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Scientific Bulletin of the Electrical Engineering Faculty – no. 2 / 2009
SENSITIVITIES COMPUTATION FOR PASSIVE
FOUR-TERMINAL NETWORKS
Horia ANDREI1, Fanica SPINEI2, Costin CEPISCA2, Ulrich L. ROHDE3, Marius SILAGHI4, Helga SILAGHI4
1
Faculty of Electrical Engineering University Valahia Targoviste 18-20 Blv. Unirii, Targoviste, Dambovita
2
Faculty of Electrical Engineering University Politehnica Bucharest 313 Splaiul Independentei, sector 6, Bucharest
3
Department of Communications, Technical University of Cottbus, Germany
4
Department of Electrical Engineering, University of Oradea
Email: [email protected]; [email protected]; [email protected];
[email protected]; [email protected], [email protected]
Abstract: - The paper proposes an original method
for determination of the sensitivities for passive
four-terminal networks in steady state nonsinusoidal regime. It is presented a computation
algorithm for determination the sensitivities using
the relative values which characterize the current
and voltage harmonics, without knowing the circuit
structure. The given example is necessary for the
discussions of some aspects related the fourterminal networks applications.
Key words: - sensitivities, four-terminal, relative
values, harmonics.
S ( k ) ( F , x) 
x k  (k ) F
F k x ( k )
U   Z  J  ,
V   Z n  J n ,
I   Y  E  ,
(3)
(4)
(5)
where the symbols are usual: Z  is the transfer branches
impedances matrix,
Y 
is the transfer branches
Z 
n
is the node-node transfer
impedances matrix, U  is the branches voltages matrix,
1. Introduction
I 
The sensitivity has been an interesting problem for a
long time in the technical literature [1], [2]. This paper
presents a method for computation of the sensitivities in
electrical steady state linear networks, correlating it to
the new four parameters which characterize the current
and respectively voltage harmonics. The given example
is used for the discussion of some aspects related to the
sensitivity problems.
The propose of this paper is to define some
sensitivities for the four-terminal networks. On consider
the steady state non-sinusoidal regime of the fourterminal networks and we calculate the sensitivities
correlated with the relative values which characterize
the current respectively the voltage harmonics.
2. The Sensitivities Definitions
is the branches currents matrix,
independent
voltages
sources
matrix,
E  is
J 
the
the
  is the nodes
independent current sources matrix, J
n
transfer matrix of sources of current and V  is the
nodes voltages matrix. Thus the variation theorems lead
in the case of the very small modifications to:
dZ kq
  Z kh Z hq ,
dYh
dYkq
 YkhYhq ,
dZh
(7)
(8)
and to two analogous relations for the magnitudes
appearing in equations (4).
Relations (7) and (8) lead to the following expressions of
the first order sensitivities, [6],
S ( Z kq , Yh )  
Classical sensitivity of a network function F on an
independent parameter x is defined, [3] or [4], by
relation
As a rule by k-order sensitivity we understand:
(2)
Consider a linear nonreciprocal network and the
following forms of the matrix equations of the network [5]:
admittances matrix,
x F  (ln F )
S ( F , x) 

F x  (ln x)
.
S (Ykq , Yh )  
(1)
Z khYh Z hq
Z kq
Ykh Z h Yhq
Ykq
,
(9)
.
(10)
By successive derivation of the relations (9) and (10)
we can obtain higher order derivatives useful for the
high-order sensitivity calculation, as in [7].
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Scientific Bulletin of the Electrical Engineering Faculty – no. 2 / 2009
Let’s consider the steady state linear network. To
illustrate the relative contributions of all harmonics
of current and voltage, we can use two formulae
equivalents, [8], of the Fourier series. If the receptor
is linear and supplied with a non-sinusoidal voltage
u(t) whose the development in a Fourier’s series is
truncated only at n terms, the characterization of the
receptor introducing a k phase angle on each
harmonic at the input terminals can be done as
follows:
u (t )  U
n
 b (k )
2 sin(kwt   k )
a (i ) 
 (i )
2
n
  (k )
a
n
U2
FU  out
U in2
(k )
 U out
(12)
k 1
where the relative values of each k - harmonics
magnitude of the voltage b(k) and current a(k) as
compared to the root – mean – square value of the
voltage, U , respectively current, I , have been
taken down as:
(k )
I
U (k )
(k )

; a
(13)
I .
U
Otherwise, [9], [10], if mark with (k) the
fractions of the magnitude of each k - harmonics of
b (k ) 
voltage compared to the fundamental value U (1) ,
U (k )
 (k ) 
,
U (1)
(14)
respectively, with (k) the fractions of the magnitude
of each k - harmonics of current compared to the
FI 
2
I out
I in2

(k )
 I out


k 1
(1)
(1)
(15)

k 1
2
b (i ) 

(i ) 2
n

2 sin(kwt   k )
( j)
S ( FU ,  in
)
(16)
k 1



, (21)

(1)
(1)
( j)
( j)
bin
2bin
FU

,
FU b ( j )
n
2
in
 bin(k )
(22)
n

k 1
( j)
( j)
 in
2 in
FU

,
n
FU  ( j )
(k ) 2
in
 in
2
( j)
S ( FI , ain
)
( j)
( j)
ain
2ain
FI

,
n
2
FI a ( j )
(
k
)
in
 ain
2
b (k )  1 ,
(18)
(23)
k 1
 ( j ) FI
( j)
S ( FI ,  in
)  in

FI  ( j )
in
;
(k )
2
ain( k )
k 1
2 n (k ) 2
 out
k 1
n
2
2
I in(1)
 in(k )
k 1
(1)
I out
k 1
These four new relative parameters verify the
conditions [6]:
2
I in2
, (20)

2
k 1
n
k 1
n
2
( k )2
U in(1)
 in
k 1
current; U out ,U in , I out , I in
represent the values of
fundamental of output and input voltage respectively
current; bout , bin ,  out ,  in represent the relative values
of harmonics of output and input voltage;
aout , ain ,  out ,  in represent the relative values of
harmonics of output and input current.
The first order sensitivity of such steady state linear
network, related to the input parameters
( j)
( j)
( j)
( j)
, can be expressed in the form of [12]:
bin
, in
, a in
, in
2 sin(kwt   k   k ) (17)
(k )
a (k ) 

2
I in(k )
k 1
n
n

2
2
(k )
k 1

n
(k )
2
I out
 aout
k 1
n
n
i (t )  I (1)
2
U in2 bin( k )
k 1
  (outk )
2
I (1)
we get the second set of formulae:

k 1
n
(1)
U out

2
( j)
S ( FU , bin
)
,
u (t )  U

2
U in( k )
k 1
2 n
2
where: U out ,U in , I out , I in represent the root-meansquare values of output and input voltage, respectively
I (k )
(1)
(k )
2
U out
 bout
n
(1)
fundamental value I
 (k ) 
n
2
 k 1
n
2 sin(kwt   k  k )
(k )
(19)
k 1
n
i (t )  I
, i  1,..., n.
In this case, for two ports networks, we can define
two magnitude voltage and current transfer function
[11]:
(11)
k 1
2
where j  1,..., n.
8
k 1
( j)2
2 in
n
(k ) 2
 in
k 1

,
(24)
(25)
Scientific Bulletin of the Electrical Engineering Faculty – no. 2 / 2009
Numerous authors show that in the electrical linear
and nonlinear networks the sensitivities satisfy the
invariant relationships or inequalities, some of which
allow deducing the lower and upper bounds of the
sensitivity index.
Generally speaking, a significant property of
linear networks is to existence of sensitivity
invariants, which have the form:
n

S ( F , xi )   ,
(26)
3. FOUR-TERMINAL NETWORKS IN STEADY STATE NONSINUSOIDAL REGIME
The four-terminal networks may be defined as a part of an
electrical system, having two pairs of terminals, 1-1’ and
2-2’, for connection of sources and loads, shown in fig.1.
There may be active and passive four-terminal. For
example, the analogues filters and the power transmission
lines may be classed as a passive four-terminal networks.
k 1
where  is a constant. A simple method for
deriving sensitivity invariants employs the concept
of homogeneity [2].
F  F ( x1 , x2 ,..., xn ) is
If
a
function
homogenous and
relationship holds:
order  ,
of
the

Ik
 Yk Z kq ,
Jq
Z1
Y2
l

1’
The passive four-terminal may be represented by an
equivalent three-element  -network or an equivalent
three-element T-network.
The two-port  -network in steady-state nonsinusoidal regime can be described by the following
equations
or equivalent
(k )
k)
k)
U in
 A( k ) U (out
 B ( k ) I (out
Z
h 1
Similarly, for the functions defined by the
relations (22),…,(25) then homogeneity given by
the relative values
(1)
(bin(1) ,., bin( n ) ), (in
,., in( n ) ), (a in(1) ,., a in( n ) )or (in(1) ,., in( n ) )
is of order 2 (  =2).
where
the
impedance
( n)
the relative values (bin ,., bin ) , the condition (27)
of the sensitivity invariants will be computed as
follows:

j 1
( j)
S ( FU , bin
)

j 1
( j)
2bin
n
 bin(k )
2
(30)
k 1
A similarly result is obtained if using the other
set of relative values
(1)
( n)
(1)
( n)
(1)
( n)
(in
,., in
), (ain
,., ain
)or(in
,.,  in
)
admittance
parameters
For example, the voltage amplification for the output
port open-circuited is defined for each frequency:
A( k ) 
(k )
U in
k ) I (k )  0
U (out
out
.
(33)
Using relation (38) we can calculate the voltage
transfer function define by (20)
2
U in
2
 2.
and
Z 1,Y 2 , Y 3 , respectively the transfer (circuit) parameters
A, B, C , D , depend of all the frequencies, k  1,.., n .
Using, for example, the parameters defined by
n
(32)
(k )
k)
k)
I in
 C ( k ) U (out
 D ( k ) I (out
 S (F , Z h )   Z k Yhkq (Z k YkhYhq )  1  0 (29)
(1)
(31)
(k )
k)
k)
I in
 (Y (2k )  Y 3(k )  Z 1(k ) Y (2k ) Y 3(k ) )U (out
 (1  Z 1(k ) Y (2k ) ) I (out
U
F  k  Z k Ykq ,
Eq
h 1
2’
Fig.1 Four-terminal network
and
l
Uout
Y3
(k )
k)
k)
U in
 (1  Z 1(k ) Y 3(k ) )U (out
 Z 1(k ) I (out
Yh
S ( F , Yh ) 
(Yk Z kh Z hq )  1  0 (28)
Y Z
h 1
h 1 k kq
l
2
Uin
(27)
The above-presented relations allow the direct
establishment of some invariant sensitivity
relationships [13].
If the network functions is a dimensionless ratio
(in which the denominator is usually given) we get
the following relationships
l
Iout
following
x
x1 F
 ...  n F   .
F x1
F x n
F 
Iin
1
FU 
2
U out
2
U in

n

(k )
bin
2
k 1 A ( k )
2
U in
2

n

(k )
bin
2
k 1 A ( k )
2
(34)
and the sensitivity related to the input parameters
( j)
bin
, j  1,.., n,
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Scientific Bulletin of the Electrical Engineering Faculty – no. 2 / 2009
( j)
S ( FU , bin
)
( j)
bin
FU

FU b ( j )
in
( j)
2bin
A
( j)
2
2 n

2
(k )
bin
k 1 A( k )
2
. (35)
(3)
S ( FU , bin
)
2
(3)
(3)
bin
2bin
FU

FU b (3)
n
2
in
 bin(k )
 23.2% (39)
k 1
2) Considering that we know, by measurements, for
input voltage the n  1 relative values of bin parameters
are
4. Example
For the linear four-terminal whose output and
input voltages and currents contain the same
number of harmonics, we can calculate the
sensitivity related to the variation of magnitude of
input harmonics, without knowing the circuit
structure.
For example, we consider a linear four-terminal
network, shown in fig.2, in steady state nonsinusoidal regime, whose input voltage contains the
first n  3 harmonics:
uin (t )  110 2 sin( t  40.10 )  10 2 sin( 2t  120.50 )  (36)
 40 2 sin( 3t ).
For calculate its sensitivity related to the parameter
(3)
of the 3-rd harmonic input voltage, we use
bin
(1)
( 2)
bin
 9364 10 4 , bin
 851 10 4
(40)
and the values of the network elements
R1  20, L1  0.4 H , C1  20F , R2  500 , L2  0.1H ,
C2  100F , R3  10, L3  5mH , C3  5F .
In this case we use a PSPICE, [14], algorithm for
calculate the following values.
According with relation (18), results
2
2
(3)
(1)
( 2)
bin
 1  bin
 bin
 3405 10  4
(41)
Using (33), (40), and (41) the voltage amplification
can be expressed
A( k ) 
two procedures.
(k )
U in
 1  Z 1( k ) Y 3( k ) 
k ) I (k )  0
U (out
out
 1  [ R1  j (kL1 
1
1
1
)][  j (
 kC3 )], (42)
kC1 R3
kL3
k  1,2,3.
From relation (35) results:
(3)
2bin
(3)
S ( FU , bin
)
A(3)
Fig. 2 Linear four-terminal network
in non-sinusoidal regime
1) Considering that we know, by measurements, for
input voltage the n  1 relative values of
bin parameters are
(1)
( 2)
bin
 93.64%, bin
 8.51%
(k ) 2
bin
2
k 1 A( k )
 23.2%
(43)
2 3

We obtain the same value of sensitivity using the both
procedures, but the second procedure it is more
complicated.
The numerical value of sensitivity proves a smaller
increase, 23 .2% , of sensitivity compared to a parameter
(3)
bin
 34.05% . Thus, we observe that the four-terminal
network functioning like an analogue filter for the third
harmonic of voltage.
(37)
According with relation (18), results
2
2
5. CONCLUSIONS
2
(3)
(1)
( 2)
bin
 1  bin
 bin
 34.05%
(38)
We can calculate, using relation (22), the first
order sensitivity of this four-terminal network
(3)
related-for example-to the relative value bin :
For the linear steady state networks whose output and
input voltages and currents contain the same number of
harmonics, we can calculate the sensitivity related to the
variation of magnitude of input harmonics, without
knowing the circuit structure.
It is of practical and efficient to express the
sensitivities of the linear steady state networks,
especially in the electronic amplifiers and filters, related
to the relative values of the input harmonics magnitude,
10
Scientific Bulletin of the Electrical Engineering Faculty – no. 2 / 2009
b, , a, and  , so that the former
should not
depend on the circuit parameters .
In this paper we define new sensitivities and we
establish relations between these sensitivities and
the new relative values of harmonics. These
relations verify the invariants general conditions.
The relative values b, , a, and  can be easily
measured and thus a number of restrictions can be
established as regards their variation limits.
The calculation of the sensitivity of the power
generator at different harmonics is of great practical
interest in order to obtain a more efficient
evaluation of the receptors affected by the
harmonics within the network.
References
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