Elektrik Devreleri 2 - Mersin Üniversitesi - 3 Fazlı Devreler

8/10/2019 Elektrik Devreleri 2 - Mersin niversitesi - 3 Fazl Devreler

1/48

=3>

CM_I@ HKSRKIKR

0.= SKLXRIKRG CX NMRCIKRIK ETXKRIJKT0.2 LCM_I@ [email protected] CM_I@ TTXKJ0.8 E IJKL0.; CM_I@ TTXKJHK E LMXTMP@T@G@G H_KIXIJKT0.3 LMR@@L RGKL VRABIKJIKR

BIJ0


2/48

=30

0. AL CM_I@ ML@JIMRMitkrgmt`c mlj rktkg okgkrmtrikrk Mitkrgmtr hkg`r. Kkr b`r mitkrgmtr ymigzb`r s`gs hmiemskli`ghk kjl rkt`rsk, = cmzimitkrgmtr hkg`r. `jh`yk lmhmr`gdkikh``j`z kikltr`l hkvrkikr`ghkl` lmygmlimr nkp = cmzilmygmlimrh. Kkr b`rmitkrgmtr 06 hkrkdk cmz cmrli `l` s`gsa`hmi kjl rkt`yarsm bugm `l` cmzimitkrgmtr hkg`r. Kkr b`r mitkrgmtr mrmimrghm =26kr hkrkdk cmz cmrlbuiugmg s`gsa`hmi kjl rkt`yarsm, byik b`r lmygmm hm cmzimitkrgmtr hkg`r.

Kgkro`g`g ìkt`j` vk rkt`j` bmljghmg al cmzis`stkj`g b`r cmzis`stkjstgilikr` ?

m-)Mygbayuttml` `l` vkym cmzimitkrgmtr b`r cmzihmg hmnm cmzim evkr`r.b-) al cmzi mitkrgmtrhk lìawmttsmmt bmgm kgkro`g`g jmi`ykt` b`r

cmzihmg hmnm hltr.d-) al cmzi kgkro` ìkt`j nmtimr b`r cmzi nmtimrhmg hmnm uduzhur.

rgk`g cmzikgkro` ìkt`j nmttgg bmlr mrikhkkr = cmzinmttg lmhmrhr.

0.= SKLXRIKRG CX NMRCIKRIK ETXKRIJKT

B`rhkg cmzim ekrì`j vk mlj lmygm vkym 2 vk 4 cmzi ekrì`j lmygmlimrg`nt`vm khkg mlj vk ekrì`j vkltrikr` ``g s`stkji` b`r estkrjk kli` lmbuikhì`rsk prabikjikr`g zjghk nmtm ympjm `nt`jmi` mzmir.

klì 0.= (m) hml` K vkltrg kik mimij. Bu vkltrg bmimged A vk ulsjhm Mhr vkltrg yg Ahmg Mym haruhur. Bu vkltr `ct nmrcik (KAM)kli`ghk estkrkbì`r`z. (AM) nmrcikr` b`zk vkltrg ygg hk estkrjàiur. Kvkltrgg tkrs` aimg (=>6 hkrkdk cmz cmrli) K vkltr `ct nmrcikrik (KMA)kli`ghk estkrì`r. B`r vkltrg mit ymgghml` nmrcikr`g`g ykr` hk``rsk vkltr=>6hkrkdk hghrijaiur.

KMA: -KAM

MIXKRGMXC ML@J HKSRK MGMI_ Xapml


3/48

=56

Ma-K K

KAMKMA

M C

H

DB

@Ma

@Ba

@aC

@aH

@aD

A

(m) (b)

klì 0.=.Skltrikr`g `ct nmrcik estkrì``

klì 0.= (b)hkl` A hj galtmsgm ekikg vk e`hkg mljimrg ygikr``mrktikgj` vk vkltryki aimrml estkrìjìkrh`r. Hj galtmsgm ekikgmljimrg vkltryki tapimj, myrimg mljimrg vkltryki tapimjgm k`tt`r.

@MA + @BA: @AD + @AH + @AC

K`ti``g b`r tmrmcgh`kr tmrmcm tmymij,

@AM+ @AB+ @AD+ @AH+ @AC: 6

H`llmt khì`rsk, mlj vkltrikr` k`ti``g b`r tmrmcgm vkym tkl` tmrmcgmmltmrihghm bmimge vkym sag nmrcikr`g ykr hk`t`rh`` erir.

klì 0.2hkl` skr` RID hkvrks`ghk MH uimrghml` ekrì`j, nkr kikjmgguimrghml` ekrì`jikr`g vkltryki tapimjgm k`tt`r.

[MH: [MB+ [BD+ [DH



4/48

=5=

[MH

[MB

[BD

[DH

M

B

D

H

klì 0.2 Tkr` RID hkvrk

0.2 LCM_I@ ML@J

GT lutupimrmrmsghm hghrikg b`r bab`ghk s`gsa`hmi b`r kjl `ghlikg`r.klì 0.4tk erih e`b` stmtarm b`r bab`g ykrikt`rì`r vk artmhml` GT lutbuhghrirsk , lutupimrg jmgykt`l mlstmrmcghmg lksìkg bab`ghk s`gsa`hmib`r kjl `ghlikg`r. Vrmt`ltk luiimgimg mitkrgmt`c mlj okgkrmtrikr`g`g bylei vk ylskl ekrì`ji` aimgimr nkp bu t`ptk ympirim ymg` bab`gikr smb`ttutuimrml lutupimr hghrir.

G T

Bab`g

klì 0.4

klì 0.8 (b) hk erih e`b` , M vk B bab`gikr` b`rb`r`gk h`l aimrml stmtarmykrikt`rì`r, artmhml` GT lutupimrhghrirsk, M vk B Bab`gikr`g`g uimrghmg06 hkrkdk cmz cmrlis`gsa`hmi `l` kjl migr.

H`llmt khì`rsk bìklk vkltrg ìl nmrc` b`r`gd`vkltrg ìl nmrc`, bìklk vkltrg `l`gd` nmrc`hkgd vkltrg `l`gd` nmrc` aihuu erir. Xapimgmgvkltrikrhkg b`r`gd` ([MB) vkltrgg `l`gd` nmrc` (B),`l`gd` ([BD) vkltrg b`r`gd` nmrc` (B) vk `l`gd` nmrc D,gd [Dvkltrgg ìl nmrc` hk Dh`r

klì 0.8 (m) hm erih e`b` GT lutupimrmrmsg b`rb`r`gk h`l `l` bab`g ykrikt`rì`r,bab`gikr`g uimr jìhkl` 8 bìkz`kbmimgrsm, lutupimr hghrihghk Mbab`g`ghk vk B bab`g`ghk `ghlikgkgs`gsa`hmi kjlimr mrmsghm hm 06mrhkrkdk cmz cmrlaiur.



5/48

=52

G

T

B

M

G T

Bab`g

M

m

b B

(m) (b)

klì 0.8l` cmzimitkrgmt`c kjlgg kihk khìjks`.

M vk B bab`gikr`ghk `ghlikgkg kjlimrg hk``j kr`s` klì 0.;tk eriyar.Bab`gikr`g smrj smyimrb`rb`r`gk k`t aiursm, kjlimrg hm ekgi`likr` haimysyimktl`g hkkrikr` b`rb`r`g`g myg aiur. klì 0.;tk M bab`g`g`g kjls Bhkg 06hkrkdk ìkrhkh`r.

6

kmkb

t06a =>6a 436a

KM KB

klì 0.;T`gsa`hmi M vk B kjlimr

klì 0.8hkl` okgkrmtrikrk `l` cmzimitkrgmtrikr vk kihk khìkg kjlikrk hk `l`

cmzi

mitkrgmt`c ml

j hkg`r. Kjl imr

g nkrnmge` b`r mghml` hkkrikr`

km : Kj . T`gwt kb : Kj . T`g (wt 06)

l` cmzi okgkrmtrhkl` M vk B bab`gikr`ghk `ghlikgkg kjlimrg ktl`ghkkrikr`g`g =26 Sait vk bab`gikr`g uimrgm bmimgmg aj`l h`rkgikr`g =2kranj aihuugu lmbui khki`j. klì 0.3 (m) hm erih e`b` cmzimrhmg =6mjpkr mlj klì`r.



6/48

=54

=2

=2

m

b

KmM:=26/6a

KbB:=26/-06a

M

B

=6 M

=6 M

=2

=2

m

b

KmM

KbB

M

RM

@B6

@M:=6M

KaM

RB

(m) (b)

klì 0.3l` cmzis`stkj

Nkr `l` cmzg ymg ymgm aimg hgìktlkgikr` ykr`gk M B uimrb`rikt`rìkrkl tklb`r nmt klì`rsk, klì 0.3 (b) kihk khì`r. KmMekrì`j` ìk @mM mlj, KbBekrì`j`ìk @bBmljmygcmzhmhr. gl, aj`l h`rkgik Das : = aimg ylikr bmihru nmihk, @mMmlj ìk @bBmljmrmsghml` cmz cmrl06hkrkdkh`r. Jtkrklaimg hgìktlkg`gk (Gtr nmtt) hkg`r. Gtr nmttghmg ekkg mlj, 06hkrkdkcmz cmrli @mM vk @bB mljimrgg vkltryki tapimjimrgm k`tt`r. l` h nmtmrmsghml` patmgs`yki cmrlhm vkltryki ìkjik buiugmbì`r.

M

KaB:=26/-06a

KMa:=26/-=>6a KaM:=26/6

aa

K MB

K MB

06a

a @aM:=6/6a

@aB

KaB

KaM

@g: 2 .@

(m) (b)

klì 0.5l` cmzis`stkjhk ekrì`j vk mlj vkltr h`ymermjimr.

l` cmzi s`stkjhk `l` h nmtimr mrmsghml` ekrì`j klì 0.5 (m) hml` vkltrh`ymermjghmg buiugur.

KaM: KaB: Kcaihuughmg, KMB: KMA+ KAB, KMB: 2 .Kc

KMB : 2 . =26 : =56 Sait .

klì 0.5 (b)hkl` vkltr h`ymermjghmg gtrhkg ekkg mljnksmpimymij.



7/48

=58

@g : 2 .@ @g : 22 BM @@ : 22 =6=6 : 2 .=6 : =8,=8 M.

u nmihk `l` cmzi b`r s`stkjhk `l` h nmt mrmsghml` patmgs`yki cmrl cmzkjlimrgg 2 lmtgm, gtr mljhm cmz mljgg 2 lmtgm k`tt`r.

l` cmzis`stkjhk e b`r cmz edgg `l` lmtgm k`tt`r.

V : 2 . K . @ . das

K : cmz ekrì`j`, @ : Cmz mlj, das: ylg e lmtsmys

0.4 CM_I@ TTXKJ

=. CM_I@ KJLG KIHK KHI

GT lutupimr mrmsgm klì 0.> (m) hm erih e`b` tmgk bab`g =26krhkrkdk cmz cmrliaimrml ykrikt`rìkrkl hghrirsk, mrmimrghm =26 kr hkrkdkcmz cmrlaimg tmgk s`gsa`hmi kjl kihk khì`r. Nkr bab`g`g uimrjìhkl` mitbìkz`k bmimgrsm, bìkz`likrk srtgkg crmimrim bab`gikrhk `ghlikgkg kjlimr hmrym migmrml ylikr bmimgmbì`r. klì 0.> (b) hk M, B vk D bab`gikr`ghk`ghlikgkg =26kr hkrkdk cmz cmrlikjlimrg hk``j krìkr` eriyar.

M

B

D

d

b

m

G

T

6 t

KM KB KD

36a 06a =26a =>6a 256a 436a

(m) (b)

klì 0.> cmzikjl`g kihk khì`` vk kjlikr`g hk``j kr`s`

M, B vk D bab`gikr`g`g smrj smyimr b`r b`r`gk k`t aiursm `ghlikgkg kjlikr`g jmls`juj hkkrikr`, haimysyim ktl`g hkkrikr` hk b`r b`r`gk k`t aiur. Bu=26 kr hkrkdk cmz cmrli vk jmls`juj hkkr` myg aimg kr` mmhmlè`b` `cmhk khì`r.



8/48

=5;

k M : Kj. T`g t

k B: Kj. T`g (t =26a

)k D: Kj. T`g (t 286a)

klì 0.> `gdkikgh``ghk, M bab`g`ghk `ghlikgkg kjl (+Kj) hkkr`g` mihltmg=26 sagrm B bab`g`ghk `ghlikgkg kjl (+Kj) hkkr`g` mir. B hkg =26sagrm D bab`g`ghk `ghlikgkg kjl (+Kj) hkkr`gk uimr. D bab`g`ghkl` kjlM bab`g`ghkl` kjl hmg 286 ekr` vkym =26 ìkr`hkh`r. D bab`g`ghkl` kjl Mbab`g`ghkl` kjl hmg 286 ekr` vmym =26ìkr`hkh`r.

2. Pihz (P) Bmis`stkj

klì 0.0 (m) hm M , B vk D bab`gikr`g`g huruju `ghlikgkg kjlikr`g vkltr ykiestkrììkr` `zìj`t`r. Bab`gikr`g sag uimr(m,b vk d) b`rikt`rì`rsk, klì =2.0(b)hk P bmis`stkj vk vkltr h`ymermjkihk khì`r.

klì 0.0 (b)hk erih e`b`, =26 hkrkdk cmz cmrli bab`g`g saguimrb`rikt`rìkrkl kihk khìkg galtmym (a) hkg`r vk M, B vk D uimrghmg vkahmg b`rkr nmt lmrrsml, MBD cmz nmtimrg vk scr nmttg ìmvk khkg b`rs`stkj kihk khkr`z. Bugm yihz s`stkj hkg`r. Cmz nmtimr ìk gtr mrmsghmlèkrì`jikr b`rb`r`gk k`t vk =26kr hkrkdk cmz cmrlihrimr. Nkrnmge` b`r cmz nmttìk gtr mrmsghml` ekrì`jk 99Cmz Ekrì`j`77 hkg`r.

D

B

Mm

b

dKmM

KdD

KbB

D

B

M6

KmM

KdD

KbB

6

M

B

D

=26a6

KaD

KaM

KaB

(m) (b)

klì 0.0 cmzikjlimrg yihz bmimgjmsvk vkltr h`ymermj

m. Cmzimr MrmsEkrì`j (Nmt Ekrì`j`) ?

M,B vk D cmzimr mrmsghml` ekrì`jikr`g vkltr h`ymermjimrg `zjkyk vkcmzimr mrmsekrì`j`g jutiml hkkr`g` cmz ekrì`j` ìk `cmhk ktjkyk mimij.



9/48

=53

klì 0.0 (b)hkl` hkvrkhk MB cmz nmtimrmrmsghml` KMBekrì`j`, KAMvk KAB

cmz ekrì`jikr`g`g vkltrki tapimjgm k`tt`r.

KMB: KMA+ KAB

klì 0.=6hm KMB vkltr KMA : -KAM vkltr ìk KAB vkltrgg bìklks`migmrml `zìj`t`r. BD cmz nmtimrmrmsghml` KBDekrì`j` KBD: KBA+ KADh`r.DM cmz nmtimrmrmsghml` ekrì`j hk

6

KBD

KDM

KMB

KaB

KMa:-KaMKaM

KBa

KaD

KDa:-KaD

=26a46a

=26a

36a

46a

klì 0.=6Pihz s`stkjhk cmz vk cmzimr mrmskjlimr vkltr h`ymermj

KDM : KDA + KAM vkltrki tapimjgm k`tt`r. klì 0.=6hm KBD vk KDM cmzimrmrms ekrì`jikr`g `zììkr` eriyar. Skltr h`ymermj `gdkikgh``ghk KMB,KBDvk KDM cmzimr mrmsekrì`jikr`g`g hk (nmt ekrì`jikr`g`g) mrmsghm =26krhkrkdki`l cmz cmrlaihuu eriyar.

KMBvkltrgg ekgi`` (jutiml hkkr`) vkltr h`ymermjghmg nksmpimgmbì`r.

KMB: 2(KAB)das46a: 2.(KAB) 4 / 2 : 4 . (KAB)

Cmz ekrì`jikr` b`rb`r`gk k`t aihuu ``g cmzimr mrmsekrì`jikrhk b`rb`r`gkk`tt`r. KMB : KBD : KDM : K. Cmzimr mrms ekrì`j cmz ekrì`j`g`g 4 lmtgmk`tt`r.

K: 4 .Kc



10/48

=55

b. Pmgiyihz bmimjm

cmzi kjlgg `ghlikgh`` M,B vk D cmz bab`gikr`g` yihz bmimrlkg saguimrg (m,b,d) b`r lpr ìk b`rikt`rjkj`z ekrkli`h`r. Pihz lprsgympmrlkg cmzimrhmg nkr nmge` b`r`g`g uimrg lmrtrjmjz gkt`dks`ghk sagudu h`yk bmimge udugu h`kr cmzimrg sag uimr ìk b`rikt`rkbì`r`z. klì0.== (m)hm erih e`b` gd cmzg uimrg lmrtrmrml bmimge uduDy` h`kr b`r`gd` vk `l`gd` cmzimrg sag uimrìk b`rikt`rki`j.B`r`gd` cmz kjly^klskg` zkr`ghk KAMvkltr ìk, `l`gd` cmz kjlsb`r`gd` cmz vkltrghk =26hkrkdk ekr`hk KAB vkltr ìk estkrì`r.gd cmzg kjls b`r`gd`hkg =26hkrkdk ìkr`hk vkym `l`gd` cmzhmg =26 hkrkdk ekr`hkh`r. gd cmzg uimrgtkrs bmimjmlim bu cmzg kjls =>6 hkrkdk hghrij aiur.

klì0.0(b)hkl` gd cmz kjlsg estkrkg KAD vkltr =>6 hkrkdkhghrigdk klì0.==(b)hkl` vkltr h`ymermjghml` e`b` KDd vkltr ìkestkrì`r. klì 0.0(b) vk klì 0.==(b) vkltr h`ymermjimrlmrimtrihghm,tkl cmrl KADvkltrgg tkrs aimrml `zìjks`h`r.

cmzi b`r mitkrgmtr yihz bmimghltmg sagrm cmzimr mrms kjlikr`g` b`rvaitjktrk ìk ikrkl yihz bmimgjmgg haru vkym ymgi aihuu nkjkgbuiugmbì`r

Haru yihz bmimjmhm cmzimr mrmskjlhm cmz kjlimrgg 4 lmtgm k`tt`r.klì 0.== (b)hkl` vkltr h`ymermjghm erih e`b` ymgi yihzbmimjmhm cmzimr mrms kjlimrhmg b`r` cmz kjlsgg 4 lmtgm k`t aihuunmihk h`kr cmzimr mrskjlimr cmz kjlimrgm k`tt`r.

KMB: 4 .Kc KMD: Kc KBD: Kc

u nmihk cmzimr mrs kjlimr cmz kjlsgg 4 lmtgm k`t `sk mitkrgmtr(mitkrgmt`c mlj okgkrmtr) haru yihz bmihr. Cmzimr mrmskjlimrhmg `l`s`cmz kjlsgm k`t `sk mitkrgmtr ymgiyihz bmihr.

D

B

Mm

b

d

6

D

B

=26a46a K

aM

KdDKaB (m) (b)

klì 0.==ymgiyihz bmis`stkj



11/48

=5>

d. Pihz T`stkjhk Mljimr

M,B vk D cmz kjlimrg klì 0.=2 (m)hm erih e`b` k`t aj`l h`rkgbmimymij. Cmzimrg hg ìktlkgikr` ykr`gk h`rkgikr`g udugu vk cmzimrgm,b vk d saguimrgb`rikt`rkrkl tkl b`r ìktlkg klki`j. Bu ìktlkgk Gtr nmtthkg`r. klì 0.=2(b)hkl` yihz s`stkj kihk khì`r. Burmhm lmygml vk yl yihzbmimgj, 6 galtmimrhm b`r nmtim b`rikt`rìj`t`r.

R=: R2: R4aihuughmg nkr cmzhmg klìkg @AM, @ABvk @ADmljimrb`rb`r`gkk`tt`r.Pl aj`l aihuu ``g nkr cmz mlj lkgh` cmzgg kjlsyim mygcmzhmhr. Cmz kjlimr =26kr hkrkdk cmrli aihulimrgm erk, cmz mljimrmrmsghm hm =26kr hkrkdk cmz cmrlvmrhr. klì 0.=4hk cmz kjlimrgg vk

cmz mljimrgg vkltr h`ymermjeriyar.

klì=2.=2 (b)hk erih e`b` yihz s`stkjhk cmz mlj nmt mljgmk`tt`r.Cmz mljimrb`r b`r`gk k`t aihuu ``g nmt mljimrhm b`r b`r`gk k`tt`r.

| @AM| : | @AB| :| @AD| : @c| @MM| : | @BB| : | @DD| : @@nmt: @cmz

D

B

M6

K mM

K dD

K bB

D

B

M R =

R 2

R 4

6 '

@BB

@MM

@DD

@A A

(b)

klì 0.=2Pihz T`stkj vk yihz hkgeki` ylg bmimg.

k`t h`rkgi` ylg yihz bmimgjms ìk jkyhmgm ekikg cmziylk 99 cmzi hkgeki` yl77 hkg`r. Pihz s`stkjhk nmt mljimr cmz mljimrgmk`tt`r.

Lmygmg scr galtms (6) ìk ylg scr galtms (6) g b`rikt`rkg ìktlkgkGtr Nmtt hkj`t`l. Bu nmttmg ekkg @aa gtr mljg (6) vkym (6) hjgaltmimrgm L`ràcug mlj lmgugugu uyeuimymrml buimbì`r`z.



12/48

=50

@AM+ @AB+ @AD: @AA

@MA + @BA+ @DA: @AA

B`r b`r`gk k`t =26acmz cmrli mlj vkltrgg bìklks` scrhr.

u nmihk hkgeki` yli yihz s`stkjhk gtrhkg ekkg mlj scrhr.

=26a

6

KaD

KaM

KaB

=26a

=26a

@aM

@aB

@aD

= 26 a

@aM

@aD

@aB

6

@ r:-@ aD

m) Cmz KJLimrvk cmz mljimrvkltr h`ymermjb)Mljimrg vkltrki tapimj

klì 0.=4

cmziP bmib`r lmygmm, cmz kjpkhmgsimrb`r b`r`gk k`t cmziyihz b`ryl bmimymij, klì 0.=8 (m).

D

B

M6

KmM

KdD

KbB

D

B

M6'

@BB:@B

@MM:@M

@DD:@D

@g:@AA

_M

_D

_B

6

K aD

K aM

K aB

m)Kjpkhmgsihkgeki` yl b)Skltr h`ymermjklì 0.=8

Nkr cmzhmg ylg klt`` cmz mljimr cmz kjlikr`ghkg kjpkhmgsg mslmhmr ekr` lmir. klì 0.=8 (b) hk cmz kjlikr`g`g vk cmz mljimrgg vkltrh`ymermj eriyar. @M, _M kjpkhmgsghmg ekkg mlj< @B, _B hkg ekkgmlj vk @Dhk _Dhkg ekkg mljhr. Gtrhkg ekkg mlj =26

acmrli@M, @Bvk@Dmljimrgg vkltr yki tapimjgm k`tt`r.



13/48

=>6

M

AMM

_K@

B

ABB

_K@

D

ADD

_K@

@g : @M+ @B+ @D

rgkl 0.=

klì 0.=8 (m) hm l` hkvrkhk ylg kjpkhmgsimr_M: _B: _D: 4646 h`r.

cmzi P bmi mitkrgmtrg cmz kjlimr 6=26SKAM , =26=26SKAB ,

=26=26SKAD h`r.

m) Nmt mljimrg, b) Gtrhkg ekkg mljg nksmpimygz. d) Cmzkjlikr`g` vk nmt mljimrgestkrkg vkltr h`ymermjg`z`g`z.

zj 0.=

m) M4684646

6=26

_

K@

M

AMM

=;6M8

4646

=26=26

_

K@

B

ABB

06M84646

=26=26

_

K@

D

ADD

b)

M6ig

6o6)8o6()2o83,4()2o83,4(ig

068=;68468@@@ig DBM

d) Skltr h`ymermj, klì 0.=8 hkl` h`ymermj bkgzkr`h`r. Pmigz, : 46amigmdmltr.

4. ekg () BmiT`stkj

klì 0.=; (m) hml` cmzi mitkrgmtrg cmz bab`gikr`g`g uimr klì 0.=; (b)hkl` e`b` bmimghghm ekg ( ) bmimgt kihk khì`r. ekg bmi b`rmitkrgmtrg kjl ikr`g`g vkltr h`ymermjklì 0.=; (d) hk eriyar



14/48

=>=

D

B

Mm

b

dKmM

KdD

KbB MB

b D

dm

KDB KMD

KBMB

M

D

=26a

=26a

=26a

KdD

KbB

KmMB

D

M

m) b)ekg bmimgt d)Skltr h`ymermj

klì 0.=;ekg bmis`stkj

cmzimitkrgmtr ekg bmimghltmg sagrm M, B vk D cmz nmtimrmrmsghml`kjl ikr`g cmz kjl ikr`gk k`t aihuu h`ymermjghmg eriyar.

u nmihk, ekg bmi b`r okgkrmtrhk cmzimr mrms kjl ikr cmz kjlikr`gkk`tt`r.

K : Kcmz

cmzib`r okgkrmtrg ekg bmimgjmsghm nkrnmge` b`r ymgiigikjkl

``g, cmzimr mrmsghml` bmimgthmg `l`s` ympir. gd bmimgtympijmhmggdk klì 0.=3 (m) hm erih e`b` ml `l` udm b`r vait jktrk bmimymrmluimr mrmsghml` ekrì`j iir. [imr mrmsghml` ekrì`j scr `sk bmimgtharuhur. Ml aimg bu `l` u b`r b`r`gk bmimgmrml ekg bmimgttmjmjimgr.Kkr vait jktrkhkg b`r cmz ekrì`j`g`g `l` lmtalugursm bmimgtklì 0.=3 (b)hkl` e`b` ymgitr. Bu `l` u bmimgrsm mitkrgmtrg smreimrmrmsghm mrb`rmlj haimr.

M

b

Bd

D

mKmM

KdD KbB

S

KdD K b

B

KmM

B

DM

(m) Haru ekg bmimgt



15/48

=>2

S

M

b

d

D

m KmM

KdD

KbB

2Kc

KbB

KdD

KmM Mm

D

d

(b)Pmgiekg bmimgt

klì 0.=3 ekg s`stkj

m.ekg T`stkjhk Mljimr

cmzib`r okgkrmtrg M,B vk D cmzimrgg uimrgm hkkrikr` b`r b`r`gk k`t tmgk aj`l h`rkg bmimymij klì =2. =5 (m) hm erikg cmzihkvrk kihkkhì`r. R=:R2:R4 cmz kjl ikr`hk b`r b`r`g`g k`t aihuu ``g nkr cmzhmg myghkkrhk mljimr klì`r. M cmzghmg klìkg @mM mlj cmz kjl ` KmM ìk myg

cmzhmhr. @bBcmz mljcmz kjl` KbB ìk, @dDgd cmz mljhm KdD cmz kjl`ìk mygcmzhmhr.

klì 0.=5 (m) hml` hkvrkhk ymg ymgm klìkg `ct ìktlkgikr` b`rikt`rkrkl tkl nmtklt``j`zhk klì 0.=5 (b) hkl` hkvrk kihk khì`r. Burmhm okgkrmtr ekg R=R2vk R4yl h`rkgikr` hk ekg bmimgjaiur. bmimitkrgmtrhkg lmg nmt (M, B vk D nmtimr) bmiR=R2vk R4h`rkgikr`gk bmihr.

R=KmMM

B

D

d

m

b

M

b'

B d'

D

m'

R2 R4

@mM

KbBKdD

@mM@dD @mM

@4

@2

@=

@bB

@dD

@dD

@mM



16/48

=>4

R =KDMM

B

DM

B

D

R 2 R4

@MM

KMBKBD

@mM@dD @mM

@4 @2

@=

@BB

@DD

klì 0.=5ekg s`stkj

klì 0.=5 (b) hkl` () ekg s`stkjhk MM ìktlkg`ghkg @MM mlj, (m)hkvrks`ghkl` `l` ìktlkghkg ekkg mljimrg vkltr yki cmrlgm k`t aihuu

erir.

@MM: @mM-@bB@MM mljgklì 0.=5 (b) hkvrks`ghkl` M hj galtmsgm l`racug mljlmgugugu uyeuimymrml hm buimbì`r`z.

@MM+ @bB: @mM @MM: @mM @bB

BB nmttghmg ekkg @BB mlj ìk DD nmttghmg ekkg @dD mlj hm mygklìhk buiugmbì`r.B galtmsgm uyeuimgmg L`rac lmgugughmg,

@BB: @bB @dD D galtmsgm uyeuimgmg l`rac lmgugughmg,

@DD: @dD @mMMitkrgmtr ylk bmimymg nmttmg ekkg @BB @MM vk @DDmljimrgm lsmdmNMX ML@J@ hkg`r. Mitkrgmtrg cmz smreimrghmg ekkg mljimrm hm CM_ML@J@ hkg`r. klì 0.=5 (m) hmg (b) yk ekìh``ghk cmz kjlikr` ìk cmzimr mrmskjlikr mrmsghm u ì`lìkr ymzimbì`r (m) hm l` e`b` bmimymij Cmz kjpkhmgsimrgg mimr`sk,



18/48

=>;

mitkrgmtrg nkr cmzghmg ekkg mlj, cmz ekrì`j`ghkg lmhmr ekr` lmir.

Mitkrgmtrg cmz vk cmzimr mrms ekrì`jikr`g`, cmz vk nmt mljimrg estkrkgvkltr h`ymermjklì 0.=> (b) hk eriyar.

KDM

M

B

DM

B

D

@M

KMBKBD

@DM@BD @MB

@B

@D

_=

_2

_2

klì 0.=> (m) Kjpkhmgsi hkgeki` ekg ylg ekg bmi mitkrgmtrhkgbksikg``

6

KBD

KDM

KMB

@dd@BD

-@BD

-@DM

@DM@MB

-@MB

-@MM

-@BB

klì 0.=> ( b)ekg s`stkjhk kjlikr vk cmz ìk nmt mljimrvkltr h`ymermj

8. CmziT`stkjhk E

m. Hkgeki` yli Pihz T`stkjhk E

B`r cmziM.M hm e V : [.@ . das vmttr. cmziP bmihkgeki` b`r yltknmrdmgmg e, b`r cmzg edgg 4 lmtgm k`tt`r.

Vy : 4 [c . @c . Das



19/48

=>3

cmzi ylk smrc khìkg ed, nmt mlj vk cmzimr mrms ekrì`jik `cmhk

khkbìjkl ``g [c : [ / 4 < @c : @ hkkrikr`g` k`ti`ltkl` ykrikr`gk ymzmij

E

4

M (R)

B (T)

D (X)

6

KDM

KMB

KBD

klì 0.2=Cmz srmsMBD

3. Hkgeki` CmziPlikr

cmzib`r M.M kbklks`gk bmimgmg hkgeki` yihz vk ekg ylikrhkg ekkgcmz vk nmt mljimrgg buiugjmsg, mlj vk ekrì`j vkltr h`ymermjimrgg`zìjks`g` vk kbklkhkg klìkg edg nksmpimgjmsg rgkl prabikjikrzkrkl mlimjmym mimij

rgkl0.2

Cmz ekrì`j` =26S aimg cmziP bmib`r mitkrgmtrk bmimgmg hkgeki` 4 cmziyl 86 lU klì`yar. Plg e lmtsmys6,>;; ekr`h`r. Nmt vk cmz mljimrgnksmpimygz.

zj0.2

S[c[ 26>=26.44

M=46@c@

M=46>;;,6.26>.4

=666.86

[Das4

V@

rgkl 0.4

klì 0.22 hkl` hkgeki` 4 cmziylg 4 cmzikbklkhkg klt`` nmt mljgvkylg cmzimrghmg ekkg mljimr nksmpimygz. Ekrì`j vk mljimr vkltrh`ymermjg`z`g`z.



22/48

=>0

[MB :==6/-=26a

[BD:==6/6a

[DM:==6/=26a ; /8; a

; /8; a

; /8; a

@M

@D

@B

@MB@DM

@BD

M

B

D

[DM

[BD

[MB

6

@d

@DM

-@DM

-@BD

@BD

@MB

-@MB

@B

@M

8;a=;a

8;a

=;a

8;a

=;a

m) (b)Skltr h`ymermj

kl`i 0.224 cmzihkgeki` yl vk vkltr h`ymermj

zj 0.4

ekg ylg nkr cmzghmg ekkg mljimrbuimij .

5,;2,2==3;228;;

=26==6oM

_

[@ MBMB

;;,=;;;,=;8;228;;

6==6oM

_

[@ BDBD

2,2=5,;5;228;;

=26==6oM

_

[@ DMDM

Plg lks`gk l`racug mlj lmgugu uyeuimymij , =4;=,4>5;22=3;22DMMBM @@@ =;=,4>=3;228;22MBBDB @@@

=6;=,4>8;225;22BDDMD @@@



23/48

=06

rgkl 0.8

rgkl prabikj 0.4 lajpikls smyimrluiimgjmhmg zgz.

zj 0.8

ekg bmi ylg nkr cmzgm uyeuimgmg ekrì`j kbklkg`g cmzimr mrmsekrì`j`gk k`tt`r.

Cmz mlj, M_

[@c 22

;

==6

Plg nkr cmzghmg 22M ekt``ghk kbklkhkg klìkg nmt mljimrb`r b`r`gkk`t vk cmz mljgg 4 lmtaimdmltr.

M[ =,4>4.22 Cmz kjpkhmgsimrgg ms8; aihuugm erk cmz mljimrekrì`jhkg 8; ekr`lmir. =26 cmz cmrlinmt ekrì`jikr`ghkg 8; kr hkrkdk ekr` lmimg cmz mljimrmrmsghm hm =26 kr hkrkdk cmz cmrlaiur. Ekrì`jikr vk cmz mljimrvkltr ykiaimrml estkrìh`ltkg sagrm `z`j yaiu ìk hk @M, @Bvk @Dnmt mljimrklì 0.22(b) hk erih e`b` `zìkbì`r.

rgkl 0.;? klì 0.24 (m) hml` hkgeki` yihz ylg, (m) kbklkhkg klt``mljimr vk gtr mljg (b) klìkg ed buiuguz. (d) vkltr h`ymermjg`z`g`z.

_

__

26/-46a

26/-46a

26/-46a

@D@D

@B

@B

@M@M

[AM:=26 /6a

[AD:=26 /=26a

[AB:=26 /-=26a

M

6

B

D

[aD

[aM

[aB

6 46a

46a

46a

@D

@M

@B

klì 0.24Hkgeki` yihz yl vk vkltr h`ymermj



24/48

=0=

zj 0.;

(m) @M cmz mlj ( nmt mlj) [AM cmz ekrì`j`ghkg (46) ìkr`hk aiur. gl,kjpkhmgsg ms(-46) aihuugm erk bu kjpkhmgs skr` R D hkg jkyhmgmekij`t`r. Plg h`kr cmz kjpkhmgsimrhm mygaihuu ``g tkl` cmz mljimrhm cmz ekrì`jikr`ghkg 46mr hkrkdk ìkr`hk aiurimr.

M@@@ DBM 326

=26

Gtrhkg ekkg mlj @6, =26 cmz cmrli@M, @Bvk @Dmljimrgg vkltr ykitapimjgm k`tt`r. Bu tapimjhm scrhr. @6: 6

b)V : 4 . [c . @c . Das: 4.=26.3.Das 46a: =>56 vmt

d)Skltr h`ymermjgg `z`j`? [AM vkltr rkckrmgs klskg`ghk migr. [AB=26 ekr`hk, [ADhk=26 ìkr`hk vkltrikrik estkrì`r. 3M i`l cmz mljimrhm cmzekrì`jikr`ghkg 46 ìkr`hk migr.

rgkl 0.3 ?klì 0.22 (m) hm l` hkgeki` ekg ylg kbklkhkg klt`` nmtmljgekg yl khkkr yihz ylk kv`rkrkl buiuguz.

zj 0.3 ? Hkgeki` ekg ylg cmz kjpkhmgs 8;;_ aihuugm

erk, khkkr yihz ylg cmz kjpkhmgsimr

8;4

;

4

__y aiur. klì 0.28

(m) hm khkkr yihz ylg kbklkyk bmimgeriyar.

klì 0.28 (m) hm l` P ylg nkr cmzgm uyeuimgmg ekrì`j cmzimr mrmsekrì`j`g(=/ 4 ) gk k`t aihuugm erk b`r cmzg khkkr hkvrks`g` klì 0.28 (b) hk lè`b` `zkbì`r`z. Bu khkkr hkvrkhkg nmt mljlaimydm buiugur.

_=

_4_2

;/4 /8;a

@D

@B

@M

[MB:==6 /-=26a

[DM:==6 /-=26a

M

6

B

D

[BD:==6 /6a

;/4 /8;a

;/4 /8;a

_=[c: 4/==6 ;/4/8;a

@

(m) (b)Pihz ylg b`r cmzgg khkkr hkvrks`

klì 0.28Khkkr yihz yl



25/48

=02

M_

[c@

8;=,4>8;

4

;4

==6

Nkr cmzg 4>,=M i`l nmt mljimr, cmz ekrì`jikr`ghkg 8; ekr` lmir.

5 . Hkgeks`z CmziPlikr

cmziM.M kbklks`gk bmimgmg hkgeks`z yihz vk ekg ylikrhkg ekkgcmz vk nmt mljimrgg vk ylg klt`` edg nksmpimgjms, vkltrh`ymermjgg `zìjks` e`b` laguimrrgkl prabikjikr zkrkl mlimymij.

rgkl 0.5

cmziekg bmib`r stdgg aj`l h`rkgikr` R=: 26, R2: 86vk R4:2;hur. Cmzimr mrms866S aimg cmzikbklkyk bmimgmg ylg klt`` nmtmljgvk ylg cmzimrghmg ekkg mljimrnksmpimygz.

zj 0.5

klì 0.2; (m) hm ylg kbklkyk bmimgeriyar. R=h`rkgd`ghkg ekkg@=mlj[RTcmzimr mrmsekrì`j` ìk mygcmzhmhr.

M@ 2626866

=

R2h`rkgd`ghkg ekkg @2mlj[TXcmzimr mrmsekrì`j` ìk mygcmzhmhr.

M@ =686

8662

R4h`rkgd`ghkg ekkg @4mljhm [XR cmzimr mrmsekrì`j` ìk mygcmzhmhr.

M@ =32;

8664



26/48

=04

866S

R2:86

@R

@X

@T

@4

@2

R

T

X

@=866S R

4:2;R

=:26

M

B D

[ X R

[R T

[ M B

6

@X

@ 2

@ 2 : = 6 M

@ 4: = 3 M

- @4

- @=

@= : 2 6 M

c

36a

36a

36a

36a

36a

36a

h

@R@T

(m) (b)Skltr h`ymermj

klì 0.2;Hkgeks`z ekg yl

[RT ekrì`j`g` rkckrmgs klskg`ghk mimrml , [TX vk [XR ekrì`j vkltrikr`g``zh`ltkg sagrm ekrì`jikrik myg cmzhm aimg cmz mljimrgg hm vkltrikr`g``zh``j`zhk klì 0.2; (b) hk erikg vkltr h`ymermjkihk khì`r.M,B vk D hj galtmimrgm L`racug mlj lmgugugu uyeuimymrml nmtmljimrgbuimij?

M hj galtms? @R: @= @4 ( vkltrki )B hj galtms? @T: @2 @= ( vkltrki )D hj galtms? @X: @4 @2 ( vkltrki )

@R: @=+ (-@4) ymzimbì`r. @=vkltr ìk (-@4) vkltrgg tapimj@Rmljgvkr`r.H`kr mljimr ``ghk myg ìkjikr ympir. klì 0.2; (b) hkl` vkltrh`ymermjghm bu vkltr yki tapimjimr estkrìj`t`r.

@Rmljg, ahc h`l ekg`gk V`smear tkarkj`g` uyeuimymrml buimbì`r`z.

2

4

2

4= )36s`g.()36das. @@@

M4,4=>;,=42>)>33,6.=3();,6.=326( 2222

@R mljgg @x vk @y bìkkgikr`g`, @= vk (-@4) vkltrikr`g`g x vk y klskg`ghkl`

bìkkgikr`ghkg buimrml , @R mljg22

yxR @@@ carji ìk hk

nksmpimymbì`r`z.

222

=

2

=2 )>33,6.26();,6.26=6()36s`g.()36das.( @@@@T



27/48

=08

M@T ;,23)42,=5(2622

222

2

2

24 )>33,6.=6();,6.=6=3()36s`g.()36das.( @@@@X

M@X 5,2233,>2=22

rgkl 0.>

klì 0.23 (m) hm erih e`b` yihz bmi hkgeks`z b`r yl, cmzimr mrmsekrì`j` 4>6S aimg cmzihrt nmtib`r kbklkyk bmimgjtr. m) Nkr cmzhmgekkg mlj b) gtr nmttghmg ekkg mlj d) Plg klt`` tapimj edbuiuguz.

zj 0.>

m)R=h`rkgd`ghkg ekkg @Mmlj[AMcmz ekrì`j` ìk mygcmzhmhr.

M@M 22=6

226

R2h`rkgd`ghkg ekkg @Bcmz mlj[ABcmz ekrì`j` ìk mygcmzhmhr.

M@B =6

22

226

R4h`rkgd`ghkg ekkg @Dcmz mlj[ADcmz ekrì`j` ìk mygcmzhmhr.

M@D 3,=5;,=2

226

[AM, [AB vk [AD cmz ekrì`jikr`g` =26kr hkrkdkhk cmz cmrli vkltrikestkrìh`ltkg sagrm, @M, @B vk @D mljimrg hm vkltr yki aimrml `zh``j`zhkklì 0.23 (b) hk l` vkltr h`ymermjkihk khì`r.

b) Gtr nmttghmg ekkg mlj @M, @B vk @D mljimrgg vkltr yki tapimjgmk`tt`r. Cmz mljimrgg x vk y klskgikr`ghkl` bìkkgikr`gk myrmij. klì 0.23(b)Gtr mljgg x klskg`ghkl` bìkkg` @xg: @M (@B+ @D) Das 36



28/48

=0;

R=:=6

@D@D

@B

@B

@M@M

[AM:226 /6a

[AB:226 /-=26a

[AD:226 /=26a

M

6

B

D

@g

R4:=2,; R2:22

@B

[AB

4>,>a

=26a

36a

46a

[AD

@D

@G

[AM@M

y

x

klì 0.23Hkgeks`z yihz yl vk vkltr h`ymermj

@xg: 22-=6.6,; =5,3.6,; : >,2

Gtr mljgg y klskg`ghkl` bìkkg`@yg: @dT`g 36

a@BT`g 36a: =5,3.6,>33-=6.6,>33 : 3,;0

M;,=6);0,3()2,>(@ 22

G

b. kbklkhkg klìkg tapimj e, nkr cmzg klt`` eikr`g tapimjgm k`tt`r.

V : V=+ V2 + V4 V : 226.22 + 226.=6 + 226.=5,3 : =60=2vmt

0.8 E IJKL

Nkrnmge` b`r ylg kbklkhkg klt`` e vmt jktrk ìk iir. B`r cmzi b`rylg klt`` ed ijkl ``g vmt jktrkg`g mlj bab`g` hkvrkyk skr`, vmtjktrkg`g ekrì`j bab`g` hk hkvrkyk pmrmiki aimrml klì 0.25 hk erih e`b`bmimgr. Smt jktrkg`g mlj bab`g`ghkg ylg klt`` mlj ekt``, ekrì`jbab`g`gk hk ylg uimrghml` ekrì`j uyeuimgh ``g i mikt`g`g `brks` vmtaimrml ylg klt`` ed estkr`r.



29/48

=03

_=

U

[

@

_M

_D

_B A

M

B

D

U

U

U

cmziyihz bmib`r ylg cmziM.M kbklks`ghkg klt`` tapimj edijkl ``g ylg nkr cmzgm klì 0.2> (m) hm erih e`b` b`rkr vmtjktrkbmimgr.

cmziekg bmib`r ylg kbklkhkg klt`` tapimj ed ijkl ``g hk

ylg nkr cmz

gm klì 0.2> (b) hk erih e`b` b`rkr vmt jktrk bmimg

r.Pihz vkym ekg bmiyl hkgeki` `sk, vmt jktrkikr`g estkrh`` cmz eikr` b`rb`r`gk k`t aiur. kbklkhkg klìkg tapimj e vmt jktrkikrhkg b`r`g`gestkrh`` hkkr`g (b`r cmzg ed) lmtgm k`t aiur. Plikr hkgeks`z `sk,kbklkhkg klìkg tapimj e vmt jktrkikr`g estkrh`` hkkrikr`g tapimjgmk`tt`r.

Hkgeki` ylg klt`` tapimj e ? V : 4.V=Hkgeks`z ylg klt`` tapimj e ? V : V=+ V2+ V4


klì0.25Smtjktrk ìk eijkl

klì0.2>m)Pihz ylg cmzeikr`g`g iijks`


30/48

=05

_4

_2

_=

M

B

D

M

B

D

U

U

U

CM_I@

PL

Tug` Gtr[

[

klì 0.2> cmziylg edg ijkl

Pihz ylg scr galtms likjkgs tmbiasugm (u bmimgt galtmsgm), ekgylg hk cmz uimr myr myr likjkgs tmbiasugm lmrijmj `sk, nkr cmzgedg ikdkl klìhk vmt jktrkikr` klì 0.2> hk erih e`b`bmimymjmyz.

Bu hurujhm yihz vk ekg ylikr`g kbklkhkg klt`` ed ijkl ``g tmgk vmt jktrk klì 0.2> (d) hkl` e`b` bmimgr. Burmhm tmgk kvmt jktrkg`gekrì`j bab`gikr`g`g b`rkr uimr cmzimrm ekrì`j bab`gikr`g`g h`kr uimr hm b`rb`r`gk bmimgjtr. Byikdk jkyhmgm ekt`rìkg sug` gtr galtms ìk nkr vmtjktrkg`g ekrì`j bab`g`gk ( [/ 4 ) ekrì`j` uyeuimgjaiur. Smt jktrk b`r tmgk`sk , k`t h`rkgd` y bmimyp MBD cmzimruyeuimghghm yihz galtmssug`gtr aiur. Smtjktrk ìk nkr cmzg ed iikbì`r.

Pl hkgeki` aiursm, vmtjktrkikr`g mlj bab`gikr`ghkg ekkg nmt mljimrmyg

aimdm

``g vmtjktrkikr`g estkrh`` hkkrikr hk b`r b`r`gk k`t aiur.Smtjktrkikrhkg b`r`g`g estkrh`` hkkr`g lmt kbklkhkg klìkg tapimjed vkr`r. Pl hkgeks`z aiursm, vmtjktrkikr`g mlj bab`gikr`ghkg ekkgmljimr cmrli aimdm ``g vmtjktrkikr`g estkrkdklikr` hkkrikrhk cmrli aiur.Smtjktrkikrhkg alugmg hkkrikr`g tapimj kbklkhkg klìkg tapimj edvkr`r.V : V=+ V2+ V4


(m)ekg ylg cmz eikr`g`giijks`

(b) cmziylg klt`` edg vmt jktrk ìk iijks`


31/48

=0>

m) l` Smt jktrk Jktahu ( Mrag Bmimgts)

cmzihkgeki` vk hkgeks`z ylikr`g kbklkhkg klt`` eikr`g iijks`ghkkg al luiimgimg `l` vmtjktrk jktahuhur. Bu jktahug h`kr jktatimrmstgilikr` ugimrhr?

m) Pihz vk ekg ylikrk mygbmimgtuyeuimgr.b) Hkgeki` vk hkgeks`z ylikr`g klt`likr` eikr` hkì `l` vmtjktrk

ìk ijkj`z` smimr.d) Hkgeki` ylikr`g e lmtsmyimrvmtjktrkikr`g estkrh`` hkkrikrhkg

buiugmbì`r.

=. Hkgeki` Pihz Pl

Hkgeki` yihz kbklkhkg klt`` ed ijkl ``g `l` vmtjktrk klì 0.20 (m)hm erih e`b` bmimgr. B`r`gd` vmtjktrkg`g mlj bab`g`ghkg ekkg mlj @Mekrì`j bab`g`gk uyeuimgmg ekrì`j hk [MB h`r. l`gd` vmtjktrkg`g mljbab`g`ghkg ekkg mlj @D, ekrì`j bab`g`gk hk [DB cmzimr mrms ekrì`jùyeuimgjtr. Smtjktrkikr`g estkrh`` hkkrikr` buijml ``g mljimr vkekrì`jikr` estkrkg vkltr h`ymermjg`zjkyk mimij. Hkgeki` yihz ylgekr` e lmtsmyi aihuugu lmbui khki`j. Pl hkgeki` aihuu ``g [MA, [BA,[DA cmz ekrì`jikr` b`r b`r`gk k`t ( [/ 4 ) vk =26 kr hkrkdk cmz cmrlihr. =26cmz cmli vkltrik cmz ekrì`jikr` `zìh`ltkg sagrm nkr cmz ml j cmzekrì`j`ghkg lmhmr ekr`hk aimrml migr. B`r`gd` vmt jktrkyk uyeuimgmg [

MB

ekrì`j` [MA vk [AB cmz ekrì`jikr`g`g vkltr yki tapimj migmrml `zì`r. l`gd`vmt jktrkyk uyeuimgmg [DBekrì`j` hk, [DAìk [ABcmz ekrì`jikr`g`g vkltr ykitapimjmigmrml `zìh``ghk klì 0.20 ( b ) hkl` vkltr h`ymermjkihk khì`r.

_M

_D

_ B A

M

B

D

U

U

@M

@D

@B

V =

V 2

46 a

46 a

@D

@M@B

[DB

[MB[DA

-[ BA

[MA

(m) l` vmt jktrkg`g bmimg(Mrag Bmimgts) (b)Skltr h`ymermj

klì 0.20Hkgeki` yihz ylg `l` vmt jktrk ìk edgg iijks`



32/48

=00

Skltr h`ymermj `gdkikgh``ghk, [MB cmzimr mrmsekrì`j` ìk [MA cmz ekrì`j`,

[DB cmzimr mrmsekrì`j` ìk [DA cmz ekrì`j` mrmsghm 46mr hkrkdki`l cmz cmrlaihuu erir. B`r`gd` vmt jktrkyk uyeuimgmg [MB ekrì`j` ìk @M mljmrmsghml` cmz cmrl (46+), `l`gd` vmt jktrkyk uyeuimgmg [DB ekrì`j` ìk @Dmljmrmsghml` cmz cmrlhm (46-) h`r.V=: [MB. @M. Das(46+)

l`gd` vmt jktrkg`g estkrh`` hkkr33 ìkr`

aiugdm 46a

aiur. E carjiikrk : 46a

ymzmij.

V=: [ .@ Das(46+ 46 ) : [.@ Das36 : [.@ / 2

V2: [.@ Das(46 46 ) : [.@ Das 6 : [.@ buiugur.

B`r`gd` vmt jktrk tapimj edg (=/4) g, `l`gd` vmt jktrk `sk tapimj edg (2/4) g estkr`r.

E lmtsmys6,>33 ekr` aiursm, : -46 aiur. B`r`gd` vmt jktrkhkg V=: [.@Das6: [.@ hkkr`, `l`gd` vmt jktrkhkg hk V2: [.@ Das36 : ( = / 2 )[.@ hkkr`

alugur.

d) Das : 6,; ekr` aimg yl ? Das :6,; , : 36 aiur. Smtjktrkikr`g estkrh`likr` hkkrikr.V= : [.@ Das (46 36 ) : [.@ Das (-46 ) : ( 4 /2) [.@V2 : [.@ Das(46 +36) : [.@ Das 06 : 6 buiugur.

l`gd` vmtjktrk 6 hkkr`g` estkr`rlkg b`r`gd` vmt jktrk ylg klt`` tapimjed estkr`r

h) Das 9 6,; aimg yl ? Plg e lmtsmys 6,; hkg llaiursm, ms36 hkg byl aimdm``g e `cmhks`ghkl` Das(46 + ) hkkr` 06 hkg byl aiur

06 hkg byl mimrg las`gs gkemt`c aihuu ``g vmtjktrkikrhkg b`r`g`gestkrh`` hkkr gkemt`ct`r. Bu, vmt jktrkikrhkg b`r`g`g tkrs smpjmshkjklt`r.B`r`gd` vkym `l`gd` vmt jktrkikrhkg nmge`s`g`g tkrs smpmdm e lmtsmysggìkr` vkym ekr` (ymg` msgg + vkym - ) aiuugm erk hk``r.

Plg klt`` tapimj ed buijml ``g tkrs smpmg vmt jktrkg`g ekrì`jbab`g`g`g uimrhk`t`rì`r. Sk vmt jktrkg`g estkrh`` hkkr alugur. H`kr vmtjktrkg`g estkrh`` hkkrhkg tkrs smpmg vmt jktrkhkg alugmg hkkrlmrimrml tapimj e buiugur.

;. l` Smtjktrk ik E Lmtsmysgijkl

Hkgeki` cmzi ylikrhk vmt jktrkikr`g estkrh`` V= vk V2 hkkrikr`ghkgylikr`g lmtsmyimrbuiugmbì`r. Smt jktrkikr`g estkrh`` hkkrikr,

V=: [[email protected] (46+ )

V2: [[email protected] (46 ) bu `l` hkkr`g cmrliul kghltmgs vk gd cmzghm 5iul b`rlmpms`tmgs vmrhr. Plg klt`` tapimj ed `l` vmt jktrk jktahu ìk ijkl``g ekrkli` bmimgtympijtr. (klì 0.4=) m) Plg kbklkhkg klt`` cmz vknmt mljimrg, b) M vk D cmz nmtimrgm bmiaimg vmt jktrkikr`g estkrh`likr`hkkrikr` nksmpimygz.

zj 0.0

Nkr cmzhmg ekkg mlj vk cmz ekrì`jikr` ìk cmz mljimr mrmsghml` mimrbuimij.

m) M@MB 26=6

266 , [MBìk mygcmzhm

M@BD 2;>

266 , [BDhkg 06 ekr`hk



38/48

26;

;5,2>5

266

DM

@ , [DMhmg 06 ìkr`hk

klì 0.4= (b) hkl` vkltr h`ymermjghm =26 cmz cmli [MB, [BD vk [DM cmzekrì`jikr` `zìh`ltkg sagrm cmz mljimrmimrgm erk `zìj`t`r.

M

B

D

U

U

V=

V2 @D

@B

@M

@MB

@DM@BD

5=6

>

266/=26a266/6a

266/-=26a

[DM -[BD

[MB

[BD

@B @BD

-@MB @MB

@D

@DM @B

-@BD@DM

46a

03,3a

>4,8a

46a

46a

36a

46a 46a46a

klì 0.4=Hkgeks`z ekg yl, mlj vk ekrì`j vkltr h`ymermj

M, B vk D galtmimrgm L`racug mlj lmgugugu uyeuimymrml nmt mljimrgnksmpimymij.

@M: @MB @DM , @B: @BD @MB @D: @DM @BD

Bu vkltryki tapimjimrvkltr h`ymermjzkr`ghk ympmij.



39/48

263

MT`gDas@M 05,83)46.;,2>()46.;,2>26(22

MT`gDas@B 83,84)46.2;()46.2;26(22

MT`gT`gDasDas@D 03,23)46.2;46.;5,2>()46.2;46.;,2>(22

b)B`r`gd` vmt jktrkg`g estkrh`` hkkr,

[MBìk @Mmrmsghml` m, te =: =8,2;/84,5 : 6,42 =: =5,5a

V=: [MB. @M. Das =: 266. 83,05 . Das =5,5a : >080 vmtt

l`gd` vmt jktrkg`g estkrh`` hkkr , =26266[[ BDDB : 36266 8,>4

,4

5,23mrdte

3,030,23@D

[DBìk @Dmrmsghml` 2:36+03,3:=;3,3

V2: 266 x 23,03 .Das=;3,3 : 266.23,03 . (-6,0=>) : - 8080vmt

Plg klt`` tapimj e , V : >080 8080 : 8666vmt.

ekg yltkl` kic`g vk laghmgsmtrg klt`` eikr scrhr. A nmihkkbklkhkg klìkg e =6iul h`rkgtk skli`ghk smrc aiur.

V : R . @MB2: =6 .(26)2: =6.866 : 8666 vmt

u nmihk, yulmrhm vmt jktrkikr`g estkrh`` hkkrikrik buihuujuz eharuhur.



40/48

265

0.; CM_I@ TTXKJHK E LMXTMP@T@G@G H_KIXIJKT

P L d a s E K R

R ( M )

T ( B )

X ( D )

A

D P

@= @

@D

@D

6@U [

@=

@

@j=

@j

(m) (b)

klì 0.42E lmtsmysgg P bmilaghmgsmtrik hzkitìjks`

E lmtsmysal ll aimg cmzihkgeki` b`r ylg e lmtsmysg, klì0.42(m) hm erih e`b`, laghmgsmtr bmimymrml hzkitjkl `st`yaruz. Plge lmtsmysDasekr` vk klt`` nmt mljhm @ hr. klì 0.42 (b) hkl` vkltrh`ymermjghm erih e`b` @ mljcmz ekrì`j`ghkg lmhmr ekr`hkh`r. @ mlj

ekrì`jik mygcmzhm aimg @U(vmtimlj vkym rkki mlj ) vk ekrì`jk 06

h`l. @j(jlgmtsimjm mlj) bìkkgikr`gk myrir. T`stkj`g e lmtsmysgDas = kylskitkbìjkl ``g hkvrkyk bmimgmg laghmgsmtrg klkdk` mlj @D: @j@j=aijmihr.Plg kbklkhkg klt``,Ergr e ( zmn`r` e) , T : 4 . [.@ . =6-4lSMNml`l` e (mlt`c e) , V : 4 . [ . @ Das. =6-4lURkmlt`c e ( lr e ) , ] : 4 [ . @ .T`g . . =6-4lSMr

] D ( l S M )

6V ( lU )

T = ( l S M )

T

] = ( l S M r )

] ( l S M r )

klì 0.44Ergr, mlt`c vk rkmlt`c eikr vkltr h`ymermj



41/48

26>

klì 0.44 hk eikr vkltr h`ymermj`zìj`t`r. Nml`l` e rkckrmgs (x klskg`)

klskg`ghk, rkmlt`c e y klskg`ghk mighltmg sagrm bu `l` hkkr`g vkltrkitapimjmigmrml ergr e buiugur.

Plg lmtsmys Das = k ylskih``ghk kbklkhkg klt`` nml`l` e myglmir. kbklkhkg klìkg rkmlt`c e ]= hkkr`gk hkr. Hkvrkyk bmimgmglaghmgsmtrikr`g vkrkdk` ]drkmlt`c ed, ylk imzj aimg ] rkmlt`c edg b`rlsjg lmrimh ``g kbklkhkg klìkg rkmlt`c etk mzmijm aiur.Haimysyim e lmtsmyshm byr.

Plg lmtsmys Das hkg Das = k lmrjml ``g ekrkli` aimg ]drkmlt`c ed (]-]=) h`r.]d: ] ]=

Pihz bmi laghmgsmtrg kbklkhkg klt`` nmt vkym cmz mlj @d, cmzimrmrms ekrì`j [ aihuugm erk laghmgsmtrikr`g vkrh`` tapimj rkmlt`c edbuimij.

]d : 4 . [c . @d: 4 .(4

[ ) .@d : 4 . [ . @d

]d : 4 . [ . @

Pihz bmilaghmgsmtr erubugug b`r cmzghmg ekkg mlj,

Dy

[

^d

[c@d

=4 , Dy@d

4

4 buiugur.

Dy[

[]d .4

..4

, ]d : [2. . Dy

Pihz bmi laghmgsmtr erubugug b`r cmzghml` laghmgsmtrikr`g lmpms`tks`yulmrhml` carjihkg buiugur.

2[

]dDy Dy : cmrmh , ]d : rkmlt`c e, SMr ( vait mjpkr

rkmlt`c): msmi nz , 2c < [ : cmzimr mrms(nmt) ekrì`j`,

vait

Laghmgsmtr erubugug vkrh`` e lvmr (lSMr) ìk `cmhk khkrskl, j`racmrmhaimrml nkr cmzg lmpms`tks`,



42/48

260

c[

]d

]y

=6.

2

0

carji ìk nksmpimg

r.

klì 0.42 hk ylg e lmtsmysg hzkitjkhk luiimgimg laghmgsmtrikryihz bmihrimr. Laghmgsmtrikr ekg bmiaihulimrgm erk, klì 0.48 hk lè`b`, laghmgsmtr erubugug b`r cmzgg lmpms`tks`g` vkrkg carji lmrtmij.SM aimrml laghmgsmtr eurubugug ed,

]d : 4 . [ .@d Nmt mlj@d: 4

^d

[

@d: 4 . [. D @d g`g bu hkkr`g` (3=) hk ykr`gk

ymzmi

j.

]d : 4 . [2. D

B`r cmzgg lmpms`tks`

24[

]dD

D: cmrmh, ]d : SM (vait mjpkr), [ : cmzimr mrms ekrì`j, vait :msmi nz, rmhymg/smg`yk,

Lmpms`tky` j`lracmrmh vk ]d y` hk (lvm) aimrml `cmhk khki`j.

c4

=6.

2

0

[

]dD

2[

]dDy carji c

4

=6.2

0

[

]dD ìk lmrimtrirsm, tapimj eikr` myg

aimg laghmgsmtrikr`g yihzhml` b`r cmzgg lmpms`tks`g`g ekg bmimjmhml`cmz lmpms`tks`g`g 4 lmtaihuu erir.

Dy : 4 D



43/48

2=6

PL das EKRT

X

A

D

@= @

@D

R

klì 0.48ekg bmilaghmgsmtrikrik e lmtsmysgg hzkitìjks`

B`r cmbr`lmgg kbklkhkg klt`` nml`l` e V lìavmt vk e lmtsmys Dasekr`h`r. Cmbr`lmgg e lmtsmysg Das= k ylskitjkl `st`yaruz. Cmbr`lmymjagtk khkdk`j`z yihz vkym ekg bmi laghmgsmtr eurubuguglmpms`tks`g` nksmpimjmym mimij.

klì 0.44 hkl` eikr vkltr h`ymermjghml` ], ]=rkmlt`c eikr`g` V ( lw), vk =bì`gkgikr` ymrhjìk buimij.

] : V . te , ]=: V . te =ekrkli` aimg laghmgsmtr eurubugug ed< ]d:]- ]=, ]d : V (te te=)

Pihz bmilaghmgsmtr eurubugug b`r cmzgg lmpms`tks`g` ( Dy) buijml ``g]d ykr`gk ymzmij.

).(.

=6.=2

0

tete[

VDy

Dy : B`r cmzgg lmpms`tks`, (j`lracmrmh)V : Cmbr`lmgg vkym nkr nmge` b`r ylg nml`l` ed, lw (lìavmt)[ : cmzikbklkg`g cmzimr mrmsekrì`j` (nmt ekrì`j`), vait: 2 c , M.M. g msmi nz, rmhymg/smg`yk: Cmbr`lmgg hl aimg e lmtsmysgg ms, hkrkdk=: T`stkj`g hzkitt`` e lmtsmysgg ms, hkrkdk



44/48

2==

ekg bmi laghmgsmtr eurubugug b`r cmzgg lmpms`tks`g` D buijml ``g

]d ykr`gk `cmhky` ymzmij.

).(.4

=6.=2

0

tete[

VD

rgkl 0.==

=666 lw, Das : 6,3 ekr` aimg b`r cmbr`lm 4>6/226 vait, ;6 Nz i` cmzi b`rkbklkhkg bksikg`yar. Cmbr`lmgg e lmtsmysg Das = : 6,0m lmrtjml``g laghmgsmtr luiimgjml `st`yaruz. m) Cmbr`lmhml` kbklk e`r``gkbmimymdmjz laghmgsmtr eurubugug lvm aimrml edg, b) Laghmgsmtreurubu (P) bmi aihuugm erk b`r cmzgg lmpms`tks`g` Dy : 1, d)Laghmgsmtrikr ekg bmimghghm b`r cmzg lmpms`tks`g` D : 1nksmpimygz.

zj 0.==

m) V : =666 lw , Das : 6,3 Das = : 6,0 ug trèagajktr`dktvki`ghkg mimrgvk tmgekgtikr`g` buimij.

: ;4a

< te ;4a

: =,44 < =: 23a

< te 23a

:6,8>>]d : V (te te=) : =666 ( =,44 6,8>>) : >42 lvm

b) c[

]dDy =>;66

;6.2.4>6

=6.>82

.

=6.2

0

2

0

d) c[

]dD 3,3=33

;6..2.4>6.4

=6.>82

..4

=6.2

0

2

0

cDy

D 3,3=33

4

=>;66

4

0.3 LMR@@L RGKL VRABIKJIKR

rgkl 0.=2

cmzib`r mitkrgmtrg cmz ekrì`j` 24= S vk cmz mlj4> M h`r. Nmt ekrì`j` vknmt mljgmitkrgmtr m) ekg b) yihz bmimghgm erk nksmpimygz.



45/48

2=2

zj 0.=2 ?

bmimgt? [ : [c : 24= S@ : 4 . @c : =,54 . 4> : 3;,> M

P bmimgt? [ : 4 . [c : =,54 . 24= : 866 S@ : @c : 4> M

rgkl 0.=4

4 cmzi b`r kikltr`li` stdgg nkr cmzgg h`rkgd` =2,; hur. 4>6 S iul nmtekrì`j`ghk stdm) Pihz b) ekg bmimghgm erk cmz ekrì`j`g`, cmz ìk nmtmljimrgvk eikr`g` nksmpimygz.

zj

P bmimgt

S2264

4>6

4

[[c M3,=5

;,=2

226

R

[c@c

@ : @c : =5,3 MVy : 4 .[.@ : =,54 . 4>6 . =5,3 : ==366 vmt

bmimgt[c : [ : 4>6 S

M8,46;,=2

4>6

R

[c@c @ : 4 .@c : =,54 . 46,8 : ;2,5 M

V: 4 . [ .@ : =,54 . >6 . ;2,5 : 48>666 vmt

rgkl 0.=8

rgkl 0.=4 hkl` cmzistdgg bksikgjks`ghk klì 0.4; hkl` e`b` nmttg b`r`lksìh`` zmjmg nmt mljimrvk eikr gk aiur1

zj

P bmimgtRk : 2 R : 2 . =2,; : 2;

M2,=;2;

4>6

Rk

[@=

V=y : [ . @=: 4>6 . =;,2 : ;553 vmt



46/48

2=4

R

RR

R

T

X

[

R

T

X

[R R

R

m)P bmimgt b)ekg bmimgt

kl`i 0.4;

bmimgt?

44,>;,=2;,=2;,=2

;,=2.2;).(Rk

RRR

RRR

M[

@ 3,8;44,>

4>6

Rk=

V=: [ . @=: 4>6.8;,3 : =542> vmt

rgkl 0.=;

224 S 4 cmzib`r kbklkhkg ekg bmib`r stdgg klt`` nmt mlj=;,8 Maihuugm erk cmz h`rkgikr`g` vk edg nksmpimygz.

zj 0.=;

=,2;

4

8,=;

224

4

@

[Rc

V : 4 . [ .@ : =,54 . 224 . =;,8 : ;0;6 vmt

rgkl 0.=3

=25/226 S, cmzib`r kbklkyk ekg bmiaimg b`r su stjm d`nmzgg nmtmlj=8,5 M hr. m) stdgg cmz h`rkgd`g` vk edg b) nmt ekr`i`j` 4>6 S aimg cmzikbklkyk bu stdgmsi bmimgr1 Nmt mljgk aiur1 d) (b) hkl`



47/48

2=8

bmimgt nmttgg b`r` lksì`gdk mlj vk e gk aiur1 h) (d) hurujughml` e

garjmi edg yzhk lmhr1

zj 0.=3

m) M@

@c 8>,>54,=

5,=8

4 [c : [ : 226 S

0,2;8>,>

226

@c

[R V : 4 .[.@ : =,54 . 226 . =8,5 :

;366 U

b) 4>6 S iul kbklkyk std yihz bmimgrsm garjmi mir. gl cmzekrì`j` y`gk 226 vaittur.

M[

V@ 8>,>

4>6.54,=

;366

.4

d) MR

[@ 44,5

0,2;.2

4>6

2= V=: [.@=: 4>6.5,44 : 2506U

h) =,;63,;

50,23,;=66. 66

=6

6

V

VVV

rgkl 0.=5

4 cmzib`r jatarug kt`lkt`ghkg u hkkrikr migjtr. =2 VT< 46 M< 226 S () ?Das : 6,>0? vkr`j 6,>35 m) jatar nmge` 4 cmzikbklkikrhk P vk mirj1 b) jatarug klt`` ergr ed vk yihz ìk ekg bmimjmimrhml` nmtmljimrgbuiuguz. d) lr vk nml`l` eikr` nksmpimygz.

zj 0.=5

m) S4>6226P < ,226

=25? S vkym 226 S nmtikbklk

T : lvM8;,==;0,6.>35,6

543,6.=2 @P: M

[

T8,=5

4>6.4

==8;6

.4

@: M[

T46

226.54,=

==8;6

.4=



48/48

b) Das 6,>0 ? : 25a< T`g : 6,8;8V : T.Das: ==,8; . 6,>0 : =6,2 lw] : T . T`g : ==,8; . 6,8;8 : ;,2 l SMR


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Elektrik Devreleri 2 - Mersin Üniversitesi - 3 Fazlı Devreler