ACTA ACUSTICA UNITED WITH ACUSTICA
Scientific Papers
Vol. 96 (2010) 403 –415
DOI 10.3813/AAA.918293
On Passive Symmetrical Tw oP or ts, Impedance
Con ver sion and Po wer Transf er
1,2) 2,1)
Pierre Cerv enka ,J acques Marchal
1)
CNRS, UMR 7190, Institut Je an le Rond d’Alembert, 78210 SaintCyr l’Ecole, France.
pierre.cerv enka@upmc.fr
2)
UPMC UnivP aris 06, UMR 7190, Institut Je an le Rond d’Alembert, 78210 SaintCyr l’Ecole, France.
jacques.marchal@upmc.fr
Summary
The theory of linear passi ve symmetrical tw oports is re vie wed. Manyr esults of practical interest can be deri ved
from the very compact analytical basis of this model. Ho we ver, such deri vations are scattered in the literature.
In addition, the demonstrations do not al wa ys takea dv antage of the generality that can be obtained by avo iding
implementations built on particular applications. Acomplete, self consistent analysis is presented here. Atten
tion is focused on po wer transfer and the relations between the impedances at each port. The speciﬁccase of
transmission lines is ﬁnally addressed.
PA CS no. 43.20.Wd, 43.20.Bi, 43.20.Ks, 43.20.Mv
theorems pro vide po werful tools for dealing with alar ge
1. Intr oduction
range of applications [20], addressing for ex ample im
pedance measurement [21, 22, 23], modeling of tubes
The po werful principles of reciprocity were formulated
[24, 25], and propag ation in layered media [26]. The un
long ago in the optical and acoustical domains by Von
derlying concern is often po wer transfer .
Helmholtz [1, 2] and Lord Rayleigh [3] although Lamb
[4] recognized that ag eneralization of these theorems This paper addresses the systems that can be described
wasalready contained in the former Lagrange’sw ork [5]. by means of apassi ve linear symmetrical tw oport black
Lorentz [6] ex tended the domain of application to electro box. Most analytical de velopments are kno wn butscat
magnetism. Extending the scope of the reciprocity princi tered in journals and te xtbooks. The authors think useful
ple is al ways an acti ve ﬁeld of research (e.g. [7] ). Among to present here asynthetic viewoft he main results that
such works, Goedbloed [8] and Potton [9] gi ve indepth re can be deri vedfrom the very fewh ypotheses that mak e
vie ws about reciprocity.The conditions of validity of the the basis of this model. In addition, the po wer transfer co
reciprocity principle are also acritical issue [10]. When eﬃcients as expressed with non dimensional parameters
properly deﬁ ned, most linear passi ve netw orks are recipro in section 3, as well as the precise conditions that vali
cal although the theoretical existence of acounter e xample date the approximations presented in section 3.5 ha ve not
has been sho wn by Telle gen [11] with the gyrator (int he been found else where in the literature. We are primarily
mechanical domain, the actual existence of apassi ve ma interested in the ener gy transfer through such asystem,
terial that wo uld allowtob uild such asystem is still ques and in the impedance con version that its presence in volv es.
tioned ). The formal analogy of the principle of reciprocity The formalism is presented without anye xplicit reference
across di ﬀ erent domains of ph ysical phenomenon has been
to propag ation equations that would be deri vedf rom the
pointed out, in particular between electrical and mechan ph ysical phenomenon actually in volv ed. The analysis is
ical systems [12, 13, 14, 15]. It justiﬁ es the interest of all performed in the conte xt of the harmonic steady state.
general properties that can be deri vedfrom formal analy
The black box that represents the system is passi ve,
sis.
which means that it does not contain anyi nternal source.
Another lar ge class of systems is deﬁ ned by symme Ener gy is exchanged with the external world in aw ay
try.The dependencyofs ymmetry on reciprocity is not a which can be described at each interf ace by apair of
straightforw ard issue and has been thoroughly addressed state variables: aforcing, extensi ve variable V ,a nd ar e
sponse, intensi ve variable P.Their product represents the
[16, 17, 18, 19]. Manyphy sical systems exhibit both the
po wer fed into or pouring out the system via each termi
reciprocity and symmetry properties. Hence the related
nal. The system is alinear tw oport: there are twointer 
faces with asingle pair of scalar conjug ate variables at
each of them; linear relations link these variables. Hence
Recei ved8J uly 2009,
accepted 21 November 2009. the scope of this paper does not extend to mechanical sys
©S.H irz el Verla g · EAA 403ACTA ACUSTICA UNITED WITH ACUSTICA Cer venka, Mar chal: On passive symmetrical tw opor ts
Vol. 96 (2010)
gation ), the restriction in the nature of the bounding media
is withdra wn. Atypical example arises when studying the
po wer transfer between atransducer and the propag ating
media through amatching layer.A mongst other ex amples
of application in the acoustic domain, the system can be
also awav eguide so that the conjug ate variables are de
ﬁned with respect to anygiv en mode.
Figure 1. Electrical scheme of atwoport.
The matrices that characterize ap assi ve symmetrical
tw oport are deri vedins ection 2.1. The interpretation
in terms of onew ay modes, as well as the introduc
tion of the characteristic impedance, are thus straightfor 
ward. The relation between the impedances at both in
terf aces is analyzed in section 2.2. Classical limit cases
are pointed out, in association with the propag ation termi
nology (half/quarter/one eighth wa velength lines ). Po we r
tr an sf er is th or ou gh ly ad dr es sedins ection 3. The analysis
is ﬁrst conducted with respect to the input impedance. The
po wer transfer coe ﬃcient exhibits acharacteristic pattern
in its dependencyont he parameters of the tw oport. This
result does not seem to be kno wn. The optimization of the
po wer transfer within di ﬀ erent situations is then discussed.
In manyparticular problems, the parameters that describe
Figure 2. Simple acoustical example of atwoport.
the media supporting the system do not depend on the di
mension that links the interf aces. Se veral results displayed
in the pre vious sections are then furthermore expanded in
tems that in vo lveg eneralized forces and displacements be
section 4when this property applies.
cause it wo uld require 6p orts per concerned interf ace [27].
We consider asymmetric system, i.e. the black box re
mains the same when both ports are permuted. Because
2. Passive symmetrical tw oport
of the symmetry,the system is necessarily homogeneous,
i.e. both interf aces ha ve the same type (e.g. mechanical
2.1. Tr ansfer matrix
mechanical or electricalelectrical ). In the acoustic do
main, alar ge variety of conﬁ gurations can be tak en into 2.1.1. Tw oport
account, pro vided ap air of acoustic pressure and normal
Port 1and Port 2are referred as input and output, re
velocity deﬁ nes the conditions at each interf ace. Wi th an
specti vely.Note the con vention of sign at the terminals
electrical system, P and V stand for voltages and currents,
in Figure 1. It implies that po wer entering the system is
respecti ve ly (Figure 1).Note that symmetry implies reci
counted positi ve.F or an acoustical system, it means that
procity when dealing with alinear passi ve tw oport.
the vectors normal to the interf aces used to deﬁ ne the nor 
Fore xample, the system can be amultilayered sym
mal velocities are oriented inw ard the system at both ports
metric media (s olids and/or ﬂuids ), whose exterior faces
(e.g. see Figure 2).Using this con vention, the follo wing
are parallel planes in contact with outside ﬂuids. The def
impedances are deﬁ ned according to
inition of the conjug ate variables calls here for the time
−1 −1
angular spectra approach (t ime and spatial Fourier formal
Z = P V ,Z=−PV . (1)
1 1 2 2
1 2
ism ). Considering the in variance of the time and spatial
frequencies in the decomposition of the pressure and nor  In the acoustical frame, Z deﬁ nes asurf ace impedance be
mal velocity ﬁe lds at each interf ace, P and V stand for cause V refers to the spectral component of anormal ve
such spectral components at gi ventime and spatial fre locity.Ine quation (1) , Z can be interpreted as the impe
2
quencies. In the simplest situation depicted Figure 2, Port dance that loads the netw ork, whereas Z stands for this
1
1and Port 2are in contact with semiinﬁ nite homogeneous load as seen from the input interf ace. If Z is an actual
2
ﬂuids. There is an incident plane wa ve ( P ,V )atP ort 1 impedance load, then e{Z }≥ 0. Note that if the exte
i i 2
that is reﬂected ( P ,V ), and atransmitted wa ve ( P ,V ) rior media that stands ne xt to Po rt 2isanhomogeneous,
r r t t
semiinﬁ nite ﬂuid de void of anys ource, Z is simply the
out of Port 2. The in variance in the frequencies translates
2
characteristic impedance of this media. Furthermore con
here through the SnellDescartes relation between θ and
a
θ .N ote that P and V refer to the conditions at the in sidering asource applied at Port 1, the pressure P and the
b 1,2 1,2 1
terf aces. Note also that if the ﬂuid b is not semiinﬁ nite nor velocity V at this interf ace cannot be independently im
1
homogeneous, the load condition at Port 2ismodiﬁ ed be posed, their ratio Z being dictated by Z as seen through
1 2
cause the conditions ( P ,V )atthe interf ace will takeinto the blackbox. One chooses to consider the source at the
2 2
account an incoming, reﬂected wa ve.Inc ase the direction input side, i.e. Port 1, and the radiating media at the out
of the plane wa vesisn ormal to the interf aces (1D propa put side, i.e. Port 2. The con verse conﬁ guration could be
404Cer venka, Mar ch al: On passive symmetrical tw opor ts ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 96 (2010)
indeed considered, i.e. source at Port 2with surf ace im
pedance Z at Port 1. Because of the linearity,the general
1
case with sources on both sides can be then handled merely
by superposition.
Hence considering that there is no ex ternal source at
Port 2, let us denote Z the shunt impedance and Y the
T T
open admittance as seen from Port 1, i.e.
Figure 3. Equi va lent electrical scheme of areciprocal tw oport
−1
Z = P V = Z ( Z = 0) ,
T 1 1 2 −1
1
P =0
2 when Z = Y .P ort 1and Port 2a re isolated from each other .
T
T
−1 −1 −1
Y = P V = Z ( Z = 0) . (2)
T 1
1 1 2
V =0
2
−1
Adegenerated case arises when Z = Y :there is al ways
In the electrical transmission line language, it corresponds
T
T
Z = Z ;both sides of the netw ork are no longer related.
to measurements on the short or open circuited “stubs”. In
1 T
There is something inside the black box that hides one port
the acoustical ﬁeld, Z is the input impedance seen when
T
to the other (e.g. perfect reﬂector ). The transfer matrix T
the output impedance is very weak; the admittance Y is
T
is not deﬁ ned. This case can be depicted as in Figure 3.
the input admittance seen when the output impedance is
Fornow,wec onsider the general case Z Y = 0.
very lar ge. In both cases, the interf ace at Port 2int hese T T
Because of (9),the transfer matrix that describes apas
idealized ex periments is close to aperfect mirror .
si ve reciprocal tw oport depends only on three comple x
The po wer that is dissipated in apassi ve tw oport cannot
parameters ( a, Z and Y ),
be ne gati ve,sothat the real parts of Z and Y are positi ve, T T
T T
−1 −1
aa Z (1 − Z Y )
T T T
e Z >0and e Y > 0. (3)
T T
T = (⇒T  = 1) . (10)
−1 −1
aY a (1 − Z Y )
T T T
Note that this property proceeds from at hermodynamic
Equivalently,t he impedance and admittance matrices read
lawand is al ways true. It does not depend on ex ternal
conditions as it is for example the case with the sign of
−1
1 a
−1
Z = Y
e{Z } or e{Z }.U sing (2),the transfer matrix T that
2 1 −1 −2 −1
T
a a (1 − Z Y )
T T
describes the tw oport reads
1 −a (1 − Z Y )
−1 T T
and Y = Z . (11)
T 2
−a (1 − Z Y ) a (1 − Z Y )
P P
1 2 T T T T
= T (4)
V −V
1 2
2.1.3. Symmetry
abZ
T
with T = , (5)
Forasymmetrical system, it would not matter which port
aY b
T
is the input port and which the output port. Hence the im
pedance Z and admittance Y matrices (7) ha ve equal di
a and b being twocomplexconstants.
agonal and antidiagonal elements: e = e and e = e .
11 22 12 21
Equation (4) can be also reordered in terms of impe
Consequently,as ymmetrical tw oport is also necessarily
dance or admittance relations according to
reciprocal because the matrices Z and Y are then symmet
rical. Hence (9) must hold, and the equality of the diagonal
P V V P
1 1 1 1
= Z and = Y , (6)
terms in (11) implies in addition
P V V P
2 2 2 2
−1
1 b (1 − Z Y ) 2
−1 T T
(12)
a = 1 − Z Y .
with Z = Y T T
T −1 −1
a a b
The transfer matrix (10) is reduced to
1 −a (1 − Z Y )
T T
−1 −1
and Y = Z = Z . (7)
−1 −1
T
−b ab
1 Z
T
T = a , (13)
Y 1
T
2.1.2. Reciprocity
The straightforw ard deﬁ nition of reciprocity is the sym and the impedance and admittance matrices read
metry of the impedance Z (oradmittance Y )matrix. Reci
−1 −1
1 a 1 −a
−1 −1
procity implies that the determinant of the transfer matrix
Z = Y and Y = Z . (14)
−1 −1
T T
a 1 −a 1
is equal to unit. In the particular case of atwoport, these
properties are ev en equi valent, as well as the follo wing
Apassi ve symmetrical tw oport is thus characterized by
particular “reciprocity relation”
atransfer matrix which depends only on twoparameters
( Z , Y ), whose determinant is unit and diagonal terms
T T
V ( P = 0,P = P ) = V (P = P, P = 0) . (8)
1 1 2 2 1 2
are equal. Note that the only kno wledge of Z and Y is
T T
not suﬃcient to select the correct determination of the root
Hence anyofthese deﬁ nitions lead to
2
of a in (12).Additional information is needed to solv e
ab 1 − Z Y = 1. (9) this problem: at av ery lowfrequency, it can be for instance
T T
405
ACTA ACUSTICA UNITED WITH ACUSTICA Cer venka, Mar chal: On passive symmetrical tw opor ts
Vol. 96 (2010)
∗
progressi ve wa vesthat tra veltow ards opposite directions,
the sign of e{P P } observ ed when V = 0w hich dic
1 2
2
tates the sign of the real part of a.T his seed being set,
Z Z
c −1 c
complementary techniques such as phase unfolding in the
T = ζ ⇒
1 1
frequencyd omain can be deplo yed.
P Z P Z
1 c 2 c
The transfer matrix T can be diagonalized according to
= V → = ζV ,
−1
V 1 −V 1
1 2
T = UDU with
Port1 Por t2
Z Z
c c
−1
T = ζ ⇒ (22)
Z Z ζ 0
c c
−1 −1
U = and D = , (15)
1 −1 0 ζ
P Z P Z
1 c 2 c
= ζV ← = V .
V −1 −V −1
1 2
in which appears the characteristic impedance Por t1 Por t2
In ot herw ords, anycondition at the output can be split into
Z = Z /Y , (16)
c T T twoprojections upon these eigen vectors,
P P Z Z
2
and the term ζ 2 c c
=
+ χ , (23)
2
−V 1 −1
Z (1 + χ )
2
c 2
1/2
Z − Z
2 c
1 − Z Y
T T
with χ = , (24)
2
ζ = . (17)
Z + Z
2 c
1 + Z Y
T T
which gi vesback to the input side:
These results can be check ed in writting
P
P Z Z
1 2 c c
−1
= ζ + χ ζ . (25)
2
V 1 −1
−1 −1 1 Z (1 + χ )
c 2
1 ζ + ζ (ζ − ζ )Z
−1 c
T = UDU =
−1 −1 −1
(ζ − ζ )Z ζ + ζ
2
c
The eigen value ζ (17) is the propag ation operator that
1 Z transforms awavetra veling from one interf ace to the other ,
T
= a . (18)
−1
Y 1 whereas the in verse value ζ in (25) accounts for retro
T
propag ation, i.e. ﬁnding back the wa ve at the interf ace
√
from where it originated. Considering that there is asource
Thorough the entire paper,the operator .. stands as usual
only on the Port 1side, the term χ in (24) can be in
for the root determination whose real part is positi ve.The
2
terpreted as the reﬂection coe ﬃcient observ ed at Port 2.
ambiguity addressed pre viously about the coe ﬃ cient a is
Equivalently the ratio between the forw ard and backw ard
nowdeferred to the root that deﬁ nes ζ in (17).This term is
wa vesatP ort 1isd eri vedfrom (25) ,
written in an ex ponential form by introducing the comple x
factor φ ,
T
Z − Z
1 c
2
χ = χ ζ = . (26)
1 2
Z + Z
1 c
ζ = exp − φ
T
The magnitudes χ  and χ  diﬀer on both sides only be
1 2
⇔ tanh φ + jkπ = Z Y (19)
T T T
cause of the losses encountered in the black box, which are
k
⇔ ζ = ( −1) exp − arctanh Z Y .
T T tak en into account by the real part of φ .Hence because
T
of the con vention stated above and the notation gi venin
The tanh function being periodic modulo π with its ima (19),that real part is always positi ve.
ginary ar gument, the ambiguity is nowm ade explicit
2.2. Characteristic impedance, input and output
through the inte ger k.W ith this notation, the transfer ma
impedances
trix reads
Looking nowatt he relation between Z and Z through
1 2
cosh φ Z sinh φ
T c T
T = , (20) the transfer matrix (13),there is
−1
Z sinh φ cosh φ
T T
c
Z + Z Z−Z
2 T 1 T
Z = ,Z= . (27)
1 2
which comes closer than (13) to the well kno wn transmis
1 + Z Y 1−ZY
2 T 1T
sion line relation.
It is easy to deri ve ag ain directly from (27) that the char 
t
From the deﬁ nition of the eigen vectors [ Z , ±1] which
c
acteristic impedance Z is the remarkable load value at the
c
build the matrix U in (15),the output impedances ±Z are
c
output side that is seen unchanged at the input side. When
left unchanged after the transform T ,
ev er Z and Z depart from this value, (27) does not gi ve
1 2
acon venient formula that links ex plicitly the di ﬀerences
Z = ±Z ⇔ Z = ±Z (21)
2 c 1 c
Z − Z and Z − Z .Nonetheless there is
2 c 1 c
Z − Z
2 1
(s ame sign in each equation ). These eigen vectors and val 2
Z Z = Z 1 + . (28)
1 2
c
ues can be interpreted in terms of wa ve propag ation as tw o
Z Z Y
c T T
406
Cer venka, Mar ch al: On passive symmetrical tw opor ts ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 96 (2010)
(28) allo ws to point out three particular cases that depend
1 j
k
⇒ T = ( −1) (35)
on the order of magnitude of Z Y .W ith the help of
T T
2 sin θ
(17),( 19 ), there is
exp( −jθ/2) Z exp( jθ/2)
c
· ,
Case 1 −1
Z exp( jθ/2) exp( −jθ/2)
c
−1
Z ≈ Z ( k ∈ Z ). Note: θ = 0because Z = Y as stated in
1 2 T
T
Z Y 1 ⇒ (29)
T T 2
ζ ≈ 1 section 2.1.2.
This situation is kno wn [29] as a λ/8i mpedance trans
k
⇒ φ = jkπ ⇒ T = ( −1) I , ( k ∈ Z)( 30 )
T
former ( arg( ζ )  = m{φ } = ±π/4mod π ).
T
in which I denotes the 2 ×2i dentity matix. Because
3. Po wer transfer
arg( ζ )  = 0mod π,this case is related to halfwav elength
lines. The input impedance is the same as the output im
3.1. Notations
pedance, although being not equal to the characteristic im
pedance Z .Iti mplies important practical consequences.
We study here the po wer transfer from as ource at the in
c
This case is thoroughly in vestig ated in section 3.5.
put of apassi ve linear symmetrical tw oport, whose trans
fer matrix is gi venin( 13 ), to wards an impedance Z that
2
Case 2
loads the output. The deﬁ nition and meaning of Z has
2
2
been already discussed at the be ginning of section 2. With
Z Z ≈ Z
1 2
c
Z Y 1 ⇒ (31)
T T 2
electrical systems or acoustical wa veg uides, the source
ζ ≈−1
supplies ener gy per time unit. With systems that in volv e
1
⇒ φ = jπ + k (32)
T plane interf aces (and therefore angular spectra ), the source
2
deli vers ener gy per time unit and per surf ace unit. In the
0 Z
c
k
⇒ T = ( −1) j , ( k ∈ Z ) follo wing, we use the terminology “po wer” for both mean
−1
Z 0
c
ings.
The source deli vers the po wer W through the input port
1
The propag ation from one port to the other in volv es a
1 1
phase rotation of aquarter of acircle (arg ( ζ ) = π/2 ∗
W = e P V = P  V  cos θ (36)
1 1 1 1 1
1
2 2
mod π ), which is the signature of the socalled quarter 
with θ = ar g{Z }.
1 1
wa velength lines. The matrix T can be also interpreted as
∗
the transfer matrix of ag yrator with gyration impedance
( X denotes the complexconjug ate of X ).
Z connected in cascade with a ±π/2phase shifting net
Let us recall that with acoustic systems, ( P ,V )refers
c
1 1
wo rk [28]: it transfers aweak impedance into av ery lar ge
to the conditions at the input interf ace. Hence in the ex
impedance and vice versa. Less drastically,a λ/4s ystem
ample gi venwith Figure 2, W should not be confused
1
can be used to perform acon version between di ﬀerent
with the po wer associated with the incoming incident wa ve
impedances. Fori nstance in the acoustic domain, a λ/4
( P ,V ).
i i
layer with impedance Z is used to match the lowimpe The po wer W that is dissipated in the load Z reads
c 2 2
dance Z of apropag ating medium with the high impe
2
W = (37)
2
2 −1
dance Z ≈ Z Z of atransducer ceramic.
1
c
2 ∗
2 2
1 + Z Y P V  cos δ − e{Y }P  − e{Z }V 
T 1 1 1 T 1 T 1
T
Case 3
21 − Z Y 
T T
Whene verthe characteristic impedance Z is real, are
c
∗
with δ = θ − ar g{1 + Z Y }. (38)
1 1 T
T
markable property arises at the transition point for which
the norm Z Y  is exactly equal to unit
T T The po wer deli vered by the source and the po wer dissi
pated by the load are related through
Z Y =1and Z ∈ R (33)
T T c
µ
W = P  V  = ηW (39)
2 1 1 1
2
∀Z ∈ R, Z  = Z ,
2 1 c
⇒
with µ = η cos θ ≤ η< 1.
if Z  = Z , ∀ar g{Z },Z ∈ R 1
2 c 2 1
The coe ﬃcient η gi vesthe relati ve amount of po wer that is
Whate verist he value of ap urely resisti ve load, the norm
actually transferred from the source to the load: the tw o
of the input impedance is equal to the norm of the charac
port dissipates the ratio 1 − η of the total po wer deli vered by
teristic impedance. Con versely,whene verthe norm of the
the source. The coe ﬃcient µ characterizes the fraction of
load is equal to the norm of the characteristic impedance,
po wer that is dissipated in the load compared to the max
the input impedance is real. The propag ation coe ﬃcient
imal po wer P  V /2t hat the source can potentially de
1 1
and the transfer matrix can be deri vedwithout assuming
li verwith the amplitudes P  and V :
1 1
that Z is real:
c
1 − Z Y e{Z }
T T 2
µ = (40)
2 1 + Z Y  Z + Z 
2 T 2 T
Z Y = exp jθ ⇒ ζ = −jtan ( θ/2) (34)
T T
∗ ∗
e ( Z − Z )(1 − Z Y )
1 T
1 T
1 θ  π
= .
⇒ φ = − ln tan + j sgn ( θ ) + kπ ,
T
1 − Z Y Z 
2 2 4 T T 1
407
ACTA ACUSTICA UNITED WITH ACUSTICA Cer venka, Mar chal: On passive symmetrical tw opor ts
Vol. 96 (2010)
3.2. Optimal input impedance
3.2.1. Optimal phase
Gi venthe amplitudes P  and V  at the input side, one
1 1
seeks ﬁrst at the phase θ of the impedance Z that maxi
1 1
mizes the po wer W dissipated in Z .Hence µ is maximal
2 2
when δ = 0( 38 ), i.e.
1
∗
θ = ar g 1 + Z Y (41)
1opt T
T
∗
1 + Z Y
T
T
.
i.e. Z = Z 
1 1
∗
1 + Z Y 
T
T
When this condition on the phase is true, the po wer that is
dissipated in the load Z reads
2
2 2 Figure 4. Po wer transfer optimization.
( R + R ) P V −P  − R R V 
min max 1 1 1 min max 1
W =
2
2( R − R )
min max
R pv 
max
= , (42)
R − R 2
min max
When the phase of Z is optimal, the transfer coe ﬃcient
1
µ is equal to
with
( R −Z )(Z − R )
max 1 1 min
−1
µ ( Z  ) = (47)
p = P − R V , v  = V − R P 
1 min 1 1 1 δ =0 1
max 1
Z  ( R − R )
1 max min
1 1 − β
and 2
= 1 − 1 − β cosh σ − 0.5 ln ,
β 1 + β
2e Z
T
R = , (43)
min
∗
in which is introduced the coe ﬃcient β that characterizes
1 + Z Y  + 1 − Z Y 
T T T
T
the relati ve di ﬀerence between the resistances R and
min
2e Y
T
−1
R ,
max
R = .
max
∗
1 + Z Y  + 1 − Z Y 
T T T
T
R − R 1 − Z Y
max min T T
It can be check ed that there is always 0 <R <R . 0 <β = = < 1,
min max
∗
R + R 1 + Z Y
max min
T
T
The locus of the solutions ( P , V )that lead to dissipate
1 1
R 1 − β
at best the gi venpow er W in the load is the hy perbola min
2
= . (48)
with ax es p  = 0and  v  = 0( see Figure 4).All the ph ys
R 1 + β
max
ical solutions are necessarily located in the sector that is
bounded by these twohalf lines: the po wer that is dissi
Whene verthe orders of magnitude (29) or (31) hold, then
pated in the load cannot be ne gati ve.Ino ther wo rds, the
the coeﬃ cient β tends to wards unity,which is equi valent
real part of the load Z being positi ve,the norm  Z  is in
2 1 to R R .
min max
the range
Figure 5sho ws the mapping of µ (β, σ ). One ob
δ =0
1
serv es the follo wing properties
R < Z  <R . (44)
min 1 max
✲ ✲
µ 0,µ 0,
Let us introduce the geometrical mean R of the resis δ =0 δ=0
1 1
m
σ →0or1 β→0
✲
tances R and R ,
min max
µ 1. (49)
δ=0
1
β→1with
0.5−σ −0.51
e{Z }
The latter approximation close to the limit β = 1s hould
T
R = R R = . (45)
m min max
not hide the sharp ev olution of µ ( β ), as well as the sharp
e{Y }
T
variation of µ ( σ)inthe vicinity of the boundaries σ = 0
Notice that R is not equal to Z .Ino rder to bro wse the and σ = 1,
m c
domain of variation of the norm  Z  (44),one uses the
1
∂µ
scalar parameter σ that is deﬁ ned in the domain [0 , 1] so δ =0
1
✲
+ ∞. (50)
β →1
that Z  ( σ = 0) = R and Z  ( σ = 1) = R :
∂β 0.5 −σ −0 .51
1 min 1 max
0.5− σ
R
min
1− σ σ
Z  ( σ ) = R R = R , (46)
1 m From (47),the po wer transfer is maximal when
min max
R
max
with 0 <σ < 1. σ = 1/2, i.e. Z  = R . (51)
1 opt m
408
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Vol. 96 (2010)
Figure 6. Relati ve po wer transfer coe ﬃcient ξ = µ/µ =
opt
1
µ (β, σ ) /µ (β, / ).
2
Figure 6e xhibits the coe ﬃcient µ relati vely to the max
imal values µ ,when the only phase is optimal,
opt
µ
δ =0
1
ξ β, σ = (56)
δ =0
1
µ
opt
1− β
2
1 − 1 − β cosh σ − 0.5 ln
1+ β
= .
2
1 − 1 − β
The follo wing limits can be observ ed in this ﬁgure
✲
ξ 1
1 β →1,
Figure 5. (a) µ (β, σ, δ = 0).Bold black: µ = µ ( σ = / ,δ =
1 opt 2 1
0. 5−σ −0. 51
0),bold red: P  constant, bold blue: V  constant. (b) Pro je c
1 1
2
✲
and ξ ξ = 1 − 4( σ − 0.5) . (57)
1 0
ti on.Solid black: µ ( β ) = µ (β, σ = / ,δ = 0),b lue/red:
opt 2 1 β →0
µ ( β ) = β/ 2.
opt
constant P  or V 
1 1
As for the function µ (β, σ ), the straightforw ard limit of
ξ when β is close to unit is misleading: there is still a
sharp change in the vicinity of the boundaries σ = 0
Gi vent he dissipated po wer W ,the po wer that is required
2
and σ = 1. Also, ξ ( β)v aries very rapidly.Ont he other
at the source is thus minimal when the load Z is equal to
2
hand, the ratio ξ is close to the function ξ (57) as long
0
Z − Z
1opt T
as β remains not too close to unit. In addition there is al
Z =
2opt
1 − Z Y ways ξ (β, σ ) >ξ (σ ), which sho ws that ξ is aw orse es
1opt T 0 0
timate of ξ.Consequently,the lack of transfer is smaller
e{Z }− j m{Z }
T 1opt
= (52)
than −3dB( halfpo wer)aslong as the norm of the in
e{Y Z }
T 1opt
put impedance remains in the range dictated by (46) with
√
e{Z }e{Y }− js in θ
T T 1opt
σ = 0.5 ± 0.5/ 2. When β is close to unit, the tolerance
= ,
e{Y exp( jθ ) }
limits are lar ger,b ut there is no elementary approximation
T 1opt
–e.g. apolynomial expression –toquantify this gain.
with
Let us recall that the optimal solution (52) can be
reached only if the source at the input interf ace can sup
Z = R exp( jθ ) . (53)
1opt m 1opt
ply the follo wing amplitudes
The coeﬃ cient µ is then equal to (see Figure 5b )
−1
P  = 2Z µ W ,
1 1opt 2
opt
µ = µ  Z  = µ β, σ = 1/2 (54)
opt δ =0 1opt δ =0
1 1
−1
−1
V  = 2Z  µ W , (58)
R − R 1 1opt 2
opt
max min
−1
2
= = β 1 − 1 − β .
R + R
max min
in which Z  and µ are gi venby( 51)and (54) .
1opt opt
It can be noticed also that
3.2.2. General case (non optimal phase )
β β
In the general case, i.e. when the phase is not optimal, the
<µ = <β ≤ 1. (55)
opt
2
2 po wer that is dissipated by the load Z (37) can still be
2
1 + 1 − β
409
ACTA ACUSTICA UNITED WITH ACUSTICA Cer venka, Mar chal: On passive symmetrical tw opor ts
Vol. 96 (2010)
written in terms of the pre viously deﬁ ned parameters δ where
1
(38) , R and R (43),and β (48) ,
min max
2
2
r cos δ − β − sin δ
min 1 1
= . (66)
W = (59)
2
r 2
2
max
cos δ + β − sin δ
1 1
2 2
( R + R ) P V  cos δ −P  − R R V 
min max 1 1 1 1 min max 1
2( R − R ) This result can be compared with (47),inwhich in partic
max min
ular R /R is gi venby( 48 ). Gi venac onstant mod
r p v 
max min max
= ,
ule Z ,t he loss of eﬃciencybecause of the non optimal
1
R − R 2
max min
phase ( δ = 0) can be quantiﬁ ed with
1
with
−1 2
µ ( Z  ) = µ ( Z  ) − 2β sin ( δ /2) , (67)
δ =0 1 δ =0 1 1
1 1
−1
p  = P − r V , v  = V − r P 
1 min 1 1 1
max
recalling that this latter expression is only meaningful as
and
long as r < Z  <r .
min 1 max
When the phase is not optimal, the maximal transfer is
1 + β
r = R ,
min min
still obtained with the same norm Z  = R as when the
1opt m
2
2
cos δ + β − sin δ
1 1
phase is optimal (see equation 51 ). The transfer coe ﬃcient
1 + β
−1 −1
is
r = R . (60)
max max
2
2
cos δ + β − sin δ
1 1
−1
2
µ ( Z  ) = β cos δ − 1 − β . (68)
δ 1opt 1
1
The geometrical mean of r and r is the same as when
min max
the phase is optimal,
to be compared with (54) .
√
r r = R R = R . (61)
min max min max m
3.3. Optimal sour ce at constant pr essureorc onstant
velocity
The phase θ of Z ,and consequently the di ﬀerence δ ,are
1 1 1
constrained by the impedance Z ,which translates into the
2
The pre vious de velopments are founded on the search for
inequality
maximal transfer coe ﬃcients without anyconstraints on
the amplitudes P  and V  at the input interf ace, butf or
1 1
 sin δ ≤ β ≤ 1. (62)
1
their ratio. This section addresses the conditions that max
imize the amount of po wer dissipated in the load, being
Being gi venthe phase diﬀ erence δ ,the locus of the solu
1
gi venaconstant amplitude V  or P  at the source. It
1 1
tions ( P , V )t hat lead to dissipate at best agiv en po wer
1 1
comes back to look at the hyperbola that are tangent to the
W in the load is the hy perbola whose ax es are  p  = 0and
2
lines V = V  and P = P .T he condition (41) about the
1 1
v  = 0. The ph ysical solutions lay in asector bounded by
phase of the input impedance is still assumed to be fulﬁ lled
twolines whose angle is nows maller than if the phase is
( δ = 0) .
1
optimal,
Gi venaconstant amplitude V ,t he conﬁ guration that
1
leads to the maximal po wer dissipation in the load is de
R ≤ r ≤ R ≤ r ≤ R (63)
min min m max max
ri vedbysolving d W /dP = 0. From (27),( 42 ), there is
2 1
and
1
Z  = R + R ⇒
1 min max
r − r 2
max min
2
−2
0 ≤ = 1 − β sin δ ≤ 1.
1 ∗
1 + Z Y
R − R T
max min T ∗−1
Z = ⇒ Z = Y . (69)
1 2
T
2e{Y }
T
The domain [ r r ]ofthe norm Z  can be bro wsed
min max 1
ag ain with the parameter σ which is nowdeﬁ ned so that
Accordingly gi venac onstant amplitude P ,t he input im
1
Z  ( σ = 0) = r and Z  ( σ = 1) = r .
1 min 1 max
pedance that leads to dissipate the maximal po wer in the
load is obtained by solving dW /dV = 0. From (27) ,
2 1
0.5− σ
r
min
1−σ σ
Z  ( σ, δ ) = r r = R , (64) (42),there is
1 1 m
max
min
r
max
1
−1 −1 −1
with 0 <σ < 1. Z  = R + R ⇒
1
min max
2
With these notations, the general expression of the
2e{Z }
T
∗
transfer coe ﬃ cient µ (40) reads
Z = ⇒ Z = Z . (70)
1 2
∗ T
1 + Z Y
T
T
( r −Z )(Z − r )
max 1 1 min
µ = (65)
In both cases, the transfer coe ﬃcient µ is reduced to
δ =0
1
Z  ( R − R )
1 max min
1 r
min
2
= cos δ − 1 − β cosh σ − 0.5 ln ,
1
2
µ = β/2 = (1 + 1 − β ) µ /2 ≥ µ /2. (71)
β r 0 opt opt
max
410
Cer venka, Mar ch al: On passive symmetrical tw opor ts ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 96 (2010)
−1
Hence µ is lar ger than half the optimal transfer coe ﬃ
0
3.5. Particular case Z  Y 
T
T
cient. The solutions (69) and (70) correspond to the fol
Se veral additional results can be deri vedint he particular
lo wing locus of the parameters ( σ, β ),
case of practical interest deﬁ ned by the follo wing order of
ln (1 + β )
magnitude:
1 − at const. V ,
1
ln (1 − β )
−1
σ = (72)
Z Y  1 . (78)
ln (1 − β )
T T
1 − at const. P .
1
ln (1 + β )
Recalling that the real parts of Z and Y are both positi ve,
T T
The curv es that represent equations (71),( 72)are dra wn
it must be kept in mind in the derivations of this section
in red and blue in Figures 5–6.
that (3),( 78)are equivalent to
When the orders of magnitude (31) or (29) apply , β ≈ 1
e Z
so that µ ≈ 1/2. T
0
0 < ≤ Z (79)
T
m Z
T
3.4. Input impedance equal to load
−1
e Y
−1 T
Y ≤ .
T −1
The equality of the impedances Z and Z with the char  m Y
1 2 T
acteristic impedance Z (21) does not satisfy ap riori the
c
It implies straightforw ard approximations on parameters
criteria that optimize the po wer transfer.From (39),( 40 ),
introduced in section 3.2,
the transfer coe ﬃcients read in that case
R ≈ e Z
min T
Z = Z = Z ⇒ (73)
1 2 c
θ ≈ δ,β→1,
1 1
−1
R ≈ e Y
max T
µ = η cos (arg{Z } ) ,
c c c
r ≈ R / cos θ ,
1 − Z Y min min 1
T T
∗
and . (80)
−1
η = = ζ ζ = exp − 2 e{φ } < 1.
c T
r ≈ R / cos θ ,
max 1
max
1 + Z Y
T T
An important consequence is that the phase of Z is op
1
When the orders of magnitude (31) or (29) apply,t here is
timal when being null. Also from the deﬁ nition (16),the
no signiﬁ cant po wer dissipation in the tw oport,
order of magnitude of the characteristic impedance Z is
c
such that
Z Y 1
T T
and Z = Z ⇒ η = 1. (74)
1,2 c c
−1
or Z Y 1
T T Z Z Y . (81)
T c
T
At the transition conﬁ guration (34),the transfer coe ﬃcient
On the other hand, it has been already pointed out that
η reads
β being close to unit does not necessarily imply ob vious
simpliﬁ cations on the transfer functions.
Z Y = 1and Z = Z ⇒ (75)
T T 1,2 c
The norms Z  or Z  cannot ha ve simultaneously the
1 2
−1
1
same order of magnitude as Y  and Z .Ina ddition,
T T
η = tan ar g Z Y .
c T Z
2 (27) sho ws that Z  and Z  share the same order of mag
1 2
−1
nitude relati vely to Y  and Z .Ino ther words, one
T T
In that case, there is no loss in the netw ork if
of the follo wing non exclusi ve propositions is necessarily
true, and depending on the case, the di ﬀerences between
ar g Z Y ≈ π/2,
T T
impedances or between admittances, as well as the trans
which can be fulﬁ lled only if Z and Y are both pure
fer coeﬃ cient µ (40) read
T T
−
imaginary parameters (equation (3) and Z Y ∈ R ).
T T
−1 −1
Z Y ⇔ Z Y
According to its deﬁ nition, the characteristic impedance 1 2
T T
Z is real only if the phases of Z and Y are equal. When
c T T
Z − Z ≈ Z ,
1 2 T
⇒ (82)
−1
this condition is met, the optimal phase of Z (41) is null
1
µ = cos θ − Z e Z ,
1 1 T
∗
( Z Y ∈ R ), and Z = Z = Z is the optimal solution
T 2 1 c
T
or
gi venby( 45 ), (52),( 53 ),
ar g Z = ar g Y ⇒ (76)
Z Z ⇔ Z Z
T T T 1 T 2
+∗
−1 −1
Z = Z = Z ∈ R .
2opt 1opt c Z − Z ≈ Y ,
T
1 2
⇒
µ = cos θ − Z e Y .
1 1 T
The expression of η is still gi venby( 73 ), but Z is no w
c c
real so that
In anyinstance, pro vided the impedance Z is not essen
1
tially reacti ve (tan θ 1),the fraction of po wer that is
1
µ = µ = η = η . (77)
c opt c opt
dissipated in the netw ork is gi venby
The eﬃciencyoft he source and the po wer transfer are
−1
1 − η = e Z e Z (83)
1 T
close to perfect ( µ = η = 1) whene ver( 74 ), or (75) to
− −1 −1
gether with Z Y ∈ R ,applies. + e Z e Y .
T T T
1
411
ACTA ACUSTICA UNITED WITH ACUSTICA Cer venka, Mar chal: On passive symmetrical tw opor ts
Vol. 96 (2010)
The results stated in (82),( 83)l ead to interesting sim
pliﬁ cations if additional conditions are introduced. Hence,
when both orders of magnitude gi venin( 82)are simulta
neously met, the load as seen through the netw ork is not
signiﬁ cantly modiﬁ ed, although the common value can de
part from the characteristic impedance Z .That is the re
c
sult already pointed out in (29).Inaddition, still pro vided
the impedance Z is not essentially reacti ve,t he coe ﬃ
1, 2
cient µ can be directly estimated from the ar gument θ of
2
the load impedance (because θ = θ ), and the po wer that
1 2
is dissipated in the netw ork is ne gligible,
−1
Z Z Y
T 1,2 T
(84)
and tan θ 1
1,2
Z = Z ,
1 2
⇒ µ = cos θ ,
1,2
Figure 7. Equi valent electrical schemes of the optimal adaptation
η = 1.
when Z Y  1, at constant input pressure (a) and constant
T T
velocity (b) .
In order to complete the discussion, it can be check ed that
when θ  is close to π/ 2, there is still Z = Z ,t he trans
1,2 1 2
is that the constraint (51) about the norm is no longer
fer coeﬃcient µ tends to ward zero, but η is not necessarily
needed to achie ve an optimal transfer.( 86)giv es straight
equal to unit.
forw ardly the imaginary component that the load impe
Starting from (82),( 83 ), another condition of great
−1
dance or admittance ( Z or Z )must ha ve for the input
practical importance that is less restricti ve than (84) and
2
2
impedance to be purely resisti ve in (87).Ifthe imaginary
that still leads to aneg ligible relati ve amount of po wer dis
part of the load does not compensate exactly for the net
sipated in the netw ork can be also deri ved,
work inﬂuence, (86) can still be used to estimate the input
−1
e Z Z e Y
T 1,2 T phase θ ,sot hat (85) gi vesthe corresponding decrease in
1
(85)
and tan θ 1
1,2 the transfer factor µ.
Concerning the optimal solutions at constant input ve
µ = cos θ ,
1
⇒
locity or pressure, results gi venins ection 3.3 are easily
η = 1.
translated when (78) applies. The transfer coe ﬃcients µ
and η are equal,
Compared to (84),the load Z and the impedance Z seen
2 1
at the input are not necessarily equal, butthe possible sig
µ = η = 0.5. (88)
niﬁ cant di ﬀ erences in the impedances Z or admittances
1, 2
− 1
Z are an imaginary component
At constant input amplitude P ,t he optimal solution is
1,2 1
reached with
−1
e Z Z Y
T 1,2 T
∗
Z = Z ⇒ Z = 2e Z . (89)
2 1 T
T
⇒ Z − Z ≈ jm Z
1 2 T
At constant input amplitude V ,t here is
1
tan θ 1and or (86)
1,2
−1
−1
−1
∗−1
Z Z e Y
T 1,2 T
Z = Y ⇒ Z = 2e Y . (90)
2 1 T
T
−1 −1
⇒ Z − Z ≈ j m Y .
T
1 2
The solutions (89) and (90) are the very classical results
sho wn in Figure 7.
The only conditions (78) and (85) lea ve alar ge range
in the choice of Z ,which thus can di ﬀer signiﬁ cantly
1
from the characteristic impedance Z ,still avoiding po wer
c
4. Transmission line
losses in the netw ork. Ne vertheless, the source does not
necessarily work in an optimal re gime because it pro vides
4.1. Mor phism
only the ratio cos θ of the total po wer that could be po
1
Let us consider af amily of passi ve linear symmetrical tw o
tentially deli vered. Fort he source to work optimally,t he
ports { T } which realizes the follo wing morphism from
x
input impedance Z must be purely resisti ve,
1
( R , +)into ( {T }, × ):
−1
e Z Z = R e Y
T 1 T
∀ x, y ∈ R, T = T T . (91)
x+ y x y
⇒ µ = η ≈ 1. (87)
This property appears in manyapplications where the real
The condition on phase (41) is indeed reduced here to θ = parameter x is alength. Equation (91) may apply when
1
0. The main di ﬀerence with the general case  Z Y  1 ev er the parameters describing the media that supports the
T T
412
Cer venka, Mar ch al: On passive symmetrical tw opor ts ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 96 (2010)
studied phenomenon do not depend on the x parameter , The ambiguity modulo π of the imaginary parts of Z and
whereas asingle pair of intensi ve and extensi ve va riables Y occurs because the tanh function is periodic with its
( P ,V )can be associated to each value of x.Ac lassical imaginary ar gument (see equation 19 ). In practical appli
x x
example is the modeling of an electrical cable: the trans cations where Z and Y are to be deri vedfrom measure
fer matrix associated with apiece of the line is equal to ments of ( Z ,Y ), this ambiguity can be solv ed by using
x x
the product of the matrices associated with all parts that for instance aphase unfolding technique in the frequenc y
build anypartition of the initial piece. In the acoustical do domain.
Kno wing the parameters Z and Y ,the transfer matrix of
main, other tri vial ex amples are obtained when studying
the tw oport corresponding to anyl ength x can be deri ved
propag ation between twop lanes (with distance x)inah o
by replacing (94) into (20) ,
mogeneous media or in atube with constant cross section.
The model is also valid whate vert he dispersi ve character
cosh Γx Z sinh Γx
c
of the propag ation is (e.g in wa veg uides ).
T = . (99)
x
−1
Z sinh Γx cosh Γx
Let us denote Z = Z and Y = Y the parameters
c
x T x T
associated with the matrix T .The commutati vity of (91)
x
Equation (99) is the well kno wn transfer matrix of atrans
implies that the ratio Z /Y does not depend on the length
x x
mission line.
x,
Whene verthe follo wing condition is met:
2
2
∀x, Z /Y = Z . (92) Γx 1, (100 )
x x
c
the ﬁr st order de velopment of the tanh function in (98)
That is why( 16)c an be said to deﬁ ne the characteristic
leads to the approximation
impedance of am edia when (91) holds.
In the decomposition (15),t here is only the diagonal
Z ≈ Z Γx = Z x,
x c
(101 )
matrix D that depends on the length x.H ence the property
−1
Y ≈ Z Γx = Y x.
x
c
(91) incorporates into this matrix according to
The terms Z and Y nowclearly appears to be respec
∀ x, y ∈ R, D = D D ⇒ ζ = ζ ζ . (93)
x+ y x y x+ y x y
ti vely the shunt impedance and the open admittance per
unit length. Hence, the transmission media is entirely de
Consequently the function ζ gi venby( 17 ), (19) has nec
scribed by either this pair ( Z,Y )ofcomplexv alues per
essarily the ex ponential form
unit length, or the set made of the characteristic impedance
Z and the propag ation coe ﬃcient Γ.
c
ζ = exp − Γx , i.e. φ =Γ x. (94)
x x
4.2. Ev olution with length
This form enlightens the comments associated with (23) ,
It has been seen in the pre vious sections that the beha vior
(25) as it looks much more familiar with the 1D propag a
of the system depends lar gely on the order of magnitude
tion operator of ah armonic wa ve: Γ is the complexprop
of Z Y .Itist herefore interesting to study the ev olution
x x
ag ation constant in which the term depending on time is
of this norm with x.R eplacing by the notation (95) in (96)
omitted. It is usually expanded as the sum of areal attenu
leads to
ation coeﬃcient and an imaginary wa ve number ,
cosh (2αx) − cos (4πx/λ )
Z Y ≈ . (102 )
2π x x
cosh (2αx) + cos (4πx/λ )
Γ= α + j , (95)
λ
Forthis analysis, it is assumed that αλ 1. In addition,
(α> 0, see con vention at end of section 2.1.3 ). Equation
the domain for the parameter x is limited to [0,x
max
−1
(19) nowreads
α ]. Both conditions are not actually restricti ve in most
practical situations. Hence, the hy perbolic terms can be
Z Y = tanh Γx . (96)
x x
replaced by the ﬁrst order de velopment close to unit. In
addition, the ar gument 2αx varies slo wly with x compared
One introduces the set of newv ariables Z and Y with
to the oscillations of the trigonometric terms whose ar gu
ment is 4πx/λ.T he numerator and denominator in (102 )
Z /Y = Z = Z /Y and Z Y =Γ. (97) takesimultaneously opposite extreme values at abscissa
c x x
x = kλ/4(k∈N). (103 )
k
By combining (92) and (96),the relations between
The ev en and odd indices k correspond to peaks of
( Z ,Y )and ( Z,Y )e xpand as
x x
−1
Z Y  and Z Y ,r especti vely.Int he vicinity of these
x x x x
extrema, there is the de velopment
Z = Z tanh Γx = Z tanh x Z Y ,
x c c
−1
−1 −1
Z Y  ( x + δx )
x x 2 m
Y = Z tanh Γx = Z tanh x Z Y ,
x
c c
or Z Y  ( x + δx )
x x 2m+ 1
−1
Z = x Z arctanh Z Y − jkπ ,
c x x
−1
2
2πδx
(98)
−2
−1 −1 ≈ (αx) 1 + , (104 )
Y = x Z arctanh Z Y − jkπ .
x x
c
αλx
413
ACTA ACUSTICA UNITED WITH ACUSTICA Cer venka, Mar chal: On passive symmetrical tw opor ts
Vol. 96 (2010)
Hence we obtain the follo wing generalization of (1 01 ):
Z Y 1
x x
Z ≈Zx−jZ mπ ,
x c
⇒∃m∈Z, (111 )
−1
Y ≈ Y x − jZ mπ .
x
c
5. Conclusion
The po wer transfer coe ﬃcient through atwoport system
has been expressed by means of twon ondimensional pa
rameters: β which is acharacteristic parameter of the tw o
port, and σ that depends on the external impedance load.
Atypical beha vior of the relati ve po wer transfer coe ﬃ
cient has been put in evidence when β is not too close to
unit (see Figure 6) .
Athorough analysis of what is interpreted as ahalf
wa velength line in most applications has been performed.
−2
Figure 8. Z Y  in log scale computed with αλ = 10 . It accounts also for the case of systems whose length is
x x
much smaller than the wa velength. Be yond the tri vial re
sult that such atwoport may ha ve then little inﬂuence in
with δx λ/ (2π ).
the po wer transfer from one port to the other,wee xhib
The order of magnitude of the maximal amplitude, at
ited simple and accurate relations between input and out
δx = 0, is
put impedances, together with the precise conditions of va
−2 −2 −2
lidity for these approximations (s ee equation 86 ). Accord
(αx) = ( k/4) (αλ) 1. (105 )
ingly,the po wer transfer coe ﬃcient can be close to unit
From either side of these peaks, the norm decreases by half
ev en though the loading impedance is di ﬀerent of the char 
for alength shift of
acteristic impedance, butthe limits for which this property
holds are asserted.
Δx αλ
Δ x = αλx/ (2π ) , i.e. ≈ 1, (106 )
As farasp assi ve,linear,symmetrical tw oports are in
x 2π
k
volv ed, the results displayed in this paper can gi ve interest
which is av ery narrowb and from the hypothesis. We are
ing insights whate verthe domain of application. Wa veg
more speciﬁ cally interested in conditions that lead to (31) ,
uides in acoustics and transmission line in the electrical
or con versely (29).T he additi ve unit constant in the sec
domain are ob vious ﬁe lds of concern.
ond term of (1 04)can thus be ne glected. One ﬁn ds
√
Refer ences
 x − mλ/ 2 < ελ/ (2π ) ⇒Z Y  <ε,
x x
or (107 ) [1] H. vonHelmholtz: Handb uch der ph ysiologischen Optik
√
−1
1 (Handbook of ph ysiological optics)L eopold Vo ss, Leipzig,
x − ( m + / ) λ/2 < ελ/ (2π ) ⇒Z Y  >ε .
2
x x
1867. §16 Satz I, 168169.
√
The relati ve width of these bands is equal to 2 ε/ ( mπ ). [2] H. vonHelmholtz: Über die ph ysikalische Bedeutung des
Princips der kleinsten Wirkung (Onthe ph ysical signiﬁ 
Hence looking at the ev olution of  Z Y  with x,t here is a
x x
cance of the principle of least action)Crelle (Jo urnal für
re gular succession of alternating extrema, the norm being
die reine und ange wandte Mathematik, Berlin ) 100 (1886 )
equal to unit in between at abscissa x (see Figure 8) .
k +1 /2
137–166 and 213–222.
Looking more closely at the minima of  Z Y ,itc an be
x x
[3] Lord Rayleigh: The theory of sound. 2nd ed. DoverP ub
noticed that (1 00)i mplies
lications, NewY ork, 1945 (1st ed. 1877 ). vo l. I, §72 p.93,
§7778 pp9799, and §107111 pp150157.
Z Y 1. (108 )
x x
[4] H. Lamb: On reciprocal theorems in dynamics. Proc. Lon
don Math. Soc. s119 (1 888)144–151.
It corresponds to the part of the curveclose to the origin in
[5] J. L. Lagrange: Analytical mechanics (Mécanique analy
Figure 8, i.e. short lengths,
tique ). MalletBachelier,P aris, 1853. T1, 300302. Re vised
√
and annotated by M. J. Bertrand.
x< ελ/ (2π ) ⇒ Z Y <ε. (109 )
x x
[6] H. A. Lorentz: The theorem of Po ynting concerning the en
Con versely (108)w hich holds for the other minima does
er gy in the electromagnetic ﬁe ld and twog eneral proposi
not imply the linear relation (101 ). Ho we vern ote wo rth y tions concerning the propag ation of light. Amsterdammer
Akademie der Wetenschappen 4 (1896)176.
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[7] C.T.T ai: Complementary reciprocity theorems in electro
2
Z Y = tanh ( Γx − jkπ ) 1
x x magnetic theory.IEEE Trans. Antennas Propag. 40 (1992 )
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⇒∃m∈Z, Γx−j mπ 1
[8] J. Goedbloed: Reciprocity and EMC measurements. EMC
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Zurich Symposium, www .ieee.or g/or ganizations/pubs/
x c
⇒ (110 )
−1
ne wsletters/emcs/summer03/jasper .pdf, 2003.
Y ≈ Z ( Γx − jmπ ) .
x
c
414
Cer venka, Mar ch al: On passive symmetrical tw opor ts ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 96 (2010)
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