The mathematical models of electronic tube and how to use them
Part 1
Why this article has appeared?
Last years, looking through the periodic literature, both Russian and English, I paid attention to set of articles devoted to the tube sound amplifiers. The range of quality of the publications is incredibly wide: from rather interesting articles of the theoretical plan (unfortunately, basically English), articles with the detailed description of designs and explanations of features and work, typical for the notable periodic editions, up to articles with the completely improbable recommendations reminding a sorcery of the shaman.
Designing, more than 20 years, the electronic devices, I had related to devices having output stages on electronic tubes, and these devices have left at me the most pleasant impressions (especially in reliability). Naturally, I had professional interest to this question, and I have wanted to study a modern condition of this question in more detail. The basic sources of the information were the periodic editions and WEB. Below, for simplicity, a word "publication"will mean as the printed editions, so and WEB.
The result had manysided and a little bit unexpected. First, I got an opinion, that the situation in this question, completely not wonderful ( especially for the Russian readers); second, I would like slightly to correct this situation; third, I have caught "a tube" infection, truth, in the easy form. Having the powerful computer and immunity of longterm work in this area, I did not begin to be snap at once for a soldering tool, though there was such desire, but old reserves has checked up, and I found a great many of tubes. It does reinforce illness.
What situation in affairs?
I agree with the remarks of many authors, that during golden age of tubes, the level close to a limit of their opportunities was achieved. But during total neglect by them, essentially have changed as technology, so and opportunity of the designers. Therefore I do not see the contradictions in joining up of the best achievements of the past years with opportunities of the present time.
To my regret, there is not enough information on this question.
Still I was surprised, with one law in distribution of attention of the authors to different parts of the circuit. Let's present the amplifier as a set of devices: active devices (electronic tubes), passive elements (resistors and capacitors), electromagnetic devices (transformers) and constructive elements (chassis and conductors).
Then such localization of attention is approximately as follous:
 The first place  the types of used tubes (stable list of preferable types).
 The second place  the output transformers (a lot of the recommendations).
 The third place  the passive components (recommendation frequently inconsistent).
 The fourth place  the constructive elements (there's no words for comments).
 The fifth place  the operation mode of a tubes (I feel a shock).
I do not deny, importance of influence of a passive components and assembling on thin sound structure, but who all the same brings in the basic portion of distortions?
At once I shall try to answer: it is amplification elements (tube) and the transformers, therefore a distortion factor from a chosen operation mode of amplification stages will be dependent. Do not forget, that regardless of the fact , the used electronic tube perhaps is very linear, the wrong operation mode will bring to nothing all of these advantages. The transformers are the special question. In this article we shall not discuss this theme. Though I hope to discuss it further.
We begin designing
I want at once to determine concept "designing". For readers, experienced in this question, it is really development of the new device.
But, for beginner, even the reiteration of the ready circuit actuallyit is process of designing, only in narrower frameworks.
Always there is deviation from the original, there is no recommended tube or the accessible output transformer differs in parameters, the static modes are not specified  actually you project the new device.
And the main result for the beginner must be completely understanding of processes occurring in the device.
If you, at the end of your work can explain, how it works, the first step to professionalism is made.
So, the circuit is approximately determined, the tubes are chosen, to begin with it is necessary to specify the static operation mode for tubes and to estimate the harmonic distortion.
Most frequently for calculation is being used a graphic method (The application of this method is shown in WEB Steve's Tube Pages. described in details in [1]).
The graphic method differs by simplicity and sufficient accuracy, but the plate characteristic presence is required.
Having looked through the handbook and having looked on the given diagrams, at me the doubts in their accuracy have appeared. For elimination of doubts I have decided to construct the diagrams myself. I have made it, using for measurement 57 tubes for taking plate characteristics and averaged all results.
Comparing the results with the handbook, I with the large satisfaction have seen, that the time is lost not in vain. It is high time to begin calculations, but the word "simplicity" in definition of a method at all does not mean easy, especially if you want to estimate the harmonic distortion by a method of five ordinates, and definition " sufficient accuracy " in this case  obvious overstatement. And I would like to find optimum for the operation mode of the tube in my circuit. Therefore, forthcoming work did not gladden me.
ВыражениеExpression " Laziness  the engine of progress " has deep psychological base, therefore at once I began to search the ways for facilitating the work. So, it is necessary to put chores on the computer, but it is necessary to have functional dependences determining the current flow in a tube from the applied voltage to electrodes.
Strictly speaking, these functional dependences are electronic model. Fortunately, this successful idea crossed the mind not to me the first, and there are some variants of electronic models. The models of electronic tubes for the SPISE and its derivatives are widely distributed. But, at once to use SPISE on the first phase it is not expedient action there is no obviousness between the chosen an operation mode and parameters of a tube.
The second part of article I completely shall dedicate to questions of use SPISE (SPISE  my love, last ten years with it I am inseparable).
Some history
The readers familiar with the kernel of the problem, this section can at once let slip. Here I want briefly to describe the electronic tubes models used.
Until recently models of electronic lamps were realized on the basis of the fundamental law of a 3/2, based on direct implementations of the idealized "3/2 power law" for current flow in an ideal electronic tube from the applied voltage to electrodes [2].
The plate characteristic for the diode is described by expression:
,
where Ip  is the plate current, А constructive factor, Up
 is the platetocathode potential.
For the triode the behavior of a plate current from applied voltage on electrodes is described by expression:
,
where Ip  is the plate current, К constructive factor,
Up  is the platetocathode potential. Ug – Ug  is the gridtocathode potential, µ
 is the amplifier factor.
For a pentode, I shall bring the model offered by Scott Reynolds [3]:
,
where Ip  is the plate current, Up  is the platetocathode potential, Ug
 is the gridtocathode potential, Ug2  is the screentocathode voltage, µ  is the amplifier factor,
K_{G1}  can be adjusted so that curves for weakly negative grid voltages match experimental data, K_{VB}  is correlated with the "knee" of plate curves.
These expressions with sufficient accuracy describe behaviors of a tube at average voltage on a control grid, however at the large positive or negative voltage on a control grid the accuracy is decreased. Partially to correct this lack possible, having broken all range of a grid voltage on separated regions. Acquaintance with this approach, possible in articles currently available at Intusoft site.
The application of these models quite is warrantable for an estimation of operational integrity in the SPISE. As in real devices of a grid voltage, in working modes, have the average magnitude. A plenty of tube models are submitted on a perfect site 
Duncan's Amp Pages.
Rather recently, Koren offered 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 [4].
For the triode, the dependence of a plate current from applied voltage on electrodes is described by expression:
,
,
The pentode models is based on the following expression:
,
,
,
where Ip  is the plate current, Ig2  is the screen current, Up  is the platetocathode potential,
Ug  is the gridtocathode potential, Ug2  is the screentocathode voltage,
µ is the amplifier factor. Coefficients X, K_{G1}, K_{G2}, Kp, K_{VB},
V_{CT}  allows fitting the equations, shown above, for a specific type of a tube and the calculations require. The equations determining a current of the plate of a tube, for convenience of record are broken in two parts. The details of the given model are provided in Norman Koren's works currently available at site
Norman L. Koren. The SPISE model may be generated independently with use of a TuParam.m Tuparam.zip program. The SPISE model library, based on the Koren's equation, currently available at the address Tubemods.zip or the extended libraries at the address NexTube. Also, Konar's article Vacuum
Tube Parameter Identification Using Computer Methods, is dedicated to this question, where a method for identifying vacuum tube SPICE model parameters using computerized optimization is presented and some models of electronic tubes are given.
The Russian variant Konar's article currently available at
NexTube.
The shown above, equations do not simulate a tube heater. For completeness of a picture I shall tell, that there are models of electronic tubes, in which the model of a heater is included.
The equations describing behavior of an electronic tube, I have shown for those readers, who want independently to try identifying vacuum tube SPICE model parameters.
At the first phase of work I also used this program. The program provides satisfactory accuracy of definition of coefficients and does not require deep knowledge MatLab (is enough launch a MatLab management window). Certainly, it is necessary to have MatLab. The student version is relatively inexpensive and is widely distributed, and I hope, that it will not be by a serious obstacle for work. All further calculations will be carrying out also in environment MatLab v. 5.2.
Below, in figure 1, the example of identifying of coefficients for a tube 6H8C (6SN7GT) with using the Tuparam program is shown.
I want, at once to warn the future users Tuparam, that the careful selection of the input data and picking up of the initial conditions is necessary for fast convergence of iterative procedure and obtaining of satisfactory results. Though the previous phrase sounds terribly, it is not necessary to be frightened. A bit later, you will understand, what it is necessary to do with the initial conditions and what points in the experimental data should be excluded.
As you can see, accuracy of modeling is high enough. On the diagram by circles are marked experimental data and continuous lines  result of modeling of the plate characteristics by the equations shown above.
The numerical values of factors are shown directly on a field of the diagram.
fig. 1.
We continue work
Now you have the model of the electronic tube, necessary for you. Its usage is possible in different ways. It is possible, at once modeling the circuit of the stage in SPISE and to look at result, but as I already spoke earlier, I consider it not the best variant, especially if the desired operation mode are not determined. It is possible to modeling in SPISE, the necessary characteristics of a tube, to print out the received curves and use of a graphic method.
I propose, the third variant  to use special program allowing quickly estimate change of a tube parameters within the limits of permissible modes, to choose some load lines, assigning a bias voltage and for each variant to calculate of parameters of the stage (the gain, the output voltage, the output power and the harmonics distortion). The process of calculations of one realization take some minutes, you can quickly try set of variants and choose best.
The program is designed as a MatLab function ( MatLab v. 5.2 or above, "student's version") to analyze an operation mode of a vacuum electronic tubes (Triode or Pentode) performing in a class A singleend stage. The requirement to knowledge MatLab is minimal, namely: launch a MatLab management window and specify a way to the program files and the model files.
As a basis for algorithm, the mentioned above, graphic method was used. Actually, the first variant of the program SeCalc.m is its machine interpretation on the basis of electronic tube model. : "Appetite comes during a meal", and the following version SEcalcS.m is expanded by means of the spectral analysis of the stage output voltage.
Program limitations
 The Program runs Norman Koren's Advanced Electronic Tube Models only (for the time being).
 An accuracy of tube operation mode defining depends on a quality of a Model in use.
 The frequency properties of the analyzed tube are not considered.
 A load impedance is assumed being active.
 For the spectrum analysis, a test signal frequency is equal to 1kHz; when displaying the calculated results, the zero and first harmonics are shown as suppressed.
 The Program is inapplicable for operation modes with control grid voltages greater than zero: though the mode parameters calculation is still available, the spectrum analysis is prohibited.
 The program is not a substitute to SPISE and, I hope, will forestall its use.
 For pentodes in a triode mode, a separate model should be employed.
If you'll detect any faults in the Program, please inform me.
As you, already have understood, I am not the professional programmer, faster curious user, therefore the program interface is strongly limited. It provides access to the basic functions of the program at a level that is not causing strongly irritations.
I strongly recommend, for increasing calculation accuracy of pentodes and beam tetrodes, to find the characteristics of a tube in a mode, which you plan for their work, and to create separate model. Especially, at the triode mode for these types of tubes.
Definition an operation mode of a tube
Before using this program, it is necessary to prepare the input data. For better convenience, input information provided as an input file. This input file contains three blocks, (i.e. service information, tube model parameters, tube and modeling mode data). The details of the input file are provided in the help file
HelpSE.
Name='6N8C'; Type='Triode';
% Model data
MU=22.87; EX=1.516; KG1=2209.8; KP=167.87;
KVB=155.4; VCT=0.70;
% Data analysis
Ipmax=.02; Ppmax=2.7; Upmax=300; Ugmax=10; Ugstep=0.5;
Start the MatLab through the menu startup and execute the program SEcalcS.m. The Program will be started, and a window with the list of Model files will appear. Select the input file 6N8C.mod.
Next, the window with two diagrams will appear. The top diagram shows a dependency of tube impedance versus tube current, the bottom one shows a dependency of a tube transconductance versus tube current (figure 2). These diagrams are built for a test load line, which are built on two points  firs Up=0, Ip=Ipmax, second Up=Upmax, Ip=0. Probably, it is not the best choice, but I have not thought up anything better. Let's specify a task: the tube will be used in the driver stage, and overall objective  achieving the maximal output voltage.
fig. 2
After the calculation is over, a window displaying a tube plate characteristics and red line of the tube maximum power will appear (figure 3).
fig 3
Now, let's build the load line. There are two opportunities to do it: first  setting the load resistance and the power supply voltage, second  to specify, on the diagram, two points, through which she should pass. Let's try the first variant.
In the literature [1], [5] frequently are recommended to choose a load resistance for the triode equal 3÷
5R_{i}, attentively having looked on figure 2, we shall determine a desirable the quiescent current 8÷9 mA. From here, having estimated R_{i} of a tube approximately equal 8000 Ohm, we shall accept the load resistance 3R_{i}=24000 Ом and the power supply voltage 400V. Having set these values, we shall see a load line (dark blue line on figure 3).
Let's continue calculations and a window displaying dynamic tube plategrid characteristics will appear (dark blue line on figure 4).
рис 4
The building of a dynamic characteristic in the plategrid system of coordinates is made for two reasons: first, is very evidently visible linearity of a tube, secondly, conveniently to choose a bias voltage (it is easy to compare assigning a bias voltage to the chosen zone of a dynamic characteristic). Out of figure 4 it is visible, that the tube is rather linear, therefore to obtain the maximal output of a voltage on an output, we shall choose bias voltage 5V (is marked by a red cross on the characteristic) and we shall look at results.
Tube 6N8C 31Dec2001
HD=1.92% HD2=1.89% HD3=0.30% HD4=0.006%
Pout=0.14W Uout=166.5V G=16.7 Rload=24000.0 Ohm
Ug0=4.97V I0=0.0085A
Totally not bad! Now we shall make more exact the results for harmonic distortions. For exception of a grid current we shall reduce amplitude of a voltage on a grid up to 4.2V and shall start the analysis. The numerical value of calculations is given below, and the spectrum of an output voltage is shown in figure 5.
Calculation spectrum:
Magnitude Ug=4.2V
Pout=0.10W Uout=135.9V peaktopeak
HD=2.35% HD2=2.34% HD3=0.17% HD4=0.053% HD5=0.042%
Totally not bad! Now we shall make more exact the results for harmonic distortions. For exception of a grid current we shall reduce amplitude of a voltage on a grid up to 4.2V and shall start the analysis. The numerical value of calculations is given below, and the spectrum of an output voltage is shown in figure 5.
Requirements are inconsistent, it is possible to try the increase of a load resistance, and accordingly increase the supply voltage, but let's try other way.
fig. 5
Look at figure 2. The parameters of a tube rather strongly depend on a flowing current, on the other hand, disposition of the plate characteristics has high equability (figure 3) at the large tube current. From here, decision directly arises: we use a current source as plate loading.
Build a new load line by use the " New R load " button and reference a load line on the diagram, by two points (green line on figure 3, the given points are marked by red crosses) and see into the result.
New set
HD=0.57% HD2=0.56% HD3=0.10% HD4=0.030%
Pout=0.01W Uout=221.7V G=22.0 Rload=462783.6Ohm
Ug0=5.03V I0=0.0084A
Results are very hopeful. Let's specify results, using the spectral analysis.
The numerical values are given below, and the spectrum of an output voltage is shown in figure 6.
Calculation spectrum:
Magnitude Ug=4.2V
Pout=0.01W Uout=183.5V peaktopeak
HD=0.66% HD2=0.65% HD3=0.05% HD4=0.053% HD5=0.042%
Certainly, the use a current source, essentially will improve parameters of the stage, but also will require loss of simplicity of the circuit. Beside a current source also the cathode follower is required in order to take these advantages.
I use in this example two extreme decisions, in order to show opportunities of improvement of the stage parameters, clearness and speed of realization of calculations.
fig. 6
One more remark: for the spectral analysis the fast Fourier transform algorithm is used. Therefore it is necessary to remember, that the level of a digital noise depends on used amount of a points. By default 1024 points are chosen and the digital noise may mislead at small levels of harmonics in an output signal.
Amount of points possibly to increase, but it will result in strong delay of calculations.
Additional results
While working with the program, the opportunity of a fast estimation of expediency of use of a tube, for sound applications (its linearity) in different modes was found out.
Especially, at of a triode mode for pentodes and beam tetrodes. Therefore. I shall investigate tubes, accessible to me, and to represent results on a site NexTube. Using these results allows you make the conclusions yourselves.
There is one more opportunity, (is included in the program SEcalcSi.m) to write an output signal of the stage, at the spectral analysis, in a file and use him as a test signal for the spectral analysis for other of a tube type. It allows easily select types of tubes and an operation modes for complementary inverse predistortion and total distortions decreasing.
Instead of the conclusion
In this part of article I have proposed to you another opportunity of use of model of an electronic tube for designing lamp amplifiers. I hope, that the program described above and library of models available at NexTube will help the beginning developers and will saved the specialists a little time.
The further plans:
The similar program for the analysis of operations mode of electronic tubes in PPstages are prepared for the publication.
The second part of article dedicated to use of tubes models in the SPISE (an example of designing) will be written.
WEB resources
I want to draw attention the readers to the following resources:
GlassWare Audio  An excellent site promoting the vacuum tube based amplifiers. Here you can find some basics of vacuum tube scheming theory, and a considerable list of old and very interesting books describing how to develop the HiEnd amplifiers.
basics of vacuum tubes  This is very nice library of concise articles explaining how the vacuum tubes actually operate. It also contains lots of interesting ideas and circuit diagrams.
Tube Pages  Here you can find the Data Sheets of electronic vacuum tubes manufactured in Europe.
Oldradio  This is the site to be visited first of all when searching for Data Sheets (in Russian) of "Made in USSR" vacuum tubes.
Audiomatica Sofia web page  This site contains useful data and samples of output characteristics of many popular electronic vacuum tubes.
Vestnik ARA (News from Association of Russian Audiophils)  This is an excellent site (in Russian) promoting the vacuum tube based amplifiers designs.
Andrea Ciuffoli  This site contains lots of links and many interesting HiEnd amplifiers circuit diagrams. It also provides lots of simulation models for electronic tubes.
References
[1] G. Tsikin, Electronic amplifiers, " Svyaz ", Moscow, 1965.
[2] N. Parol, et al, Receivingamplifier tube, " Soviet radio", 1966.
[3] S. Reynolds, “VacuumTube Models for PSpice Simulations,” Glass Audio, vol. 5, no. 4, 1993.
[4] N. Koren, "Improved VacuumTube Models for SPICE Simulations," Glass Audio, vol. 8, no. 5, (1996).
[5] E. Fedoseeva, Amplifiers devices, "Art", Moscow, 1961.
