janusz walczak, seweryn mazurkiewicz, dariusz grabowski resistor

JANUSZ WALCZAK, SEWERYN MAZURKIEWICZ, DARIUSZ GRABOWSKI
SILESIAN UNIVERSITY OF TECHNOLOGY, POLAND
RESISTOR STOCHASTIC MODELS
LOSOWE MODELE REZYSTORA
RU (t1 , t2 ) = E[U (t1 )U (t2 ))] = E[R2 ] · RI (t1 , t2 ),
INTRODUCTION
Many works, including monograph [4], have been devoted
to analysis of stochastic phenomena in electrical and electronic circuits. They often deal with determination of probabilistic
characteristics for stochastic processes observed in systems.
Works in the field of stochastic system analysis can be divided
into two main topic groups [3]. The first one concerns deterministic systems in which some stochastic signal sources are
present. The analysis of such systems is usually carried out
by means of stochastic differential or integral equations. The
second one concerns systems in which sources as well as basic elements require probabilistic description. In this case problems under consideration include model construction either
for elements whose lumped parameters R, L, M, C are random variables or for time-varying elements described by functions R(t), L(t), M (t), C(t) which are deterministic functions
or stochastic processes [1]. Such models can be built among
others with the aid of stochastic moments [5]. This paper deals with determination of the first and the second order moments for the voltage stochastic process observed in the case
of random resistor element R assuming that the current stochastic process moments are given. It is a continuation of the
previous works devoted to determination of stochastic process
moments for deterministic elements R and C supplied by stochastic current source [2], [9] and for nonlinear inertiales elements described by random polynomials [8].
1.
where:
R – random variable with given distribution,
U (t) – voltage stochastic process of the resistor,
I(t) – current stochastic process of the resistor.
= E[R ] · RI (t, t) −
m2U (t),
mI (t) – expected value of the process I(t),
mR – expected value of the variable R,
2
σU
(t) – variance of the process U (t),
RU (t1 , t2 ) – autocorrelation function of the process U (t),
RI (t1 , t2 ) – autocorrelation function of the process I(t),
E[R2 ] – second moment of the random variable R.
The analysis of equations (2), (3) and (4) leads to a conclusion that if the given current process and the random variable
representing the resistance R are independent, then the complete description of the resistor can be made with the aid of the
first and the second moments of the current process as well as
the random variable R.
In the second case, equation (2) is not valid and the following relation can be written:
mU (t) = E[U (t)] = E[R · I(t)].
(5)
There are few methods to expand formula (5) [6]. The simplest one consists in application of the expected value definition:
g(x, y)fXY (x, y, t1 , t2 ) dxdy,
−∞ −∞
(6)
where:
fXY (x, y, t1 , t2 ) – joint probability density function of the
random variables X(t1 ) and Y (t2 ) defined by the stochastic processes X(t) and Y (t) at moments t1 and t2 ,
respectively,
g(x, y) – deterministic function of two variables.
The following two cases can be considered:
– the current process I(t) and the random variable R are
statistically independent,
– the current process U (t) and the random variable R are
not statistically independent.
In the first case, assuming that the resistor current is
known and applying expected value operator [7] to equation
(1) results in closed form formulae expressing the first and the
second moments of the voltage across the resistor:
2
mU (t) – expected value of the process U (t),
E[g(X(t1 ), Y (t2 ))] =
Stochastic current and voltage processes in the case of a
resistor, whose resistance is a random variable, are related
by:
U (t) = R · I(t),
(1)
2
σU
(t)
where:
Z∞ Z∞
RESISTANCE AS A RANDOM VARIABLE
mU (t) = E[U (t)] = E[R] · E[I(t)] = mR · mI (t),
(4)
(2)
(3)
Equation (6) can be applied to express moments of the voltage
stochastic process across the resistor in the the case of statistical dependence between the resistance R and the current
stochastic process:
Z∞ Z∞
mU (t) =
rifRI (r, i, t) drdi,
(7)
r2 i2 fRI (r, i, t) drdi − m2U (t),
(8)
−∞ −∞
2
σU
(t)
Z∞ Z∞
=
−∞ −∞
Z∞ Z∞ Z∞
RU (t1 , t2 ) =
3.
r2 i1 i2 fRI (r, i1 , i2 , t1 , t2 ) drdi1 di2 ,
−∞ −∞ −∞
(9)
where:
fRI (r, i, t) – joint probability density function of the random
variable R and the stochastic processes I(t) at a moment t.
fRI (r, i1 , i2 , t1 , t2 ) – joint probability density function of the
random variable R and the stochastic processes I(t) at
moments t1 and t2 .
In this case, moments of the input current process and the
random variable describing the element parameter are not sufficient to express moments of the output voltage process. Joint
probability density functions must be given.
2.
RESISTANCE AS A STOCHASTIC PROCESS
Stochastic models of a time-varying resistor can be also introduced. Such a resistor is described by the following equation:
U (t) = R(t) · I(t).
(10)
Three special cases can been considered. If R(t) is a deterministic function the first and the second moments of the
output voltage process are expressed by:
mU (t) = R(t) · mI (t),
2
σU
(t)
2
= R (t) RI (t, t) −
m2I (t)
(11)
,
(12)
RU (t1 , t2 ) = R(t1 )R(t2 )RI (t1 , t2 ).
(13)
If the stochastic process R(t) is independent of the current
processes I(t), the first and the second moments of the output
voltage process are expressed by:
mU (t) = mR (t) · mI (t),
(14)
2
σU
(t) = RR (t, t)RI (t, t) − m2U (t),
(15)
RU (t1 , t2 ) = RR (t1 , t2 )RI (t1 , t2 ).
(16)
where:
mR (t) – expected value of the process R(t),
RR (t1 , t2 ) – autocorrelation function of the process R(t).
If the stochastic process R(t) is not independent of the
current processes I(t), the first and the second moment equations are similar to (7), (8) and (9):
Z∞ Z∞
mU (t) =
rifRI (r, i, t) drdi,
(17)
−∞ −∞
∞
∞
Z Z
2
σU
(t) =
r2 i2 fRI (r, i, t) drdi − m2U (t),
−∞ −∞
r1 r2 i1 i2 fRI (r1 , r2 , i1 , i2 , t1 , t2 ) dr1 dr2 di1 di2 ,
Methods which enable calculation of expected values, variances and correlation functions for stochastic processes observed in the case of a random resistor element R have been
described. If the random variable describing the resistance
and the input stochastic process are independent, then the
output stochastic process moments can be determined only
on the base of the moments of the input process and the resistance random variable. Otherwise, the joint probability density
functions must be known. Random models of LM C elements
have been given in the work [9].
4.
REFERENCES
1. Banchuin R., Chaisricharoen R.: Stochastic Inductance
Model of On Chip Active Inductor, 2nd Int. Conf. on Education Technology and Computer, June 2010, Vol. 5, pp.
V5-1 – V5-5.
2. Grabowski D.: Moments of Stochastic Power Processes
for Basic Linear Elements, Int. Conf. of Fundamentals of
Electrotechnics and Circuit Theory, IC-SPETO 2009, pp.
83-84.
3. Kadlecova E., Kubasek R., Kolarova E.: RL Circuits Modeling with Noisy Parameters, Conf. on Applied Electronics, Pilsen 6-7 Sept. 2006, pp. 217-220.
4. Skowronek K.: Obwody elektryczne w ujeciu
˛
stochastycznym, Monografia. Wyd. Pol. Pozn., Poznań 2011.
5. Socha L.: Równania momentów w stochastycznych układach dynamicznych, PWN, Warszawa 1993.
6. Soong T. T.: Random Differential Equations in Science
and Engineering, Math. in Science and Eng., Vol. 103,
Academic Press, New York 1973.
7. Swiesznikow A. A.: Podstawowe metody funkcji losowych, WNT, Warszawa 1965.
8. Walczak J., Mazurkiewicz S.: Transformacje momentów
sygnałów stochastycznych w losowych układach nieliniowych, Kwartalnik Elektryka Pol. Śl., nr. 3, 2011.
9. Walczak J., Mazurkiewicz S., Grabowski D., Stochastic
models of lumped elements, Int. Symposium on Theoretical Electrical Engineering, Pilsen 2013, in press.
5.
SHORT ABSTRACT
W artykule wyprowadzono równania momentów pierwszego i drugiego rzedu
˛
losowych modeli rezystancji. Analize˛ przeprowadzono dla dwóch przypadków. Pierwszy z
nich dotyczył sytuacji, gdy wymuszenie w postaci stochastycznego sygnału pradowego
˛
było niezależne wzgledem
˛
losowych parametrów rezystora. W drugim przypadku
uwzgledniono
˛
zależność procesu pradowego
˛
od parametrów deterministycznych lub losowych rezystora.
(18)
RU (t1 , t2 ) =
Z
SUMMARY
(19)
R4
where:
fRI (r, i, t) – joint probability density function of the stochastic process R(t) and the stochastic processes I(t) at a
moment t,
fRI (r1 , r2 , i1 , i2 , t1 , t2 ) – joint probability density function of
the stochastic process R(t) and the stochastic processes
I(t) at moments t1 and t2 .
prof. dr hab. inż. Janusz Walczak
mgr inż. Seweryn Mazurkiewicz
dr inż. Dariusz Grabowski
Politechnika Ślaska
˛
Wydział Elektryczny
Instytut Elektrotechniki i Informatyki
ul. Akademicka 10
44–100 Gliwice
Email: [email protected]
[email protected]
[email protected]