<p> TESLA data </p>

d448a577-2c31-435d-b601-8c1597ed6ce7 14 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

TESLA Triode Project Reconciliation

3 appVersion 0.98.6179 ≡

vector of guesses X0 0.01 1000 0.001 1002 0.5 1.5 17 mat transpose :

0.01 1000 0.001 1002 0.5 1.5 17 mat

fake zero equalities factor m 0.001 :

AlgLib Solver for a system of nonlinear equations(Uni) sol k 1 β f rows range u k el β k el : for β Jac u f u Jacobian : StepMax 0 : ε 1014 -^: 12 mat u X0 StepMax ε β f β Jac al_nleqsolve : 41 line :

β 4 el Ua β 5 el^+ (1 β 5 el /^Ua β 5 el^β 4 el + β 5 el nthroot ≡

Ua b β ia β 3 el 1 β 2 el / Ua * 1 β 2 el β 1 el β 6 el b + Ua β 5 el^β 4 el + β 5 el nthroot / + (* exp + ln * (β 7 el^* :

Solver function β f 400 β ia 1.044 m * - 1200 β ia 2.679 m * - 2100 β ia 4.749 m * - 20 0.5 - β ia 0.18 m * - 120 0.5 - β ia 1.66 m * - 240 0.5 - β ia 4.3 m * - 401 - β ia 0.132 m * - 71 mat :

β sol : 0.009893 999.999938 0.000911 100.002413 1.57414 0.812437 1.56712 71 mat

<p> TESLA data </p>

a1 0010 0.227 20 0.537 30 0.796 40 1.044 50 1.279 60 1.495 70 1.698 80 1.894 90 2.085 100 2.278 110 2.476 120 2.679 130 2.889 140 3.103 150 3.324 160 3.549 170 3.779 180 4.015 190 4.225 200 4.5 210 4.749 222 mat :

b1 0010 0.088 20 0.18 30 0.277 40 0.393 50 0.5 60 0.631 70 0.776 80 0.932 90 1.1 100 1.28 110 1.46 120 1.66 130 1.86 140 2.07 150 2.28 160 2.5 170 2.71 180 2.93 190 3.15 200 3.38 210 3.6 220 3.84 230 4.07 240 4.3 252 mat :

c1 0010 0.031 20 0.063 30 0.096 40 0.132 50 0.17 60 0.216 70 0.262 80 0.324 90 0.4 100 0.5 110 0.622 120 0.764 130 0.923 140 1.098 150 1.278 160 1.466 170 1.659 180 1.857 190 2.059 200 2.265 210 2.474 220 2.687 230 2.902 240 3.119 250 3.338 260 3.558 270 3.779 282 mat :

d1 0060 0.032 70 0.064 80 0.097 90 0.133 100 0.17 110 0.211 120 0.26 130 0.321 140 0.4 150 0.5 160 0.626 170 0.773 180 0.935 190 1.105 200 1.278 210 1.449 220 1.62 230 1.8 240 1.991 250 2.2 260 2.428 270 2.668 280 2.907 290 3.132 300 3.333 262 mat :

e1 00110 0.027 120 0.055 130 0.086 140 0.12 150 0.16 160 0.206 170 0.262 180 0.327 190 0.406 200 0.5 210 0.61 220 0.736 230 0.876 240 1.028 250 1.2 260 1.362 270 1.543 280 1.734 290 1.936 300 2.15 310 2.375 320 2.611 330 2.853 340 3.1 252 mat :

f1 00160 0.036 170 0.073 180 0.112 190 0.154 200 0.2 210 0.251 220 0.309 230 0.376 240 0.456 250 0.55 260 0.661 270 0.787 280 0.928 290 1.081 300 1.25 310 1.428 320 1.614 330 1.807 340 2.003 350 2.2 360 2.396 370 2.591 375 2.687 242 mat :

i1 00310 0.016 320 0.035 330 0.061 340 0.1 350 0.15 360 0.218 370 0.3 375 0.347 92 mat :

h1 00260 0.025 270 0.053 280 0.088 290 0.132 300 0.19 310 0.263 320 0.346 330 0.444 340 0.546 350 0.65 360 0.755 370 0.859 375 0.911 142 mat :

g1 00210 0.033 220 0.067 230 0.106 240 0.149 250 0.2 260 0.26 270 0.329 280 0.408 290 0.498 300 0.6 310 0.713 320 0.837 330 0.968 340 1.11 350 1.25 360 1.4 370 1.55 375 1.62 192 mat :

a1 b1 c1 d1 e1 f1 g1 h1 i1 19 mat

a1 a1 2 a1 rows range 12 range el : b1 b1 2 b1 rows range 12 range el : c1 c1 2 c1 rows range 12 range el : d1 d1 2 d1 rows range 12 range el : e1 e1 2 e1 rows range 12 range el : f1 f1 2 f1 rows range 12 range el : g1 g1 2 g1 rows range 12 range el : h1 h1 2 h1 rows range 12 range el : i1 i1 2 i1 rows range 12 range el : 91 sys

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

e1 11 mat 110 0.027 120 0.055 130 0.086 140 0.12 150 0.16 160 0.206 170 0.262 180 0.327 190 0.406 200 0.5 210 0.61 220 0.736 230 0.876 240 1.028 250 1.2 260 1.362 270 1.543 280 1.734 290 1.936 300 2.15 310 2.375 320 2.611 330 2.853 340 3.1 242 mat 11 mat

d1 2 col d1 1 col augment 11 mat 0.032 60 0.064 70 0.097 80 0.133 90 0.17 100 0.211 110 0.26 120 0.321 130 0.4 140 0.5 150 0.626 160 0.773 170 0.935 180 1.105 190 1.278 200 1.449 210 1.62 220 1.8 230 1.991 240 2.2 250 2.428 260 2.668 270 2.907 280 3.132 290 3.333 300252 mat 11 mat

Sanity wrt Mathcad β's ==============

j 1 :

sanity

TESLA a1 b1 c1 d1 e1 f1 g1 h1 i1 19 mat : tesla TESLA j el : XY tesla 1 col tesla 2 col m * augment : data XY o 5 black plot : 41 line

Triode voltage [V] Ua Aanode voltage (x) ≡ b Grid voltage ≡ 21 sys

recast the system model Ua b β ia β 3 el 1 β 2 el / Ua * 1 β 2 el β 1 el β 6 el b + Ua β 5 el^β 4 el + β 5 el nthroot / + (* exp + ln * (β 7 el^* :

Mathcad Given/Find β 0.009728 783 0.001113 475.8 2 0.6425 1.49 17 mat transpose :

β 0.009893 999.999938 0.000911 100.002413 1.57414 0.812437 1.56712 71 mat :

About this project . . .

This Smath document exemplifies the non-linear solver al_nleqsolve.

As given,from the guesses X0, it preserves the 1rst TESLLA [a1],

compromised wrt to other TESLA [b1, c1, ...]. The data sets do not

represent a clean family of functions. Tentatively, one may want to

reconstruct the TESLA family from the example below. To ease

explore the family, we have isolated β from sol that you can disable.

]]>

Solver function x β nleq x 0 β ia x 0.5 - β ia x 1 - β ia x 1.5 - β ia x 2 - β ia x 2.5 - β ia x 3 - β ia x 3.5 - β ia x 4 - β ia 91 mat :

x β nleq 1 el x β nleq 2 el x β nleq 3 el x β nleq 4 el x β nleq 5 el x β nleq 6 el x β nleq 7 el x β nleq 8 el x β nleq 9 el data 10 1 sys

160 β nleq 4 el 0.000786

slope s 0.05 x * :

shift λ 1 :

'n 'curvature n 1 :

x n f s 1 x λ - (exp + ln n^(* :

j 4 :

sanity

m 0.001 :

x 1 f x 1 - ( n f x 2 - ( n f x 3 - ( n f x 4 - ( n f x 5 - ( n f 6 1 sys

TESLA a1 b1 c1 d1 e1 f1 g1 h1 i1 19 mat : tesla TESLA j el : XY tesla 1 col tesla 2 col m * augment : data XY o 5 black plot : 41 line

's*x' incline
'λ'      shift
'n'      curvature s 0.0000000185 x * : λ 60 : n 1.17 : 31 sys

x n f s 1 x λ - (exp + ln n^(* :

data x n f 2 1 sys

400 β ia 1.044 m * - 1200 β ia 2.679 m * - 2100 β ia 4.749 m * - 20 0.5 - β ia 0.18 m * - 120 0.5 - β ia 1.66 m * - 240 0.5 - β ia 4.3 m * - 401 - β ia 0.132 m * - 71 mat

The nonlinear system 7 equalities 7 parameters

size 4 :

data a1 o size black plot b1 o size black plot c1 o size black plot d1 o size black plot e1 o size black plot f1 o size black plot g1 o size black plot h1 o size black plot i1 o size black plot 91 sys :

a1 40 1.044 12 mat 120 2.679 12 mat 210 4.749 12 mat 31 sys

nleq 40 1.044 . 14 red 15 mat 120 2.679 . 14 red 15 mat 210 4.749 . 14 red 15 mat 20 0.18 . 14 red 15 mat 120 1.66 . 14 red 15 mat 240 4.3 . 14 red 15 mat 40 0.132 . 14 red 15 mat 71 sys :

b1 20 0.18 12 mat 120 1.66 12 mat 240 4.3 12 mat 31 sys

c1 40 0.132 12 mat 11 sys

data nleq 2 1 sys