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

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 Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. DAC ’14 June 01 - 05 2014, San Francisco, CA, USA Copyright 2014 ACM 978-1-4503-2730-5/14/06$15.00. 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 x ˆ of a real interval [x] is given by x ˆ = x0 + x1 ǫ 1 + x2 ǫ 2 + · · · + xn ǫ n ,

(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 x ˆ t = x0 + x1 ǫ 1 , b−a where x0 = a+b and x = . Conversely, if we know 1 2 2 x ˆ = 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 x ˆ = 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) con˜ corresponding to the nodal tains only the variables {X} voltages that connect to the s BJTs directly. The model ˆ correis therefore reduced by deleting the variables {X} sponding to currents through the voltage sources and the

nodal voltages which are not connected with the BJTs di˜ are computed, the rectly. When later the solutions for {X} ˆ values for {X} 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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