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1-5. A ferromagnetic core is shown i n Figure PI --2. The depth of the core is 5 crn. The other dirnensiclns of the corc ;we as shown in thc. I'igurc. f;intl the value of the curwit that will producc a Tlui of 0.005 Wh. With tllis current. c\lhat is the ILIA dcnsily al lhc top of the corc'? Wlictt is the [lux densily kit the riglit side of the
. 1^4^ ssumc.^. tlial ~ h crelative pcrnie:~bilityof [hc core i s 1000.
I -. I ... i Cole depth 5 C I I I Sor,rrrrox There are three regioils in this core. The top and hottoirl form one region. the left side Ihrms a second region, and the sight side forms a ~hirdregion. If we assume that the mean path length of the flux is in the center of each leg of the corc, and if we ignore spreading at the corners of the corc, then the path lengths are 1 , = 2(37.5 cm) = 55 cnl. 1, = 30 cm, and l3 = 30 crn. The reluctances of these regions are:
The ~ o t a lreluctance is ~ h u s
and the ~rragnetornotiveforce requtred to prcxluce a flu\ of 0.003 U'h i s
aircl the required current is
Tlre flux den\ity 011 the lop of the corc ib
The flux (lensit! on the right side of the core is
1-6. A Serro~nagneticcore \villi a relative p e r l n ~ ~ h i l i r y 01' 1500 i s sho\vn in Figurc P 1-3. The dirnelision< are as slio\:n in the diagram, and thc deprl~ol'the rare IS 7 cl.11. The air gaps on [he lef't ant1 right hicles of lhe core are 0.070 21nd 0.020 cm, respectively. I3ecause of' i'ri~igingeffect>, the effective m a of (lie air gaps is 5
percent larger than their physical cire. if' tllere art.. ,. ' tul-11sin 111t. coil wrltppecl around the center leg of
the core arltl iS the current in the coil is I .O A. \:hat is ~ h cflux in each ol' the Irf'~,center. and right legs of'
the core'? What is the flux del~sityin each air fap'!
+ I I
Core depth = 7 cm
Sol.r.rrro~'This core can bc divicicil up into five regions. 1,t.t. '2, he thz relucta~iceif tlie left-hand pc3rtion
of the core. ' R , be the relucrance of the left-hand air gap. ; f f ; be the reluctalice of the righr-hand portio~lof
the core. .$?, be tlie reluctance of the right-hand air gap. ~ i n d'i?, be the ~.eluctanceof the centel. leg of rht: core. Then the total reluctance of the core is
cp = 1 :-=^ .. 1.1^ I^ rn. =90,1 k A. t I\ ' b
I , , A ( 2 o o o j i 4 ~ ~I o 7 H I ~ I ' ) ( U. O ~n1)(0.07 rn)
7 I- lie total reluctaiice is
I (^) In the fir\t printing, this value 1\35 given ~ncorrcctlqa s 300.
So~rrrrosThe two coils on this core are \ ~ ; o ~ ~ l dSO that theii. mag~~etumotiveli)rces are atidi~i:e;so rhe totill ~nagnetornoti~.eforce on this cixe is
The total reluctance in [lie core I >
and the flux ill the core is:
1-12. (^) Thc core shown in Figure 1'1-4 i b madc of a steel whose magcti~alioncul-vc i s h o ~ \ ~ nin Figure PI-9. Repeat Probhr> 1-7, 1,111 lhis time do not assume a constant vallrc o f ; I ,. f-Iovv much Sl~rxis producrd in the core by the currents spwiSiecl'? What is [he relative permeability of tllis core under these contlitio~ls'?Was the assumption ill Problem 1-7 thdt the relative perlneabilily was q u a 1 to 1000 ;I good assumption for these conditions'? I!, i t a gc.~,dassumption in general'.?
-50 crn
15 crn
Sor.rrrrov The ~nagnetl~ationcurve for ttu5 ccwc ~j \ho\til be lo^^.
IMJ (^167 ) Magneti~lng i n r e n 5 1 ~ H ( A tun1~1111)
The lwo coils 011 this core arc woulld so th:i~ heir rnagl~eton~otivt:I'orces are atlditive. so the total rnagneton~oti\iefhrce on this core is
Therefore. thc magnclizi~igintensi~yH is
From the magncti~ationcurvc. H-0.1.5 7' and the total f l u x in the core is
Thc relative pcrlncithilily of Lhc core c a n hr iolrnd i'r.on~the rcluc~anccas lollows: L?=-~-^ >.',(^ r,^ I &n.r P,PoA Solving for p , yields
The assumption thu1 p, = I O O O is not very good here. 11 15 not va.y good i n senel-al.
-~ ~^ - ~-^ --
1-14. ,A,^ t\vo-leggrc{^ mgnctic^ core^ wilh^ an air gap^ is^ shown in^ Fiyitrc^ 1'1-1^ I.^ The^ dep~lior^ thi:^ corc^ is^5 am,llie
length or the air gap in the core is 0.06 c n. and thc ni~rnheror turns on tllc coil is 100. The rnq~lrtiiation
curve or thr: corc m2r~crialis shown in Figure PI-9. Assilnle a 5 percent incrmsc in erfcctive air-gap area lo
account for fringing. How much curvenr is required to prcxluce an air--gapt l \ ~ xclrnsity of 0.5 T'? kt'hat are
the flux densities of the four sides ofthe core at tliat ct~vrent'l What i the total flux present ill the ail. gap'?
The rnaglietizi~~gintensity 1.ccli1iredto ~ ~ l - ~ d u c eo flux density of'0.5 T i l l tlie air gap can be found from the
eiluat ion :- {c,, H ,!: :
The ~nagnetirlngililt'nsity required to produce a flux density 01 0 524 'r In he nght-hanti leg of the core can
be found ftom Figure P 1-9 to he = 3 10 4 tlm
The nlagnetizitig intensity required to produce a t l u x density of 0.262 T in the top. left, arid bottom legs of the core can be found from Figure F'l-O ro he
The total R4MF required to produce the 11 u is
'IOT = H:>g',%? 'HrjLht it$!.k,l + Lsy1t,17 + 'IcfT+ H;,..;r>l;,1;,<,t::3,r,
,T:, ', = (,398kA - t/m j(0.0006 m)+ (410 :I t/ln)(O.ilO n~) + 3(240 :"\Jm)(0.40 m)
7,.,,.,=?78.6+ I64+?_XLI=601A. t
and the required corrent is
The flux densities in the four aides of the core and the total flux present in the nil- gap were calculated above.
