理 工 学 部 研 究 報 告 第39号23 39 (2003) 1-4
Eng. Kinki Univ.
J. School Sci.
藤 本 英 昭**
中 野 雄 二*
Synthesis of a class of lumped and distributed mixed networks which consist of capacitors and a coupled parallel transmission line
Hideaki FUJIMOTO**
Yuuji NAKANO*
In this paper we present necessary and sufficient conditions for realizing a class of 2-port networks which consist of five capacitors and a coupled parallel transmission line. The 2-port networks may be used in a microwave filter which is installed in telecommunication equipment in cellular or cordless telephones.
d Network, Maicrowave Filter.
Key words: Circuit Theory, Lumpled and Distributed Mixe
1 Introduction
transpose of any denotes the
The superscript "T"
matrix.
Assume y13 = 924 and 9-14 = Y23, then the following is obtained as for elements of W.
As one of the interesting problems in circuit theory there is the realization of multivariable immittance matrices. In the past several authors have reported as for such problems in literature.[1],—,[4]
In this paper we present necessary and sufficient con- ditions for realizing a class of 2-port networks which consist of five capacitors and a coupled parallel trans- mission line.(Fig.1) The 2-port networks can be used in a microwave filter which is installed in telecommu- nication devices in cellular or cordless telephones.
2 Circuit Analysis
In this section, relationships with respect to the ad- mittance matrix of a 2-port network shown in Fig.2 are derived.
Figure 2 shows a 2-port network which is a configu- ration of a 4-port network N terminated in a 2-port network N. Its 2-port network is described as N here- after. We assume N and N are linear, passive, time-
invariant, lossless and reciprocal. Let W={wii } be
a 2 x2 admittance matrix of N between ports 1 and 2
in Fig.2. Furthermore, let Y = {yij } and Y = {y2~ } be 4x 4 and 2x 2 admittance matrices of Si and N, respectively. Then, after some matrix manipulation, the matrix W may be expressed in the form
平 成15年5月17日 受 理
d distributed mixed 2-port net- Lumped an
Figure 1:
works.
*近 畿 大 学 大 学院 総合 理 工 学研 究科
**電 気 電 子 工 学 科
constant vector of order 2.
can have the following:
_ [yk] be any real for 1Hl2 and H3,we
Let y Then,
QED
following matrix.
We thus complete the proof.
Lemma 2: Let W(s) be the
where 0 < a, b. Then W(s) is the reactance matrix.
Proof: W(s) has the following properties:
(1) WT(—s) = —W(s)
(2) W(s) is holornorphic in the open right-half of s- plane.
(3) Poles of W(s) at the origin and infinity of s- plane are simple, and the associated residue matrices 11110 and IR„, where
Ho = lim sW(s) and IHh = lirn 1 W(s),
5-40 s
are hermitian and nonnegative-definite.
The properties (1), (2) and the first half of (3) are obvious. Furthermore, the latter half of (3) is clear from Lemma 1. Let Wo(s) = Ho/s, and let W00(s)=slHla,. Then Wo(s) and W,,(s) are reac- tance matrices. Therefore, the matrix W(s)
W(s) = Wo(s) + Wco(s) is reactance. QED
Theorem 1: The necessary and sufficient con- ditions for a reactance 2x2 symmetric matrix W (p, s) = [wu (p, s)] to be realized as the admittance matrix of the 2-port network between ports 1 and 2 in Fig.1(a) are that
(1) wrr (p, s) = w2 2(p, s) (2)
c12, 0</31 <02 and0<72 <
1 must be adapted, and when where 0 < cx1 =
When "+", k=
k=2.
the upper or only the lower signs are to where only
be used.
3 Realizablity Conditions
In this section necessary and sufficient conditions are presented for the realizability of 2-port networks shown in Fig.1 in a standpoint of functions of two complex variables "p" and "s". The symbols "p" and
" s" are used to express the Richards transformation tanh Ts (T > 0) and complex frequency, respectively.
following matrices IHI4 be the
Let IHI1,
Lemma 1:
Figure 2: A 2 port network N in which a 4-port net- work N is terminated by a 2-port network N.
(2) Y(s) is holomorphic in the open right-half of s- plane.
(3) A pole of 1Y(s) at the infinity of s-plane is simple, and the associated residue matrix IHI(= 1Y(s)/s), is hermitian and nonnegative-definite.
The properties (1),(2) and the first half of (3) are obvious. Therefore, it is necessary to prove the latter half of (3). However, it is clear from Lemma 1. Thus, we can see that 1Y(s) is the reactance matrix. Let us determine capacitances in Fig.1(a). As a result, we can get the following:
then we can get
and Y12 from eq.(7), Find yi i
Fig.2 be- of N in
Y(p)
matrix
admittance Thus, an
comes
Then 1Y(p) has the following three properties.
(1) 1YT(—p) = —117(p)
(2) C7(p) is holomorphic in the open right-half of p-plane.
(3) A pole of1Y(p) at the origin of p-plane is simple, and the associated residue matrixIHI(= p1Y(p)), as is seen from Lemma 1, is hermitian and nonnegative- definite.
Thus,
Therefore,Y(p) is the reactance matrix in p.
The above is the impedance matrix of the equivalent 2-port network between ports 3 and 4 of the coupled parallel line in Fig.1(a). From this Z(p), the even- Proof: The proof of necessity will be omitted here.
Therefore, the proof is presented only for the suffi- cient conditions. The right hand side of eq.(2) can be written in the form of the right hand side of eq.(3).
2
w11 (p, s) ± w12(p, s) = (5ks— ---(Sks)ry k(3) (I k+6k)s+—
P where bk = 1/ak. By comparing eq.(3) with eq.(1), the following relationships are obtained:
eq.(5), we will and 9-14 from
to obtain y13 _ —1 . Then, In order
adopt E
2/13 = —(SI s
llowing properties:
The above 1Y(s) has the fo (1) YT(—s) = —1Y(s)
Fig.2 is
Thus, a 2 x 2 admittance matrix Y(p) of N in given as
reactance
(p), i e.,
the of
From Lemma 2 we see that Y(p) is matrix. Calculate the inverse matrix the impedance matrix Z(p). Then,
The above is the impedance matrix of the equivalent 2-port network between ports 3 and 4 of the coupled parallel line in Fig.1(c). From this l. (p) we can get the following two mode admittances.
Ye — 72' Yo = 1'r We complete the proof of Theorem 3. QED 4 Conclusion
In this paper, the necessary and sufficient conditions have been presented on three lumped and distributed mixed networks
References
[1] T.Koga, "Synthesis of finite passive n-ports with prescribed positive real matrices of several vari-
ables," IEEE Trans. Circuit Theory, CT-15, pp.2-23, March 1969
[2] H.Fujimoto, J.Ishii and H.Ozaki, "l~Iultr- variable Richard's transformations," Proc. of
the IEEE International Symp. on Circuits and
Systs., pp.501- 504, Tokyo, 1979, or Trans. IE-
ICE, vol.E62, No.8, pp.529-535, Aug. 1979 [3] S.Okabe and H.Ozaki, "Realization of a class of homogenous positive real matrices," Proc. of the IEEE International Symp. on Circuits and Systs., pp.505- 508, Tokyo, 1979.
[4] H.Fujimoto,"Darligton-type Realization of a
class of 2-variable 2-wire-lines networks", J.
School Sci. Eng. Kinki Univ., vol.36, pp.179-l85, 2000.
and odd-mode admittances of coupled parallel line under consideration, ye and yo, become as follows:
We complete the proof of Theorem 1. QED
Theorem 2: The necessary and sufficient conditions for a reactance 2 x2 symmetric matrix W (p, s) = [w0 (p, s)] to be realized as the admittance matrix of the 2-port network between ports 1 and 2 in Fig.1(b) are that
(1) wi i (p, s) = w22 (p, s) (2)
Proof: The proof of necessity will be omitted here.
The proof of sufficiency can be done by replacing p yr and 72 in Theorem 1 with l/p, ry., and yr, respectively. As a result, the capacitances in Fig.1(b) are the same to those in Theorem 1. However, the even and odd mode admittances ye and yo become
as follows:
QED
Theorem 3: The necessary and sufficient conditions for a reactance 2 x 2 symmetric matrix W(p, s) = [wu (p, s)] to be realized as an admittance matrix of the 2-port network between ports 1 and 2 in Fig.l(c) are that
(1) w11 (p, s) = W22 (P1 s) (2)
where 0<ar =a2, 0<01 <)2 and 0<7i <72.
When "+", k = 1 and q = 1 must be adapted, and when "— ", k = 2 and q = —1 .
Proof: The proof of necessity will be omitted here.
Therefore, only the sufficiency is proved below. A method for determining capacitances in Fig.l (c) is same into that stated in Theorem 1. We must de- termine two mode admittances of the coupled line.
From the condition (2) we can have
