In this paper, calculation of parameters of low-pass ladder networks with mixed lumped and\ud
distributed elements by means of artificial neural networks is given. The results of ANN are compared with the values that are desired. It has been observed that the calculated and the desired values are extremely close to each other. So this algorith can be used to obtain the parameters that\ud
will be used to synthesize such circuits

Ozmen, Atilla

Sengul, Metin

Yilmaz, Melek

DSpace@Kadir Has

 
ISTANBUL UNIVERSITY –  
JOURNAL OF ELECTRICAL & ELECTRONICS ENGINEERING 
 
 
YEAR     
VOLUME 
NUMBER  
 
 
: 2003 
: 3 
: 2 
 
 
(955-959) 
 
 
 
  
 
 
Received Date : 05.01.2003 
Accepted Date: 06.06.2003 
  
 
DESIGN OF LOW-PASS LADDER NETWORKS WITH MIXED 
LUMPED AND DISTRIBUTED ELEMENTS BY MEANS OF 
ARTIFICIAL NEURAL NETWORKS 
 
 
Metin ŞENGÜL1  Atilla ÖZMEN2  Melek YILMAZ3 
 
 
Kadir Has University, Engineering Faculty, 34230 Cibali, Fatih, İstanbul 
Tel: +90 212 534 10 34 
 
1 E-mail:  msengul@khas.edu.tr    2E-mail:  aozmen@khas.edu.tr  
3E-mail:  myilmaz@khas.edu.tr  
 
 
ABSTRACT 
 
In this paper, calculation of parameters of low-pass ladder networks with mixed lumped and 
distributed elements by means of artificial neural networks is given. The results of ANN are 
compared with the values that are desired. It has been observed that the calculated and the desired 
values are extremely close to each other. So this algorith can be used to obtain the parameters that 
will be used to synthesize such circuits. 
 
Keywords: Lumped and distributed elements, artificial neural networks, back propagation 
 
 
 
I. INTRODUCTION 
One of the fundamental problems in the design 
and development of communication systems is to 
match a given device to the system via coupling 
circuits so as to achieve optimum performance 
over the broadest possible frequency band. This 
problem inherently involves the design of an 
equalizer network to match the given complex 
impedances, and usually referred as impedance 
matching or equalization. 
 
It has been appreciated for quite long time that 
networks containing both lumped and distributed 
elements offer many agvantageous associated 
with the present day emphasis on integrated 
circuits and more suitable especially for 
microwave filter applications, than those 
containing only lumped or distributed elements. 
The design of mixed lumped-distributed 
structures has been studied by several authors, by 
using as well as practical design techniques [1] 
and also by multivariable network synthesis 
techniques [1]. The problem has gained renew 
interest because of the possible application of the 
Design of Low-pass Ladder Networks with Mixed Lumped and Distributed Elements 
by Means of Artificial Neural Networks 
956
 
 
Metin ŞENGÜL, Atilla ÖZMEN, Melek YILMAZ 
mixed element structures as the reference 
networks for the design of multidimensional 
Wave Digital Filters [1]. 
 
In the second part of this paper, characterization 
of two-dimensional lossless two-ports by 
scattering approach will be given. The third part 
belongs to artificial neural networks. In the 
succeeding part, the calculation of the 5th degrees 
circuit (3 lumped and 2 distributed elements) 
parameters that will be used to synthesize the 
circuit and the obtained results can be found. 
 
2. CHARACTERIZATION OF TWO-
DIMENSIONAL LOSSLESS TWO-
PORTS BY SCATTERING 
APPROACH 
In the theory of one-dimensional lossless two-
ports, the so called scattering matrix and the 
corresponding scattering transfer matrix have 
turned out to be the most appropriate tools for 
establishing the synthesis theory. Of particular 
interest is the use of the Belevithc canonic forms 
of these matrices [1]. It is already known that, in 
the study of two-dimensional lossless two-ports, 
the scattering approach can be adopted as basis 
for the investigations. 
 
Fettweis has shown that, the scattering matrix of 
a multidimensional reactance two-port with 
possible unit elements, as well as the 
corresponding scattering transfer matrix can be 
expressed in terms of canonic polynomials f, g 
and h [1]. In arriving at this description, the 
derivation follows essentially the same lines as 
in the one dimensional case, with the 
introduction of a particular type of 
multidimensional Hurwitz polynomials, namely 
Scattering Hurwitz or Principle Hurwitz 
polynomials, which plat yhe same role in 
multidimensional characterization as the 
classical Hurwitz polynomials so in one-
dimensional case [1]. 
 
Before we consider the description of lossless 
two-ports by canonic polynomials, let us give 
first some basic definitions and the notation 
which will be used for the two-variable 
polynomials in the succeeding sections. 
 
A polynomial g in two variables λ,p  will be 
denoted by g(p,λ) or simply by g. g can be 
written as a polynomial in p and λ as 
∑∑
= =
=
pn
k
n
l
lk
kl pgpg
1 1
),(
λ λλ   (1) 
where  and  are the partial degrees of g in 
the variables p and λ respectively. 
pn λn
Definition 1: A two-variable polynomial g(p,λ) 
is said to be of degree m, if 
)(max 0 lkm klg += ≠
pn λ
λnnm p
. In terms of the partial 
degrees  and n  it is defines as 
+= . 
 
Arranging with respect to one of the variables 
with coefficient polynomials in the other 
variable, we obtain the recursive canonic form, 
∑∑
=
−
=
== λ λλλ
n
l
k
n
k
k pGpGpg
p
11
)()(),(  (2) 
Another form that we adopt for the 
representation of the two-variable polynomials is 
in matrix notation, 
T
g
Tppg λλ Λ=),(    (3) 
where [ ]pnT pppp K21=  and [ ]λλnK λxnn pλλλT 21=
g
 and  coefficient 
matrix Λ  is defined by 










=Λ
λ
λ
λ
nnnn
n
n
g
ppp
ggg
ggg
ggg
L
MLM
L
L
10
11110
00100
  (4) 
The paraconjugate of the two-variable 
polynomials is defined as ),(* λ−−= pgg . 
Definition 2: A two-variable polynomial g(p,λ) 
is called as a mth-degree polynomial with no 
missing terms if it is expressible as 
Design of Low-pass Ladder Networks with Mixed Lumped and Distributed Elements 
by Means of Artificial Neural Networks 
957
 
 
Metin ŞENGÜL, Atilla ÖZMEN, Melek YILMAZ 
ij
∑ ∑
=
−
=


=
m
k
k
km
l
l
kl pgpg
0 0
),( λλ   (5) 
where  are nonzero real constants. In matrix 
notation, this corresponds to the nonzro upper-
triangular coefficient matrix form. 
g
 










=Λ
00
0
0
0
1110
00100
L
LM
L
L
pn
n
g
g
gg
ggg λ
  (6) 
Now, let us consider two-variable 
characterization of lossless two-ports with mixed 
lumped-distributed elements and state the 
properties of canonic polynomials. 
 
Consider the generic form of a lossless two-port 
formed with cascade connections of lumped and 
distributed two-ports as shown in Fig. 1. 
 
 
            Z1,ι  Z2,ι 
 
            N1   N2              Nn 
 
Figure 1 Generic form of cascaded lumped and 
distributed two-ports 
 
The scattering matrix describing the mixed 
element two-port can be expressed in the 
Belevitch canonical form as [1], 




−−−
−−=
),(),(
),(),(
),(
1),( λσλ
λσλ
λλ phpf
pfph
pg
pS  (7) 
where 
• ),( λpf  is a monic real polynomial 
that consist of transmission zeros, 
• ),(),( λλ pgveph  are real 
polynomials in the complex variables 
jwp += σ  and Ω+= jελ  
( ττλ ),tanh( p=  being the delay 
length of unit elements), 
),( λpg
σ
,(),(), λλ −=−− phphp
βµ
αµ
)22
,
/,
()2
)1
)22
300120112011
2/1
30103010
2
20
10212112
303030
12
2/1
20
2
10
11
2/12
02
11
2/12
0102
gghhgg
hhggh
gghg
gghh
ggh
hhh
gghg
+−−
−+
==
==
=+
=+
=++
•  is a Scattering Hurwitz 
polynomial, 
•  is a unimodular constant; 1=σ , 
• Losslessness of the mixed element two-
port requires that  
),(),()(),( λλλλ −−+− pfpfgpg
. 
 
From the physical implementation point of view, 
one practical circuit configuration is that of 
simple low-pass ladder sections connected with 
unit elements (LPLUE) as shown in Fig. 2. 
 L1   L2 
 
          Z1,ι Z2,ι             Z3,ι 
    C1              C2 
Figure 2 Low-pass ladder with unit element 
 
In the Table 1, the explicit formulas for 5th order 
LPLUE are given. 
n Coefficient Relations 
5 [h01,h02,h10,h20,h30] independent coefficients 
β
µµ
µ
α
βααβ
/)(
(
/)(
//(
(
300121
20
1012
010130
0211021110
200202
0110011001
hhg
g
hh
hg
hhggg
hg
hhgg
=
=
−=
−=
−=
+=
−=
 
 
3. NEURAL NETWORKS 
Artificial neural networks (ANN) which consist 
of simplified neurons connected to each other are 
the models of nervous system. Although each 
neuron has a simple function alone, they can be 
used to solve complex problems, when they are 
used together. 
Design of Low-pass Ladder Networks with Mixed Lumped and Distributed Elements 
by Means of Artificial Neural Networks 
958
 
 
Metin ŞENGÜL, Atilla ÖZMEN, Melek YILMAZ 
 
Artificial neural networks are adaptive systems 
which have learning capabilities. ANNs adapt 
and organize themselves to the changing 
conditions, improve a function and make the 
calculation by learning. ANNs can produce the 
correct response even though missing or 
corrupted input is given to them. They are more 
suitable for the daily life problems because of 
their nonlinear characteristics [2]. 
 
In Figure 3, a neuron can be seen which consists 
of a summing junction and a non-linear 
activation function. Here, x1, x2, …,xn are inputs; 
w1, w2,…, wn are synaptic weight coefficients 
and y is output. 
 
 
 in
 
 
 
 
 
 
Figure 3 Neuron model 
A neural network model can be seen in Figure 4. 
Each neuron has many inputs and only one 
output, and this output is the input for the other 
neurons, so system is formed in parallelly. 
 
Inp  uts Outputs
Hidden Layer 
 
Figure 4 ANN with one hidden layer 
ANNs can be used in signal processing, image 
processing, pattern recognition, medical, military 
systems, finance systems, artificial intelligence 
and power systems. 
 
3.1. Learning in Artificial Neural 
Networks 
Learning in artificial neural networks is based on 
the calculation of the synaptic weight 
coefficients suitable for the problem. Learning 
rule is an equation set by which all or some of 
the synaptic weight coefficients change so as to 
modify the response of each neuron in time. By 
this way ANN can adapt itself to get the desired 
response. 
 
ANNs are learnt by example data instead of 
programming. Learning process can be divided 
into two groups; supervised and unsupervised 
learning. 
 
In supervised learning, both the input and the 
response are given to the system. For each input, 
obtained response and desired response are 
compared. To get the minimum difference, 
synaptic weight coefficients are changed. When 
an acceptable error is obtained, learning process 
is stopped and then these synaptic weight 
coefficients can be used with the data that are not 
used in learning process. 
 
Σ  
x 1 
x 2 
x 3 
x n 
b 
w 1 w 2 
w 3 
w n 
u=w 0 x 0 +w 1 x1+ w 2 x 2 +...+w n x n 
put 
bias 
output 
y 
weight  
coefficients  non - linear  
function 
 
y=f(u+b) 
3.2. Back Propagation Algorithm 
In this paper, back propagation algorithm is used 
as the learning algorithm, which has emerged as 
the most widely used and successful algorithm 
for the design of multiplayer feedforward 
networks [3]. 
 
In learning process, first of all, an error is 
obtained by subtracting the result from the 
desired value. Then the error is squared. In this 
algorithm, it is desired to realize a learning 
process with an error whose square is minimum. 
22 )( yt −=ε     (8) 
)))((( wuyεε =    (9) 
Then delta values are calculated at the output 
nodes. 
ww ∂
∂≡∇
2
2 εε     (10) 
And by using back propagation, all the values at 
the output nodes are calculated. 
Design of Low-pass Ladder Networks with Mixed Lumped and Distributed Elements 
by Means of Artificial Neural Networks 
959
 
 
Metin ŞENGÜL, Atilla ÖZMEN, Melek YILMAZ 
x
xusgm
w
u
u
y
yw
δ
ε
εε
2
)(2 '
2
2
=
−=
∂
∂⋅∂
∂⋅∂
∂=∇
  (11) 
)
x
(' usgmεδ −≡    (12) 
Finally, the components of gradient are 
calculated and desired synaptic weight 
coefficients are obtained. 
w δµ2−=∆     (13) 
w




ww oldnew ∆+=    (14) 
 
4. EXAMPLE AND CONCLUSION 
In this section, we solve the classical double 
matching problem. Here, an LPLUE network of 
degree five, (np=3, nλ=3) will be constructed 
over the normalized frequency band of 0≤w≤1. 
In the course of the learning process, the 
coefficients 
{h00,h01,h02,h10,h20,h30,g00,g01,g02,g10,g20,
g30} are chosen as the input parameters, the 
other coefficients are derived by means of 
designed neural network. 
We obtained the following coefficient matrices 
that completely describe the scattering 
parameters of the matching network under 
consideration. Also you can see the matrices that 
are derived via the explicit expressions given in 
Table 1. 
Ah= 




−
−
−−
−
009960.0
05316.31827.0
61081.13420.01042.0
6281.17310.10
Ag=  








009960.0
05316.30746.2
6108.18760.50396.2
9107.19694.21
Ah=  Ag=   








−
−
−−
−
009960.0
03089.21827.0
4744.56353.51042.0
6281.17310.10






009960.0
03454.60746.2
3960.47664.60396.2
9107.19694.21
h and g matrices derived by ANN and explicit 
expressions, respectivelly. 
 
ZS   C1      Z1     L    Z2      C2   ZL 
 
Fig 5. Double matching example with lumped-
distributed elements [L=2.1439, C1=0.8825, 
C2=1.0528, Z1=1.0356, Z2=3.6650, ι=0.2191], 
[ZS=RS+jwLS, ZL=(RL//(1/jwCL)+jwLL)] 
 
 
Fig 6. Gain performance of double 
matchingexample 
 
5. REFERENCES 
[1] Aksen, A., “Design of Lossloss Two-
ports with Mixed Lumped and 
Distributed Elements for Broadband 
Matching”, Doktora tezi, Ruhr 
Üniversitesi, Bochum, 1994 
[2] HAYKIN, S., “Neural Networks” 
Macmillan College Publishing 
Company Inc., 1994, ISBN: 0-02-
352761-7 
[3] Şengül.M, Özmen.A, Yılmaz.M, “Toplu 
ve Dağınık Elemanlı Alçak-Geçiren 
Merdiven Tipi Devre Parametrelerinin 
Yapay Sinir Ağları Kullanılarak 
Hesaplanması”, Eleco 2002, Bursa 
 


Design of low-pass ladder networks with mixed lumped and distributed elements by means of artificial neural networks

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Design of low-pass ladder networks with mixed lumped and distributed elements by means of artificial neural networks

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