Vacuum Tube Parameter Identification Using Computer Methods
In this paper we present a method for determining vacuum tube SPICE model parameters using computerized optimization. The parameters of interest in the work presented are those which affect the characteristic plate curves for vacuum tube triodes; interelectrode capacitances, heater characteristics, grid current, influences of additional grids, etc. will not be considered further.
A review of the models used in this paper
where ip is the instantaneous plate current, vG is the instantaneous grid voltage, vP is the instantaneous plate voltage, µ is the tube's amplification factor, and KG is a constant of proportionality called the perveance. (In this and all subsequent relations it is assumed that the cathode is at ground potential. In addition, the above and subsequent relations are valid only when their respective quantities between the parenthesis are positive. When this quantity is negative, no plate current will flow.)
Expanding on the 3/2 power law, Koren presented phenomenological models for vacuum tube triodes and pentodes which were shown to effectively model the behavior of tubes over a wide range of plate voltages and currents . To model triodes, Koren used the following equations:
Here µ and kG1 are essentially the amplification factor and twice the inverse of the perveance from the 3/2 power law, x is the exponent in the 3/2 power law, and kP and kVB control the nature of the "bending" of the curves near cutoff. (The (1+sgn(E1)) portion of equation (2b) forces the plate current to zero when E1 is negative.)
Koren  states his approach "takes advantage of the fact that log(1+exp(x)) approaches x when x>>1, and 0 for x<<1." Thus, at high plate currents this model behaves very much as the 3/2 power law--with the exception that the 3/2 exponent is now treated as a variable parameter. However, at low plate currents, the log(1+exp(x)) relationship results in behavior that effectively mimics the behavior of real-world tubes at low currents. In addition, these performance improvements are achieved with relatively low computational complexity and with only three additional parameters.
Parameter identification using analytic methods
Solving for µ using two data points yields:
Since the 3/2 power law isn't 100% accurate, the values resulting from the computations above will depend somewhat on the selected data points. More-or-less acceptable results typically ensue if the first point is taken at zero grid bias and relatively high plate current and the second at a fairly low grid voltage and relatively low plate current--taking care to avoid the region where the tube enters cutoff.
Unfortunately, the simple parameter identification method used above becomes unmanageable in the case of Koren's (and most others') triode models because of the large number of parameters involved--and hence the large number of equations to simultaneously solve. To find parameters for Koren's model, its inventor suggests using an iterative method starting with parameters appropriate for the 3/2 power law . Unfortunately, the tedium and uncertainty of open-ended iterations over five (or more in the case of other models) variables makes this a rather unappealing prospect.
Computer-based optimization applied to parameter identification
At its simplest, a numeric computation and visualization environment can be used to easily plot a model's plate characteristics and compare those curves against a set of reference data points. This can speed up an iterative parameter hunt considerably. However, the real power of a tool such as MATLAB lies in the ability to automatically find model parameters using pre-programmed optimizing algorithms. To this end, we have implemented an extensible set of MATLAB scripts and functions (refereed to as m-files) and have successfully used them to determine model parameters for a number of triodes. The code was designed such that wherever possible speed is maximized and specificity minimized. Thus, any of a number of user-implemented tube models or optimizing functions found in MATLAB's Optimization Toolbox can be used with very little code modification.
Within the m-files, model parameters are found by minimization of an error function. The error function can be selected to return the mean fractional error over all reference points or the RMS fractional error. Because most minimization algorithms do not constrain variables and because negative values for parameters used in the Koren model (and many others) are unacceptable, logarithmic transformation of variables was used to constrain parameters to positive values without losing the ability to use unconstrained optimizers. While this increases the calculation burden, it prevents the optimizer from converging on unacceptable final parameters.
Another characteristic of the fmins() function (and something that is typical of minimization algorithms in general) is that it will find only a local minimum of a function and not necessarily the function's absolute minimum. Thus, it is important to define the initial parameters as close to the expected final parameters as possible or to perform numerous optimizations beginning from different points.
For the optimizations presented below, initial parameter values were set as follows: µ as calculated for the 3/2 power law, kG1 to 1/(2KG) (where KG again comes from the 3/2 power law model), x to 1.5, and kP and kVB to 500. After some experimentation, it was found that reference data consisting of approximately 50 carefully selected points was, in a sense, ideal for the triodes investigated. Fewer points resulted in rising error while additional points yielded little improvement.
Given the somewhat lengthy time it takes for the MATLAB m-files to converge and the burden of preparing a data file with 50 or so points, one might feel that this method merely replaces one kind of tedium for another. While this may be a legitimate sentiment, it should be kept in mind that the method presented here represents more of a closed solution than the hunt-and-peck of manual iteration.
The performance of the Koren model using parameters determined by computerized optimization yields a worst-case mean error of 6.5% for the reference points used. In general it performs with at least a five-fold reduction in error compared to the 3/2 power law models, and improvements can be seen throughout the operational range of the tube--not just in the cutoff region as may be expected. It is interesting to note the significant divergence from the 3/2 exponent for the 5842 tube. This is likely the result of the somewhat unconventional grid construction used in this tube; however, it may also be due to inaccurate performance data reported in the databooks.
 "A SPICE Model For Vacuum Tubes," Intusoft Newsletter, (1989 Feb.).
 S. Reynolds, "Vacuum-Tube Models for PSpice Simulations," Glass Audio, vol. 5, no. 4, pp. 17-23 (1993).
 "Modeling Vacuum Tubes: Parts I and II," Intusoft Newsletter, (1994 Feb. and Mar.).
 N. Koren, "Improved Vacuum-Tube Models for SPICE Simulations," Glass Audio, vol. 8, no. 5, pp. 18-27+ (1996).
 W. Sjursen, "Improved SPICE Model For Triode Vacuum Tubes," J. Audio Eng. Soc., vol. 45, pp. 1082-1088 (1997 Dec.).
 W. M. Leach, Jr., "SPICE Models for Vacuum-Tube Amplifiers," J. Audio Eng. Soc., vol. 43, pp. 117-126 (1995 Mar.).
 J. Maillet, "Algebraic Technique For Modeling Triodes," Glass Audio, vol. 10, no. 2, pp. 2-9 (1998).
 MATLAB, The MathWorks, Inc., Natick, MA.
 Audiomatica WWW site, http://www.mclink.it/com/audiomatica/sofia/.
 currently unidentified source.
FIGURE 1: 6DJ8 plate curves using Koren's model with parameters listed in TABLE 1; grid voltage = 0 to -10 volts. (Reference data points are indicated by circles.)
FIGURE 2: 12AT7 plate curves using Koren's model with parameters listed in TABLE 1; grid voltage = 0 to -6 volts. (Reference data points are indicated by circles.)
FIGURE 3: 5842 plate curves using Koren's model with parameters listed in TABLE 1; grid voltage = 0 to -3 volts. (Reference data points are indicated by circles.)
FIGURE 4: 6DJ8 plate curves using a 3/2 power law model with parameters listed in TABLE 2; grid voltage = 0 to -10 volts. (Reference data points are indicated by circles.)
FIGURE 5: 12AT7 plate curves using the 3/2 power law model.with parameters listed in TABLE 2; grid voltage = 0 to -6 volts. (Reference data points are indicated by circles.)
FIGURE 6: 5842 plate curves using the 3/2 power law model with parameters listed in TABLE 2; grid voltage = 0 to -3 volts. (Reference data points are indicated by circles.)
r 0.05 26 Dec 2003
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