Готман В.И. Короткие замыкания и несимметричные режимы в электроэнергетических системах

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ɜɪɟɦɟɧɢ t = 0,01 ɫ. ɋ ɭɱɟɬɨɦ ɫɤɚɡɚɧɧɨɝɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɭɞɚɪɧɨɝɨ ɬɨɤɚ

ɄɁ ɡɚɩɢɲɟɬɫɹ ɬɚɤ:

0,01

ɩ max ɩ max ɩ max ɩ

yyy

iI I e I K IK





, (3.10)

ɝɞɟ

0,01





– ɭɞɚɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ;

– ɞɟɣɫɬɜɭɸɳɟɟ ɡɧɚ-

ɱɟɧɢɟ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɬɨɤɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɤɨɪɨɬɤɨ-

ɝɨ ɡɚɦɵɤɚɧɢɹ (t = 0).

Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɜɵɪɚɠɟɧɢɹ (3.10),

ɩɨɤɚɡɵɜɚɟɬ ɩɪɟɜɵɲɟɧɢɟ ɭɞɚɪɧɨɝɨ

ɬɨɤɚ ɄɁ ɧɚɞ ɚɦɩɥɢɬɭɞɨɣ ɩɟɪɢɨɞɢɱɟɫɤɨɣ

ɫɥɚɝɚɟɦɨɣ. ȿɝɨ ɜɟɥɢɱɢɧɚ ɡɚɜɢɫɢɬ ɨɬ ɩɨ-

ɫɬɨɹɧɧɨɣ ɜɪɟɦɟɧɢ

(ɪɢɫ. 3.5) ɢ ɧɚɯɨ-

ɞɢɬɫɹ ɜ ɩɪɟɞɟɥɚɯ

K

. ɑɟɦ ɛɨɥɶɲɟ

ɜɟɥɢɱɢɧɚ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɰɟɩɢ,

ɬɟɦ ɛɵɫɬɪɟɟ ɡɚɬɭɯɚɟɬ ɚɩɟɪɢɨɞɢɱɟɫɤɢɣ

ɬɨɤ ɢ ɬɟɦ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɦɟɧɶɲɟ ɭɞɚɪ-

ɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ. Ɉɧ ɞɨɫɬɢɝɚɟɬ ɩɪɟ-

ɞɟɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɭɫɥɨ-

ɜɢɹɯ: ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɜ ɰɟɩɢ ɪɟɚɤɬɢɜɧɨɝɨ

ɫɨɩɪɨɬɢɜɥɟɧɢɹ (

) ɚɩɟɪɢɨɞɢɱɟɫɤɨ-

ɝɨ ɬɨɤɚ, ɢ

; ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ (

= 0)

ɚɩɟɪɢɨɞɢɱɟɫɤɢɣ ɬɨɤ ɧɟ ɡɚɬɭɯɚɟɬ, ɢ

. ɉɚɪɚɦɟɬɪ

ɩɨ ɜɵɪɚɠɟɧɢɸ

(3.10) ɞɚɟɬ ɩɨɝɪɟɲɧɨɫɬɶ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɬɨɱɧɨɦɭ ɜɵɪɚɠɟɧɢɸ ɜ ɩɪɟɞɟ-

ɥɚɯ 0,1…1,4 % ɩɪɢ

xrt

ɢ ɞɨ 2,5 % ɩɪɢ

/1,3

ɍɞɚɪɧɵɣ ɬɨɤ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɩɪɨɜɟɪɤɢ ɷɥɟɤɬɪɨɬɟɯɧɢɱɟɫɤɨɣ ɚɩ-

ɩɚɪɚɬɭɪɵ ɧɚ ɷɥɟɤɬɪɨɞɢɧɚɦɢɱɟɫɤɭɸ (ɦɟɯɚɧɢɱɟɫɤɭɸ) ɩɪɨɱɧɨɫɬɶ.

3.2. ȿɠɤɬɭɝɮɹɴɠɠ ɢɨɛɲɠɨɣɠ ɭɩɥɛ ɥɩɫɩɭɥɩɞɩ ɢɛɧɶɥɛɨɣɺ

Ⱦɟɣɫɬɜɭɸɳɢɦ ɡɧɚɱɟɧɢɟɦ ɬɨɤɚ ɄɁ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟ-

ɧɢ t ɧɚɡɵɜɚɸɬ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɧɨɟ ɡɧɚɱɟɧɢɟ ɩɨɥɧɨɝɨ ɬɨɤɚ ɡɚ ɨɞɢɧ ɩɟɪɢ-

ɨɞ T, ɜ ɫɟɪɟɞɢɧɟ ɤɨɬɨɪɨɝɨ ɧɚɯɨɞɢɬɫɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɦɨɦɟɧɬ, ɬ. ɟ.

Iidt





. (3.11)

ȼɵɪɚɠɟɧɢɟ ɩɨɥɧɨɝɨ ɬɨɤɚ



ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɫɥɨɠ-

ɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɢ ɜ ɬɟɱɟɧɢɟ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢɡɦɟɧɹɟɬɫɹ, ɫɥɟɞɨ-

ɜɚɬɟɥɶɧɨ, ɢ ɞɟɣɫɬɜɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɬɨɤɚ ɧɟ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ ɜɨ ɜɪɟ-

Ɋɢɫ. 3.5 Ɂɚɜɢɫɢɦɨɫɬɶ ɭɞɚɪɧɨɝɨ

ɤɨɷɮɮɢɰɢɟɧɬɚ ɨɬ ɩɨɫɬɨɹɧɧɨɣ

ɜɪɟɦɟɧɢ

ɦɟɧɢ. Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɜɵɱɢɫɥɟɧɢɹ

I ɩɪɢɧɢɦɚɸɬ, ɱɬɨ ɞɟɣɫɬɜɭɸɳɢɟ

ɡɧɚɱɟɧɢɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɢ ɚɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫɥɚɝɚɸɳɢɯ ɡɚ ɪɚɫɫɦɚɬɪɢ-

ɜɚɟɦɵɣ ɩɟɪɢɨɞ

ɧɟ ɢɡɦɟɧɹɸɬɫɹ, ɢ ɤɚɠɞɚɹ ɢɡ ɧɢɯ ɪɚɜɧɚ ɫɜɨɟɦɭ ɡɧɚɱɟ-

ɧɢɸ ɜ ɞɚɧɧɵɣ

ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ

, ɬ. ɟ.

ɩ(0)

ɚ t ɚ t

Ii Ie



ɋ ɭɱɟɬɨɦ ɩɪɢɧɹɬɵɯ ɞɨɩɭɳɟɧɢɣ ɞɟɣɫɬɜɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɩɨɥɧɨɝɨ ɬɨɤɚ ɄɁ

ɜ ɦɨɦɟɧɬ

ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɪɚɠɟɧɢɹ (3.10) ɨɩɪɟɞɟɥɢɬɫɹ ɬɚɤ:

ɩttɚt

III 

. (3.12)

Ɇɟɬɨɞ ɪɚɫɱɟɬɚ

ɩ t

, ɤɨɝɞɚ ɢɫɬɨɱɧɢɤɨɦ ɩɢɬɚɧɢɹ ɹɜɥɹɟɬɫɹ ɝɟɧɟɪɚɬɨɪ

ɤɨɧɟɱɧɨɣ ɦɨɳɧɨɫɬɢ, ɢɡɥɨɠɟɧ ɜ ɪɚɡɞ. 5.3; ɞɥɹ ɦɨɳɧɨɝɨ ɢɫɬɨɱɧɢɤɚ, ɧɚ-

ɩɪɹɠɟɧɢɟ ɤɨɬɨɪɨɝɨ ɜ ɦɨɦɟɧɬ ɄɁ ɧɟɢɡɦɟɧɧɨ, ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɫɥɚɝɚɟɦɚɹ ɜ

ɩɟɪɟɯɨɞɧɨɦ ɪɟɠɢɦɟ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ.

ɩɩ(0)t

Ʉɚɤ ɩɪɚɜɢɥɨ, ɧɚ ɩɪɚɤɬɢɤɟ ɪɚɫɫɱɢɬɵɜɚɸɬ

, ɧɚɢɛɨɥɶɲɟɟ ɞɟɣɫɬɜɭɸ-

ɳɟɟ ɡɧɚɱɟɧɢɟ ɩɨɥɧɨɝɨ ɬɨɤɚ ɄɁ.

ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɩɟɪɜɵɣ ɩɟɪɢɨɞ ɩɟɪɟɯɨɞɧɨ-

ɝɨ ɩɪɨɰɟɫɫɚ, ɜ ɫɟɪɟɞɢɧɟ ɤɨɬɨɪɨɝɨ ɧɚɯɨɞɢɬɫɹ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t = 0,01 ɫ.

ɋ ɭɱɟɬɨɦ ɫɨɨɬɧɨɲɟɧɢɹ (3.10) ɜɵɪɚɠɟɧɢɸ (3.12) ɦɨɠɧɨ ɩɪɢɞɚɬɶ ɜɢɞ:



ɉ(0)

12 1

II K 

. (3.13)

Ɂɞɟɫɶ ɩɪɢɧɹɬɨ, ɱɬɨ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɡɚ ɜɪɟɦɹ

0,01 ɫt ɧɟ ɢɡɦɟɧɹɟɬɫɹ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɤɚɡɚɧɧɵɦɢ ɜɵɲɟ ɩɪɟɞɟɥɚɦɢ

ɢɡɦɟɧɟɧɢɹ

ɨɬɧɨɲɟɧɢɟ

ɩy

ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɪɟɞɟɥɚɯ

1/ 3

II

ɉɨ ɩɚɪɚɦɟɬɪɚɦ

I ɢ

ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɟɪɦɢɱɟɫɤɨɟ ɜɨɡɞɟɣɫɬɜɢɟ ɬɨ-

ɤɚ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚɧɢɹ ɧɚ ɷɥɟɤɬɪɨɬɟɯɧɢɱɟɫɤɭɸ ɚɩɩɚɪɚɬɭɪɭ.

Ʌɩɨɭɫɩɦɷɨɶɠ ɝɩɪɫɩɬɶ

1. ɑɟɦ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ «ɲɢɧɵ ɛɟɫɤɨɧɟɱɧɨɣ ɦɨɳɧɨɫɬɢ»?

2. Ʉɚɤɢɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɫɬɨɹɧɧɚɹ ɜɪɟɦɟɧɢ

ɤɚɤɨɜɚ ɟɟ ɮɢɡɢɱɟɫɤɚɹ ɫɭɳɧɨɫɬɶ?

3. Ⱦɥɹ ɤɚɤɢɯ ɭɫɥɨɜɢɣ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚɧɢɹ ɪɚɫ-

ɫɱɢɬɵɜɚɟɬɫɹ ɭɞɚɪɧɵɣ ɬɨɤ?

Ⱦɦɛɝɛ 4

ɊȻɋȻɇɀɍɋɖ Ƀ ɋɀɁɃɇɖ ɘɆɀɅɍɋɃɒɀɌɅɃɐ ɇȻɓɃɈ

4.1. ɋɠɡɣɧɨɶɠ ɬɩɬɭɩɺɨɣɺ ɧɛɳɣɨ

Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɫɢɧɯɪɨɧɧɨɣ ɢ ɚɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧ (ȺɆ)

ɢ ɢɯ ɩɚɪɚɦɟɬɪɵ ɡɚɜɢɫɹɬ ɨɬ ɰɟɥɟɣ ɪɚɫɱɟɬɚ ɪɟɠɢɦɚ (ɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɢɥɢ

ɩɟɪɟɯɨɞɧɵɣ), ɟɝɨ ɫɬɚɞɢɢ, ɬɪɟɛɨɜɚɧɢɣ ɤ ɬɨɱɧɨɫɬɢ ɪɚɫɱɟɬɚ ɢ ɨɬ ɜɥɢɹɧɢɹ

ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɦɚɲɢɧɵ ɧɚ ɢɫɫɥɟɞɭɟɦɵɣ ɩɪɨɰɟɫɫ. ȼ ɷɬɨɦ ɫɦɵɫɥɟ ɛɭ-

ɞɟɦ ɪɚɡɥɢɱɚɬɶ ɫɥɟɞɭɸɳɢɟ ɪɟɠɢɦɧɵɟ ɫɨɫɬɨɹɧɢɹ ɦɚɲɢɧ:

1. ɇɨɪɦɚɥɶɧɵɣ ɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɪɟɠɢɦ, ɜ ɤɨɬɨɪɨɦ

ɫɢɧɯɪɨɧɧɚɹ

ɦɚɲɢɧɚ (ɋɆ) ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɢɧɯɪɨɧɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɞɥɹ

(0)t 

2. ɍɫɬɚɧɨɜɢɜɲɢɣɫɹ ɪɟɠɢɦ ɄɁ, ɜ ɤɨɬɨɪɨɦ ɦɚɲɢɧɚ ɬɚɤɠɟ ɩɪɟɞɫɬɚɜ-

ɥɹɟɬɫɹ ɫɢɧɯɪɨɧɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɞɥɹ

t f

3. ɉɟɪɟɯɨɞɧɵɣ ɪɟɠɢɦ ɞɥɹ ɦɨɦɟɧɬɚ

0t . Ɂɞɟɫɶ, ɤɚɤ ɩɪɚɜɢɥɨ, ɡɚɞɚ-

ɱɚ ɨɝɪɚɧɢɱɟɧɚ ɪɚɫɱɟɬɨɦ ɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫɥɚɝɚɟɦɨɣ

ɬɨɤɚ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚɧɢɹ. Ⱦɥɹ ɷɬɢɯ ɭɫɥɨɜɢɣ ɦɚɲɢɧɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ

ɩɟɪɟɯɨɞɧɵɦɢ ɢɥɢ ɫɜɟɪɯɩɟɪɟɯɨɞɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ.

4. ɉɟɪɟɯɨɞɧɵɣ ɪɟɠɢɦ ɫ ɩɨɥɧɵɦ ɨɬɪɚɠɟɧɢɟɦ ɜɵɧɭɠɞɟɧɧɵɯ ɢ ɫɜɨ-

ɛɨɞɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɬɨɤɨɜ ɧɚ ɜɫɟɦ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ. Ⱦɥɹ ɪɟɲɟɧɢɹ

ɷɬɨɣ ɡɚɞɚɱɢ ɪɟɠɢɦ ɦɚɲɢɧɵ ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɉɚɪɤɚ–Ƚɨɪɟɜɚ.

4.2. Ɍɰɠɧɛ

ɢɛɧɠɴɠɨɣɺ ɣ ɪɛɫɛɧɠɭɫɶ ɬɣɨɰɫɩɨɨɩɤ ɧɛɳɣɨɶ

ɝ ɮɬɭɛɨɩɝɣɝɳɠɧɬɺ ɫɠɡɣɧɠ

ɍɫɬɚɧɨɜɢɦ ɷɥɟɤɬɪɢɱɟɫɤɭɸ ɫɯɟɦɭ ɡɚɦɟɳɟɧɢɹ ɢ ɩɚɪɚɦɟɬɪɵ ɫɢɧ-

ɯɪɨɧɧɨɣ ɦɚɲɢɧɵ ɞɥɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɠɢɦɚ. ɇɚ ɪɢɫ. 4.1, ɚ ɩɪɟɞɫɬɚɜɥɟɧ

ɪɚɡɪɟɡ ɋɆ ɢ ɩɪɹɦɨɭɝɨɥɶɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ

, dq

, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɚɹ

ɫ ɪɨɬɨɪɨɦ. Ɉɫɶ

d ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɞɨɥɶɧɨɣ ɨɫɶɸ ɦɚɲɢɧɵ, ɨɫɶ q – ɩɨɩɟ-

ɪɟɱɧɨɣ. ȼ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɜɟ ɦɚɝɧɢɬɧɨ ɫɜɹɡɚɧɧɵɟ

ɨɛɦɨɬɤɢ (ɫɦ. ɪɢɫ. 4.1, ɛ): ɨɛɦɨɬɤɚ ɜɨɡɛɭɠɞɟɧɢɹ (ɪɚɫɩɨɥɨɠɟɧɚ ɫɥɟɜɚ) ɢ

ɨɛɦɨɬɤɚ ɫɬɚɬɨɪɚ. ȼ ɩɨɩɟɪɟɱɧɨɣ ɨɫɢ ɪɚɫɩɨɥɚɝɚɟɬɫɹ ɬɨɥɶɤɨ ɤɨɧɬɭɪ ɫɬɚ-

ɬɨɪɧɨɣ ɨɛɦɨɬɤɢ. Ɋɟɚɤɬɢɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɤɚɠɞɨɝɨ ɤɨɧɬɭɪɚ ɜ ɫɨɨɬɜɟɬ-

ɫɬɜɢɢ ɫ ɮɢɡɢɤɨɣ ɹɜɥɟɧɢɣ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ ɜɢɞɟ ɞɜɭɯ ɫɨɫɬɚɜɥɹɸɳɢɯ:

– ɪɟɚɤɬɚɧɫɨɜ ɪɚɫɫɟɹɧɢɹ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ ɢ ɨɛɦɨɬɤɢ

ɫɬɚɬɨɪɚ;

– ɢɧɞɭɤɬɢɜɧɵɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɩɪɨɞɨɥɶɧɨɣ ɢ ɩɨɩɟɪɟɱ-

ɧɨɣ ɪɟɚɤɰɢɣ ɫɬɚɬɨɪɚ.

Ɋɢɫ. 4.1: ɚ – ɪɚɡɪɟɡ ɫɢɧɯɪɨ

ɯɨɞɭ; ɛ – ɦɚɝɧɢɬɧɨ

ɜ – ɤɨɧɬɭɪ ɨɛ

ȼ ɧɟɹɜɧɨɩɨɥɸɫɧɨɣ

ɫɬɚɬɨɪɨɦ ɩɨ ɜɫɟɣ ɨɤɪɭɠɧɨ

ɫɹ ɜ ɪɚɜɟɧɫɬɜɟ

ad aq

, ɢ

ɩɨɩɟɪɟɱɧɨɟ

, ɫɢɧɯɪɨɧɧ

ɤɨɜɵ, ɩɪɢ ɷɬɨɦ

ɉɨ ɫɭɳɟɫɬɜɭ,

ɬɢɜɥɟɧɢɹ ɫɬɚɬɨɪɧɨɣ ɨɛɦɨɬ

ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ.

ȼ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪ

ɫɨɡɞɚɟɬ ɩɨ ɨɫɢ

ɦɚɝ

ɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɦɦɭ

ɝɞɟ

fff

ɎɎ

– ɩɨɬɨɤ

ɦɨɞɟɣɫɬɜɭɟɬ ɬɨɥɶɤɨ ɫ ɷɬɨ

ɫɬɪɚɧɫɬɜɭ;

– ɤɨɷɮɮɢɰ

ɩɨɥɟɡɧɵɣ ɩɨɬɨɤ.

ɀɢɪɧɵɦ ɲɪɢɮɬɨɦ ɩɪɹɦɨɝɨ ɧɚɱɟ

ɧɨɣ ɦɚɲɢɧɵ ɢ ɦɚɝɧɢɬɧɵɟ ɩɨɬɨɤɢ ɧɚ ɯɨ

ɫɜɹɡɚɧɧɵɟ ɤɨɧɬɭɪɵ ɜ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ d;

ɨɬɤɢ ɫɬɚɬɨɪɚ ɜ ɩɨɩɟɪɟɱɧɨɣ ɨɫɢ q

ɚɲɢɧɟ ɜɨɡɞɭɲɧɵɣ ɡɚɡɨɪ ɦɟɠɞɭ ɪɨ

ɬɢ ɪɚɫɬɨɱɤɢ ɫɬɚɬɨɪɚ ɨɞɢɧɚɤɨɜ, ɱɬɨ ɨ

ɩɨɷɬɨɦɭ ɞɥɹ ɬɚɤɨɣ ɦɚɲɢɧɵ ɩɪɨɞɨɥɶ

ɟ ɢɧɞɭɤɬɢɜɧɵɟ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɬɚɤɠ



ıqaq

xxx 

ɩɪɟɞɫɬɚɜɥɹɸɬ ɩɨɥɧɵɟ ɢɧɞɭɤɬɢɜɧ

ɢ (ɪɢɫ. 4.1, ɛ, ɜ), ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɹ

ɠɢɦɟ ɩɨɫɬɨɹɧɧɵɣ ɬɨɤ ɨɛɦɨɬɤɢ ɜɨɡɛ

ɢɬɧɵɣ ɩɨɬɨɤ

(ɪɢɫ. 4.1, ɚ), ɤɨɬɨɪ

ɜɭɯ ɫɨɫɬɚɜɥɹɸɳɢɯ:

ffd

ɎɎ Ɏ

ɪɚɫɫɟɹɧɢɹ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ;

ɨɛɦɨɬɤɨɣ, ɡɚɦɵɤɚɹɫɶ ɩɨ ɜɨɡɞ

ɲɧ

ɟɧɬ ɪɚɫɫɟɹɧɢɹ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢ

ɬɚɧɢɹ ɨɛɨɡɧɚɱɟɧɵ ɜɟɤɬɨɪɧɵɟ ɩɟɪɟɦɟɧɧɵɟ.

ɨɫɬɨɦ

ɬɨɪɨɦ ɢ

ɪɚɠɚɟɬ-

ɨɟ

ɨɞɢɧɚ-

ɟ ɫɨɩɪɨ-

ɠɞɟɧɢɹ

ɣ ɦɨɠ-

(4.1)

ɜɡɚɢ-

ɦɭ ɩɪɨ-

;

–

ɉɨɥɟɡɧɵɣ ɩɨɬɨɤ

ɫɬɚɬɨɪɚ ɫɢɧɯɪɨɧɧɭɸ ɗȾɋ,

ȼ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚ

ɨɫɢ

. ȼ ɧɚɝɪɭɠɟɧɧɨɣ ɦɚ

ɱɟɫɤɨɣ ɫɭɦɦɨɣ ɜɟɤɬɨɪɚ ɧɚ

ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɟɝɨ ɪɟɚɤɬɚ

ɞɥɹ ɧɟɹɜɧɨɩɨɥɸɫɧɨɝɨ ɝɟɧɟ

ɬ. ɟ. ɧɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɪ

ɩɨɩɟɪɟɱɧɭɸ ɫɨɫɬɚɜɥɹɸɳ

ɦɚɲɢɧɭ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢ

Ɋɢɫ. 4.2.

ɟɤ

ɩɪɢ ɜɪɚɳɟɧɢɢ ɪɨɬɨɪɚ ɢɧɞɭɰɢɪɭɟɬ ɜ

ɨɬɨɪɚɹ ɨɬɫɬɚɟɬ ɨɬ ɩɨɬɨɤɚ

ɧɚ 90°:

E Ɏ

(ɪɢɫ. 4.2) ɜɟɤɬɨɪ

ɚɫɩɨɥɚɝ

ɢɧɟ ɜɟɤɬɨɪ

ɭɪɚɜɧɨɜɟɲɢɜɚɟɬɫɹ ɝ

ɪɹɠɟɧɢɹ

ɧɚ ɡɚɠɢɦɚɯ ɝɟɧɟɪɚɬɨɪɚ ɢ

ɫɚɯ. Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɞɢɚɝɪɚɦɦɵ (ɪɢ

ɚɬɨɪɚ ɫɩɪɚɜɟɞɥɢɜɨ ɜɟɤɬɨɪɧɨɟ ɭɪɚɜɧɟ

jx EUI

ɡɥɚɝɚɬɶ ɬɨɤ ɢ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɩɪɨɞɨ

ɟ. ɉɨɷɬɨɦɭ ɧɟɹɜɧɨɩɨɥɸɫɧɭɸ ɫɢɧ

ɶ ɫɯɟɦɨɣ, ɩɪɢɜɟɞɟɧɧɨɣ ɧɚ ɪɢɫ. 4.3, ɚ.

ɨɪɧɵɟ ɞɢɚɝɪɚɦɦɵ ɧɟɹɜɧɨɩɨɥɸɫɧɨɣ (ɚ)

ɢ ɹɜɧɨɩɨɥɸɫɧɨɣ (ɛ) ɦɚɲɢɧ

ɨɛɦɨɬɤɟ

(4.2)

ɚɟɬɫɹ ɧɚ

ɨɦɟɬɪɢ-

ɩɚɞɟɧɢɹ

. 4.2, ɚ),

ɧɢɟ

(4.3)

ɶɧɭɸ ɢ

ɪɨɧɧɭɸ

Ⱦɥɹ ɹɜɧɨɩɨɥɸɫɧɵɯ

ɧɟɨɞɢɧɚɤɨɜɵ:

xx

(ɨɛ

ɤɨɜɵɦ ɜɨɡɞɭɲɧɵɦ ɡɚɡɨɪ

ɜ ɩɨɩɟɪɟɱɧɨɣ ɨɫɢ ɨɧ ɛɨɥɶ

ɫɬɪɨɝɨ ɝɨɜɨɪɹ, ɧɟ ɩɨɡɜɨɥɹɟ

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɪɟɚɥɶɧɭɸ

ɪɟɚɤɬɚɧɫɚɦɢ ɩɨ ɨɫɢ

ɠɟ ɧɚɩɪɹɠɟɧɢɢ

ɢ ɭɝɥɟ

ɚɤɬɢɜɧɚɹ ɦɨɳɧɨɫɬɢ, ɬɨ ɫɨ

ɧɹɬɶ

. ɗȾɋ ɬɚɤɨɣ ɦɚɲɢ

ɫɨɜɩɚɞɚɟɬ ɫ ɩɨɥɨɠɟɧɢɟɦ ɜ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɜɟɞɹ

ɩɨɥɸɫɧɭɸ ɦɚɲɢɧɭ ɷɤɜɢɜ

ɩɪɨɬɢɜɥɟɧɢɟɦ

. ɗɬɨɦɭ

ɧɚɹ ɧɚ ɪɢɫ. 4.3, ɜ.

Ɋɢɫ. 4.3. ɋɯɟɦɵ ɡɚɦɟ

ɦɚ

ɋ ɧɟɡɧɚɱɢɬɟɥɶɧɨɣ ɩ

ɲɢɧɭ ɩɪɟɞɫɬɚɜɥɹɬɶ ɷɤɜɢɜɚ

ɢ ɗȾɋ



(ɫɦ. ɪɢɫ. 4.2, ɛ).

Ⱦɢɚɩɚɡɨɧ ɫɢɧɯɪɨɧɧ

ɥɹɟɬ:

ɞɥɹ ɧɟɹɜɧɨɩɨɥɸɫɧɵ

ɞɥɹ ɹɜɧɨɩɨɥɸɫɧɵɯ

ɋɨɝɥɚɫɧɨ ɜɟɤɬɨɪɧɨɣ

ɗȾɋ

ɦɨɠɧɨ ɜɵɱɢɫɥɢ

ɜɟɪɲɢɧɚɦɢ: «ɧɚɱɚɥɨ ɤɨɨɪ

ɩɨ ɫɥɟɞɭɸɳɟɦɭ ɜɵɪɚɠɟɧɢ



ɚɲɢɧ ɢɧɞɭɤɬɢɜɧɵɟ ɫɨɩɪɨɬɢɜɥɟɧɢɹ

ɱɧɨ

0,6

xx|

). ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ

ɦ ɩɨ ɞɥɢɧɟ ɨɤɪɭɠɧɨɫɬɢ ɪɚɫɬɨɱɤɢ

ɟ, ɱɟɦ ɜ ɩɪɨɞɨɥɶɧɨɣ. ɇɟɪɚɜɟɧɫɬɜɨ

ɡɚɦɟɫɬɢɬɶ ɦɚɲɢɧɭ ɨɞɧɢɦ ɫɨɩɪɨɬɢɜ

ɚɲɢɧɭ ɡɚɦɟɧɹɸɬ ɮɢɤɬɢɜɧɨɣ ɫ ɨɞɢɧ

. ȿɫɥɢ ɢɫɯɨɞɢɬɶ ɢɡ ɬɨɝɨ, ɱɬɨ ɩɪɢ ɨɞɧ

ɭ ɨɛɟɢɯ ɦɚɲɢɧ ɫɨɜɩɚɞɚɥɢ ɢɯ ɚɤɬɢɜ

ɪɨɬɢɜɥɟɧɢɟ ɮɢɤɬɢɜɧɨɣ ɦɚɲɢɧɵ ɫɥɟɞ

ɵ ɛɭɞɟɬ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜɟɤɬɨɪɨɦ

ɤɬɨɪɚ

(ɫɦ. ɪɢɫ. 4.2, ɛ).

ɮɢɤɬɢɜɧɭɸ ɗȾɋ

, ɦɨɠɧɨ ɡɚɦɟɧ

ɥɟɧɬɧɨɣ ɧɟɹɜɧɨɩɨɥɸɫɧɨɣ ɫ ɪɟɚɤɬɢɜ

ɨɨɬɜɟɬɫɬɜɭɟɬ ɫɯɟɦɚ ɡɚɦɟɳɟɧɢɹ, ɩɪɟɞ

ɟɧɢɹ ɧɟɹɜɧɨɩɨɥɸɫɧɨɣ (ɚ) ɢ ɹɜɧɨɩɨɥɸɫɧɨ

ɢɧ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ ɪɟɠɢɦɟ

ɨɝɪɟɲɧɨɫɬɶɸ ɞɨɩɭɫɬɢɦɨ ɹɜɧɨɩɨɥɸɫ

ɟɧɬɧɨɣ ɧɟɹɜɧɨɩɨɥɸɫɧɨɣ ɫ ɪɟɚɤɬɢɜɧɨ

Ɋɚɡɥɢɱɢɟ



ɫɨɫɬɚɜɥɹɟɬ 1…2

ɯ ɪɟɚɤɬɢɜɧɵɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɦɚɲɢ

0,95...2,55

;

0,6...1,45

;

0,4...1

ɞɢɚɝɪɚɦɦɟ (ɪɢɫ. 4.2, ɚ) ɦɨɞɭɥɶ ɫɢɧ

ɶ ɧɚ ɨɫɧɨɜɟ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɬɪɟɭɝɨ

ɢɧɚɬ», «ɩɪɹɦɨɣ ɭɝɨɥ», «ɤɨɧɟɰ ɜɟɤɬ





cos sin

UIx



ɟɨɞɢɧɚ-

ɫɬɚɬɨɪɚ:

ɟɧɢɟɦ.

ɤɨɜɵɦɢ

ɦ ɢ ɬɨɦ

ɚɹ ɢ ɪɟ-

ɭɟɬ ɩɪɢ-

ɨɬɨɪɵɣ

ɬɶ ɹɜɧɨ-

ɧɵɦ ɫɨ-

ɫɬɚɜɥɟɧ-

(ɛ, ɜ)

ɭɸ ɦɚ-

ɫɬɶɸ

ɫɨɫɬɚɜ-

ɯɪɨɧɧɨɣ

ɶɧɢɤɚ ɫ

ɪɚ

(4.4)

ɉɨ ɜɵɪɚɠɟɧɢɹɦ, ɚɧɚɥɨɝɢɱɧɵɦ (4.4), ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ



ȼ ɡɚɤɥɸɱɟɧɢɟ ɞɥɹ ɫɪɚɜɧɟɧɢɹ ɩɪɢɜɟɞɟɦ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɨɜ ɞɥɹ

ɞɜɭɯ ɫɢɧɯɪɨɧɧɵɯ ɝɟɧɟɪɚɬɨɪɨɜ:

x ɧɟɹɜɧɨɩɨɥɸɫɧɵɣ ɝɟɧɟɪɚɬɨɪ ɫ

1, 4 5

ɩɪɢ ɧɨɦɢɧɚɥɶɧɨɣ

ɧɚɝɪɭɡɤɟ ɢɦɟɟɬ

1U ,





cos 0,85 31,8

; ɢɦ ɫɨɨɬɜɟɬɫɬɜɭɟɬ

2,15

ɢ 34,9

;

x ɹɜɧɨɩɨɥɸɫɧɵɣ ɝɟɧɟɪɚɬɨɪ ɫ

1, 4 5

x ,

ɩɪɢ ɧɨɦɢɧɚɥɶɧɨɣ

ɧɚɝɪɭɡɤɟ ɢɦɟɟɬ

1U ,

, cos 0,85

2,14

ɢ 29

; ɷɤɜɢɜɚ-

ɥɟɧɬɧɵɟ ɟɦɭ ɧɟɹɜɧɨɩɨɥɸɫɧɵɟ ɝɟɧɟɪɚɬɨɪɵ ɢɦɟɸɬ

2,15



, 1, 4 5

ɢɥɢ

1, 7 5

=1.

4.3. Ɋɠɫɠɰɩɟɨɶɠ ɘȿɌ ɣ ɫɠɛɥɭɣɝɨɩɬɭɣ ɬɣɨɰɫɩɨɨɩɤ ɧɛɳɣɨɶ

Ɉɛɪɚɬɢɦɫɹ ɤ ɫɢɧɯɪɨɧɧɨɣ ɹɜɧɨɩɨɥɸɫɧɨɣ ɦɚɲɢɧɟ ɛɟɡ ɞɟɦɩɮɟɪɧɵɯ

(ɭɫɩɨɤɨɢɬɟɥɶɧɵɯ) ɨɛɦɨɬɨɤ. ɉɪɢ ɄɁ ɜɨɡɧɢɤɚɟɬ ɩɟɪɟɯɨɞɧɵɣ ɩɪɨɰɟɫɫ,

ɩɪɢɜɨɞɹɳɢɣ ɤ ɢɡɦɟɧɟɧɢɸ ɬɨɤɨɜ ɢ ɧɚɩɪɹɠɟɧɢɣ ɩɪɟɞɲɟɫɬɜɭɸɳɟɝɨ ɪɟ-

ɠɢɦɚ. ȼɵɹɫɧɢɦ, ɤɚɤɢɦɢ ɗȾɋ ɢ ɪɟɚɤɬɢɜɧɨɫɬɹɦɢ ɦɨɠɧɨ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ

ɫɢɧɯɪɨɧɧɭɸ ɦɚɲɢɧɭ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ ɫ ɰɟ-

ɥɶɸ ɪɚɫɱɟɬɚ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɬɨɤɚ ɄɁ ɞɥɹ

0t .

ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɨɛɭɫɥɨɜɥɟɧɚ ɬɟɦ, ɱɬɨ ɫɢɧɯɪɨɧɧɚɹ ɗȾɋ (

ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɦɚɲɢɧɭ ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ, ɜ ɦɨɦɟɧɬ ɄɁ

ɫɤɚɱɤɨɨɛɪɚɡɧɨ ɢɡɦɟɧɹɟɬɫɹ. ȼ ɫɢɥɭ ɷɬɨɝɨ ɨɧɚ ɧɟɢɡɜɟɫɬɧɚ ɢ ɧɟɩɪɢɟɦɥɟɦɚ

ɞɥɹ ɪɚɫɱɟɬɚ ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ, ɪɚɜɧɵɦ ɨɛɪɚɡɨɦ ɤɚɤ ɢ

, ɫɜɹ-

ɡɚɧɧɵɟ ɫ

Ⱦɥɹ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɨɛɪɚɬɢɦɫɹ ɤ ɛɚɥɚɧɫɭ ɦɚɝɧɢɬ-

ɧɵɯ ɩɨɬɨɤɨɜ ɜ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ

d ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ ɞɥɹ ɧɨɪɦɚɥɶɧɨɝɨ

ɪɟɠɢɦɚ (ɫɦ. ɪɢɫ. 4.4, ɚ). ȼ ɭɤɚɡɚɧɧɨɣ ɨɫɢ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɜɟ ɦɚɝɧɢ-

ɬɨɫɜɹɡɚɧɧɵɟ ɨɛɦɨɬɤɢ: ɨɛɦɨɬɤɚ ɜɨɡɛɭɠɞɟɧɢɹ ɢ ɨɛɦɨɬɤɚ ɫɬɚɬɨɪɚ.

ȼ ɪɟɠɢɦɟ ɯɨɥɨɫɬɨɝɨ ɯɨɞɚ ɬɨɤ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ

ɫɨɡɞɚɟɬ

ɦɚɝɧɢɬɧɵɣ ɩɨɬɨɤ

, ɫɨɫɬɨɹɳɢɣ ɢɡ ɩɨɬɨɤɚ ɪɚɫɫɟɹɧɢɹ ɪɨɬɨɪɚ

ɩɨɥɟɡɧɨɝɨ ɩɨɬɨɤɚ

Ɏ :

ɹɜɥɹɟɬɫɹ ɬɨɤɨɦ

ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ, ɩɪɢɜɟɞɟɧɧɵɦ ɤ ɫɬɚɬɨɪɭ [1].

ɎɎ

ɝɞɟ

–

ɟɚɤɬɢɜɧɨɫɬɶ ɪ

ɬɢɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɪɟɚ

ɧɚɹ ɢɧɞɭɤɬɢɜɧɨɫɬɶ ɨɛɦɨɬɤ

ɉɨɥɟɡɧɵɣ ɩɨɬɨɤ

ɜ ɫɬɚɬɨɪɧɨɣ ɰɟɩɢ ɫɢɧɯɪɨ

ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɟɞɢɧɢɰ

ȼ ɧɟɧɚɫɵɳɟɧɧɨɣ ɦɚ

ɫɬɨɹɧɧɭɸ ɞɨɥɸ ɩɨɬɨɤɚ

ɪɚɫɫɟɹɧɢɹ ɨɛɦɨɬɤɢ ɜɨɡɛɭ

Ɋɢɫ. 4.4: ɚ – ɫɨɫɬɚɜɥɹ

ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦ



df fad ff

xx x

  Ɏ II

ɫɫɟɹɧɢɹ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ;

–

ɰɢɢ ɫɬɚɬɨɪɚ ɩɨ ɨɫɢ

;

xx x



ɜɨɡɛɭɠɞɟɧɢɹ.

fad

x I

ɩɪɢ ɜɪɚɳɟɧɢɢ ɪɨɬɨɪɚ ɨɛɭɫɥ

ɧɭɸ ɷɥɟɤɬɪɨɞɜɢɠɭɳɭɸ ɫɢɥɭ

. ȼ

**dq

ɎǼ

ɲɢɧɟ ɩɨɬɨɤ

ɫɨɫɬɚɜɥɹɟɬ ɧɟɤɨɬ

, ɤɨɬɨɪɚɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɤɨɷɮɮɢ

ɞɟɧɢɹ

ff f

ffad

VV V



ɳɢɟ ɩɨɬɨɤɨɜ ɩɨ ɨɫɢ d ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢ

ɟ; ɛ – ɜɟɤɬɨɪɧɚɹ ɞɢɚɝɪɚɦɦɚ ɹɜɧɨɩɨɥɸɫɧ

ɜ ɪɚɛɨɱɟɦ ɪɟɠɢɦɟ

(4.5)

–

ɢɧɞɭɤ-

– ɩɨɥ-

ɜɥɢɜɚɟɬ

ɫɢɫɬɟɦɟ

ɪɭɸ ɩɨ-

ɢɟɧɬɨɦ

(4.6)

ɧɵ

ɣ ɋɆ

ɉɪɢ ɡɚɦɤɧɭɬɨɣ ɰɟɩɢ ɫɬɚɬɨɪɚ ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ ɩɨ ɧɟɣ

ɩɪɨɬɟɤɚɟɬ ɧɟɢɡɦɟɧɧɵɣ ɩɨ ɚɦɩɥɢɬɭɞɟ ɬɨɤ

, ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨ-

ɠɟɧ ɧɚ ɩɪɨɞɨɥɶɧɭɸ

I ɢ ɩɨɩɟɪɟɱɧɭɸ

ɫɨɫɬɚɜɥɹɸɳɢɟ. Ɍɨɤ

I (ɩɨ-

ɫɤɨɥɶɤɭ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɩɨɬɨɤɢ ɬɨɥɶɤɨ ɩɨ ɨɫɢ

d ) ɫɨɡɞɚɟɬ ɩɨɬɨɤ ɪɟɚɤ-

ɰɢɢ ɫɬɚɬɨɪɚ

ad d ad

Ɏ I , ɤɨɬɨɪɵɣ ɩɪɨɧɢɡɵɜɚɟɬ ɨɛɦɨɬɤɭ ɜɨɡɛɭɠɞɟɧɢɹ.

ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɷɬɢɦ ɩɨɥɧɨɟ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɟ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ ɜ

ɧɚɝɪɭɡɨɱɧɨɦ ɪɟɠɢɦɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ

ȌɎɎ

fad6



. (4.7)

Ȼɥɚɝɨɞɚɪɹ ɦɚɝɧɢɬɧɨɣ ɫɜɹɡɢ ɦɟɠɞɭ ɨɛɦɨɬɤɚɦɢ ɫɬɚɬɨɪɚ ɢ ɪɨɬɨɪɚ

ɩɪɢ ɜɧɟɡɚɩɧɨɦ ɢɡɦɟɧɟɧɢɢ ɬɨɤɚ ɫɬɚɬɨɪɧɨɣ ɰɟɩɢ ɧɚ

'I ɜ ɨɛɦɨɬɤɟ ɜɨɡ-

ɛɭɠɞɟɧɢɹ ɢɧɞɭɰɢɪɭɟɬɫɹ ɩɨɫɬɨɹɧɧɵɣ (ɚɩɟɪɢɨɞɢɱɟɫɤɢɣ) ɬɨɤ

ɬɚɤɨɣ

ɜɟɥɢɱɢɧɵ, ɱɬɨ ɪɟɡɭɥɶɬɢɪɭɸɳɟɟ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɟ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ

ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɵɦ. ɗɬɨ ɭɫɥɨɜɢɟ ɹɜɥɹɟɬɫɹ ɫɥɟɞɫɬɜɢɟɦ ɡɚɤɨɧɚ

Ʌɟɧɰɚ. Ɏɢɡɢɱɟɫɤɢ ɷɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɄɁ ɩɨɬɨɤɢ



0ad

ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɢɯ ɡɧɚɱɟɧɢɹ ɜ ɧɨɪɦɚɥɶɧɨɦ ɪɟ-

ɠɢɦɟ (

Ɏ ) ɩɥɸɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɢɦ ɩɪɢɪɚɳɟɧɢɹ



'Ɏ



0ad

'Ɏ

, ɨɛɭɫɥɨɜɥɟɧɧɵɟ ɬɨɤɚɦɢ

'I . Ɉɞɧɚɤɨ ɩɪɢɪɚɳɟɧɢɹ ɭɤɚ-

ɡɚɧɧɵɯ ɩɨɬɨɤɨɜ ɤɨɦɩɟɧɫɢɪɭɸɬ ɞɪɭɝ ɞɪɭɝɚ ɬɚɤ, ɱɬɨ

 

ɎɎ0

fad

''

ɨɫɬɚɜɥɹɹ ɧɟɢɡɦɟɧɧɵɦ ɪɟɡɭɥɶɬɢɪɭɸɳɟɟ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɟ ɨɛɦɨɬɤɢ ɜɨɡ-

ɛɭɠɞɟɧɢɹ (

) ɫɨɝɥɚɫɧɨ ɜɵɪɚɠɟɧɢɸ (4.7).

Ⱦɥɹ ɪɟɲɟɧɢɹ ɪɚɧɟɟ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɣ ɡɚɞɚɱɢ (ɧɚɯɨɠɞɟɧɢɟ ɩɚɪɚ-

ɦɟɬɪɨɜ ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɧɚɪɭɲɟɧɢɹ ɪɟɠɢ-

ɦɚ) ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɧɟɢɡɦɟɧɧɨɫɬɶɸ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɹ

. Ɉɬɦɟɬɢɦ, ɱɬɨ

ɱɚɫɬɶ ɩɨɬɨɤɚ

, ɨɩɪɟɞɟɥɹɟɦɚɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ

, ɨɫɬɚɟɬɫɹ ɫɜɹɡɚɧ-

ɧɨɣ ɬɨɥɶɤɨ ɫ ɨɛɦɨɬɤɨɣ ɜɨɡɛɭɠɞɟɧɢɹ. Ɂɧɚɹ ɤɨɷɮɮɢɰɢɟɧɬ ɪɚɫɫɟɹɧɢɹ

ɜɵɞɟɥɢɦ ɬɭ ɱɚɫɬɶ

, ɤɨɬɨɪɚɹ ɜɡɚɢɦɨɞɟɣɫɬɜɭɟɬ ɫɨ ɫɬɚɬɨɪɨɦ:





1 ı

dff6

ȌȌ

, ɝɞɟ

ad f



. (4.8)

ɂɦɟɧɧɨ ɷɬɨ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɟ

ɢ ɨɛɭɫɥɨɜɥɟɧɧɚɹ ɢɦ ɗȾɋ ɜ ɨɛ-

ɦɨɬɤɟ ɫɬɚɬɨɪɚ

ɫɨɯɪɚɧɹɸɬ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫ-

ɫɚ ɫɜɨɢ ɩɪɟɞɲɟɫɬɜɭɸɳɢɟ ɡɧɚɱɟɧɢɹ.

Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɡɚɩɢɫɚɧɵ ɫɤɚɥɹɪɧɵɟ ɜɵɪɚɠɟɧɢɹ, ɜ ɤɨɬɨɪɵɯ ɭɱɬɟɧɵ ɮɚɡɵ ɜɟɤɬɨɪɨɜ.

ɉɪɢɞɚɞɢɦ ɜɵɪɚɠɟɧɢɸ (4.8) ɛɨɥɟɟ ɧɚɝɥɹɞɧɵɣ ɜɢɞ:





222

ȌɎɎ

ad ad ad

dd f ad f f d ad f ad d

ff f

ad ad ad

qd qdd d qdd q

fff

xx x

Ix Ix Ix I

xx x

xxx

EI UIx I UIx E

xxx

  

§·

    

¨¸

©¹

ȼ ɨɤɨɧɱɚɬɟɥɶɧɨɣ ɮɨɪɦɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ

ɡɚɩɢɲɟɬɫɹ ɬɚɤ:

qqdd

EUIx



, (4.9)

ɝɞɟ

dd ad f

xxxx

 

(4.10)

– ɩɪɨɞɨɥɶɧɚɹ ɩɟɪɟɯɨɞɧɚɹ ɪɟɚɤɬɢɜɧɨɫɬɶ, ɜɟɥɢɱɢɧɚ ɩɚɫɩɨɪɬɧɚɹ;

– ɪɟ-

ɚɤɬɢɜɧɨɫɬɶ ɪɚɫɫɟɹɧɢɹ ɫɬɚɬɨɪɧɨɣ ɨɛɦɨɬɤɢ;

– ɩɟɪɟɯɨɞɧɚɹ ɗȾɋ (ɪɚɫ-

ɩɨɥɚɝɚɟɬɫɹ ɩɨ ɨɫɢ

q ɋɆ).

Ɍɟɪɦɢɧɵ «ɩɟɪɟɯɨɞɧɚɹ ɗȾɋ», «ɩɟɪɟɯɨɞɧɚɹ ɪɟɚɤɬɢɜɧɨɫɬɶ» ɫɥɟɞɭɟɬ

ɨɬɧɨɫɢɬɶ ɤ ɬɨɦɭ, ɱɬɨ

ɜɦɟɫɬɟ ɫ

ɩɨɡɜɨɥɹɸɬ ɨɰɟɧɢɬɶ ɩɟɪɟɯɨɞ ɨɬ

ɧɨɪɦɚɥɶɧɨɝɨ ɪɟɠɢɦɚ ɤ ɪɟɠɢɦɭ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚɧɢɹ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɭɤɚ-

ɡɚɧɧɵɟ ɩɚɪɚɦɟɬɪɵ (

) ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɋɆ ɛɟɡ ɞɟɦɩɮɟɪɧɵɯ ɨɛɦɨ-

ɬɨɤ. ɋɥɟɞɭɟɬ ɨɫɨɛɨ ɩɨɞɱɟɪɤɧɭɬɶ, ɱɬɨ ɧɚɱɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ



,ɫɨɝɥɚɫɧɨ

ɜɵɪɚɠɟɧɢɸ (4.9), ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɩɚɪɚɦɟɬɪɚɦ

I , ɫ ɤɨɬɨɪɵɦɢ

ɦɚɲɢɧɚ ɪɚɛɨɬɚɥɚ ɞɨ ɧɚɪɭɲɟɧɢɹ ɪɟɠɢɦɚ. ȼ ɷɬɨɦ ɫɦɵɫɥɟ



ɧɚɡɵɜɚɸɬ

ɪɚɫɱɟɬɧɨɣ, ɩɨɫɤɨɥɶɤɭ ɟɟ ɧɟɥɶɡɹ ɢɡɦɟɪɢɬɶ. ɇɟɢɡɦɟɧɧɨɫɬɶ

ɩɪɢ 0t

ɨɩɪɟɞɟɥɹɟɬ ɟɟ ɩɪɚɤɬɢɱɟɫɤɭɸ ɰɟɧɧɨɫɬɶ ɢ ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɱɢɬɵɜɚɬɶ ɧɚ-

ɱɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫɥɚɝɚɟɦɨɣ ɬɨɤɚ ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ.

ȼ ɞɚɥɶɧɟɣɲɟɦ, ɩɪɢ

0t ! ,

ɢɡɦɟɧɹɟɬɫɹ ɞɨ ɡɧɚɱɟɧɢɹ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟ-

ɝɨ ɧɨɜɨɦɭ ɭɫɬɚɧɨɜɢɜɲɟɦɭɫɹ ɪɟɠɢɦɭ ɦɚɲɢɧɵ (ɫɦ. ɪɢɫ. 4.6).

ɇɚ ɪɢɫ. 4.4, ɛ ɩɪɢɜɟɞɟɧɚ ɜɟɤɬɨɪɧɚɹ ɞɢɚɝɪɚɦɦɚ ɹɜɧɨɩɨɥɸɫɧɨɣ ɦɚɲɢ-

ɧɵ, ɪɚɛɨɬɚɸɳɟɣ ɜ ɧɚɝɪɭɡɨɱɧɨɦ ɪɟɠɢɦɟ ɫ ɨɬɫɬɚɸɳɢɦ ɬɨɤɨɦ. ȼɟɤɬɨɪ

ɫɨɜɩɚɞɚɟɬ ɫ ɜɟɤɬɨɪɨɦ

ɢ ɦɟɧɶɲɟ ɟɝɨ ɧɚ ɜɟɥɢɱɢɧɭ ɩɚɞɟɧɢɹ ɧɚɩɪɹɠɟ-

ɧɢɹ



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ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɢɡ ɫɬɪɭɤɬɭɪɵ ɜɵɪɚɠɟɧɢɹ (4.10) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɟ-

ɪɟɯɨɞɧɨɦɭ ɫɨɩɪɨɬɢɜɥɟɧɢɸ

ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɯɟɦɚ ɡɚɦɟɳɟɧɢɹ ɦɚɲɢɧɵ,

ɢɡɨɛɪɚɠɟɧɧɚɹ ɧɚ ɪɢɫ. 4.5, ɜ. Ɉɤɚɡɵɜɚɟɬɫɹ, ɷɬɚ ɫɯɟɦɚ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟ-

ɧɚ ɢɧɵɦ ɩɭɬɟɦ – ɩɨɫɪɟɞɫɬɜɨɦ ɷɤɜɢɜɚɥɟɧɬɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɦɚɝɧɢɬɧɨ

ɫɜɹɡɚɧɧɵɯ ɰɟɩɟɣ.