On Passive Symmetrical Two-Ports, Impedance Conversion and ...

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ACTA ACUSTICA UNITED WITH ACUSTICA
Scientific Papers
Vol. 96 (2010) 403 –415
DOI 10.3813/AAA.918293
On Passive Symmetrical Tw o-P 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 Saint-Cyr -l’Ecole, France.
pierre.cerv enka@upmc.fr
2)
UPMC UnivP aris 06, UMR 7190, Institut Je an le Rond d’Alembert, 78210 Saint-Cyr -l’Ecole, France.
jacques.marchal@upmc.fr
Summary
The theory of linear passi ve symmetrical tw o-ports 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 o-port 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 in-depth 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 o-port: 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 o-por 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 atwo-port.
The matrices that characterize ap assi ve symmetrical
tw o-port are deri vedins ection 2.1. The interpretation
in terms of one-w 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 o-port. 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 atwo-port.
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 o-port
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 electrical-electrical ). In the acoustic do-
main, alar ge variety of confi gurations can be tak en into 2.1.1. Tw o-port
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 o-port.
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 semi-infi 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
semi-infi 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 Snell-Descartes 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 semi-infi 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 (1-D propa- put side, i.e. Port 2. The con verse confi guration could be
404Cer venka, Mar ch al: On passive symmetrical tw o-por 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 o-port
−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 o-port depends only on three comple x
The po wer that is dissipated in apassi ve tw o-port 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 o-port 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 o-port 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 atwo-port, these
properties are ev en equi valent, as well as the follo wing
Apassi ve symmetrical tw o-port 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 o-por 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 o-por 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 o-port, 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 so-called 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
2|1 − 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 o-por 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 −0.5|1


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 .5|1
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




Cer venka, Mar ch al: On passive symmetrical tw o-por ts ACTA ACUSTICA UNITED WITH ACUSTICA
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. 5|1
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( half-po 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 | = 2|Z |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 | = 2|Z | 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 o-por 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 o-por 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 o-port,
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 o-por 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 o-por 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 o-port 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 1-D 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 o-por 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 atwo-port system
has been expressed by means of twon on-dimensional 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 atwo-port 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 o-ports 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, 168-169.

The relati ve width of these bands is equal to 2 ε/ ( mπ ). [2] H. vonHelmholtz: Über die ph ysikalische Bedeutung des
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x x
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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
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Z Y 1. (108 )
x x
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Figure 8, i.e. short lengths,
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ne wsletters/emcs/summer03/jasper .pdf, 2003.
Y ≈ Z ( Γx − jmπ ) .
x
c
414












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Vol. 96 (2010)
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