Электротехника и электроника

81

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

Çàäàíèå

Íàéòè òîêè â ñõåìå, ïðèâåäåííîé íà ðèñóíêå.

=

2

1,8

j

57,0

;

)

j40,813,619

+=

;

4

0

.

Ðåøåíèå

Ðåøèòå ñàìîñòîÿòåëüíî ñëåäóþùèå çàäàíèÿ

¹

ï/ï

Àëãîðèòìû Êîíêðåòíîå ñîîòâåòñòâèå çàäàíèÿ

ïðåäëîæåííîìó àëãîðèòìó

12 3

1Ðàñ÷åò íåçàâèñèìûõ

íà÷àëüíûõ óñëîâèé

Èç äîêîììóòàöèîííîé ñõåìû íàõîäèì

íà÷àëüíûå óñëîâèÿ:

()

0i0i

RR

E

0i

LL

21

L

==

+

=

+−

;

()

0u0u

RR

ER

0u

cc

21

2

c

==

+

=

+−

.

82

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

12 3

2Ñòðîèòñÿ îïåðàòîðíàÿ

ñõåìà çàìåùåíèÿ ñ

èñïîëüçîâàíèåì òàáëèöû

1

3 Íàõîäèòñÿ îïåðàòîðíîå

èçîáðàæåíèå òîêîâ I

2

(P) è

I

3

(P)

Òàê êàê â ñõåìå çàìåùåíèÿ èìååòñÿ âñåãî

äâà óçëà, íàïðÿæåíèå ìåæäó êîòîðûìè

çàäàíî, èñïîëüçóåì îáîáùåííûé çàêîí Îìà:

()

[]

()

2

L

2

RpLp

0iLpE

pI

+

⋅⋅+

=

;

()

()[]













+

−

=

CR

1

pR

0uE

pI

3

c

3

.

4Íàõîäÿòñÿ îðèãèíàëû

òîêîâ i

2

, i

3

, i

1

Òîê i

2

(t) ðàññ÷èòûâàåì ñ ïîìîùüþ òåîðåìû

ðàçëîæåíèÿ:

()

























+













⋅+













=

L

R

pp

0ip

L

E

pF

2

L

;

()

























+=

L

R

ppF

2

2.

;

p

1

= 0; p

2

=

L

R

2

−

;

() ()

pFppF

32

⋅=

;

()

0ip

L

E

pF

L1

⋅+=

;

() ()

L

E

0FpF

111

==

;

() ()

0i

L

R

L

E

.pF

L

2

21

⋅













−=

;

()

L

R

ppF

2

3

+=

;

()

L

R

0F

2

3

=

;

()

1pF

3

=

′

.

83

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

Çàäàíèå 1

Íàéòè òîê â ñõåìå ïîñëå êîììóòàöèè.

_________________________________________________________________________________________________

12 3

Èñïîëüçóÿ âòîðóþ ôîðìó çàïèñè òåîðåìû

ðàçëîæåíèÿ, ïîëó÷àåì:

()

()[]

⋅⋅−+=⋅

′

⋅

+=

0iRE

R

E

l

pFp

pF

0F

ti

L2

t

2

p

232

21

3

1

2

()

t

L

2

R

L

t

L

2

R

2

R

t

L

2

R

l0il1

R

E

l

−−

−

⋅+













−⋅=⋅

.

Òîê i

3

(t) íàõîäèì ñ ïîìîùüþ òàáëèöû 2.

()

CR

1

p

1

R

0uE

CR

1

pR

0uE

pI

3

c

3

c

3

+

⋅

−

=













+

−

=

.

Èñïîëüçóÿ òàáëè÷íîå ïðåîáðàçîâàíèå

()

t

k

P

k

lá

pp

á

−

⋅=

+

, çàïèñûâàåì:

()

C

3

R

t

3

c

3

l

R

0uE

ti

−

⋅

−

=

.

Òîê i

1

(t) îïðåäåëÿåì êàê ñóììó îðèãèíàëîâ

òîêîâ i

2

(t) è i

3

(t):

() () () ()

+⋅+













−===

−−

t

L

2

R

L

t

L

2

R

2

321

l0il1

R

E

tititi

+

()

C

3

R

t

3

c

l

R

0uE

−

⋅

−

.

84

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

_________________________________________________________________________________________________

Çàäàíèå 2

Èñòî÷íèê ïîñòîÿííîé ÝÄÑ äëèòåëüíîå âðåìÿ âêëþ÷åí â öåïü. Â ìîìåíò

âðåìåíè t = 0 êëþ÷ ïåðåâîäèòñÿ èç ïîëîæåíèÿ 1 â ïîëîæåíèå 2. Íàéòè çàêîí

èçìåíåíèÿ ïåðåõîäíîãî òîêà â ýëåìåíòå R

2

.

_________________________________________________________________________________________________

85

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

Ïðèëîæåíèå 1.

Òàáëèöà 1. Îïåðàòîðíûå ñõåìû çàìåùåíèÿ.

1) Ïðè íóëåâûõ íà÷àëüíûõ óñëîâèÿõ îïåðàòîðíûå ñõåìû çàìåùåíèÿ

èíäóêòèâíîãî è åìêîñòíîãî ýëåìåíòîâ óïðîùàþòñÿ è ñîäåðæàò òîëüêî îäèí

ïàññèâíûé ýëåìåíò - èíäóêòèâíîñòü pL èëè åìêîñòü

pC

1

.

Òàáëèöà 2. Òàáëèöà èçîáðàæåíèé è îðèãèíàëîâ ïî Ëàïëàñó

Èçîáðàæåíèå Îðèãèíàë

1

/

ð 1

1/

ð

2

t

1/

p

n

()

!1n

t

1n

−

(n - öåëîå ïîëîæèòåëüíîå ÷èñëî)

()

àð

1

±

at

e

±

()

2

àð

1

+

at

et

−

⋅

()

n

àð

1

+

at1-n

et

1)!-(n

1

−

⋅⋅

(n - öåëîå ïîëîæèòåëüíîå ÷èñëî)

()[]

àðp

1

+⋅

a)e-(1

at

−

()()[]

bpàð

p

+⋅+

)ae-(be

a-b

1

atbt

−−

⋅

()()[]

bpàðp

1

+⋅+⋅

)

a

e

-

b

e

(

a-b

1

ab

1

atbt

−−

⋅+

86

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

Òàáëèöà 3. Èçîáðàæåíèå ïàðàìåòðîâ ýëåìåíòîâ

ýëåêòðè÷åñêîé ñõåìû

()

[]

2

b

àð

a

p

++

+

btcose

at

⋅

−

()

22

à

ð

p

+

atcos

(

)

22

à

ð

1

+

atsina1

⋅

(

)

[]

22

à

ð

p

1

+⋅

()

atcos-1

a

2

⋅

1

(

)

2

22

àð

p

+

atsin

2a

t

⋅

Îðèãèíàë Èçîáðàæåíèå

e(t) = E = const E(p) = E/p

j(t) = j = const j(p) = j/p

e(t) = E

m

sin(

ω

t +

α

) E(p) = (p

⋅

sin

α

+

ω⋅

cos

α

)/(p

2

+

ω

2

)

j(t) = j

m

sin(

ω

t +

β

) j(p) = (p

⋅

cos

β

-

ω⋅

sin

β

)/(p

2

+

ω

2

)

U

R

(t) = R

⋅

i(t) U

R

(p) = R

⋅

I(p)

dt

di

L (t)U

L

⋅=

U

L

(p) = p

⋅

L

⋅

I(p) - L

⋅

i(0

+

)

∫

∞

=

t

-

C

i(t)dt

C

1

(t)U

U

C

(p) = I(p)/(p

⋅

C) + U

C

(0

+

)/p

Îïåðàòîðíîå ñîïðîòèâëåíèå

ïîñëåäîâàòåëüíîãî ñîåäèíåíèÿ RLC

Z(p) = R + p

⋅

L + 1/(p

⋅

c)

Îïåðàòîðíîå ñîïðîòèâëåíèå

ïàðàëëåëüíîãî ñîåäèíåíèÿ RLC

Y(p) = 1/R + 1/pL + pC

87

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

5. Ïðèìåð âûïîëíåíèÿ óïðàæíåíèÿ òðåíèíãà íà óìåíèå ¹5

Çàäàíèå

Îïðåäåëèòü ÌÄÑ êàòóøêè, ðàñïîëîæåííîé íà ìàãíèòîïðîâîäå, êîòîðûé

âûïîëíåí èç ëèñòîâîé ýëåêòðîòåõíè÷åñêîé ñòàëè, ñ òåì ÷òîáû ïîëó÷èòü

ìàãíèòíûé ïîòîê Ô íà ïåðâîì ó÷àñòêå ìàãíèòîïðîâîäà ðàâíûì 4,8⋅10

-3

Âá.

Ðàçìåðû ìàãíèòîïðîâîäà: a

1

= 50ìì; a

2

= 60ìì; a

3

= 190ìì; a

4

= 60ìì; a

5

=

60ìì; a

6

= 60ìì; a

7

= 380ìì. Ñòûêè èìåþò âîçäóøíûé çàçîð, ðàâíûé 0,1 ìì.

¹

ï/ï

Àëãîðèòìû Êîíêðåòíîå ñîîòâåòñòâèå çàäàíèÿ

ïðåäëîæåííîìó àëãîðèòìó

1 Îïðåäåëåíèå çíà÷åíèé

ìàãíèòíîé èíäóêöèè ïî

èçâåñòíîìó ìàãíèòíîìó

ïîòîêó è ïëîùàäè

ïîïåðå÷íîãî ñå÷åíèÿ

ó÷àñòêà ìàãíèòîïðîâîäà

Ïðîâîäèì ñðåäíþþ ìàãíèòíóþ ëèíèþ è ïî

èñõîäíûì äàííûì îïðåäåëÿåì äëèíó

ñðåäíåé ëèíèè äëÿ êàæäîãî ó÷àñòêà:

l

1

= à

7

- à

6

- à

5

= 380 - 60 - 60 = 260 ìì

l

2

= l

1

+ 2

⋅

(190 - 30 - 25) + 60 + 60 = 650 ìì

l

3

= l

4

= 0,1 ìì

Ïëîùàäü ïîïåðå÷íîãî ñå÷åíèÿ ïåðâîãî

ó÷àñòêà S

1

= a

1

⋅

à

4

= 3000 ìì

2

;

S

2

= a

2

⋅

à

4

= 3600 ìì

2

;

S

3

= a

1

⋅

à

4

= 3000 ìì

2

;

S

4

= a

2

⋅

à

4

= 3600 ìì

2

.

Ìàãíèòíàÿ èíäóêöèÿ:

B

1

= Ô

1

S

1

= 4,80

⋅

10

-3

Âá

⋅

3

⋅

10

-3

ì

2

= 1,6 Òë;

Â

1

S

1

= Â

2

S

2

B

2

= Â

1

S

1

/S

2

= 1,6

⋅

10

-3

/3,6

⋅

10

-3

= 1,33 Òë;

B

3

= Â

2

S

2

/S

3

= 1,6 Òë;

B

4

= Â

3

S

3

/S

4

= 1,33 Òë.

Ðåøåíèå

l

88

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

Ðåøèòå ñàìîñòîÿòåëüíî ñëåäóþùèå çàäàíèÿ

Çàäàíèå 1

Îïðåäåëèòü ÌÄÑ êàòóøêè, ðàñïîëîæåííîé íà ìàãíèòîïðîâîäå, êîòîðûé

âûïîëíåí èç ëèñòîâîé ñòàëè, ñ òåì, ÷òîáû ïîëó÷èòü íà ïåðâîì ó÷àñòêå

ìàãíèòíóþ èíäóêöèþ B

1

= 2Òë. Ðàçìåðû ìàãíèòîïðîâîäà: a

1

= 60ìì; a

2

= 70ìì;

a

3

= 200ìì; a

4

= 70ìì; a

5

= 70ìì; a

6

= 60ìì; a

7

= 400ìì. Ñòûêè èìåþò

âîçäóøíûé çàçîð, ðàâíûé 0,2 ìì.

_________________________________________________________________________________________________

2 Îïðåäåëåíèå

íàïðÿæåííîñòè

ìàãíèòíîãî ïîëÿ ïî

çàäàííîé çàâèñèìîñòè

ìàãíèòíîé èíäóêöèè îò

íàïðÿæåííîñòè

ìàãíèòíîãî ïîëÿ

Íàïðÿæåííîñòü ïîëÿ íà ó÷àñòêàõ 1 è 2

îïðåäåëÿåì ïî êðèâîé íàìàãíè÷èâàíèÿ.

Ñîãëàñíî êðèâîé íàìàãíè÷èâàíèÿ äëÿ

ëèñòîâîé ýëåêòðîòåõíè÷åñêîé ñòàëè

H

1

= 40 À/ñì; H

2

= 17À/ñì.

Íàïðÿæåííîñòü ïîëÿ â çàçîðàõ îïðåäåëÿåòñÿ

ïî ôîðìóëå: H

0

= B/

µ

0

= 0,8

⋅

10

6

Â;

H

3

= 0,8

⋅

10

6

⋅

Â

3

= 0,8

⋅

10

6

⋅

1,6 = 12800 À/ñì;

H

4

= 0,8

⋅

10

6

⋅

1,33 = 10640 À/ñì.

3 Îïðåäåëåíèå ÌÄÑ ñ

ïîìîùüþ çàêîíà ïîëíîãî

òîêà

Ïî çàêîíó Êèðõãîôà, ñóììà ìãíîâåííûõ

íàïðÿæåíèé íà îòäåëüíûõ ó÷àñòêàõ ðàâíà

èñêîìîé ÌÄÑ:

F = I

⋅

w = H

1

⋅

l

1

+ H

2

⋅

l

2

+ H

3

⋅

l

3

+ H

4

⋅

l

4

= 40

⋅

26 +

17

⋅

65 + + 12800

⋅

0,01 + 10640

⋅

0,01 = 2379,4

À.

89

Ñîâðåìåííûé Ãóìàíèòàðíûé Óíèâåðñèòåò

Çàäàíèå 2

Îïðåäåëèòü ÌÄÑ êàòóøêè, ðàñïîëîæåííîé íà ìàãíèòîïðîâîäå, êîòîðûé

âûïîëíåí èç ëèñòîâîé ýëåêòðîòåõíè÷åñêîé ñòàëè, ñ òåì, ÷òîáû ïîëó÷èòü

ìàãíèòíûé ïîòîê Ô íà ïåðâîì ó÷àñòêå ìàãíèòîïðîâîäà ðàâíûì 6⋅10

-3

Âá (ðèñ.

1, óìåíèå 5). Ðàçìåðû ìàãíèòîïðîâîäà: a

1

= 40ìì; a

2

= 60ìì; a

3

= 210ìì; a

4

= 70ìì; a

5

= 40ìì; a

6

= 60ìì; a

7

= 350ìì. Ñòûêè èìåþò âîçäóøíûé çàçîð,

ðàâíûé 0,15 ìì.

_________________________________________________________________________________________________

Ðåäàêòîð Í.Â. Äðóõ

Îïåðàòîð êîìïüþòåðíîé âåðñòêè Å.Ì. Êóçíåöîâà

___________________________________________________________________________________

Èçä. ëèö. ËÐ ¹ 071765 îò 07.12.1998 Ñäàíî â ïå÷àòü

ÍÎÓ Ñîâðåìåííûé Ãóìàíèòàðíûé Èíñòèòóò

Òèðàæ Çàêàç

ÝËÅÊÒÐÎÒÅÕÍÈÊÀ È ÝËÅÊÒÐÎÍÈÊÀ

ÞÍÈÒÀ 1

ÝËÅÊÒÐÈ×ÅÑÊÈÅ È ÌÀÃÍÈÒÍÛÅ ÖÅÏÈ

Электротехника и электроника

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