Download An Efficient Two-level DC Operating Points Finder for Transistor

An Efficient Two-level DC Operating Points Finder for
Transistor Circuits
Jian Deng, Kim Batselier, Yang Zhang and Ngai Wong
Department of Electrical and Electronic Engineering
The University of Hong Kong
{dengjian, kimb, yzhang, nwong}@eee.hku.hk
ABSTRACT
DC analysis, as a foundation for the simulation of many
electronic circuits, is concerned with locating DC operating
points. In this paper, a new and efficient algorithm to find
all DC operating points is proposed for transistor circuits.
The novelty of this DC operating points finder is its twolevel simple implementation based on the affine arithmetic
preconditioning and interval contraction method. Compared
to traditional methods such as homotopy, this finder offers
a dramatically faster way of computing all roots, without
sacrificing any accuracy. Explicit numerical examples and
comparative analysis are given to demonstrate the feasibility
and accuracy of the proposed approach.
Categories and Subject Descriptors
EDA7.3 [Analog Design and Simulation]: Analog, mixedsignal, RF, electromagnetic, substrate noise modeling and
simulation
General Terms
Algorithms, Design, Performance, Theory, Verification
Keywords
DC analysis, nonlinear equations, transistor circuits simulation, inclusion method
1. INTRODUCTION
The direct current (DC) analysis is an essential step for
designers to simulate the behavior of circuits [3]. To describe the DC behavior of a circuit, a set of equations can
be derived from Kirchhoff’s current law and the constitutive relations of components. The solutions for the system
of equations are called DC operating points. In real circuit design, locating all DC operating points is reduced to a
problem concerning solving a system of nonlinear algebraic
equations. This problem has attracted much attention from
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http://dx.doi.org/10.1145/2593069.2593087.
both academia and industry due to its difficulty and importance [10, 16]. The difficulty stems from the fact that in
transistor circuits, the circuit model contains many strongly
nonlinear terms. Thus ordinary piecewise-linear approximation may fail in transistor circuits, especially when there
exist multiple solutions. A system of nonlinear equations is
given as
F(x) = 0, F : Rn → Rn .
(1)
Nonlinear building blocks, such as bipolar junction transistor (BJT), are the major contributors of nonlinearities.
The time complexity for determining all the solutions of (1)
dramatically increases as the size of nonlinear components
grows. Traditionally, electronic design automation (EDA)
tools such as SPICE can utilize the Newton-Raphson (NR) algorithm or its variants [2] to find the roots. Although
the NR method can provide a robust and quadratic convergence, it suffers from the strict requirement of the initial
guess, which often needs to be sufficiently close to the real
solution. Moreover the NR method is not suited for finding
all the solutions since the number of DC operating points is
generally uncertain [14].
To alleviate the computational complexity of finding multiple DC operating points in transistor circutis, several methods have been proposed in the past decades. A method
named homotopy and its variants are widely adopted and
recognized as a robust and accurate numerical method [8,
17]. Homotopy methods have the advantage that it is easy
to come up with an initial guess. Although the homotopy
method is a powerful tool to find multiple roots, it also encounters the danger of ill conditions, such as bifurcations [5]. Additionally, the computational and implementation
complexity are its shortcomings. In [7], an algorithm is introduced to deal with multiple electronic models, by partitioning the original circuit into subcircuits. However, this
method can not guarantee to find all solutions. In short,
an efficient and generic framework to find all DC operating
points in a simple and flexible manner is lacking.
A major problem for above lies in that the possible intervals of roots are usually very large due to the exponential nature of the BJT, which can be described by EbersMoll transistor model [6]. Thus, huge computational effort
is spent on searching for smaller intervals until the roots
are found. The idea of interval contraction in [1] provides
the foundation for the proposed root finder in this article. A
kind of contraction method was used in [15], which was able
to shrink the possible range of roots effectively. However, it
was just a preliminary and pre-processing step for locating
the intervals which contain the solutions, no detailed com-
parison between this contraction method and homotopy has
been given.
The key contribution of this paper lies in providing a simple and fast method that is guaranteed to find all DC points,
based on affine arithmetic preconditioning and interval contraction method. Compared with the widely used homotopy
approach, the proposed approach offers faster convergence
speed with the same level of accuracy. More importantly,
theoretical analysis is provided to show how the proposed
method guarantees to find all DC operating points.
The remainder of this paper is organized as follows. First,
a brief introduction of interval analysis and affine arithmetic
is provided in Section 2. Based on these two theories, the
interval contraction method and its global convergence analysis are provided. Then, Section 3 describes the implementation of the framework. In Section 4, three numerical examples are given to verify the proposed scheme, as well as
to compare it with the homotopy method in [16]. Finally, in
Section 5 we give some remarks and conclusions.
First of all, necessary facts about interval analysis and
affine arithmetic are presented to build the theoretical foundation of our method. Next, we provide an interval contraction method for a typical kind of nonlinear equation, which
is well suited to finding DC operating points in transistor
circuits. Finally, a theoretical analysis for the global convergence of this interval contraction method is proposed.
2.1 Interval analysis and affine arithmetic
Interval analysis is a mathematical tool to represent the
uncertainty of a value as an interval. For example, the real quantity x is presented as a real compact interval [x] =
[xl , xu ], meaning that xl ≤ x ≤ xu . Thus the computation
in interval analysis is connected with the operation on intervals, such as addition, subtraction, multiplication, division
and so forth. Operations on two real intervals are described
by the following rules:
[x] + [y] = [xl + y l , xu + y l ],
[x] − [y] = [xl − y u , xu − y l ],
[x] · [y] = [min{xl y l , xl y u , xu y l , xu y u },
max{xl y l , xl y u , xu y l , xu y u }].
Next, we provide the rule for matrix vector multiplication
when the entries of the vector are intervals. Suppose A =
[aij ]n×n . [X] and [Y] are both Rn×1 interval vectors. This
means that the ith entry [xi ] in the interval vector [X] is an
interval, represented as [xli , xui ]. The same applies for the
interval vector [Y]. Then
(2)
satisfies
yil
=
n
X
aij βj , βj =
j=1
yiu
=
n
X
j=1
aij δj , δj =
(
xlj
xuj
aij > 0
,
aij ≤ 0
(
xuj
xlj
aij > 0
.
aij ≤ 0
Φ([x]) = {Φ(x)|x ∈ [x]}.
(3)
Obviously, Φ(x) are continuous, monotone or piecewise
monotone on the given interval [x]. This property is called
inclusion monotonicity [1] and is described by
[x] ∈ [y] ⇒ Φ([x]) ∈ Φ([y]).
(4)
It is easy to prove that a function f : R → R, which is composed of elementary operations +, −, × and some inclusion
monotone functions Φ(x), is also inclusion monotone. This
means that
[x] ∈ [y] ⇒ f ([x]) ∈ f ([y]).
(5)
Based on equation (5), we get
2. THEORETICAL BACKGROUND
A · [X] = [Y ]
Interval analysis was introduced as a self-validating numerical algorithm, and quickly developed as a systematical theory and application for the evaluation of nonlinear
functions after Moore’s book [13]. Operations on the interval can also be extended to standard nonlinear functions
Φ = {sin, cos, exp, ln · · · } as,
x ∈ [x] ⇒ f (x) ∈ f ([x]),
R(f, [x]) ∈ f ([x]),
(6)
(7)
where R(f, [x]) is the range of f (x) on the interval [x]. As
a consequence, if f (x∗ ) = 0 6∈ f ([x0 ]), then x∗ 6∈ [x0 ]. The
following theorem gives a necessary condition to determine
whether two functions f (x) and g(x) have an intersection on
the interval [x].
Theorem 1. Let f (x) and g(x) be both inclusion monotone functions on an interval [x], Yf = f ([x]), Yg = g([x]).
If Yf ∩ Yg = ∅, then there is no intersection point for f (x)
and g(x) on the interval [x].
Theorem 1 states that if the intersection of two inclusion
monotone functions, f (x) and g(x), is empty on the interval
[x], then h(x) = f (x) − g(x) has no root on the interval
[x] according to (6). In Section 3, we utilize Theorem 1 to
eliminate intervals that do not contain the roots for a system
of nonlinear equations efficiently.
Although interval analysis is very suitable to handle nonlinear problems, it still suffers from overestimation. This
is the phenomenon in which the results of interval operations are much wider than the exact range, namely, f ([x]) =
β · R(f, [x]), β ≫ 1. The situation would get worse when the
starting interval is very large, which often leads to the failure
of interval contraction method to find the roots of nonlinear
equations. We therefore introduce affine arithmetic, which
allows us to shrink the starting interval. According to [4],
the affine form x̂ of a real interval [x] is given by
x̂ = x0 + x1 ǫ1 + x2 ǫ2 + · · · + xn ǫn ,
(8)
where x0 is the center point of [x], xi is a finite floatingpoint number, and ǫi is in the range [−1, 1]. Each ǫi stands
for an independent component of uncertainty for the interval
[x], and xi is the magnitude of this component. For an
interval [xt ] = [a, b], its affine form is hence x̂t = x0 + x1 ǫ1 ,
where x0 = a+b
and x1 = b−a
. Conversely, if we know
2
2
x̂ = x0 + x1 ǫ1 + x2 ǫ2 + · · · + xn ǫn , then P
the corresponding
interval [x] is [x0 − r, x0 + r], where r is n
i=1 |xi |.
The benefits of affine arithmetic over interval analysis are
apparent. Let us, for example, evaluate the function f (x) =
x(5−x) on the interval [2, 3]. The exact range of f (x) should
be [6, 6.25]. Using the rules for basic operations in interval
analysis, we obtain [4, 9] as the range of the function, which
is twenty times wider. However, using affine arithmetic the
interval [2, 3] is converted to x̂ = 2.5 + 0.5ǫ1 , evaluating this
into f (x) = x(5 − x) and converting the result back to an
interval we can get the correct range [6, 6.25] without any
overestimation.
2.2 Interval Contraction Method
An iterative interval contraction method is usually applied
to decrease the computational complexity when solving nonlinear scalar equations such as
F (x) = L(x),
(9)
where F (x) is the nonlinear part and L(x) is a linear function. Considering the DC operating points problem in this
paper, F (x) = etx with t 6= 0. L(x) can be represented as
ax + b. Equation (9) therefore has the following form,
etx = ax + b.
(10)
etx
ax + b
Y
m0 x + cu0
m0 x + cl0
cu0
xl0
cl0
x⋆
xl1
xu1 xt0
xu0
X
Figure 1: Interval Contraction Method
We now introduce an iterative contraction method to solve
(10). The key idea of this method is to use two linear enclosure lines to approximate the exponential function over
an interval [x0 ]. This approximation is called Chebyshev (or
minimax) affine approximation [4]. As shown in Figure 1,
x⋆ is the root and lies in the interval [xl0 , xu0 ]. We can use
two lines y = m0 x + cu0 and y = m0 x + cl0 to enclose the
nonlinear function y = etx . The values of m0 , cl0 and cu0
are determined by y = etx and the initial interval [xl0 , xu0 ].
y = m0 x + cl0 is also the tangent to y = etx at the point xt0 .
We will use this tangent point to split the interval into two
parts, [xl1 , xt0 ] and [xt0 , xu1 ], only if the new interval [xl1 , xu1 ]
contains xt0 . In this case, (10) has more than one root.
In general, the initial interval could be chosen randomly
and sufficiently large such that it contains all the roots. For
the application of finding all DC operating points, the initial
interval is obviously given by [Vee , Vcc ] where Vcc is the positive voltage supply and Vee is the negative voltage supply
or ground.
Figure 1 shows that the interval contraction method decreases the size of the interval in each iteration. The iterations are stopped when the new computed interval is small
enough to obtain the solution. The new interval is small
enough as soon as |xu − xl | < τ is satisfied, where τ is a
user-defined tolerance. The two bounds of the new interval,
xl1 and xu1 , are obtained from solving the following equations
m0 x + cu0 = ax + b,
m0 x + cl0 = ax + b.
Equation (10) can be easily extended to systems of nonlinear
equations as


 t1 x(1) 
e
x(1)
t
x(2)
2
 x(2) 

e




(11)
E ·  .  = A ·  .  + B,
 .. 
 .. 
x(m)
etn x(n)
where E is an m × n matrix, A is an m × m matrix, B is an
m × 1 vector, and n < m. Similar to the univariate method,
we can get the new interval by solving the following system
of linear equations,




x(1)
x(1)
 x(2) 
 x(2) 




E · (M ·  .  + [C]) = A ·  .  + B.
(12)
 .. 
 .. 
x(m)
x(m)
The exponential part of (11) is replaced by the Chebyshev
approximation. Then the new interval of xi (1 ≤ i ≤ m) is
found by applying interval operations on (12), which results
in the new interval vector [Xi+1 ]. The next step is to compare [Xi+1 ] with [Xi ]. This is done by computing the intersection [X∩ ] between [Xi ] and [Xi+1 ]. If some entry in [X∩ ]
is empty, we should immediately delete this guess because
no solution is in this interval. Otherwise, we should assign
[X∩ ] to [Xi+1 ]. At this point we need to check whether for
some entry the tangent point is in the new interval [Xi+1 ]. If
this is the case, [Xi+1 ] needs to be split into two parts. Two
execution threads can then be started to continue with the
iterations for each of the two intervals. If the tangent point
is not in the new interval, then we proceed with the iterations on [Xi+1 ]. The complete interval contraction method
is summarized in Algorithm 1.
Algorithm 1 Iterative Contraction Method
Input data:
An m × 2 Interval matrix [X0 ] = [X0l , X0u ],
Coefficient matrix E, A, B according to equation (11),
Tolerance τ ,
[Xresult ] = Function IC([X0 ], E, A, B, τ )
1: For each interval [x(i)] in interval vector [X0 ]
l
x(i)
2:
compute m(i), cu
0 (i) and c0 (i) for each e
3: Build and solve equation like (12), and get [Xnew ]
4: Find [X∩ ] = [X0 ] ∩ [Xnew ]
Pm
u
l
5: If
j=1 (x(j) − x(j) ) < τ
6:
return [X∩ ]
7: elseif there is some empty entry in [X∩ ]
8:
return ∅
9: elseif the tangent point x(j)t ∈ [X∩ ]
10:
split [X∩ ] into [Xs1 ] and [Xs2 ] according to x(j)t
11:
IC([Xs1 ], E, A, B, τ )
12:
IC([Xs2 ], E, A, B, τ )
13: else
14:
IC([X∩ ], E, A, B, τ )
etx
Y
cu0
ax + b
cl0
Figure 3: Ebers-Moll Transistor Model
x⋆
xl0
xt0
m0 x +
xu0
xl1
cu0
xu1 X
m0 x + cl0
Figure 2: Special case: ax + b is tangent to etx
BJT. However, there is no simple and tractable mathematical model for the field effect transistor (FET). Nonetheless,
some results of this paper related to BJT can be extended
to FET in the future [16, 18]. The model is given by
1
−αr
fe (Ve )
Ie
=
,
(13)
−αf
1
fc (Vc )
Ic
where
fe (Ve ) = me (etVe − 1)
2.3 Global Convergence
The objective of Algorithm 1 is to find all intervals, each
of which contains only one root for the equation etx − (ax +
b) = 0, t 6= 0. This root-finding problem can be transformed
into the mathematically equivalent problem of finding the
intersection point between f (x) = etx , t 6= 0 and g(x) =
ax + b over the interval [xl0 , xu0 ].
Figure 1 and Figure 2 show two circumstances where the
interval contains only one root. As depicted in Figure 1,
suppose g(x) = ax + b has an intersection point x′ with
k(x) = m0 x + cu0 (x′ could be xl1 or xu1 depending on the
sign of t and a). Therefore we know that g(x) must have an
intersection point x′′ with j(x) = m0 x + cl0 , since x′ exists.
Using the intermediate value theorem it is trivial to prove
that if x′ ∈ (xl0 , xu0 ), then g(x) has only one intersection
point with f (x) over the interval [xl0 , xu0 ]. Since the new
interval [x1 ] is given by
[x1 ] = [x0 ] ∩ [min(x′ , x′′ ), max(x′ , x′′ )],
it is guaranteed to be smaller than [x0 ]. This guarantees
convergence to the root x⋆ .
Another case is shown in Figure 2, where g(x) is a tangent line of f (x) = etx . g(x) has only one intersection point
with f (x) on [xl0 , xu0 ]. Under this circumstance, x′′ which is
the intersection point between g(x) and j(x) should be in
(xl0 , xu0 ). The intersection point x′ between g(x) and k(x)
is not in (xl0 , xu0 ). This does not, however, affect the contraction of the interval. The proposed method will therefore
still converge to the root x⋆ .
3. IMPLEMENTATION FRAMEWORK
In this section, we discuss the Ebers-Moll transistor model and derive the circuit’s modified nodal analysis (MNA)
equations [11]. The MNA equations are then transformed
into the form of (11). Then it is shown how affine arithmetic
is used as a pre-processing step to shrink the initial interval
before running Algorithm 1.
3.1 DC model transformation
The Ebers-Moll transistor model [6], shown in Figure 3,
is frequently used for describing the DC behaviour of the
and
fc (Vc ) = mc (etVc − 1), (14)
and
me αf = mc αr .
(15)
The first step of the DC analysis is to apply MNA to the
given circuit, which contains only passive elements and independent voltage sources. The passivity of the BJT in DC
analysis has been proven in [9]. This property results in the
following MNA matrix equation:
P x = 0,
(16)
where P ∈ R(n+m)×(n+m) . x is a (n + m) × 1 vector and
contains the n nodal voltages and m currents through the m
independent voltage sources. All equations are derived from
Kirchhoff’s current law at the n non-reference nodes and
Kirchhoff’s voltage law across the m independent voltage
sources.
Suppose that there are s BJTs in the circuit, and that
between each two of them there are w common nodes in
total (Normally w is less than s). The first step to write
(16) into the form of (11) is to use the Ebers-Moll transistor
model to obtain
 t v1 


e1 e
v1
 t1 vc1 


v2

e


 . 


.


..
(17)
E·T · .  = A·
 + B,
.


 t vs 
v(3s−w−1) 
e s e 
s
v(3s−w)
ets vc
with (17), E ∈ R(3s−w)×2s , T ∈ R2s×2s , A ∈ R(3s−w)×(3s−w) ,
B ∈ R(3s−w)×1 . In addition, T is a block diagonal matrix
where each 2 × 2 diagonal block has the following form
mei
−αri mci
Ti =
.
(18)
−αfi mei
mci
Each vep and vcp (p ∈ [1, s]) is equal to the respective nodal
voltage difference vi − vj , i, j ∈ [1, 3s − w].
Compared to the MNA matrix equation (16), (17) contains only the variables {X̃} corresponding to the nodal
voltages that connect to the s BJTs directly. The model
is therefore reduced by deleting the variables {X̂} corresponding to currents through the voltage sources and the
nodal voltages which are not connected with the BJTs directly. When later the solutions for {X̃} are computed, the
values for {X̂} are then easily obtained from the MNA matrix equation (16). This reduction of redundant variables
increases the computational efficiency.
The next step is to write out the base-emitter and basecollector voltages of each of the s BJTs in terms of the nodal
voltages as
 1


ve
v1
vc1 


v2
 


.


.
⋆
..
V =  ..  = Q · 
(19)
 = Q · V0 .
 


xse 
v(3s−w−1) 
v(3s−w)
xsc
4.
NUMERICAL EXPERIMENTS
Vcc
Figure 4: Schmitt Trigger Circuit
Vcc1
The final step of the model transformation is then to introduce the vector X ∈ R(3s−w)×1 of nodal voltages. Notice
that since w < s, we can let the first 2s entries in X be
equal to V ⋆ , and the remaining (s − w) entries of X are the
same as the last (s − w) entries of V0 . Thus we have
Q
,
(20)
X = R · V0 , R =
0
I
where I is the R(s−w)×(s−w) unit matrix. We can use (20)
to write V0 = R−1 X and substituting this into (17) results
in
 t1 x(1) 


e
x(1)
 et1 x(2) 


 x(2) 



..
−1 
E·T ·
 + B.
 = A·R ·
..
.




.
ets x(2s−1) 
x(3s
−
w)
ets x(2s)
(21)
3.2 Preconditioning by affine arithmetic
It is important to realize that at this point Algorithm 1
cannot be applied directly to (21). First, we need to convert
X into an interval vector. Usually, the initial interval is
very large due to the exponential terms. Affine arithmetic
can now be used as a preconditioing step to obtain smaller
initial intervals. Generally speaking, the initial interval for
all unknown nodal voltages in V0 are the same. The upper
bound and lower bound are the largest positive and negative
voltage ( or ground voltage), among all independent voltage
sources. In order to obtain the interval vector X we need
to apply the matrix vector operation (2) to (20). Then the
obtained intervals are converted into their affine forms as

et1 x(1)
et1 x(2)
.
.
.







 =



ets x(2s−1) 
ets x(2s)

x0 (1) + xr (1)ǫ1
x0 (2) + xr (2)ǫ2




(E·T )† ·A·R−1 ·
.
 + (R·T )† ·B,
.


.
x0 (3s − w) + xr (3s − w)ǫ(3s−w)
(22)
Vcc2
Figure 5: Four-transistor Benchmark Circuit
In this section, the proposed two-level scheme is applied
to three numerical examples. Our approach is implemented
using Matlab [12], and compared with a Matlab implementation of the homotopy method from [16] for the first two
examples. The last example illustrates that the proposed
method is also capable of solving general nonlinear equations. All three experiments are performed on an Intel Core
I7 desktop PC with 2.6GHz CPU and 4GB RAM.
The first two examples focus on real circuits. The first circuit in Figure 4 has two BJTs, and possesses three DC operating points. The second one in Figure 5 has four BJTs, and
the number of DC operating points is nine. For both circuits
a tolerance τ = 10−8 was used. Both the proposed method
and the homotopy method can find all solutions. The results have been verified by comparing them with the solutions
in [16]. The proposed method is more than 10 times faster
than the homotopy method with the same accuracy (Table
1). The accuracy of the two methods is expressed in terms
of their respective residuals r, computed by evaluating the
obtained roots into the MNA equations (16). The reported
worst residual is then the maximal absolute value |r| over
all solutions. The time complexity of the proposed method
is O(s3 log τ1 ), where s is the number of BJTs, and τ is the
tolerance.

where (·)† represents the pseudo inverse. By taking the logarithm of each value in the right hand side of (22) and dividing each of these values by their respective ti , the smaller
intervals are obtained. This concludes the preconditioning
step after which Algorithm 1 can now finally be applied.
15
10
5
y
0
−5
−10
y = (2 + x2 ) − (2x + e
−15
−20
−1
0
1
2
3
4
5
6
7
2xlog2
3
8
)
9
10
x
Figure 6: Univariate Nonlinear Function
Table 1: Numerical Results (Tolerance τ = 10−8 )
Homotopy
Proposed
Method [16]
Method
Example 1 Run time
0.018639
0.32064
(sec)
(Figure 4)
Example 2
(Figure 5)
Example 3
(Figure 6)
Worst
Residual
1.2305 × 10−5
2.281 × 10−5
Run time
(sec)
0.06111
0.96957
Worst
Residual
4.2361 × 10−7
9.295 × 10−7
Run time
(sec)
0.007380
NA
Worst
Residual
8.103 × 10−10
NA
The last example in Figure 6 demonstrates the potential of
the proposed method to solve more complicated nonlinear
functions. The univariate nonlinear function in Figure 6
contains an exponential and a quadratic function. It has
three roots, all of which are located with small residuals by
our proposed method. It is difficult for the NR method to
find these three root without a good initial guess.
5. CONCLUSION
This paper has presented a simple and efficient approach
for finding all DC operating points of transistor circuits.
Based on the interval analysis and affine arithmetic, this
proposed approach give rise to remarkable computation efficiency in locating all DC operating points. Compared to
the widely used homotopy method, it provides better performance in terms of convergence speed. Furthermore, the
whole implementation framework of this method does not
rely on any special EDA tools.
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