CS
Transient Sensitivity Analysis in Circuit Simulation
Z. Ilievski, H. Xu, A. Verhoeven, E.J.W. ter Maten, W.H.A. Schilders
Department of Mathematical and Computer Science, TU Eindhoven, The Netherlands
Philips Research laboratories, Eindhoven, The Netherlands
E-mail [email protected]
Abstract— Sensitivity analysis is an important tool that can be
used to assess and improve the design and accuracy of a model
describing an electronic circuit. The method described in section
II shows the application of the adjoint method to transient
sensitivity analysis. The cost of calculating the sensitivity is
ˆ (t , p)
reduced by successfully replacing a high dimension term x
faults in analog circuits [5]. However, in studying sizing
problems (in which for instance the physical area of a
capacitor has to be taken into account), it appears that
especially parameters of capacitors give rise to terms that
require additional investigation. Here the effect of the index of
the related DAE shows up.
with a much lower dimension term  (t ) . Although successful in
reducing the cost, there are some weaknesses that I will point out
as areas in need of further attention.
Keywords - Sensitivity Analysis; Transient Analysis; Adjoint
Method
II. TRANSIENT SENSITIVITY ANALYSIS
Fig. 1. shows a typical circuit layout, it has four nodes
(n0…n3), five branches (b1…b5) each with a component. A
source s(t), resistors R1 & R2 and an inductor L1.
I. INTRODUCTION
Given a model description in the form of a set of differential
algebraic equations it is possible to observe how a circuit's
output reacts to varying input parameters, which are
introduced at the requirements stage of design. In this way it is
possible to determine which parameters or components of a
given circuit are sensitive. It may be the case that a relatively
large change has been detected in the output for a small change
of a parameter value, this then identifies exactly which
parameters are in need of further adjustment to ensure the
quality of a model. A model may even be in need of a
complete redesign to reduce this sensitivity.
The approaches to DC and AC sensitivity analysis have
already proved to be satisfactory and the DC case is an
important starting point for the transient analysis. In both cases
the adjoint method is efficient when the number of parameters
is larger than the number of output variables.
A typical time domain problem in circuit analysis is the
product of voltage difference times the current through an
electronic component (power) and, when integrated of time,
this reflects the total power that is dissipated. Another time
domain problem is the determination of the time moment when
a certain unknown crosses a particular value. Such a moment
can be the moment at which synchronization is required in cosimulation between a circuit simulator and another simulation
tool.
In transient analysis, the adjoint method can be formulated
as a convolution of the circuit equations with a carefully
constructed function, that, by its nature, requires a backward
integration in time of a related DAE (and for which a proper
initial value has to be determined). The method has been
popularized in [1,2] for linear problems. For more general
DAEs the method has been studied in [3] in a more
mathematical way.
In [4] the application to the nonlinear DAEs of circuit
equations was studied more closely. Nice applications can be
derived for the problem of finding optimal sources in detecting
Fig. 1. Circuit layout
Equation (1) is a general equation that can be used to describe
how any circuit behaves over a period of time. x(t ) is the
state vector and represents the node voltage, j & q describe the
current and charge behavior.
d
q(x(t ))  s(t )
(1)
dt
Because x(t ) can be sensitive to a parameter p it can be
rewritten as x(t , p ) . The current behavior is dependant on
x(t , p) but also on p so we can rewrite (1) as
j(x(t )) 
j(x(t , p), p) 
d
q(x(t , p), p)  s(t , p) .
dt
(2)
Equation (2) includes the influence of these parameters.
p is used to tune and optimize some circuit functionality, we
assume this functionality to be described as an Observation
Function. The Observation Function depends on the type of
analysis being performed and in the case of transient analysis a
basic Observation Function can be expressed as
F(x(t , p), p) , from which other Observation Functions can
be obtained, like
CS
T
G(x(p), p)   F(x(t , p), p) dt .
(3)
0
x(t , p)
p
(4)
When applying this to a transient analysis case and
differentiating through the integral in (3) we produce
T  F(x(t , p), p)
dG(x(p), p)
F(x(t , p), p)  . (5)
dt
  
xˆ (t , p) 
0
dp
x
p


ˆ (t , p) and
The initial problem here is the cost of calculating x
[1,2,3] show in detail how (5) can be re-expressed with a term
 (t ) which is of much lower order than xˆ (t , p) . The
following is an overview of how it is done.
Differentiating (2) w.r.t. p,
dj(x(t , p), p) d dq(x(t , p), p) ds(t , p)


dp
dt
dp
dp
Rearranging this, multiplying by a term
gives

T
0
 (t ) and
0
 d dq(x(t , p), p) dj(x(t , p), p) ds(t , p) 
dt  0


dp
dp
dp 
 dt
* (t )
(7)

d (t ) 
q(x(t , p), p) 
 *
 
c xˆ dt  * (t ) Cxˆ 
  (t )g 
dt 
p


0

T
0
 * j(x(t , p), p) d* (t ) q(x(t , p), p)
  (t )


p
dt
p

* (t )
q(x 0 , p) 
 q(x 0 , p)
dG (x(p), p)

 * (0)
xˆ 0 
dp
x
p 

0 
T
d* (t ) q(x, p) F (x(t , p), p)


dt
p
p
 ds(t , p) j(x, p) 
 dt

p 
 dp
* (t )
(13)
Although the current method for calculating the transient
sensitivity of a circuit has a reduced cost, some disadvantages
are introduced when including the analysis of parameters for
the capacitances. Because of this
zero and so
(8)
q(x, p)
is not equal to
p
d (t )
must be calculated. It is not ideal to have
dt
to do this as the calculation of its value by a finite difference
method introduces errors. Another point of concern is the
calculation of the integral in (13), as well as introducing an
error in its own evaluation,
ds (t , p) 
dt
dp 
d (t )
is calculated many times
dt
between t=0 and t=T. We will be describing ways on how to
calculate sensitivities in a stable and an efficient way.
IV. REFERENCES
[1]
Now if we compare the left hand side of (8) with the first term
on the right hand side of (5) as long as  (t ) satisfies
F(x(t , p), p)
d* (t )
  * (t )G 
C,
x
dt
[2]
(9)
[3]
or
d (t )
 F(x(t , p), p) 
C
 G *  (t )  

dt
x


the initial condition  (T )  0 by solving equations (9), (10)
as described in [1,2,3].
ˆ (0, p) is
Now that all  (t ) have been obtained and x
available from the DC analysis the sensitivity equation (13)
can now be calculated with much less cost than if we had used
(5).
III. OUTLINE
integrating
T

condition of a circuit, and so (5) is now expressed as (13).
The values  (t ) for 0  t  T are calculated in reverse from
(6)
Using partial integration on the first term and substituting it
back in to (7) and after some further manipulation we
eventually derive (8).
T
Only the boundary values at t  0 and t  T for
ˆx(t , p) remain. The calculation of xˆ (T , p) is avoided by
ˆ (0, p) is easily found from the DC
setting  (T )  0 , x
(13).
Applying the adjoint method and differentiating (3) w.r.t p we
can find an expression for the sensitivity. In both AC and DC
analysis the adjoint method is efficient.
xˆ (t , p) 
ˆ (t , p) and finally express the sensitivity as
we can eliminate x
*
*
,
(10)
where
q(x(t , p), p)
j(x(t , p), p)
C
, G
.
x
x
[4]
[5]
(11) , (12)
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Hong Xu, Transient Sensitivity Analysis in Circuit Simulation,
MSc-Thesis, Department of Mathematics and Computing Science, TU
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