Biuletyn Instytutu Spawalnictwa No. 01/2012

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Combined models of arc with constant column length The series connection of the Cassie-Berger and Mayr-Kulakov models makes it possible to obtain the Habedank model, where substitute conductance fulfils the dependence (22) (23) If one now implements the simplified Mayr-Kulakov model taking into consideration the virtual static characteristics of the arc component U Mstat (i), instead of (21) one receives and after reduction 1 1 1 Similarly, instead of (23) the resistance = + (16) (16) g g form of the model will be M g C 1 r C 1 r M drC dt dr dt M 1 = θ C ⎡ u ⎢1 − ⎢⎣ U 2 2 C ⎛ ⎜ ⎝ rC r 2 ⎞ ⎟ ⎠ 2 1 ⎡ u r = ⎢ − M 1 θ ⎣ rP r M M ⎤ ⎥ ⎥⎦ ⎤ ⎥ ⎦ 1 g M 1 g M dg dt M dg dt M 2 1 ⎡ u g g ⎤ = ⎢ −1⎥ θ Ms ⎣i ⋅U Mstat () i g M ⎦ 1 ⎡ u g ⎤ = ⎢ −1⎥ θ ⎣ U () i g ⎦ Ms Mstat M (24) (25) 2 and resistance is 1 dr ⎡ ⎤ M 1 u rM = ⎢1 − ⎥ (26) rM dt θ Ms r = r M +r C (17) ⎣ r ⋅i ⋅U Mstat () i r ⎦ As the same current flows through both elements and after taking into consideration that 1 dr ⎡ ⎤ and after reduction M 1 u r = ⎢ − M 1 ⎥ (27) g C = i / u C , g M = i / u M and g = i / u it can be r ⎣ () r M dt θ Ms U Mstat i ⎦ stated that where g rC uC = u = u (18) U Mstat (i) (18) = U stat (i) - U 0 sign(i), U C = f(U 0 ). gC r The voltage of U 0 corresponds to ranges of and strong arc currents. g r The Habedank model (20)-(23) is sometimes used to simulate commutation processes M uM = u = u (19) (19) g M r in electric circuits with high voltage electrical Then on the basis of (18) and (19), one can devices; its expansion being the series connection of as many as three models (1 – Cas- express the Habedank model in the conductance form: sie-Berger, 2 – Mayr-Kulakov) [7]. Known as 2 ⎡ 2 1 dg 1 ⎛ ⎞ ⎤ KEMA, the model was even implemented as a C u g = ⎢ ⎜ ⎟ −1⎥ (20) (20) 2 gC dt θC ⎣⎢ U C ⎝ gC ⎠ ⎥⎦ blackbox in simulation programmes [8, 9]. In the TWV hybrid arc model [2], the values of (21) currents flowing through two parallel 2 1 dg 1 ⎡ ⎤ M u g g = ⎢ −1⎥ (21) g M dt θ M ⎣ PM g M ⎦ nonlinear conductances, corresponding to the The resistance form of the formulas is as follows: Mayr-Kulakov and Cassie-Berger models, depend on their resultant value and therefore can (22) be presented as follows: 2 ⎛ 2 ⎛ i ⎞ ⎛ () (23) ⎟ ⎞ ⎜ i ⎞ i t = u ⎜ ⎟ + ⋅ ⋅ − ⎜ − ⎟ kol g = ukol ⋅ gM exp − u 1 exp 2 kol gC 2 ⎝ I0 ⎠ ⎝ ⎝ I0 ⎠⎠ (28) Hence one receives g ⎛ i − 2 ⎞ ⎞⎞ () t = g () () ⎜ ⎜ ⎟ + ⋅ − ⎜ ⎟ ⎟ M t ⋅ exp g 2 C t 1 exp 2 ⎝ I0 ⎠ ⎝ ⎝ I0 ⎠⎠ ⎛ ⎛ i − 2 (29) 18 BIULETYN INSTYTUTU SPAWALNICTWA NR 01/2012
The conditions of the selection of models are as follows: - Mayr-Kulakov model In practical considerations [2] one usually assumes that ϴ (|i|) = const and G min = 0. Then, formula (32) can be simplified to g 2 i dg 1 dg 1 ⎧ u ⎫ koli ukoli ≈ θ (30) (30) = ⎨[ 1− ε () i ] + ε () i −1 2 ⎬ P dt g dt θ ⎩ gU C PM ⎭ M M () t g M () t = − M , if i I0 In welding [10, 11] and furnace [2] arcs, the value of limiting current I0 is approximately 5 A. In the case of the application of the TWV model for the approximation of the characteristics of high-pressure arc lamps, the value of I0 is almost 10 times lower [12]. The hybrid model of the arc column in the conductance form is [2] g kol ⎡ ⎛ i ⎞⎤ u i ⎡ ⎛ i ⎞⎤ i dgkol ⎢1 ⎜ − ⎢ ⎥ I ⎟⎥ U ⎜ − C ⎣ I ⎟ (32) 2 2 2 ⎣ ⎝ 0 ⎠⎦ ⎝ 0 ⎠⎦ PM dt 2 2 2 () t = kol G + − exp⎜ ⎟ + exp⎜ ⎟ −θ ( i ) min (32) where G min – constant conductance dependent on the distance between electrodes, their shape and arrangement as well as on the gas and the temperature of the environment in currentless moments; I 0 - transition current between the Cassie-Berger and Mayr-Kulakov models. In a general case, the suppression function ϴ depends on current i If the current is relatively low, one can assume that ϴ≈ϴ 1 , and when current is high one can assume that ϴ≈ϴ 0 . If ϴ→0, the static characteristic results from adopted assumptions related to the participation of individual constituent models: • if │i│ I 0 and dg/dt = 0, then u = U C sign(i). The creation of the resistance form of the hybrid model, analogous to the conductance TWV, is difficult for computer recording and implementation. For this reason, the resistance form is not subject to consideration. The TWV model is successfully used in simulations of stationary processes in welding and electrothermal devices as well as in systems with discharge light sources [2, 10-12]. If here, like previously, one introduces the Mayr-Kulakov model, taking into consideration the virtual static characteristic of the arc component Ustat(iM), instead of (34) one receives At time intervals when current |i| values are low, the Mayr-Kulakov conductance constituent plays an important role and the dependence i (33) M ≈i takes place. Thus, one can approximately 0 + θ1 ( −α i ) (33) write that U stat (i M ) = U stat (i) and then θ = θ exp 1 g 1 g dg dt dg dt 1 ⎧ ukoli ukoli = ⎨[ 1−ε () i ] + ε () i −1 2 θ ⎩ gU C iM ⋅U stat M 1 ⎧ ukoli ukol = ⎨[ 1−ε () i ] + ε () i −1 2 θ ⎩ gU U () i ⎭ ⎬⎫ C stat ( i ) ⎭ ⎬⎫ (36) (37) Representation of the disturbance of arc column length with modified Cassie-Berger and Mayr-Kulakov An increase in the geometric size of the plasma arc column is accompanied by an increase in energy necessary to generate an additional NR 01/2012 BIULETYN INSTYTUTU SPAWALNICTWA 19
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Page 3 and 4: Summaries of the articles J. Dworak
Page 5 and 6: Jerzy Dworak Research Impact of las
Page 7 and 8: cess being carried out at a specifi
Page 9 and 10: pulses up to several dozen millisec
Page 11 and 12: ectangular pulse, P i max =702 W, p
Page 13 and 14: ectangular pulse, P i max =1752 W s
Page 15 and 16: Antoni Sawicki Modified Habedank an
Page 17: 1 dr r dt - in the resistance form
Page 21 and 22: Similarly as previously (37), one c
Page 23 and 24: Eugeniusz Turyk, Agnieszka Żydzik-
Page 25 and 26: gical weldability, confirmed by the
Page 27 and 28: Damage was also caused to the welde
Page 29 and 30: Initial selection of method for wel
Page 31 and 32: Fig. 19. Assembly table with fixed
Page 33 and 34: Agnieszka Kiszka, Tomasz Pfeifer Va
Page 35 and 36: Fig. 4. Course of changes in curren
Page 37 and 38: extreme, the process was less stabl
Page 39 and 40: Fig. 13. Macrostructure of T-shaped
Page 41 and 42: L=1 mm, I=25 A L=1 mm, I=50 A L=1 m
Page 43 and 44: case of TIG welding (for similar we
Page 45 and 46: Other, equally important research i
Page 47 and 48: tion of a function connected with t
Page 49 and 50: The intensity of arc radiation was
Page 51 and 52: On the basis of calculations (Table
Page 53 and 54: NR 01/2012 BIULETYN INSTYTUTU SPAWA
Page 55: INSTYTUT SPAWALNICTWA The Polish We

Biuletyn Instytutu Spawalnictwa No. 01/2012

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