Montre´al, Montre´al, analytical transfer functions. It also presents corresponding
QC,Canada. fabricated prototypes along with measured results, validating
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 2
the analytical models and confirming the various group de- where in (2) a and b are the through and coupled transfer
lay characteristics. Section III presents the wave-interference functions of the coupler without the end connection. The C-
mechanisms that may be used to construct a general transfer section transfer function is a particular case of this network
functionwithcoupled-lineall-passphasers.Thesemechanisms with S =1.
0
are then used to explain some unique group delay features of Thesecond-ordertransmission-lineall-passphaseristheD-
LC and HC coupled-line phasers. Finally, Sec. IV provides section, which consists of a coupled-line coupler terminated
conclusions. with a C-section, as shown in Fig. 1(b). Using the general
transfer function form (2) with S given by (1a), the transfer
0
II. COUPLED-LINE ALL-PASSPHASERS function of a D-section is found as
A. Basic Transfer Functions 1 ρ tan2θ jρ tanθ
S (θ)= − a − b , (3a)
All-pass transfer functionsare based on two building-block 21 1 ρ tan2θ+jρ tanθ
transfer functions, the C-section and the D-section transfer (cid:18) − a b (cid:19)
with
functions [29].
A C-section transfer functioncan be realized by a two-port
1 k 1+k
transmission-linestructure,consistingofa(four-port)coupled- ρa = − 1 1 (3b)
1+k 1 k
line couplerwith itsthroughandisolatedportsinterconnected r 2r − 2
and
by an ideal transmission-line section, as shown in Fig. 1(a).
Its transfer function can be derived by applying the intercon-
1 k 1 k
nectionboundaryconditionbetweenthe ingoingandoutgoing ρ = − 1 + − 2 (3c)
b
1+k 1+k
waves at the two end ports of the coupler. The C-section r 1 r 2
transfer function is given by
where k are the coupling coefficients of the two sections.
1,2
ρ jtanθ Upon the transformation s = jtanθ, the D-section is seen
S (θ)= − , (1a)
21 ρ+jtanθ to be a second-order phaser, with two pairs of complex para-
(cid:18) (cid:19)
conjugate zeros and poles in the s plane.
1+k −
with ρ= , (1b)
1 k
r − B. Phaser Topologies
where k is the coupling coefficient between the coupled lines
ThegroupdelayprofilesofaC-sectionandaD-sectionhave
forming the structure. Its all-pass nature, S = 1, θ,
| 21| ∀ a specific and restricted shape that depend on the length and
is immediately verified by noting that the magnitudes of
coupling coefficients of the sections involved. However,com-
the numerator and denominator in (1a) are equal, and it
bining several coupled-line sections in an appropriate fashion
may also be verified that the function provides a frequency-
allows synthesizing virtually arbitrary prescribed group delay
dependentgroupdelayresponsereachingamaximumvalueat
responses between the input and the output ports within a
θ =mπ/2,wheremisaninteger.Thisconditioncorresponds
given frequency range. Such phasers can be conceptualized
to frequencies where the length of the coupled-line section is
and categorized based on how the different coupled-line sec-
an odd multiple of a quarter wavelength. Upon the high-pass
tions are connected together. The three basic configurations
to low-pass transformation s = jtanθ, the C-section is seen
shown in Fig. 2 are possible, and are next described.
tobeafirst-orderphaser,withonerealpoleandonerealzero,
1) Transversally-cascaded(TC)coupled-linephaser.Inthis
placedsymmetricallyabouttheimaginaryaxisinthes plane.
− configuration several C-sections are cascaded in the
vin vin directionthatistransversetotheaxisofthetransmission
lines forming the C-sections, as shown in Fig. 2(a).
ℓ=λg/4@ω0 ℓ=λg/4@ω0 ℓ=λg/4@ω0 The corresponding transfer function is then simply the
product of the individual C-section transfer functions
k1 k1 k2 and thus reads
vout (a) vout (b) N ρ jtanθ
STC(θ)= i− , (4)
21 ρ +jtanθ
Fig.1. Transmission-typeall-passphasertopologies.a)C-sectionandb)D- i=1(cid:18) i (cid:19)
Y
section, with ω0 being the quarter-wavelength frequency of a transmission
line. where ρi = (1+ki)/(1 ki), ki being the coupling
−
coefficient of the ith section.
p
2) Longitudinally-cascaded (LC) coupled-line phaser. In
The derivation of a generalized transfer function corre-
this configuration,severalcoupled-linesections are cas-
spondingtoacoupled-linecouplerterminatedwithanarbitrary
caded in the direction of the transmission lines forming
all-passloadwithtransferfunctionS isprovidedinApp.V-A,
0
the coupled-lines, as shown in Fig. 2(a), with the last
and reads
sectionbeingaC-section[30].Thecorrespondingtrans-
a2S ferfunctioncanbederivedbyiterativelyconstructingthe
S (θ)=b+ 0 , (2)
21 1 bS loadtransferfunctionsstartingfromthelastcoupled-line
0
−
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 3
Transversally-cascaded Longitudinally-cascaded
sectiontowardstotheinputport,asderivedinApp.V-B. configuration configuration
vin vin
The result is
ℓ1,k1 ℓ1,k1 ℓ3,k3
a2S (θ)
SLC(θ)=S (θ)=b + 1 2 , (5)
21 1 1 1−b1S2(θ) ℓ2,k2 ℓn,kN
where S2(θ) is the overall transfer function of the ℓ2,k2 vout Hybrid-cascaded
structure starting from the 2nd to the Nth coupled-line configuration
section.ItistobenotedthataD-sectionistheparticular ℓ3,k3 vin ℓ2,k2
case of an LC coupled-line phaser with N =2. ℓ3,k3
3) Hybrid-cascaded (HC) coupled-line phaser. This con- ℓn,kN ℓ1,k1 ℓ4,k4
figurations consists of a combination of TC and LC
coupled-line sections, as illustrated in Fig. 2(c). This
ma aTyCbecosuepelneda-slianecopuhpalseedr.-liTnheecocuoprrleesrptoenrmdiinngatetrdanwsfitehr vout vout ℓn,kN ℓN−1,kN−1
functioncanbewrittenusingtheTCcoupled-linetrans- (a)
fer function as S in (2), leading to 1
0 ω ℓ1=ℓ2=ℓ3=ℓ4=ℓ TCcoupled-lines
S2H1C(θ)=b1+ 1 a21bS10S(0θ()θ), (6a) dθ/d×00..68 LCcoupled-lines
−
HCcoupled-lines
y
S0(θ)= N ρρi−+jjttaannθθ . (6b) dela 0.4 k1 =0.7
iY=2(cid:18) i (cid:19) up 0.2 k2 =0.6
The diversity of possible transfer functionsprovidedby the Gro k3 =0.5
0 k =0.4
TC, LC and HC coupled-line phasers for general parameters 0 0.2 0.4 0.6 0.8 1 4
Electrical length θ π (rad)
is too great to be tractable in such a paper. Therefore, for
×
the sake of simplicity and comparability, we shall restrict (b)
our analysis to the case of commensurate coupled-line sec-
Fig. 2. Cascaded coupled-line all-pass phasers. a) Generic topologies
tions (i.e. coupled-line sections having all the same length). for a transversally-cascaded (TC), longitudinally-cascaded (LC) and hybrid-
However, it should be kept in mind that the above analytical cascaded (HC)coupled-line phasers. b)Typical group delay response ofthe
three phaser topologies in(a)overthe lowest coupled-line section harmonic
transfer functions are general and that the corresponding
frequencybandwithN =4coupled-line sectionsofidenticallengthℓi=ℓ.
phasers exhibit much richer synthesis possibilities than those
presented next.
Figure 3 shows the typical group delay responses of these
C. Group Delay Response
three configurations for different number of coupled-line sec-
The three configurations in Fig. 2(a) exhibit very different tionsN. While the TC coupled-linephaser maintainsa single
group delay responses, as shown in Fig. 2(b) for the case of delay peak across the bandwidth for all N’s, the number of
N =4andℓ =ℓ =ℓ =ℓ =ℓ.Underthelattercondition, delay peaks in the LC and the HC case scales with N. In
1 2 3 4
the TC coupled-line phaser has a delay shape similar to that general, for N coupled-linesections, the group delay exhibits
of a regular C-section, with a single delay maximum over N delaypeakswithina periodicband.Thetrendsobservedin
the lowestcoupled-linesection harmonicfrequencyband,θ Fig. 2(b) are confirmed:
∈
[0,π].Incontrast,theLCconfigurationexhibits4delaypeaks, 1) While the delay values of the peaks in the LC case are
within the same bandwidth. These peaks are quasi-uniformly near identical, except for the 1st and the Nthe peak, a
spacedandquasi-equalinmagnitude,withgroupdelayswings, strong variation of the delay peak values occurs in the
∆τ = τmax τmin, that are larger than those obtained in HC case, with the central peak exhibiting the largest
−
the TC case1. Similar to the LC case, the HC coupled-line group delay swing.
phaser exhibits 4 delay peaks, but with dramatically different 2) The HC coupled-line phaser provides the largest group
characteristics. While the outermost delay peaks are lower, delay swing, and the largest absolute delay values,
the center oneshave a steep slope within a narrow bandwidth among the three cases, at the cost of locally reduced
and thus exhibit a very large delay swing ∆τ. Moreover, bandwidth around the peak values.
it is observed that the two central peaks are closer to the 3) Compared to the LC coupled-line phaser, the spacing
each other compared to the LC case. Thus, for a given betweenadjacentpeaksacrossthebandwidthisstrongly
maximum achievable coupling k, the HC coupled-line phaser non-uniform in the case of HC coupled-line phasers,
provides the largest group delay swing ∆τ, among the three with more crowding near the centre of the periodic
configurations. bandwidth.
Despite the very different group delay characteristics, the
1Note that the response can be drastically different in the case of non-
commensurate sections. following theorem is common to the three phaser configu-
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 4
N =7 N =11 N =15
0.1 reasonable agreement within the design bandwidth, as seen
0.08 (TC in Fig. 6(b), with excellent agreement between the measured
0.06 co andsimulatedgroupdelayresponses.Itshouldbeparticularly
u
p
0.04 led noted that, despite their simplicity, the analytically derived
/dω 0.020 -lines) tirsatincssfeirnfaunccotinovnisncpirnegdimctatnhneervainrioaulsl gcraoseusp. dTehlaeysecfhoarrmacutlears-
θ 0.15
d
× (L may thus be deemed appropriatefor fast and efficient designs
0.1 C
y co of coupled-line phasers.
roupdela 0.00.550 upled-lines) TMFhEeeMaosr-uyHreFdSS THLCCCcccooouuupppllleeeddd---llliiinnneeesss SS2111((ddBB))
G 0000....1234 (HCcoupled-lin (0.2ns/div) −−21000
0 es) τ
0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 ay −30
Electrical length θ π (rad) del
× p −40
u
o
Fig. 3. Typical group delay responses for the different cascaded coupled- gr −50
line phasers in Fig. 2(a) with different numbers N of coupled-line sections 0 1 2 3 0 0.5 1 1.5 2 2.5 3
of length. Here all the phaser sections have the same length ℓ while k1 > frequency(GHz) frequency(GHz)
k2>k3...>kN.
Fig.5. MeasuredS-parametersandgroupdelayresponsesoftheprototypes
showninFig.4comparedwithfull-wave andtheoretical results.
rations: Given N sections, the total area under the τ θ
−
curve is Nπ regardless of the configuration. This theorem
is demonstrated in App. V-C. As a result, based on the
aforementioned considerations regarding the number of delay
peaks, compared to the weakly-dispersive TC coupled-line
phasers, LC and the HC phasers make a more efficient use of
theirdelay-frequencyarea byshapingthedelaycurvesoas to
producehigherdispersionregions.Inotherwords,theylocally
enhances the group delay peaks around specific frequencies
by reducing the other peaks, so as to keep the total area
constant.Adetailedwave-interferencemechanismwillbeused
in Sec. III to explain the other delay characteristics of these
phaser configurations.
(a)
D. Experimental Illustrations Meas. FEM-HFSS Theory 0 S21 S11
v)
In order to confirm the delay characteristics of the three s/di B) −10
csctoorunippslilisentdeo-lfitnetwechoncooRnlOofig4gy0u,0ra3atCsionssush,boswtthrnaetierinlapyrFoeitgros.,tye4pa.echTwh2ee0rempribolustoitlthty,ipceikns. ay(1nτ meters(d −−3200
tTrahnessittiroipnlianteeaischfeindptuhtroanudghouatpcuotpplaonrat.r-Awnavaerrgauyidoefctoonsdtruipctliinnge oupdel S-para −40
gr −50
vias following the signal layer profile, were used to maintain 0 1 2 3 0 0.5 1 1.5 2 2.5 3
frequency(GHz) frequency(GHz)
the same potential between the top and ground planes.
Figure 5 shows the measured S-parameters and the group (b)
delay responses for the three prototypes of Fig. 4, corre-
Fig. 6. HC coupled-line phaser with N = 9 coupled-line sections. a)
sponding to the TC-, LC- and HC coupled-line phasers for Photographs. b)Measured group delay response. c)Measured S-parameters.
N = 4. All prototypes are well-matched across the entire Linewidth, line-gap and length of every section are 20 mils, 8 mils and
1000mils,respectively.
bandwidth of interest with S < 10 dB in all cases. An
11
−
excellent agreement between measured results, full-wave and
the analyticalresults of Sec. II-B, is observed in all cases. As
III. WAVE-INTERFERENCE EXPLANATION
concluded from Fig. 2(b), HC coupled-line phasers provide
the largest group delay swing among all three configurations. A. Signal Flow Analysis
Fig. 6(a) shows another prototype for the HC coupled-line The transfer function of a coupled-line coupler terminated
phaser with N = 9. Again, measured S-parameters exhibit with anall-passtransferfunctionS was derivedin App.V-A
0
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 5
Fig. 4. Photographs of coupled-line phaser prototypes in stripline technology consisting of N =4 coupled-line sections in three different configurations.
a)TCcoupled-line phaser.b)LCcoupled-line phaser,andc)HCcoupled-line phaser.Thepictures showthegroundingviasandoutline ofthecoppertraces
sandwiched between thetwosubstrate layers.Linewidth andline-gap ofeachcoupled-line section are:ℓi∈[22222222]milsandgi∈[2217127]mils,
whichcorrespondstothecouplingcoefficients ki∈[0.0650.0950.1450.215]extracted fromfull-wavesimulation(FEM-HFSS).Lengthofeachsectionis
1000mils.(a)TCphaser. (b)LCphaser.(c)HCphaser.
using a scattering matrix approach. An alternative approach summed up to build the overall transfer function S as
21
to obtain such a transfer function is using the signal flow
graphanalysisinconjunctionwithwaveinterferenceconsider-
directcoupled directthrough
ation[31],[32].Thesignalflowgraphanalysisprovidesdeeper
S (θ)= b +(a S2 a)
insightintothewavepropagationmechanismsinvolvedandis 21 × ×
thereby instrumental to unveil the group delay characteristics z}|{ zfirstloo}p| {
of coupled-line phasers. +(a S2 b S2 a)
× × × ×
Let us consider an HC coupled-line phaser with N =3,
z }|secondloop {
composed of ideally matched coupled-line sections with infi-
+(a S2 b S2 b S2 a)+...
nite isolation, as shown in Fig. 7(a). Let us also assume that
× × × × × ×
all coupled sections have the same length and coupling coef- =b+az2S2[1+bS2+}b|2S4+b3S6+{...]
ficient. The structure may be seen as a coupled-line coupler a2S2
=b+ , (7)
terminated with a pair of TC C-sections. The corresponding
1 bS2
signal flow graph is shown in Fig. 7(b). In the terminating C- −
which is identical to (2) obtained using the scattering matrix
sections, the through signal component is coupled back into
method with S = S2, where S may represent an arbitrary
thestructureviatheendconnection,resultingintheformation 0 0
all-passfunction.Sincetheoveralltransferfunctionisall-pass,
of signal loops, which providesthe necessary variation in the
it may be expressed as ejφ (ejφ = 1, φ), and (7) may be
delay across the design bandwidth [32]. The problem can be | | ∀
compactly rewritten as
simplifiedbyconsideringa nettransferfunctionS2 represent-
ing the complete termination, as indicated in Fig. 7(b).
S (θ)=ejφ =b+a2S S (8a)
21 0 loop
with
∞
k k a S = bnSn. (8b)
b b S2 loop 0
b b nX=0
ℓ ℓ a
Differentiating(8a), usingthedefinitionτ(θ)= dφ/dω,and
−
re-arranging the terms, yields
b b
TermI TermII
ℓ k a a a a
b b db a2 db dS
τ(θ)=je−jφ + S +b 0 Γ
vin vout vin vout vin vout zd}|θ{ zb (cid:18) 0dθ }| dθ (cid:19) loo{p
(a) (b)
Fig. 7. Wave interference phenomenology in HC coupled-line phasers.  TermIII
tacr)oaunPpshflieanrsgefsrunlkac.ytioobun)tbCfooanrrrdeNsthpeo=nthd3rinogucgoshuipgtrlneaadnl-slfflienorewfusnegccrttaiipoohnnsain.ofSteiirdsmetnshteiocofavltehlreeanllgctotrhuaspnlsaefndedr- +zSloop(cid:18)2aS0}dd|aθ +a2ddSθ0(cid:19){, (9a)
function ofasingleC-section.
with ∞ 
Based on the simplified signal flow of Fig. 7(b), the con- Γ = nbnSn. (9b)
loop 0
tributions of the different waves along the structure may be
n=0
X
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 6
In the particular case S0 = 1, the HC coupled-line phaser S0 τ0(θ) τ0(θ)
reduces to the single C-section with the group delay [32] )s
tin
u
τ(θ)=jejφ db + a2 dbΓloop+2adaSloop . (10) .bra( S0
dθ b dθ dθ ]1
(cid:18) (cid:19) S2
Comparing the delay expression (9a) of HC phasers to [τ
the group delay (10) of a C-section, it appears that both
Term II and Term III have an extra factor proportional to
1
dS /dθ contributing to the delay swing when the termination
0
is dispersive, i.e. dS /dθ =0. This reveals that the greatness 0.5
0 6 π
ofthegroupdelayswingsobservedintheLCandHCcoupled- ×
) 0
line phasers in Sec. II are essentially due to the dispersive θ
(
0
nature of the terminations.In contrast, in the TC coupled-line φ−0.5
phasercase,whichhaveregularC-sectionswithnon-dispersive
−1
terminations, i.e. S = 1, the coefficients proportional to
0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
dS /dθ in Terms II and III vanish, resulting in smaller delay
0 electricallengthθ π electricallengthθ π
swings. × ×
(a) (b)
B. Wave Interference Phenomenonology in HC and LC Fig. 8. Typical groupdelay response ofacoupled-line phaser forthe case
Coupled-Line Phasers ofa)anon-dispersive andb)adispersive load,S0.
Consider again a coupled-linecouplerterminatedwith load
of arbitrary all-pass transfer function and a corresponding
from Case II above, and vice-versa for the group delay
signalflowgraphofthe typeshownin Fig. 7(b).Withoutloss
ofgenerality,considertheterminationtobeS insteadofS2to minima. This is exactly the case for the highest peaks
0
and progressively less the case for lower peaks.
represent a general termination. Depending on S , the phaser
0
maybeeitheranLCoranHCcoupled-linephaser.Following • The group delay peaks are uniformly spaced, which is
anexpectedresultfromthe factthatS isnon-dispersive
the signal flow graph, the total group delay, τ(θ), is given by 0
(9a), with S = ∞bnSn and Γ = ∞nbnSn. (linear phase φ0).
loop 0 0 loop 0 0
Since the load transfer function, S , is assumed to be Now consider a dispersive termination S as in an HC
0 0
P P
all-pass and, in general, dispersive, it may be written as coupled-line phaser. Figure 8(b) shows the typical delay
S (θ) = ejφ0(θ) = e−jRτ0dθ. Two extreme cases may then response in such a case. The dispersive characteristics of
0
be distinguished: S manifests itself in the non-constant nature of the delay
0
1) Case I: if φ0 = 2mπ and θ = nπ (ensuring b = 0), τ0(θ)andinthestrongwavelengthcompressionregionaround
where m, n are integers, S =6 +1, so that 6 θ =π/2.Again,thelocationsofthevariousgroupdelaypeaks
0
of the phaseroccuraroundthe regionswherethe phase ofthe
S =1+b+b2+b3+b4+b5+b6+...
loop termination φ is 2mπ, and vice-versa for the group delay
0
Γ =b+2b2+3b3+4b4+5b5+6b6+.... minima. Furthermore, the adjacent delay peaks are now non-
loop
uniformly spaced, with closely packed peaks in the highly
Thisrepresentsaconstructiveloop-interferencesituation
dispersive region of S . In conclusion, while the locations of
where both S and Γ are maximized, resulting in 0
loop loop 2mπ phase values in φ determine the locations of the delay
a maximum group delay according to (9a). 0
peaks of the phaser, the dispersive nature of the termination
2) Case II: if φ = (2m+1)π and θ = nπ, where m, n
are integers,0S = 1, so that 6 S0 controls the spacing between the adjacent peaks.
0 − Finally, consider an LC coupled-line phaser, as illustrated
S =1 b+b2 b3+b4 b5+b6... in Fig. 9. In this case, the transfer function of the load of the
loop
− − −
Γ = b+2b2 3b3+4b4 5b5+6b6.... i = 0 coupled-line section is the transfer function of another
loop
− − − LC coupled-line phaser formed between the 1st section and
This represents a destructive loop-interference situation the 6th section. Compared to the HC coupled-line phaser, the
where bothSloop and Γloop are minimized,resulting in a closedformexpressionofS isnotreadilyavailable,asitmust
0
minimum group delay according to (9a).
constructed iteratively, as done in App. V-B. In this situation,
Let us examine the different types of coupled-line phasers a more qualitative approach may be followed to deduce the
at the light of these observations.First, consider the case of a effectsofS inanLCconfiguration.Theterminationfunction
0
non-dispersivetermination,S0i.eaterminationwithaconstant S0 may be seen as a regular C-section, having a transfer
non-zero group delay. Figure 8(a) shows the typical delay function SC, of same size with small perturbations in the
0
response τ[S21] of such a phaser. Two observations can be couplingcoefficientsalongthestructure.Since,aconventional
made: C-sectionhasaperiodicdelayresponsewithuniformlyspaced
• The group delay peaks occur around the regions where peaks,theLCcoupled-linephasermustexhibitasimilardelay
the phase of the load φ is a multiple of 2π, as expected pattern in the small perturbation limit. Figure 9(a) confirms
0
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 7
Sc achievablecoupling-coefficient,theHCconfigurationprovides
0
the largest group delay swing, at the expense of reduced
i = 0 i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 bandwidth, around the peak locations. This follows from the
factthattheareaunderthegroupdelaycurveisNπregardless
the configuration.
S2L1C S0 S1 S2 S3 S4 S5 Basedonthetypicalgroupdelaycharacteristics,ithasbeen
1 shown that the TC configuration is best suited for broad-
)stinu C-section )stinu.b 0)θ(φ bsuaintadbplehafsoerrsn,awrrhoiwle-bthaendLCappalnidcaTtioCnscornefiqguuirriantgionlasrgaeregmroourpe
.bra( LCcoupled-line ra()θ 0 .bra( dapelpalyiedswtiongtsh.eAcroiugpolreodu-sliwneavpeh-iansteerrsferseoncaesatnoalpyrsoisvihdaes bdeeeenp
)θ(τ0 CL(]S[τ12 −1)stinu icnhsairgahctteinritsotitchseodfeLlaCymanedchHanCiscmosupalnedd-tlhineeunpihqauseergsr,obuapseddeloany
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 wave propagation phenomenology.The TC, LC and HC con-
electricallengthθ π electricallengthθ π
× × figurationsrepresentthethreefundamentalcascadingschemes
(a) (b) upon which more complex phasers maybe constructed to ob-
tain diverseand rich dispersioncharacteristics. An illustration
Fig.9. TypicalgroupdelayresponseofanLCcoupled-linephaser.b)Group
delay ofS0 inanLCcoupled-line phaser, compared to that ofaregular C- of such a construction is shown in Fig. 10, which combines
section of equivalent length. b) Group delay of the LC coupled-line phaser TC, LC and HC topologiesinto a generalcoupled-linephaser
related to the transmission phase of S0. All the lengths are assumed to be configuration.Suchaconfigurationenablesvirtuallyunlimited
equalandki<ki+1∀iinthecaseofanLCcoupled-line phaser.
group delay responses, and are anticipated to be useful in
synthesizing efficient transmission-line phasers for various R-
ASP applications.
this prediction, as τ (θ) closely resembles the delay response
0
of a regular C-section, τC(θ), exceptat the two extreme ends
0
where the peaks are slightly smaller. Once the termination
S is known, the delay response of the overall phaser can
0 e
be easily found. Figure 9(b) shows the overall group delay da T-cascade
c
s
of the phaser, and as described above, the locations of the ac
vreagriioounsswgrhoeurepthdeelapyhapseeaokfsthoeftethrmeipnhataisoenrφo0ccisuraamrouultnipdlethoef -H vin L-cascade
2π. Moreover, since these 2mπ locations of φ (θ) are quasi-
0
L-cascade
uniformlyspaced, the resulting delay peaksin τ[S ] are also
21
quasi-uniformly separated in θ. ed
a
c
TheLC coupled-linephaserresultsin Fig. 9 mightgivethe sa H-cascade
c
impression that the LC configuration provides little benefit -T
over a simple C-section, since responses shown are very
similar.However,theresultsofFig.9areobtainedforcoupled-
vout
line sectionsof identicallengthandsmallcouplingvariations,
forthesakeofthephenomenologicalexplanation.However,if Fig. 10. Generalized coupled-line all-pass phaser consisting of unlimited
combinations ofTC-,LC-andHCcoupled-line structures.
thesectionlengthsareallowedtobedifferentfromeachother
and if the coupling coefficients are allowed to vary more, the
LC configuration may synthesize a great diversity of group
delay functions, which is clearly impossible using the single V. APPENDIX
C-section of corresponding length [22]. A. Derivation of the C-Section Transfer Function [Eq. (1a)]
Consider an ideal lossless, perfectly matched and perfectly
IV. CONCLUSION isolated TEM backward-wave coupled-line coupler, shown in
Generalized coupled-line all-pass phasers, based on Fig. 11(a). The 4-port scattering matrix of such a coupler
transversally-cascaded(TC),longitudinally-cascaded(LC)and is [33]
hybrid-cascaded(HC)coupledtransmissionlinesections,have
been presented and demonstrated using analytical, full-wave ψ+ 0 b(θ) a(θ) 0 ψ−
1 1
andexperimentalresults.Thecorrespondinganalyticaltransfer ψ+ b(θ) 0 0 a(θ) ψ−
functionshavebeenderivedusingmatrixmethodswhichhave  ψ2+ = a(θ) 0 0 b(θ)  ψ1− , (11)
3 1
been found to accurately model the unique group delay char-  ψ+   0 a(θ) b(θ) 0  ψ− 
 4    1 
acteristics of these phasers. It has been shown that in contrast where    
tothe TCphaser,LCandHCcoupled-linephasers,consisting
of N coupled-sections exhibit N group delay peaks within jksinθ
b(θ)= , (12a)
a harmonic frequency band. Moreover, for a given maximum √1 k2cosθ+jsinθ
−
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 8
√1 k2 Similarly, the transfer function of the structure looking into
a(θ)= − , (12b)
√1 k2cosθ+jsinθ the (N 2)th section can be written in terms of the (N 1)th
− − −
section as
and k is the coupling coefficient. The connection via a two-
port of transfer function S0 of ports 3 and 4 corresponds to SN−2(θ)=bN−2+ a2N−2SN−1 . (18)
the relationship 1 bN−2SN−1
−
Generalizingthisprocedure,thetransferfunctionof thestruc-
ψ− 0 S ψ+ turelookingintotheith sectioncanbeiterativelyexpressedin
ψ3− = S 00 ψ3+ . (13) terms of (i+1)th-section as
(cid:20) 4 (cid:21) (cid:20) 0 (cid:21)(cid:20) 4 (cid:21)
Subsequently prescribing ψ3− = S0ψ4+ and ψ4− =S0ψ3+ in Si(θ)=bi+ a2iSi+1 , (19a)
(13)transformsthefour-portcoupled-linecouplerintoa(two- 1 biSi+1
−
port) C-section described by the following set of equations: with
ψ+ =bψ−+aS ψ+, ψ+ =bψ−+bS ψ+, ψ+ =aψ−/(1 jk sinθ
tbwS1o0-)paonrtd2trψa4+nsf=er0afuψ4n2−c/ti(o12n−bS01). Thes0e r3elati3ons lead1to th−e bi(θ)= 1−ki2ciosθi+i jsinθi! (19b)
and p
S21(θ)= ψψ12+− =b+ 1a−2SbS00. (14) ai(θ)= 1−kip2c1os−θik+i2 jsinθi!. (19c)
For S =1, this function may be explicitly written as Followingthisiterativepprocedurefromendtotheinputofthe
0
structure, the overall transfer function of an LC coupled-line
phaser is found as
√1+kcosθ j√1 ksinθ
S (θ)= − −
21 (cid:18)√1+kcosθ−j√1−ksinθ(cid:19) S1(θ)=b1+ a21S2(θ) . (20)
ρ jtanθ 1+k 1 b1S2(θ)
= − , where ρ= . (15) −
ρ+jtanθ 1 k
(cid:18) (cid:19) r − C. ProofoftheConstantAreaundertheGroupDelayCurves
for TC-, LC and HC Coupled-Line Phasers
ψ− ℓ,k ψ− The proof the the area under the group delay curves for
1 1 1 the TC-, LC and HC coupled-line phasers is constant may
ψ+ ψ+
2 1 ψ1− 3 ψ1− S0 be conveniently in the low-pass Laplace domain. Consider
ψ2+ ψ4+ first the transfer function of C- and D-sections, given by (1a)
(a) and (3a), respectively, which become under the highpass-to-
θ1,k1 θ2,k2 θ3,k3 θi,ki θN−1,kN−1 θN,kN lowpass transformation s=jΩ=jtanθ
ρ s H (s)
SC-section(s)= − = 1 , (21a)
21 ρ+s H ( s)
S1 S2 S3 Si SN−1 SN ρ s(cid:18) ρ∗ (cid:19)s 1 −H (s)
(b) SD-section(s)= i− i − = 2 , (21b)
21 ρ +s ρ∗+s H ( s)
(cid:18) i (cid:19)(cid:18) i (cid:19) 2 −
Fig.11. Coupled-linephaserconfigurations.a)C-sectionterminatedwithan
arbitrary loadoftransferfunction S0.b)LCcoupled-line phaser. where H (s) and H (s) are first- and second-order Hurwitz
1 2
polynomials, respectively, and ρ is a function of ρ and ρ ,
i a b
which are defined in (3).
A general Nth-order all-pass phaser can be expressed in
B. Derivation of the Longitudinally-cascaded (LC) Coupled-
terms of an Nth-order Hurwitz polynomial, H (s) = sN +
line Phaser Transfer Function [Eq. (5)] N
cN−1SN−1 + ... + c1s + c0, where cn , which can
Consider the LC coupled-line sections of Fig. 11(b). The always by decomposed as a productof first∈andℜsecond order
transfer function SN of the last coupled-section is given by polynomials,
ρ jtanθ
SN(θ)= N − N , (16) H (s) p ρ s q ρ sρ∗ s
(cid:18)ρN +jtanθN(cid:19) S21(s)= H N( s) = ρm+−s ρn+−sρn∗ −+s. (22)
whereρ = 1+k /1 k .Using(2),withS =S ,the N − mY=1 m nY=1 n n
N N N 0 N
transfer function of the st−ructure looking into the (N 1)th Thismeans thatany Nth-orderall-pass phasercan be realized
section is foupnd as − using p C-sections and q D-sections, such that N = p+2q.
Consequently, the area under the τ θ curve of a Nth-order
SN−1(θ)=bN−1+ 1 a2Nb−N1−S1NSN. (17) asellc-tpioasnscnoentswtiotrukenctasn. be deduced fro−m those of its C- and D-
−
IEEETRANSACTIONSFORMICROWAVETHEORYANDTECHNIQUES2013 9
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