ThammasatInt. J. Sc. Tech.,Vol. 13. No. 2, April-June2008
Modelingof an RC Circuit [Jsinga
StochasticDifferential Equation
Tarun Kumar Rawat and Harish Parthasarathy
Division of Electronicsand CommunicationEngineering
Netaji SubhasInstituteof Technology
Sector-3,Dwarka,New Delhi-l10075,India.
com
E-mail: tarundso(dsmai1.
Abstract
circuit: external noise and internal noise. The
in
an
electrical
noises
can
exist
Two types of
effectsof both types of noisesin electricalcircuits are ignoredwhen using a deterministicdifferential
equationfor their modeling.The deterministicmodel of the circuit is replacedby a stochasticmodel by
adding a noise term in both the potential source(externalnoise) and the resistance(internal noise).
Stochasticmodels are more appropriateto describefor instancethe outcomeof repeatedexperiments.
The noise addedin the potentialsourceis assumedto be a white noise and that addedin the resistance
is assumtd to be a correlatedprocess.A first-order ordinary differential equation and its stochastic
analoguesis used for the DC responseanalysisof an RC circuit. The first-orderdifferential equation,
which describesthe concentrationof charge in an the capacitor, is solved explicitly in both the
deterministicand stochasticcase.This method gives an explicit relation expressingthe sensitivityof
the performancewith respectto input parametersand their tolerancesso it can be used for design
optimization.Numerical simulationsin MATLAB are obtainedusing the Euler-Maruyamamethod.
Keywords: RC electrical circuit, ordinary differential equation, stochasticdifferential equation,
Brownian motion process,Euler-Maruyamamethod.
the input and intemal parameters in the
deterministic model by random processes.
Random differential equationsof this type can
be interpreted as ,stochastic differential
equations, following Ito's basic work in the
early 1940s [a][5]. Solutionsof such equations
represent Markov diffusion processes, the
prototype of which is the Brownian motion
processalternativelycalled the Wiener process
1. Introduction
Two types of noises can exist in an
electrical circuit: external noise and internal
noise. Extemal noise denotesfluctuationsin an
otherwisedeterministicsystemto which external
forces are applied. An example that has wide
applicationsin engineeringconsistsof adding a
noise term to the right side of a deterministic
equation. Langevin formulated the first such
model in 1908 to describe the velocity of a
particle moving in a random force field. A well
known exampleof internalnoise in an electrical
circuit is thermal noise, caused by the
discretenessof electric charges [][2]. Many
other types of internalnoise in electricalcircuits
are: shot noise,low frequencynoise,burst noise
etc. [3]. Obviously, at a finite temperature,there
is no dissipative electrical circuit without
thermalnoise.
The effects of both types of noises in
electrical circuits are ignored when using a
deterministic differential equation for their
modeling. Random effects due to both external
and internal noise can be included by replacing
t6l-tel
The literaturemainly distinguishesbetween
sampling and non-sampling methods for the
purposeof stochasticanalysis.Samplingis done
with the Monte Carlo method tl0l tlll
Stochastic information on circuits without
sampling is obtained with Hermite-Polynomia
chaos ll2l. Our method falls in the later
category.
In this paper, the ordinary differentiaequation and its stochastic analogous, which
describes the concentration of charge in the
capacitorof a RC circuit, is solvedexplicitly for
both the deterministicand stochasticcases.This
method is based on results from the theory of
stochasticdifferential equations(SDE) [5]. The
40
ThammasatInt. J. Sc. Tech.,Vol. 13, No. 2, April-June2008
method is general.in the following sense.Any
electrical circuit which consists of resistor,
inductor and capacitor can be modeled by an
ordinary differential equation in which the
parameters of the differential operators are
functions of circuit elements.The deterministic
ordinary differential equation can be converted
into a stochasticdifferential equationby adding
noise to the input potential source and to the
circuit elements. The noise added in the
potential sourceis assumedto be a white noise
and that added in the parametersis assumedto
be a correlatedprocessbecausetheseparameters
changevery slowly with time and hencemust be
modeled as a correlatedprocess(Appendix A).
The resulting SDE is solved by using a nonMonte Carlo method.MATLAB simulationsare
usedto verify analyticalresults.
In the next section,the stochasticcalculus
theory is reviewed.Sections3 and4 describethe
modeling of an RC circuit using an ordinary
differential equation (ODE) and an SDE,
respectively,and their analyticalsolution is also
presented.In Section 5, the Euler-Maruyama
schemeis reviewedfor the numericalsolutionof
SDE. Finally, the analytical results obtained in
Sections 3 & 4 are verified with MATLAB
simulations in Section 6 followed by the
c o n c l u s i o ni sn S e c t i o n7 .
Stochastic integrals and differential
equationswere introduced by Ito [4] and are
being widely used and, hence,are called the Ito
integral. Stratonovich [3] proposed another
representation for stochastic integrals and
differential equations under rather restrictive
conditions,which has a numberof advantagesin
this
computational techniques. Using
representation,we can work with stochastic
integrals in the same way as with the ordinary
integrals of smooth functions, such as
integrationby parts and changing of variables.
The Ito integral, however, has a mathematical
expectation which can be written more
concisely. Simple formulas for the transition
from one integral to the other allow at all times
for the selectionof the representationwhich is
most convenientfor any particular purpose.In
this section, we only explicitly summarizethe
difference and the conversionbetween Ito and
rigorously
Stratonovich integrals. More
mathematicaltreatmentsof this issue are given
in Ual-[6].
A. Stochasticlntegrals
L e t / u < t , 3 " ' l t n = 7 b e a p a r t i t i o no f
. r -l
/.
\
t h e i n t e r v a lf 0 . / I a n d d , = l r l ? X ( l ,- I , t I .
The
l:
Definition
Ito
integral
T
Stochastic Differential Equation and
Stochastic Calculus
Considera SDE;
2.
JC1r,
is definedas the limit in the
quadraticmean (qm)
T
dX(t1= t(x(t),t)dt+ G(x(r),/)dB(/) (l)
X1';1an1';
Y,(t)= JC1r,
where f is an r - vector valued function, G is
an nxm matrix valued function, B(/) is an
rn - dimensional Brownian motion process or
Wiener process, and the solution X(/) is an
Io
def
n
qm-lim
= d n - - )o ) c 1 x , , , t , r ) ( 8 , -, 8 , ,, )
(3)
i=l
If the integrandG is jointly measurableand
n-vector process.By a solution X(l) of the
stochasticdifferential equation (1), is meant a
processX(l) , for all t , in some intervaf [t' f ]
T
l' )ds< oo
Jr1lc1r,x1s))
(4)
I0
must satisfythe integralequation:
then the limit in (3) existst14l[15].If G
insteadof (4),theweakercondition
satisfies,
T.
x(/) : X(to) + Jf(s,X(s))ds
,,,
x1r;;an(s)
T
(21
Jlctt,x{t))
T
+ JG1s,
x(s))dB(s)
l t , l t < c t t , a l m o s st u r e l v - ( 5 )
the stochasticintegral in (3) is defined as the
hmit in probabili4t.
DeJinition 2; The Stratonovich integral is
definedby :
where X(rn ) is a specifiedinitial value,the first
integral in (2) is an ordinary integral, and the
secondintegralin (2) is the stochasticintegral.
4l
Thammasat
"'- -:
x . ( r ) = x , ( r o )J+, t r r x( r ) . r ) + ] t ' t 1 .
2_t=,_^_,
r, (r)= Jc1",
x1";;ar1";
t0
( G , ,( X , ( r ) . r ) ) r G * , ( X , (r ) . r l ) d r
der qm-rim -\
( X,
= E nr O ) G l - t f
i
=
l
+ X,
\
)
4
' t , - ' l ( 8 ,-B',
,)
,
(l l)
+(1)fc(x ,(r),r)dB,
)
(6)
the
to
the
conditions
on
In addition
existenceofthe Ito integral,it is requiredfor the
existenceofthe Stratonovichintegral in (6) that
the G(X(t),t) function be continuousin / and
have continuous partial derivatives with respect
where the last integralsof right side of (10) and
(11) are Stratonovich and Ito integrals,
the
respectively. From (10) and (1 l)
Staratonovichdifferential equation:
(
\
to X, [3]-[6].
Under the existence of (6), Stratonovich
[13] provided the connectionbetweenthesetwo
stochasticintegralsby :
,
*l [IT(c-
^
n
z/=lr=l
)
\r
tr G" ,r ,I l d r + G d B .
)
/r2)
correspondsto the Ito differential equation
(13)
dX, =fdt +GdB,
Y,0) : Y,(/)
lT-,
1
d x, . =| rl - -, Il uFu . r c
where )',li=,{c*)rc*,
correction term.
(\ 7
' I)
r X . s ) )c , , ( X " , s v s
is the so-called
-) J - ? ^ ?K\ = t
t l=t
wherethe n-vector (G,^ ), is the /th column
\ae ,f axr). To
clearly illustrate these integration concepts, the
simple examplein [13] using a one-dimensional
of
nxm
matrix
G,,
FT
considered.
l,.w(s)dw,,is
Example1. By using (3), we have [9]:
stochasticintegral
T
|
,
.
)
) , l^
y t Q ) : l w ( s l d w=, ( w i - w i , ) f2 - ( T - t o ) r 2
,o
(8)
By using(6), we have[9]:
T
r
'
/ , ( 1 )= l w ( s ) d w , =
that dB, is always independentof 8,, and,dBi
( w ,? - r r, ., ,2, \, l -)
{gt
tt
is approximatelydt .
In contrast to the above Ito description, in
(l) the white noise driving term is consideredas
an approximation of a very wide band but
smooth colored process. The differential
equationcan be solved exactly for each sample
function of the smooth processusing classical
calculus. As driving noise approacheswhite, the
solved processdoes convergeto the solution in
the Stratonovich sense. Thus, the rule of
ordinary calculus can be applied to a
Stratonovichequation.However, if the noise in
SDE is not statemultiplicative,i.e., G in (l) is
independent of states, and only a nonrandom
function of time t , the correction term
disappears,which can be seen from (10)-(13).
That is, Ito interpretation and Stratonovich
intemretationcoincides.
r0
which aan be also obtained by a direct
integration by partsjust as for ordinary integrals.
Since stochasticintegralscan be computed
in two distinct ways, there are two different
solutions,Ito and Stratonovichsolutionsto (2),
and hence to (l). However these two different
solutions can be converted into each other by
using simple formulas [13] of :
I
rr.
x , ( r )= X , ( / n )J|+
[ r 1 x , ( r )-,+r2 z)t- /-i = l / - /tI -"- l
,'-
. (G,(,X ,( r ) , ) ) , Gr,(X ,(r),)p r
d
+ ( S )I G ( X /(r \ , r l d B ,
B. Interpretation
The paradox of obtaining two different
stochasticprocessesas solutions to the same
stochastic differential equation arises from the
pathological nature of white noise. Since all
sample functions of a Wiener process are
nowhere differentiable and of unbounded
variation,and hence,are not smooth,we can not
interpret the Ito integral as an ordinary
Riemann-Stieltjes integral. Therefore, Ito
calculus does not conform to the rules of
ordinary calculus,which can be easily seen in
example 1. The correction term is due to the fact
(10)
or
A'
Thammasat
The processdefined by the Ito integral in (3)
is a Martingale, i.e., the conditional
whilethe
e x p e c t a t i o n E l lYY, , , r ( s < l ] = { ,
Stratonovich integral { (l) is not. This property
provides the expectation computation advantage
for an Ito equation,which is very convenient for
theoretic moment stability analysis. Moreover,
because of more restrictive conditions on the
existence of Stratonovich integrals, an Ito
equation is suited for more general applications
than a Stratonovich equation.
o . 2 0.3
0 1
Modeling of an RC
3. Deterministic
Circuit
Let QQ) be the charge on the capacitor and
V (t) be the potential source applied to the input
04
0.5
I
06
0.9
07
Fig. 1 Brownian motion processB, (l)
of a RC circuit. Using Kirchoff s secondlaw,
: - o f t \
(14)
L ' 1 t 1 =l ( r ) R + = : L
and since IQ) = a91,114,, the following
equationholds:
(15)
R Q ' Q ) +C - ' Q Q )= V ( t )
or
tVQ)
(16)
Q ' Q ) +( R C ) - ' Q Q )= R
function,
continuous
lf V (t) is a piecewise
the solutionof the first order linear differential
(16)is:
equation
=0(o)"-e[-#)
Q(t\
.*i.*(-?)nu,'"
where p(0)
the capacitor.
, , i ! '-'n,
1nr
,,*1i
06
\
i
,li'
j
it't
4
,,o1,ii;ri'
I rr*iiI
1
-0
-0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0I
1
Fig. 2 Brownian motion processB, (l)
known as the intensity of noise. Their
magnitudes determine the deviation of the
stochastic case from the deterministic one. The
correlated process w(t) , sometimes called
colored noise, can be generatedby a linear
stochastic differential equation forced by white
noise.which is givenas:
(20)
d,{t) = -pw(t)dt + o@B,(t)
(l7)
is the initial charge stored in
4. Stochastic Modeling of RC Circuit
The resistanceand potential sourcemay not
be deterministic but of the form :
R'=R+"noise"=R+aw(t)
, l
0.4
where B, (l) is a Brownian motion processwith
unit variance parameter, P,o > 0 are fixed
(18)
constants
and
and
wo - N(0,o2 pf 2)
is
where w(l) is a zero mean, exponentially
independent of B, (l) . As we explained in
section 2, (20;) is the same in the Ito and
Stratonovich sense (since the coefficient of
dB,(t) is nonrandom),and can be manipulated
correlated stationary process, and N, (l) is a
white noise process of mean zero and variance
one,and a, B are nonnegative constants,
by formal rules as if Bt(t) were continuously
differentiable. Therefore equation (20) can be
rewritten as:
V' (t) = V(t) +"noise"
:V(t) + BNr(t) (19)
+J
ThammasatInt. J. Sc. Tech.,Vol. 13, No. 2, April-June2008
dw(t\
(8,(r)\
B(r)=l
I
(30)
dt
where 1y',(r) is a zero mean, white Gaussian
noise with E[Nr (/)Nr G)]= 5(t - r) . The
function
correlation
of
w(t)
\8,(/),
The analytical solution of (25) is a random
process.
i s R ". ( r ) : o 2 12e - o '
For large values of
+
process that
approximateswhite noise. Substituting(18) and
( 1 9 )i n ( 1 5 ) ,w e g e t :
If
(n + aw(t))Q'
Q)+ C' QQ)= v (t) + BN,(t)
(22)
differentialequation:
riM(t\
or
dt
Taking the expectationof (3 1), we get:
p,w(t)
is
dQV)
a
,.
the expectation
E[X(0)Xr (0)] < - ,
= M(/) is the solution of the ordinary
= A M ( r )+ E I Z ( r \ l
(32)
EIX(/)I= EIX(g)Iexp(Ar)
R+aw(t)
Q3)
+ [exp(A(r- s))E[z(s)lds
;
O(t)
''''
dt
C(R + dw(t))
Q4)
,
BdBr(t)
dt+
R+aw(t)
R+aw(t)
where dBr(t) = Nz(t)dt and Br(t) is the
Brownian motion process, independentof
B,(r) . Writing equations(20) and (24) in
matrix-vector
form,we obtain:
dx(t) : A(t)x(t)dt + z(t)dt + K(/)dB(/)
-r
( 31 )
t[X(l)]
v(t)
, BN^t)
R + rxw(t)
or
dO(r\:-
_
+ K(s)dB(s)])
ll4slas
Jexp(-As
wide-band
QU)
C(R+aw(t))
dt
X(r)= exp(ar){x1o;
v(t)
(2s)
(33)
for every / > 0. If the randomvariable X(0) is
constantthen the expectationof the stochastic
solution is equal to the deterministicsolution of
the circuit. The tunction M(/): E[X(/)] is
independentof the fluctuationalpart of the SDE.
5. Numerical Solutions of SDEs
The Euler-Maruyamanumerical method is
used for the simulationof X(r) [5]. The EulerMaruyama scheme is based on numerical
methods for ordinary differential equations.
Let the stochasticprocess X,,t, < t < T
be the solution of the scalar SDE with M
Brownian motion processes,
M
where
dx, = f (X,,t)dt+Zs, (x,,t)dB,j
/wff))
X(r):l
I
\Qtttl
(- nw(t)
(26)
I ,,,,
A(1)=lo
C(R + dw(t)) )
\
,?,, I
z1r1=[
K ( / ) =ol
I
o
P
with an initial value X ,n = X o. Let us consider
an equidistantdiscretizationof the time interval
I n = ! o + n h , w h e r eh = ( T - r o ) fn = t n , t - l n
tn-l
:
(28)
the
t R .- " ( r ) l
(op
e4)
j=l
)a,
and the correspondingdiscretizationof
motion
Brownian
process
t,,t
)
I
LB!, = B,',., B;i, =
. to be able to
,)dB!
apply any stochasticnumericalscheme,first one
has to generate, for all j,
the random
(2e)
R+aw(t))
and
incrementsof the Brownian motion process .Br
44
Fig. 4 The deterministic solution and sample
path of the stochastic solution with
stochastic resistance (noise intensitv
a:l\.
as independentGaussianrandom variableswith
meanEfLBll = 0 and Et(LB;;'l: h.
The Euler-Maruyamaschemefor this SDE has
the form:
M
x n , t= x , , + f ( x , , t , ) h + l g , { x , , , t , ) L B 1
25_
(35)
For measuring the accuracy of a numerical
solution to an SDE, we use the strong order of
convergence.A stochastic numerical scheme
convergeswith strong order y, if thereexist real
c o n s t a n tKs > 0
a n dd > 0
Ellx, - xin< Khl
2-
1 5
a
[5],sothat:
/ze(0,d)
I
(36)
xi
where the numericalsolution is denotedby
The Euler-Maruyama scheme converges with
y:
strongordsr
1
l
1 :
l
I
0
0
/
/Z
5
0
Fig. 5 The deterministic solution and sample
path of the stochastic solution with
stochastic potential source (noise
intensitP
Y:l).
6. Simulation Results
The Brownian motion processesthat we
use for the simulation of the stochasticsolution
are shown in Fig. I and Fig. 2. Assume
R=10O, C=0.1F, V(t)=V =20V, and
X(0) = 0 .The resultsof the stochasticsolution
l
o
0
of the circuit with both stochasticresistanceand
is
s t o c h a s t i cp o t e n t i a l s o u r c e ( a = 7 , B = l )
shown in Fig. 3. A random behavior which
dependson the intensity of noise is observedin
Fig. 3. If the resistance is modeled as a
superpositionof mean value plus a zero mean
correlatedprocess,then an impact on the DC
response is observed in Fig. 4. A random
behavior is observedin Fig. 5, if the potential
source is modeled as a superpositionof mean
value plus a noise, where the noise is modeled
as a white Gaussiannoise.
Fig. 3 The deterministic solution and sample
path of the stochastic solution with
stochastic resistance and stochastic
potential source (noise intensity
a =l,F:l).
..-,.",...'l
7. Conclusion
We have seenthat deterministicmodels are
alvi'aysaccompaniedby stochasticones, which
arise whenever we plug in random variables.
Thesestochasticmodels are more appropriateto
describe,for instance,the outcome of repeated
experiments.This paper showsan applicationof
the Ito stochastic calculus to the problem of
modelinga seriesRC electricalcircuit, including
45
ThammasatInt. J. Sc. Tech.,Vol. 13. No.2. April-June200tt
both analytical and numerical solutions. The
stochastic model is obtained from the
deterministicmodel by adding noise in both the
potential source and the resistance.The noise
addedin the potential sourceis assumedto be a
white noise and that added in the resistanceis
assumed to be a correlated process. The
resulting SDE is solved using a non-Monte
Carlo method. A monte Carlo method does not
give explicit relation an expressing the
sensitivity of the perfoffnancewith respect to
input parametersand their tolerances.So, our
methodcan be usedfor designoptimization.
by formal rules as if Btft) were continuously
differentiable.Therefore equation (39) can be
rewrittenas:
dw,(t\ - p , w , ( t l + o , P ,N
(40)
,(t)
dt
where ly',(l) is a zero mean,white Gaussian
n o i s e w i t h E I Nt Q ) N t G ) l = 6 ( t - r ) . T h e
w,(t)
of
function
correlation
i s R , . ( r ) = o ,'t 42
e
p,,w(t)
Appendix
A. Generalizationo./the Method
Considera linear circuit that is built out of
resistors.capacitorsand inductors. v(f) is the
a
F o r l a r g e v a l u e so f
wide-band process that
approximateswhite noise.Let X(t) be the state
vector,which is definedas :
X(r) : [w,(r) ]r:(/).. .w,,(t) r, (/)' . .x,,(/)l'
(41)
input pr6cessand ,t(t) the output process.r(t)
satisfiesthe differentialequation:
If the potentialsourceis modeledas a
of meanvalueplusa whitenoise,
superposition
then this state vector satisfiesthe following
SDEs:
dw,(t): *ptwt|)dt + o,p,dB,(t)
(37)
s t . a . , " ' , a u a r et u n c t l o n s
T h e p a r a m e t e rQ
of the circuit elements. These parameters
becomerandom variablesif they are influenced
by random parameters such as surrounding
temperature.Theseparameters,with the random
variationscan be written as :
(38)
u) = a, + a,w,(t)
where w,(/)s are zero mean exponentially
con'elatedprocessesand a, s are nonnegatlve
constantsknown as the intensityof noise.Their
magnitude determines the deviation of the
stochasticcase from the deterministicone. The
corelated proccss u', (t) , sometimes called
colorednoise,can be generatedby a linear SDE
tbrcedby white noise[9].
thr,(t)= -pr||iQ)dt + o,p,dB,(t) 1< i< n
(39)
where B,(l) is a Brownianmotion process
with unit varianceparameter,p,,o,>0
is
" '
are
f i r e d c o n s t a n t as n d w , ( 0 ) - - 1 V ( 0o. t p , f 2 1 ; t
i n d e p e n d e not f B , ( t ) . A s w e c x p l a i n e di n
section 2. (39) is thc same in the lto and
Stratonovichsense (since the coefficient of
dB (t) is nonrandom).and can be rnan:ipulated
46
dw,,(t)= | p,,*,,(t)dt+ o, p,dB,(t)
dx,(t)= x.(t)dt
dxr(t): x1ft)dt
:
dx,,,(t) = x,,(t)dt
(t ) = -bi r,,(t) + al x,,, ( t ) +' " + ai,x, ft \pt
clx,,
+ v(t)dt+ BdB(t)
8. References
tll M. Vargas and R. Pallas-ArenY,On
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t996.
t3l P. J. Fish, ElectronicNoise and low noise
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I4l K.
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I.
Fujishiro,
Hiroki
and
Takei,
Determinationof Small Signal Parameters
and Noise Figures of MESFET's bY
Circuit SimulatorEmploying
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