IEEE Circuits and Systems Magazine - Q1 2023 - 41

The overall equations/variables balance for the
model of Fig. 1 is given in Table 1. As can be seen, the
individual component models are not closed, while
thanks to the connection equations the compound
one is.
Though we have just scratched the surface of EBM,
and Modelica in particular, we can observe that constitutive
equations-besides being in declarative form-
do not depend on connections, while connection equations
do not depend on the behavior of the connected
models. This allows us to point out a few relevant facts
for the following discussion.
■ There is a clear interface/behavior separation,
hence models are interchangeable as long as the
interface (i.e., the connector structure) is preserved.
This is particularly useful when the detail
of a model (or part of a model) needs tailoring for
the simulation study to conduct.
■ Viewed differently, the separation is between
internal behavior and boundary conditions. A
model is written independently of how it will be
connected to others. Together with the declarative
approach, this makes models resemble very
closely the way they appear, e.g., as and equations
in a textbook.
V. Declarative and Imperative PROS/CONS
In Section III-B we anticipated the importance of the
D2I translation process. We now revisit the matter to
evidence pros and cons of declarative and imperative
modeling, coming at the end of this section to set the focus
on ICs and their cooling systems. To do so, we need
to introduce some details on how declarative models
are translated into imperative ones by an EBM language
automated translator-or how a modeler would have
to translate models by hand if not using EBM tools. To
this end we consider and discuss a couple of D2I translations,
here too referring to simple purposed cases.
A. D2I-Example 1
In this example we consider a one-dimensional heat
transfer problem in a solid, namely a rod heated at one
side by a prescribed power Pp(t) − t is time-and connected
to a prescribed temperature Tp(t) on the other
side, the lateral surface being adiabatic. We denote by S
the rod uniform section, by L its length, and by ρ, c, λ its
constant density, specific heat and thermal conductivity.
Approximating the continuous rod with a sequence
of volume lumps, each one with its own temperature
and exchanging heat with the previous and the following
one, we readily get to the declarative model schematised
in Fig. 2 and written as the system of differential
equations




CT tG Tt Tt TtNN Np()
CT tP tG Tt Tt
CT tG Tt Tt Tt
 ()


11 2
112
()
ii ii (
()
()
()
() =... −
=− +()
()
() ()
()
iN
p()
=− −()
=− +−+ )
−1 32
21 (2)
where the dot indicates derivative with time, N is the
number of volumes (or lumps) into which the rod is divided,
Ti is the temperature (assumed spatially uniform)
of the i-th lump, and
Figure 1. EBM introductory example-a complete model.
C = ρcSL/N, G = λSN/L
Table 1.
EBM introductory example-variables and equations (CONSTitutive and
CONNection).
Entity
S
G
C
gnd
Running total
red set
green set
blue set
Total
FIRST QUARTER 2023
Variables introduced
6 (a.v,a.i,b.v,b.i,v, i)
6 (a.v,a.i,b.v,b.i,v, i)
6 (a.v,a.i,b.v,b.i,v, i)
2 (a.v,a.i)
20
none
none
none
20
Equations introduced
4 (1+3 from TwoPin)
4 (1+3 from TwoPin)
4 (1+3 from TwoPin)
1
13
2
2
3
20
C
O
N
S
T
C
O
N
N
(3)
are respectively the heat capacity
of one lump and the center-tocenter
inter-lump thermal conductance.
Since (2) is linear in Ti, Pp and
Tp, we can write it in the compact
matrix form
T˙ (t) = AT (t) + Bu(t)
where
Tt
(4)
() =




Tt
Tt
1
N
()
()
 ,( )ut = 






Pt
Tt
p
p
()
()



(5)
IEEE CIRCUITS AND SYSTEMS MAGAZINE
41
IEEE Circuits and Systems Magazine - Q1 2023

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