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Unformatted text preview: workshop wrz "Nnnlinnnr CAD and Modeling" Friday. June 12, 1987
1987 IEEE NIT—S International Microwave Symposium, Las Vegas, NV 5/9—5/12/87 [)1 the Basic Algorithms of SPICE with Application to Microwave Circuit Simulation
by Stephen E. Sussman—Fort
Dept. of Electrical Engineering
State University of New York
Stony Brook, NY 11794—2350 Abstract: The famous circuit simulator, SPICE, has been in use for many years, and recently
there has been some discussion over the program's applicability to microwave networks. This paper
presents a tutorial exposition of the basic algorithms used in SPICE to analyze nonlinear networks
in the time domain. Miile this topic is of interest in its own right, an important conclusion is
that SPICE's accuracy is dependent much more on the Validity of its physical device models, rather
than on the numerical methods used within the program. we illustrate how SPICE 28.6 was expanded by us to include microwave regime models for the GaAs FET, Schottky diode, and the lossy
transmission line. I . INTRODUCTION SPICE performs the transient analysis of networks consisting of linear and nonlinear dynamic
elements. It also performs DC and frequency domain analysis of such circuits. The DC problem may be nonlinear, whereas frequency domain analysis is, by definition, small—signal within SPICE and
hence is a linear operation. Circuits may include elements such as (1) R, L, C, M, T, nonlinear R, and transmission lines:
(2) device models for transistors and diodes which include nonlinear capacitance, inductance, and
resistance and temperature effects: (3) both "ﬁlinear and linear controlled sources; and (4) independent sources. New models are constantly being added to SPICE to handle new devices and
circuit applications. we shall discuss how SPICE goes about solving for a set of voltage time functions in a general
nonlinear dynamic network. Nonlinear and linear DC problems and linear frequency domain analysis
will not be discussed explicitly since these procedures are simpler and are, in some sense, a
subset of the more general nonlinear dynamic case. 1
II. Mathematical Description of the Problem and Conventional Approachesto Solution
A nonlinear dynamic network is described by a set of nonlinear differential equations in terms
of a set of unknowns X = [X11 X2, NH Xn] “ which may be a set of voltages and/or currents. The equations can often be put into normal form
as i = f(x,t)
or can be written more generally in the implicit form
f(i,x,t) = 0.
For general nonlinear networks, it is difficult to obtain the normal form, so the implicit form is usually preferred. Note that for linear networks, the normal form is the same the familiar linear
state equations. In terms of a network formulation, one way of performing transient analysis is to construct the implicit set of nonlinear differential equations from the circuit and to solve this system as
follows. 1. Start with the branch definitions of the elements. 2. Apply Kirchhoff's voltage and current laws (KUL and KCL) systematically throughout the
circuit so as to formulate the equations f(i,x,t)=0. 3. Apply a solution algorithm which converts the nonlinear differential equations to set of nonlinear algebraic equations (e.g. the backward difference formula which approximates
x(t ) in terms of x and k past values x , x n—k+1)' ‘ q 'n+1 n+1 n n1
a. Apply a solution algorithm which solves the nonlinear algebraic equation for x the
solution vector at the time point t (8.9. The Newton—Raphson algorithm). ’ 000' X n+1’
n+1 5. Repeat for all time points of interest. This method is a general and modern way of handling the network problem, and has become more
popular since (1) the advent of the tableau approach to network analysis, which gives us an
orderly procedure for developing the implicit set of nonlinear differential equations; and (2) the
development of efficient solution methods for solving these equations. Nevertheless, the method is complicated and does have its shortcomings — e.g. it is difficult
to relate lack of convergence to a property of the circuit. It has advantages as well, most
notably the ease with which it is possible to automatically adjust the step size or time increment
for analysis. then SPICE was first written, however, this method had not been fully developed, so
a different tack was taken at that time which is discussed below. III. SPICE's Approach to the Transient Analysis Problem The equation formulation and solution method is the same as that of Sec. 11 except for one
profound difference: the order of the steps is changed slightly. '"1? We start with the branch definitions. .25 The solution algorithm is applied to the branches of the network. This converts the nonlinear differential equation of the branch to a linear relation at a given step in the
ii, iteration for solving for the branch quantities at a specific time point. 3, Apply KVL and KCL systematically throughout the network to obtain node voltages, for example, again at a given step in the iteration for solving for these quantities at a
specific time point. 9. Repeat over all time points of interest.
a... Note that the basic difference is to initially apply the solution algorithm to each circuit
element individually rather than to the entire network at once. KVL and KCL are then used to
effectively apply the algorithm to the entire network. ,This.approach,hasi been shown to be identical to that of Sec. II, and leads to the concept of l"companionyvmodeling" or "associated
discrete models" for the elements. 7 “ SPICE has successfully implemented this approach in its transient analysis algorithm. we now
illustrate more concretely how a network of nonlinear RLC elements can be analyzed in the time domain with this method. (Please refer to the viewgraphs and explanations presented during the
actual talk for details.) IV. Conclusion The algorithms of SPICE are seen to depend on the methodology for constructing and solving a set
nonlinear differential equations, and not explicitly upon frequency ranges or type of components.
what this means is that as long as SPICE's models truly reflect the correct physical behavior of the device, the results of the program should be accurate. This places an enormous responsibility upon the user to make sure that the models in the program are appropriate for the application and,
in particular, the frequency range of interest. SPICE has successfully been expanded to include microwave components. For example, our own work
has enhanced SPICE 28.6 to include models for the GaAs MESFET, the Schottky diode, and the lossy
transmission line with skin~effect. nt least one commercially available version of SPICE
(Microwave SPICE) has included these and other microwave circuit elements, which now allows the
program to successfully model linear and nonlinear networks in the microwave regime. SPICE may not be the most efficient or fastest method of analyzing nonlinear networks in the
time domain, especially uhen the nonlinearity of the circuit is restricted to a limited number of
elements. Certain classes of networks may be analyzed more quickly by such novel approaches as
Harmonic—Newton, which is discussed by other speakers of this session. With circuits suoh as NUS
digital networks, it is also possible to trade off the detail of SPICE's transient analysis with
processing speed by employing relaxation based methods. we expect, however, that SPICE will remain, for the indefinite future, a reliable servant to designers at least for cirtuits or
subcircuits of manageable size. 1. 3. 4. 5C 8. Bibliography L. D. Chua and P. M. Lin, Comguter—Aided Analysis 9E Electronic Circuits: Algorithms and
Comgutational Technigues, Prentice~Hall, NJ, 1975. J. Vlach and K. Singhal, Comguter Methods f9; Analysis and Design, Van Nostrand Reinhold,
NY, 1983. D. A. Calahan, ComEuter—Aided Network Design, McGraw—Hill, 1972. Microwave SPICE User's Guide, EEsof, Inc. westlake Village, CA. S. E. SussmanFort, F. L. Huang and J. C. Hantgan, "A SPICE Model for Enhancement and Depletion Model GaAs FETs," IEEE Trans. on Microwave Theory and Technigues, vol. MIT34,
pp. 1115—1119, NOV. 1985. S. E. Sussman—Fort and J., Hantgan, "SPICE Implementation of Lossy Transmission Line and
Schottky Diode Models," IEEE Trans. on Microwave Theory and Technigues, submitted. 0n the Basic algorithms of SPICE
with Application to Microwave Circuit Simulation Smmm[.&$mnhn
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1mm min/orb at eacL Jumped B CONCLUSIONS SPICE performs transient analgsis of nonlinear dgnamic networks by solving the corresponding nonlinear dill.
eqns. via the technique of discretization and lineari nation (companion modeling). The technique is applicable to hNY lreqoencg range as
long as the models are PHYSlCﬁLLY hCCURhTE. Other algorithms provide greater speed with possible
loss of detail and/or generalitg of allowable non
linearities. a; 1* NewtonRaphson Method
Solve equation y = F(x) = O for x:
i 1. Plot y=F(x) r 2. Select starting guess for solution, x=xO l ' 3. Draw a line tangent to y=F(x) at the point
h Y0=F(Xo)
r ‘ i 4. The point where the tangent line crosses the xaxis, x=x1 should be an
improvement over the initial guess Repeat the cycle until convergence Analytically: write equation of tangent
line to curve F(x)=0 using the point—slope
formula: (Y‘Yo) / (X'Xo) = F’(Xo) where F’(x0) is the slope of the tangent
line at x=x0. Set y=0, solve for x=x1, and
repeat cycle. ; . m ; x1= x0 — F(x0) l F’(x0) derived from Taylor Series NewtonRaphson méiibh F(x) as a Taylor series about x0 and truncate the senes
tive term and remainder term as follows: F(x) = F (xo) + F’(xo)(x  xo) + \ Let x be a root and x; the ﬁrst approximation Foo = Fm) + mo) . (x1 x0) + ”90;! " ”‘9’ od approximation to I: so that F(x1) is nearly zero Let us assume 3:; is a go
we have , and the remainder is also very small. Now, replacing F(x1) by zero,
‘49 o = Foo) + F'(xo)(x1— x»)
m Solving for x1, we have ;WM
9
Q which we recall as the NewtonRaphsom: obtained from a different
. nongeometrical approach. ...
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