# Composite Tube Models

Copyright © 2003,2005. Author: Dmitry Nizhegorodov (dmitrynizh@hotmail.com). My other projects and articles

## 1.   Abstract

In this article we outline the difficulties of modeling real vacuum tubes and develop a practical modeling approach we call "composite tube models". Triodes are discussed most, yet the same idea works for tetrodes, pentodes, etc. We present several benefits of composite tube models and then concentrate mainly on tube linearity aspects and properties of directly heated cathodes. Mathematical foundation of composite models is presented, and the shape of definitive integrals for varying triode parameters is provided. There is apparent similarity between that and phenomenological Koren tube models, which helps to understand why Koren formulas are useful in modeling real-world plate curve families that often exhibit peculiar "tuck under" pattern in the low-mu, high plate voltage region.

## 2.   The Problems

It is a well-known fact that Child's law of 3/2 does not correctly describe real families of plate curves. The main problem is what John Harper calls "tucking under", see . It is a peculiar crowding of plate curves in the high-plate-voltage, low current region that is impossible to match with a straightforward power-of-1.5 model and sometimes difficult to match with more complex models such as Koren Improved ones. Even when a good match happens, the phenomenological nature of Koren equations makes us wondering what is the physical justification for the need for parameters Kp, Kvb, or Ex not equal to 1.5. Also, how a model which is at least as useful as Koren's can be derived from "first principles"?

## 3.   The Main Idea

The main idea behind our approach is to slice a tube, and then model he whole as collection of models each representing one of several similar but not always or necessarily identical parts.

The main technique is as follows: a tube is represented as a parallel set of smaller devices. The sum of transfer functions of all such devices is the transfer function of the tube:

F = f1 + f2 + ...

Consider a triode. A triode's transfer function is represented by electron flow, and is influenced by the distances between the cathode surface and the grid, geometry of the grid, grid-anode distance, cathode area, anode area, The anode is most often a rectangular plate, often folded as a cylindrical or rectangular profile. A cathode is a wire, cylinder, less often a plate. It is therefore a valid assumption that triode geometry extends considerably into one dimension, often in two, where electrons typically flow orthogonally to that; this can be represented as a collection, or a composite, of smaller parts having the same inter-electrode distances as the original tube. here a triode is split onto 4 parts. We pretend that boundary effects do not escalate and the electrons alone the cuts continue flowing "upwards", as if the field is not disturbed. This is a valid assumption, as slicing is performed for the sake of mathematical modelling, not physical construction. In the latter case the boundary effects would dominate if too many slices were done.

We'll be using the composites to investigate how geometry of "real" tubes affect their linearity, and vice versa, how to find a composite model for a real device that can not be described with a simple model. When triode geometry is involved, representing a complex construction as several slices can result in accurate modeling. Another area where composites appeared useful - directly heated triodes. Here is the main idea: here the geometry is assumed equal (although electrode distances can be made varying as well, but each "triodelet" takes cathode AC signal developed across the filament wire and thus amplifying it. This leads to accurate simulation of hum in DHTs (harmonic hum) and to some qualitative and quantitative techniques of DHT hum cancellation.

## 4.   Two Linear devices combined.

It is almost obvious that when two transfer functions, f1 and f2, combine, the sum is f1 + f2. It is easy to assume same happens when two linear transconductance devices, such as very idealized tubes run in parallel. Yet is is not so simple. Surprisingly, distortion creeps in - because many physical devices such as vacuum tubes have usable transfer function is limited to the 1st quadrant. Here is why.

Let's consider a very simplified tube model,

(1) Ip = K * (Vg + Vp/Mu)

and let's now assume 2 devices with non-equal K and Mu are connected in parallel: Ip = Ip1 + Ip2, or

(2) Ip = K1 * (Vg + Vp/Mu1) + K2 * (Vg + Vp/Mu2)

after some rewriting it becomes

(3) Ip = K12/(K1Mu2 + K2Mu1) * Vg + Vp)/Mu12

where K12 = K1 + K2 and Mu12 = Mu1 + Mu2

The function seems very much the same, Where is the catch? Alas, we forgot an important restriction: Ip must be positive. In other words, corrected (1) is

(4) Ip = (Vg + Vp/Mu > 0) ? K * (Vg + Vp/Mu) : 0

This changes the situation fundamentally. One slice can be conducting, while another still in cut-off! The simple algebra we used to derive (3) and (4) no longer works.

Here are pictures to illustrate that: It is now clear that the combined family of curves will distort, and that it happens because there are regimes when one element is conducting and the other one is not yet.

I've developed a special tool, a java application, that can be used to explore this in details. The tool allows to set two triode models with specific parameters and see the combined plate curve families. You can download the tool here.

Here is how two ideal triodes with Mu=5.5 and Mu=4.5 combine: the blue line is the loadline. We may have hoped that the loadline for the idealized triodes be distortion-free, but it is not so. Distortion on loadlines is change of density of grid lines - exactly that happens here.

Similar pattern happens when 3 or more slices with different mu combine, with two or more kinks visible on plate lines. when the number of slices is taken to the limit, the plate lines for large Vg are no longer straight but visibly curved.

## 5.   Exponential models

Same situation exists for more realistic exponential models, such as classical Child's law triode models with (Vg + Vp/Mu)^3/2 or using other values for the exponent. Plate lines become bent or rounded up in the low-left corner.

## 6.   Koren Models

When several slices modeled with Koren Improved formulas combine, the result is a plot that si less linera than each of the parts. Below I use a 3-element composite, each based on a 2a3 triode model. Each element represents a fraction of a 2a3 tube (set by Kg1) and has a different Mu.  This is the "tuck under" pattern. This is what often separates old and linear tubes such as 2a3 or 300b from newer yet less linear such as el34 or kt88 triode-connected. Compare these two curves (taken from audiomatica's site). The first one is 300b, the second one is el34 triode-connected:  ## 7.   Composites to model directly heated tubes

Once upon a time, contemplating filament wire with AC flowing into it, I realized that a directly heated tube can be modeled as a composite if small tubes, each having its cathode potential modulated proportionally to the potential modulation of the corresponding fragment of filament wire.

Lets's take a look at a composite with filament wire taps again: If one cathode pin is grounded or attached to the cathode bias circuitry, then all cathode segments develop varying, alternating potentials, proportional to the distance of each segment from the grounded pin. This way AC hum equal to AC main frequency develops on the plate. If the cathode-bias circuitry is balanced, then for each elementary segment there is exactly one that works "against" it. Thus, the mains frequency is cancelled. But we hear a hum that is twice of mains frequency. It dawned on me that second harmonic distortion of two segment (or slices) that are perfectly out-of-phase (180 degrees shift) will be in-phase. A picture on the right helps to visualize this effect.

This is how an approach of modeling DHTs using composites was born. Two composies, out-of-fase, form a device amplifying 2nd harmonic. Taken to an extreme, into class B area, It is a lot like rectification, full-wave detection or frequency doubling. In class A area, it is the way how small but bothersome levels of hum develop in DHT zero-feedback stages. An AC-heated DHT with balanced cathode circuitry is a device with many pairs of filament segments working as tinu differential amplifiers, dutifully generating hum on the plate.

Thus, the composite tube approach appears to be a lot of fun, and it allows to model a DHT pretty well. From a purely theoretical standpoint, a better approximation is 3-triode composite, the one in the middle receives no cathode AC, and yet better approximation is 5-segment composite, with 2nd and 4th receiving 1/2 of AC, 1st and 5th receiving full AC, and 3rd - none. Obviously, 1st and 2nd are out of phase with the 4th and 5th. When the number of slices is taken to the limit, we get real DHT. Of course, each "slice" must become wimpier and wimpier otherwise we produce a tube with unrealistically low Rp and high Ip. From a practical standpoint, a 2-tube composite is oten good enough, though.

SPICE-ing such triode composites is fairly simple, as for instance, the parameter KG1 for the Koren models must be divided by the number of "slices" in the composite, which results in curves identical to the "base" tube, no other transfer function changes needed (I tried that, I've developed java applets to match the tubes). Here is one of my 2-slice composite DHT models, where RFIL_LEFT, RFIL_RIGHT and RFIL_MIDDLE represent the filament, and K1 and K2 are filament pins:

```  .SUBCKT 2a3-composite  1 2 3 4 ; P G K1 K2
+ PARAMS:  RGI=2000
** audiomatica
+ MU=4.58 EX=1.512 KG1=1710.0 KP=40.8 KVB=1188.0 VCT=-2.24 ; Vp_MAX=600.0
Ip_MAX=0.2 Vg_step=12.0
** ??old + MU=4.2 EX=1.4 KG1=1500 KP=60 KVB=300 RGI=2000
+ CCG=7.5P  CGP=16P CCP=5.5P
* cathode resistor is 1 ohm, the pins K1 and K2 are .25 ohms from the ends of it
RFIL_LEFT    3 31 .25
RFIL_RIGHT   4 41 .25
RFIL_MIDDLE  31 41 .5
E11 32 0
VALUE={V(1,31)/KP*LOG(1+EXP(KP*(1/MU+V(2,31)/SQRT(KVB+V(1,31)*V(1,31)))))}
E12 42 0
VALUE={V(1,41)/KP*LOG(1+EXP(KP*(1/MU+V(2,41)/SQRT(KVB+V(1,41)*V(1,41)))))}
RE11 32 0 1G
RE12 42 0 1G
G11 1 31 VALUE={(PWR(V(32),EX)+PWRS(V(32),EX))/(2*KG1)}
G12 1 41 VALUE={(PWR(V(42),EX)+PWRS(V(42),EX))/(2*KG1)}
RCP1 1 3 1G
RCP2 1 4 1G
C1 2 3 {CCG} ; CATHODE-GRID
C2 2 1 {CGP} ; GRID=PLATE
C3 1 3 {CCP} ; CATHODE-PLATE
D3 5 3 DX ; FOR GRID CURRENT
D4 6 4 DX ; FOR GRID CURRENT
RG1 2 5 {RGI} ; FOR GRID CURRENT
RG2 2 6 {RGI} ; FOR GRID CURRENT
.MODEL DX D(IS=1N RS=1 CJO=10PF TT=1N)
.ENDS
*\$
```
As it can be seen, the triode now is a 4-pin device, and it does generate residual HUM 120Hz if 60Hz AC runs between K1 and K2 !! Obviously, the filament must have been added to the circuit as a resistor. To approximate the distribution of the AC potential across the filament, I used the avg(ACampl,0) which is ACampl/2 and which means the cathode pins are half-way into the cathode "resistor". Another way to say it is that RFIL_MIDDLE is twice of RFIL_RIGHT or RFIL_LEFT.

## 8.   Parallel stages as composites

In , parallel connection of triodes and "triodelts" is studied. In particular, levels of harmonic hum generated by DHTs is explained.

## 9.   Comparing two-, five-, nine-slice composites

SPICE experiments with 2-element, 5-element and 9-element composites revel that in many cases 2-element approach is "good" enough. Details coming soon.

## 10.   Mathematics

Few models of technical devices are perfectly accurate. More, even when a model reflects the basic operational principle of a device, it is very common that some parameters deviate, and various sections of a device work differently; the summed output function is thus different.

In mathematical terms, here we talk about a Riemann sum of individual transfer functions, and when the number of parts approaches infinity, an integral. here x is Mu. This integral assumes that Mu is the variable that will vary. Notice how the integral reminds certain parts of Koren Improved triode formulas, . An interactive tool for finding Koren coefficients is in .

Each symbol listed in the library has "Implementation" property referring to a Dual Composite model name from dmitry_composites.lib file. Feel free to browse this file, which is in text format. It contains some additional comments. The file provides both 2-element and 5-element composite models for every DHT. If at any time during simulation you'd like to switch to a 5-element model for a particular part, then double click on the part on your simulation window, which will open its properties editor. Find the "Implementation" property and add suffix _5 to it and close the properties window. The part now runs the 5-element model.

## 12.   References

 John Harper, Tube 201

 filament-ac-harmonic.htm Harmonic hum in directly heated tubes.

 parallel-triodes.htm SPICE experiments with parallelled tube stages.

 dht-filament-hum-cancel.htm Techniques of DHT hum cancellation in SET amps.

 http://www.normankoren.com/Audio/Tube_params.html Norman Koren tube models

 tubeparams_image.htm Coefficient finding tool

Author: Dmitry Nizhegorodov (dmitrynizh@hotmail.com). My other projects and articles