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               A coupled system for electrical circuits. Numerical simulations.
               Monica Selva Soto∗
               University of Cologne, Mathematical Institute, Weyertal 86-90, 50931 Cologne, Germany


               We are interested in the numerical simulation of electrical circuits modelled by a coupled system of differential algebraic
               equations and partial differential equations. The partial differential equations describe the semiconductor devices in the
               circuit. When solving this problem numerically we discretize in space the partial differential equations in the system and
               solve the resulting differential algebraic equation. In this paper a brief description of the model is given and some of its
               properties are presented. Some numerical simulations are also shown.
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               1    Introduction
Nowadays semiconductor devices in an electrical circuit are modelled by small circuits containing basic network elements
(capacitors, resistors, inductors, voltage and current sources) described by algebraic and ordinary differential equations. But
these equivalent circuits may depend on hundreds of parameters and their correct adjustment has become a very difficult task
for the network design. This has motivated the idea of using distributed device models, represented by a system of partial
differential equations, to describe the behavior of the semiconductor devices within the circuit [1]. The resulting mathematical
models couple the differential algebraic equations (DAEs) modelling the electrical circuit and the partial differential equations
(PDEs) for the semiconductor devices in it.
   In this work we are interested in the numerical solution of the system that results when the Modified Nodal Analysis (MNA)
equations for electrical circuits are coupled to drift-diffusion equations modelling the semiconductor devices in the circuit. In
section (2) this model is briefly described. The DAE that results from the coupling of the MNA equations and the discrete
drift-diffusion equations is constructed in section (3) and results about its index are presented. The knowledge about the DAE
index allows us to determine the conditions that consistent initial values must satisfy and which numerical methods are feasible
for its solution. Finally, in section (4) some simulation results are shown.
   For the sake of simplicity in the sections below we assume that the circuit contains only one semiconductor device with
nS + 1 contacts to the circuit.

               2    Coupled system for electrical circuits
The Modified Nodal Analysis equations for an electrical circuit containing capacitors, resistors, inductors, independent voltage
and current sources and semiconductor devices have the form
                    d
               AC      qC (AT e, t) + AR g(AT e, t) + AL jL + AV jV + AI iS (t) + AS jS = 0,
                            C               R                                                                                                       (1a)
                    dt
                                                                     d
                                                                        φ(jL , t) − AT e = 0,
                                                                                     L                                                              (1b)
                                                                     dt
                                                                            T
                                                                          AV e − vS (t) = 0, t ∈ (ta , tb )                                         (1c)
where AC , AR , AL , AV , AI and AS are the element related reduced incidence matrices. They describe the branch-node
relationships for capacitors, resistors, inductors, voltage and current sources and semiconductor devices respectively. The
independent variable t represents the time. The functions qC (u, t), φ(j, t), g(u, t), vS (t) and iS (t) above describe the charge
of the capacitors in the circuit, the fluxes of inductors, the current through resistors as well as the independent voltage and
current sources respectively. The unknowns are the node potentials, excepting the mass node e(t) : R → RnN and the currents
through inductors, voltage sources and semiconductor devices jL (t) : R → RnL and jV (t) : R → RnV and jS : R → RnS
respectively.
   The coupling between these equations and the drift-diffusion model, presented in the next section, is given in two ways:
first, the current jS through the semiconductor’s contacts depends on the drift-diffusion variables and secondly, the boundary
conditions for the drift-diffusion variables depend on the node potentials e(t) of the circuit.
   ∗   e-mail: mselva@math.uni-koeln.de




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               2.1   Drift-Diffusion equations
           d
Let Ω ⊂ R with 1 ≤ d ≤ 3 be a nonempty, open and bounded domain with regular boundary Γ = ∂Ω such that Ω describes     ¯
the range of the semiconductor device, including its contacts. Suppose x ∈ Ω represents the space variable and t is the time
variable, t ∈ (ta , tb ).
   The drift-diffusion model is given by the following set of Partial Differential Equations for the electrostatic potential ψ(x, t)
and the electrons and holes densities n(x, t) and p(x, t) respectively,
                                 ∂
                   · Jn + Jp − ε    ψ            =      0,                                                                                           (1d)
                                ∂t
                           ∂n 1
                         −    +    · Jn          =      R,                                                                                           (1e)
                           ∂t   q
                           ∂p 1
                              +    · Jp          =    −R,                                                                                             (1f)
                           ∂t   q
where q is the elementary charge and ε, the dielectric constant. The current densities caused by electrons and holes, Jn and
Jp respectively, can be described as a composition of a drift and a diffusion current,
               Jn (x, t) = qµn (UT n − n ψ) , Jp (x, t) = − qµp (UT p + p ψ) .                                                                       (1g)
The function R = R(n, p) in (1e)-(1f) describes the balance of generation and recombination of electrons and holes. The
functions µn and µp in (1g) are the mobilities of electrons and holes, we assume them to be nonnegative and bounded
functions of x. The constant UT represents the thermal voltage.
   The boundary of the semiconductor devices we consider in this work can be divided into two disjoint parts, Γ = ΓO ∪ ΓA ,
with ΓO = ∪nS +1 Γk . The semiconductor device is a (nS + 1)-terminal element, it is completely defined by nS currents
              k=1
and nS voltages called the branch currents and branch voltages of the semiconductor. Let us choose the contact at ΓnS +1 as
reference contact and as branch currents
                                                ∂
               jSk = −            Jn + Jp − ε      ψ · ν ds,         k = 1, 2, . . . , nS .                                                          (1h)
                             Γk                 ∂t

Suppose further that the k-th semiconductor contact is attached to the node ik of the circuit. Then the matrix AS ∈ RnN ×nS
in (1a) is such that
                          
                           1, if i = ij
              AS (i, j) =   −1, if i = inS +1
                          
                              0, else.

and the vector AT e conforms the branch voltages of the semiconductor device. Its boundary conditions are then the following
                S

               ψ(x, t) = ψD (x, AT e(t)) + ψbi (x),
                                 S                           n(x, t) = nD (x), p(x, t) = pD (x),                  on ΓO                               (1i)
                                                             ψ · ν = 0, Jn · ν = 0, Jp · ν = 0,                   on ΓA .                             (1j)
where ψbi (x) is the built-in potential of the semiconductor device and the function ψD (x, AT e) is such that
                                                                                             S

                                      δk AT e, x ∈ Γk , k = 1, 2, . . . , nS ,
                                       T
                                          S
               ψD (x, AT e) =
                       S              0,       x ∈ ΓnS +1 .
if δk represents the k-th unit vector.
    The properties of coupled systems related to the one presented here have been recently studied, see e.g. [1, 2, 3, 5, 6, 7].
    Before discretizing the drift-diffusion equations in space, let us introduce the functions wk (x) ∈ H 2 (Ω), k = 1, 2, . . . , nS
as the solutions of
               ∆wk = 0 in Ω,            wk · ν = 0, on ΓA ,        wk (x) = δki , x ∈ Γi , 1 ≤ i ≤ nS + 1                                              (2)
where δij = 1 if i = j, otherwise δij = 0. These are a basis of the linear space of functions
                                                        W = { u ∈ H 2 (Ω) : ∆u = 0 in Ω,
               ( u · ν)|ΓA = 0, u|Γj = aj , aj ∈ R, j = 1, 2, . . . , nS , u|ΓnS +1 = 0. }
Because (u, v)W =        Ω
                              u·      v dx is a scalar product in W, the matrix
                                                                                        
                                  Ω
                                       w1 ·   w1 dx    ...      Ω
                                                                     wnS ·     w1 dx
                                         .             .               .                
               M     =                   .
                                          .             .
                                                        .               .
                                                                        .                
                                  Ω
                                      w1 ·    wnS dx . . .     Ω
                                                                     wnS ·     wnS dx
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is positive definite. Note also that with the help of these functions, the current through the semiconductor’s contacts can be
rewritten as

             jSk (t) =     −        J · ν ds = −        J · ν wk ds = −          J·   wk dx
                               Γk                   Γ                        Ω
                                                               d
                       =   −       (Jn + Jp ) ·    wk dx + ε              ψ·     wk dx                                                                  (3)
                               Ω                               dt   Ω


             3    Discretization of the drift-diffusion equations in the model
Let (ψh (x, t), nh (x, t), ph (x, t)) denote an approximation to the weak solution of (1d)-(1j) obtained by the finite element
                                                                        T
                                           T
                                                 ˆ      ˆ     ˆ
method. If vh (t) = (Ψ(t) N (t) P (t)) and vh = Ψ(t) N (t) P (t)   ˆ      where the vectors Ψ(t), N (t) and P (t) contain the
                                                                                            ˆ     ˆ          ˆ
approximations to ψ(xi , t), n(xi , t) and p(xi , t) for mesh-points xi ∈ Ω ∪ ΓA while Ψ(t), N (t) and P (t) are the approxima-
tions to ψ(xi , t), n(xi , t) and p(xi , t) with xi ∈ ΓO , the Differential Algebraic Equation, obtained after discretization in space
of the drift-diffusion equations in the coupled system can be written as [4],
                  d
             AC      qC (AT e, t) + AR g(AT e, t) + AL jL + AV jV + AI iS (t) + AS jS = 0,
                          C               R                                                                                                           (4a)
                  dt
                                                                   d
                                                                      φ(jL , t) − AT e = 0,
                                                                                   L                                                                  (4b)
                                                                   dt
                                                                          T
                                                                        AV e − vS (t) = 0,                                                            (4c)
                                                                                 qS +   εMh AT e
                                                                                             S      =   0,                                            (4d)
                                                                                              dqS
                                                            jS + jS,h (AT e, vh , vh ) +
                                                                  c
                                                                        S         ˆ                 =   0,                                            (4e)
                                                                                               dt
                                                                         dvh
                                                                    Sh       + r(AT e, vh , vh )
                                                                                  S         ˆ       =   0,                                             (4f)
                                                                          dt
                                                                                            v
                                                                                        h(ˆh )      =   0                                             (4g)
                                                                                                                    c
where Sh is a nonsingular matrix, Mh is a symmetric and positive definite matrix and the functions r, h and jS,h are con-
tinuously differentiable. Equations (4d) and (4e) result when (3) is discretized in space, (4f) is the spatial discretization of
(1d)-(1f), while (4g) corresponds to the discretized boundary conditions.
   It can be proved [4, 5] that this DAE has always index less or equal to two and it is two if and only if the circuit contains
loops of capacitors, voltage sources and semiconductor devices with at least one voltage source or one semiconductor device
or cut sets of inductors and current sources.

             4    Numerical simulations
The simulation results shown in this section were obtained with a M ATLAB program developed by us based on the approach
presented here. One and two dimensional models for semiconductor devices have been included in this program.
   Consider the electrical circuit in figure (1). The semiconductor devices in it were modelled in two spatial dimensions with
length, width and gate length equal to 210nm, 350nm and 70nm respectively. The input signals are those identified with VSet
and VReset . They are shown in figure (2) as well as the output signal (node potential e3 ). For small frequencies the behavior
of the circuit is as expected, when the set signal is active the output signal goes to one and when the reset signal is active the
output goes to zero. If neither is activated the previous state is maintained. As one can see in figure (2), as the frequency
increases the behavior of the circuit moves away from its ideal behavior. If the spatial dimensions of the semiconductor
devices in the circuit (length, width, gate length) are then reduced it behaves again as expected. Such kind of experiments can
not be made if the semiconductor devices are replaced by equivalent circuits because there is no known relationship between
the dimensions of the semiconductor device and the value of the parameters the equivalent circuit depends on.

             5    Conclusions
Electrical circuits containing semiconductor devices can be modelled as a coupled system of differential algebraic and partial
differential equations. An approximate solution of such a system can be obtained, as proposed here, by discretizing the partial
differential equations in space and solving numerically the resulting DAE. In order to gain information about how to choose
consistent initial values, what type of numerical methods may be used for the solution of this DAE, etc., it is important to
determine its index. In the last section simulation results using a M ATLAB program for the simulation of electrical circuits
based on this approach have been shown.


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Fig. 1 Flip-flop circuit

                               Set Signal                                                            1 MHz
       6                                                                   6
       4                                                                   4
                                                                           2
       2
                                                                           0
       0
                                                                               0      0.5        1            1.5             2                 2.5
           0       0.5        1          1.5        2          2.5                                                                           −5
                                                                                                                                         x 10
                              Reset Signal                                                           100 MHz
       6                                                                   6
       4                                                                   4
       2                                                                   2
                                                                           0
       0
                                                                               0      0.5        1            1.5             2                 2.5
           0       0.5        1          1.5        2          2.5                                                                           −7
                                                                                                                                         x 10
                                                                                                      1 GHz
                                                                           6
                                                                           4
                                                                           2
                                                                           0
                                                                               0      0.5        1            1.5             2                 2.5
                                                                                                                                             −8
                                                                                                                                         x 10


Fig. 2 Input (set and reset) and output signals in the flip-flop circuit


    As mentioned in the previous section, when the semiconductor devices in the circuit are described by distributed models it
is possible to observe experimentally the relationship between the spatial dimensions of the semiconductors and its behavior
to different frequencies. These kind of experiments are not possible with the traditional approach, where the semiconductor
devices are replaced by equivalent circuits containing basic network elements.

Acknowledgements This work is partially financed by the German ministry BMBF, grant 03GUNAVN, as part of the internal consortium
”Numerical Simulation of Multiscale Models for High-Frequency Circuits in Time Domain”. The teams of this consortium formed the young
researcher’s minisymposium on ”Multiscale Systems in Refined Network Modeling: Analysis and Numerical Simulation” at GAMM 2006:
                                                                                                            ¨
M. Bodestedt and M. Selva Soto (BMBF grant 03TINAVN by C. Tischendorf), M. Brunk (03JUNAVN by A. Jungel), T. Sickenberger and
                                  o                                                      u
R. Winkler (03RONAVN by W. R¨ misch), and A. Bartel and S. Knorr (03GUNAVN by M. G¨ nther). The GAMM committee is especially
acknowledged for the possibility to organize this minisymposium.



               References
                           u
 [1] Ali G., Bartel A., G¨ nther M., Tischendorf C., Elliptic Partial Differential Algebraic Multiphysics Models in Electrical Network
     Design, Math. Models Meth. Appl. Sci., 13(9), 1261-1278 (2003).
                          u
 [2] Ali G., Bartel A., G¨ nther M.: Parabolic Differential-Algebraic Models in Electrical Network Design, SIAM J. Mult. Model. Sim.,
     4:3, 813–838 (2005).
 [3] Bodestedt M., Tischendorf C.: PDAE models of integrated circuits and perturbation analysis, to appear in Math. Comput. Model. Dyn.
     Syst., Preprint 2004-08, Institute of Mathematics, Humboldt University of Berlin (2004).
 [4] Guhlke C., Selva Soto M., Tischendorf C.: A new approach for the simulation of electrical circuits, in preparation.
 [5] Selva Soto, M., Tischendorf C., Numerical Analysis of DAEs from coupled circuit and semiconductor simulation, APNUM, (53)2-4,
     471-488 (2005).
 [6] Selva Soto, M., An Index Analysis from Coupled Circuit and Device Simulation, Proceedings of the Conference Scientific Computing
     in Electrical Engineering (2005).
 [7] Tischendorf C., Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation, Ha-
     bilitation Thesis, Humboldt University of Berlin (2003).

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