Myth or fact? You can increase the force of your solenoid by removing turns. (Part 2 of 3) Paul H. Schimpf Eastern Washington University

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1 Myth o fct? You cn incese the foce of you soenoid by emoving tuns. (Pt of ) Pu H. Schimpf Esten Wshington Univesity Fom Enegy to Foce Foce is the deivtive of enegy with espect to the position of the mtue. Fo this we e going to need n expession of the vition of inductnce with position, (x). et s begin by defining coodinte system, s shown in Figue : x Figue : Amtue coodinte system At x the mtue is the wy inside the coi. At x the mtue is t the enty edge of the coi. These e two positions fo which the inductnce is esy to define using the stndd ppoximtion fo the inductnce of coi: N A () N A () ( ) ( ) In ctuity, you might wnt to define () s somewht fthe outside the coi in ode fo the mtue to void the finge of the B fied, theeby impoving the ccucy. Fo the ske of simpicity, howeve, we use the ength of the coi in () nd povide n tentive wy to mke tht djustment beow. () is much ge thn () becuse, the etive pemebiity of the ion mtue, is much ge thn, the pemebiity of fee spce. (x) wi vy monotonicy between these two extemes, nd the pecise shpe of this vition with position depends on the constuction nd shpe of the soenoid. A pecise mode of this vition equies deveopment of detied geometic mode (such s finite eement mode) of the soenoid s mgnetic fied. Fo ou puposes hee, we equie is esonbe ppoximtion fo cyindic soenoid, nd fo now we wi ssume n exponenti decy:

2 ( x) e (5) As equied, this hs the vue t x, nd we wnt to choose the pmete such tht () / : e e n n ( ) (6) A vue fo somewht ess thn this, but ge thn unity, wi hve the sme effect s defining () outside the end of the coi. We cn now deive the foce, s foows: d F E de F ( x) ( x) R d R ( x) e d R e ( x) ( x) e R d (7) At this point we mke seve substitutions in ode to get the foce in tems of coi design pmetes. We stt by computing R fom the esistnce pe unit ength of the wie, γ, nd the tot ength of wie, which is computed fom the numbe of tuns times the vege cicumfeence of the tuns: R π Nγ (8) It shoud be noted tht is not the sme s the inside dius of the coi, which we sh c. The diffeence between these is iustted in Figue 5. Figue 5: Coi dius vs. vege tun dius

3 The pmete γ my be obtined fom tbe of wie guges (Refeence nd Refeence 5), nd is so computbe fom the esistivity of coppe, ρ, nd the coss-section e of the wie, : ρ γ (9) We so substitute into (7) the inductnce t the stop position,, ong with the coss-section e of the coi opening, A: N A () A π () Substituting () into (), nd substituting the esut ong with (8) into the st ine of (7), esuts in: F πγ 8 W f e 8πγ e () In () we hve defined the squed tio between nd s new pmete, the winding fcto, W f. The winding fcto is wys ess thn unity, nd epesents eduction on wht might othewise be consideed the nomin foce: W F f F nom W f () Discussion A compison between the pedictions of () nd commeciy-vibe cyindic soenoid e iustted in Figue 6. The simuted coi used the foowing pmetes: 7 mm,. mm,.5 mm, AWG, N57. Both the simuted nd ctu coi esistnces wee 5. Ω. As cn be seen, the simuted cuve mtches the high powe cuve fo the soenoid quite we. At owe powes, the foce of the ctu soenoid dops off t highe exponenti te. This coud be simuted by substituting highe powe of x in the inductnce fomu of (5).

4 Foce (N) 6 8 Stoke (mm) Figue 6: Simuted (eft) vs. ctu (ight) foce It is woth noting t this point tht ssuming ine vition in inductnce fom () to () woud esut in foce tht is constnt vs. stoke. Whie thee e some ctu soenoids tht ppoximte tht, the moe usu cse is the exponenti vition shown in Figue 6. This is why we modeed the vition in inductnce s n exponenti decy, the thn ine, in (5). As stted eie, the pecise shpe of (x) depends on the geometic constuction of the soenoid. Notice tht N does not ppe in () t. An N tem shows up in the denominto s pt of the squed esistnce, but N so shows up in the numeto s pt of the inductnce, nd the two effects thus cnce out. This mens tht whie emoving tuns wi incese the cuent, it wi so decese the mgnetomotive foce of the B fied by the sme fcto. Anothe wy to ecognize this cnceing effect is to ook t the estimtion of the mgnetic fied in (). In tht eqution, emoving tuns deceses B, but it so inceses I due to popotion eduction in R, nd the two effects cnce ech othe. Thee is, howeve, nothe fcto tht we must conside. Eq. () does not ccount fo the fct tht is ctuy function of N. Fo fixed ength coi, incesing N wi eventuy incese. But since emins unchnged, the tio between the two is educed. Since this tio is wys ess thn unity, this mens tht dding tuns wi eventuy decese the foce. In ode to exmine the extent of this effect, we must substitute into () mode fo the dependence of on N. Figue 7 shows coss-section of wies coming ound the soenoid coe. ( - ) centeine of coe Figue 7: Tun pcking The spce vibe fo these N tuns is ( - ). The e tken up by the wies is popotion of this defined by the pcking density, λ. The theoetic mximum pcking density fo the

5 ttice ngement shown bove is π/.97 (Refeence 6). If the wies e stcked in gid insted of ttice, the pcking density is esiy shown to be π/.785. We cn now expess s function of N, s foows, whee is the coss-section e of wie, vibe fom n AWG tbe [Refeences nd 5]: λa A λ ( ) vi N wies N + λ βn + λ Fo bevity, we hve copsed λ,, nd into singe pmete, β. Substituting () into (), nd futhe simpifying by exmining the foce t x, we obtin: π () F 8πγ ( βn + ) (5) This eves efinement on the winding fcto in (): W f (6) β + ( N ) When N is sm, W f wi ppoch the squed tio of to in (). As N inceses, the tio becomes sme. Thus, N does py oe in the foce, but it is secondy effect hving to do with the incese in, not the incese in esistnce. The te t which n incese in N wi decese the foce depends on the vue of β, which depends on the pcking density, the wie guge, nd the ength of the coi. In ode to get fee fo this vition, we now ook t the effect of educing N on the winding fcto, the etive foce (comped to nomin), nd the etive powe, fo coi with the foowing pmetes: 6 mm, 7 mm, AWG 6,.9 mm, N nom 5 (Refeence 7). Figue 8 shows the vition in ong with the winding fcto. Fom to 5 tuns, the vege tun dius inceses iney fom 7. mm to 9.5 mm. Athough the denominto is qudtic, in this cse the shpe of the winding fcto in the egion of inteest is ney ine.

6 .8 9 winding fcto.6 8 (mm) N Figue 8: Winding fcto nd vege tun dius vs. N Figue 9 shows the vition in etive foce nd etive stedy-stte powe s consequence of vying N. The etive foce hs the sme shpe s the winding fcto bove, except tht it is mde etive to the nomin foce t N 5. The etive powe is invesey popotion to R π Nγ, whee is function of N. As n exmpe, these cuves pedict tht educing the numbe of tuns fom 5 to wi incese the foce to 5% of nomin, but t the cost of n incese in powe to % of nomin. A decese of tuns fom 5 to 7 wi incese the foce to % of nomin, t n incese in powe to % of nomin. Figue pots the tio of etive foce to etive powe s n efficiency mesue. At tuns, the efficiency is 8% wht it is t 5 tuns. This shows tht N shoud be teted s secondy fcto in designing coi fo desied foce. Refeing to () nd (5), othe coi pmetes tht e impotnt in the foce ccution e the esistnce pe unit ength of the wie, γ, nd the tot ength of the coi,..8.9 etive foce etive powe etive efficiency N Figue 9: Retive foce nd powe vs. N. 6 8 N Figue : Retive efficiency vs. N As shown in Figue nd Figue, it is much moe efficient to incese foce by decesing γ (decesing wie guge). These figues show the sme quntities s Figues 9 nd, fo

7 tun coi with γ (nd ) nging fom the vues fo AWG6 though AWG5. Note especiy tht unike decesing N, decesing γ not ony inceses the foce, it so inceses efficiency etive foce etive powe etive efficiency esistnce pe unit ength (mohm/m) Figue : Retive foce nd powe vs. γ 6 8 esistnce pe unit ength (mohm/m) Figue : Retive efficiency vs. γ ying offes nothe sttegy fo incesing foce. Figue nd Figue show these mesues gin fo tun coi using AWG wie, with vied fom 5 to 5 mm. As withγ, decesing inceses both foce nd efficiency. In this cse, howeve, the etive foce cuve offes moe stedy, nd modest, inceses..5 etive foce.95.9 etive powe etive efficiency coi ength (mm) Figue : Retive foce nd powe vs coi ength (mm) Figue : Retive efficiency vs. Figue 5 shows the mesued foce vs. tuns fom n ctu soenoid using the sme pmetes s those simuted in Figues 8, 9, nd. The foce ws smped times t ech of seve decements in N, nd est-sques ppoximtion ws used to find the best-fitting invese qudtic in N, which is supeimposed on the dt. The esuts gee quite we with the simution in Figue 9. Fo exmpe, decesing the numbe of tuns fom 5 to 7 incesed the foce by 6%. As in the simution, this equied twice the powe. Whie this ws commeci soenoid, the cois wee cey hnd-wpped s they did not come off in unifom yes, which is the ikey expntion fo why the dt does not spn the best-fit cuve t smpes of N.

8 ..5. foce (N) tuns Figue 5: Foce vs. tuns fo n ctu soenoid If tuns shoud not be pimy design fcto fo foce, then how shoud one detemine the nomin numbe of tuns to use fo given ppiction? Note tht (), in theoy, ows the design of coi fo given foce with vey few tuns. In ctuity, howeve, the imited cuentcying cpcity of the wie wi pevent the use of too few tuns. Fo DC coi, the stedy-stte cuent is cn be obtined fom ou estimte of tot esistnce in (8). We equie tht to be ess thn some imit bsed on the cuent-cying cpcity of the wie, which cn be obtined fom tbe of wie guges. A fction sfety fcto is pobby desied hee, nd some coi designes simpy estbish ue-of-thumb fo mximum cuent bsed on the coss-section e of the wie. Hee we use the tte ppoch nd c tht pmete η (Amps pe sque mete of coss-section): I DC R η (7) Substituting (8) fo R nd soving (9) fo esuts in: π γn N ρ η γ πρη (8) This gives nice owe imit fo the poduct of the numbe of tuns nd the vege coi dius. Notice so tht γ hs been cnceed out, though the esistivity of coppe emins in the eqution.

9 It hs been suggested to the utho tht η.5 A/mm.5 x 6 A/m is esonbe vue, but the utho knows of t est one cde mchine design tht goes substntiy ove tht vue (nd gets the hot), though it emins beow the ecommended wie mpcity with insution. To chieve desied foce, esonbe design sequence woud be to fist use (), with the ssumption of some esonbe winding fcto, sy.7, in ode to detemine n ppopite γ (wie esistnce pe unit ength) nd coi ength,. These two pmetes cn be tded off ginst ech othe such tht owe esistnce wie coud be used by engthening the coi, o shote coi coud be used by incesing the wie esistnce. The numbe of tuns, N, nd coi dius, (o ) coud then be detemined fom (8). These two pmetes cn so be tded off ginst ech othe such the numbe of tuns cn be educed with coesponding incese in the dius, nd vice-ves. Of couse, the mgnetic fied nd inductnce ppoximtions in () nd () e bsed on ong cois, so one woud wnt to be wy of mking coi too shot (), in compison to width ( nd ). These bseine pmete vues coud then be substituted into (5) which wi incude n estimte of the ctu winding fcto, nd the pmetes coud then be tuned incementy whie ensuing tht (8) emins stisfied. About the utho Pu H. Schimpf hods the B.S., M.S., nd Ph.D. degees in Eectic Engineeing fom the Univesity of Wshington, with espective degee specities in digit eectonics, embedded sign pocessing, nd modeing eectomgnetic fieds in compex domins such s the humn body. He peviousy woked in the eectonics industy fo yes t Honeywe Mine Systems in Mukiteo Wshington (which te becme division of Aint Techsystems, Hughes Aeospce, nd Rytheon), whee he ttined the eve of Senio Pincip Deveopment Enginee, pio to enteing cee in cdemics. He ws peviousy tenued Associte Pofesso of Eectic Engineeing nd Compute Science t Wshington Stte Univesity, nd is pesenty Pofesso nd Chi of Compute Science t Esten Wshington Univesity. One of his hobbies is the estotion nd epi of cde gmes, of which he pesenty owns thee: Chicgo Coin Twin Rife (eecto-mechnic), Sten Meteo (ey soid stte), nd Wiims Fish Tes (te soid stte). Pu ecenty designed pinted cicuit bod to intefce n Aduino Meg to 85 diffeent By/Sten pinb mchines so tht his embedded systems students cn pogm pinb gme ogic using moden pocesso. The bod incudes the cpbiity to py sound effects fom WA fies stoed on n SD cd.