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Файл:Физика лазеров. Часть 2. Основы теории лазеров. Учебное пособие
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ɩɨɩɟɪɟɱɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ, ɬɨ ɜɫɥɟɞɫɬɜɢɟ
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ɛɨɥɶɲɟɣ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɨɩɬɢɱɟɫɤɨɝɨ ɩɨɥɹ ɩɪɨɞɨɥɶɧɨɣ ɦɨɞɵ ɜ
ɰɟɧɬɪɟ ɡɟɪɤɚɥɚ, ɩɨɩɟɪɟɱɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ
ɛɭɞɟɬ ɢɦɟɬɶ ɦɢɧɢɦɭɦ ɜ ɰɟɧɬɪɟ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ (ɫɦ. ɪɢɫ. 9.4). ɋɥɟɞɭɟɬ
ɡɚɦɟɬɢɬɶ, ɱɬɨ ɷɬɨɬ ɦɢɧɢɦɭɦ ɞɨɥɠɟɧ ɛɵɬɶ ɧɢɠɟ ɩɨɪɨɝɨɜɨɝɨ ɡɧɚɱɟɧɢɹ,
ɩɨɫɤɨɥɶɤɭ ɩɪɢ ɧɟɤɨɬɨɪɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɰɟɧɬɪɚ ɪɚɡɧɨɫɬɶ ɧɚɫɟɥɟɧɧɨɫɬɟɣ
ɨɤɚɡɵɜɚɟɬɫɹ ɜɵɲɟ ɩɨɪɨɝɨɜɨɝɨ ɡɧɚɱɟɧɢɹ
ɜɫɥɟɞɫɬɜɢɟ ɭɦɟɧɶɲɟɧɢɹ
ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɩɪɨɞɨɥɶɧɨɣ ɦɨɞɵ ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, - ɨɫɥɚɛɥɟɧɢɹ
ɷɮɮɟɤɬɚ ɧɚɫɵɳɟɧɢɹ.
TEM
00
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ɩɨɪ
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ɩɨɪ
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10
Ɋɢɫ. 9.4.
Ɋɚɫɩɪɟɞɟɥɟɧɢɹ ɜ ɩɨɩɟɪɟɱɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɨɩɬɢɱɟɫɤɢɯ
ɤɨɥɟɛɚɧɢɣ ɞɥɹ ɩɪɨɞɨɥɶɧɨɣ ɢ ɩɟɪɜɨɣ ɩɨɩɟɪɟɱɧɨɣ ɦɨɞ, ɚ ɬɚɤɠɟ -
ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ ɫ ɭɱɟɬɨɦ ɜɥɢɹɧɢɹ ɷɮɮɟɤɬɚ ɧɚɫɵɳɟɧɢɹ ɩɨɞ
0
1
,
ɞɟɣɫɬɜɢɟɦ ɩɨɥɹ ɩɪɨɞɨɥɶɧɨɣ ɦɨɞɵ:
ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɫɢɥɟɧɢɹ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɞɥɹ ɩɪɨɞɨɥɶɧɨɣ ɢ ɩɟɪɜɨɣ
ɩɨɪ
- ɩɨɪɨɝɨɜɵɟ
ɩɨɪ
ɩɨɩɟɪɟɱɧɨɣ ɦɨɞ.
Ʉɚɤ ɞɨɥɠɧɨ ɛɵɬɶ ɹɫɧɨ ɢɡ ɷɬɨɝɨ ɪɢɫɭɧɤɚ, ɞɥɹ ɩɟɪɜɨɣ ɩɨɩɟɪɟɱɧɨɣ
ɦɨɞɵ, ɢɦɟɸɳɟɣ ɦɚɤɫɢɦɭɦɵ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɩɨɥɹ, ɫɞɜɢɧɭɬɵɟ ɨɬ
ɰɟɧɬɪɚ, ɩɨɪɨɝɨɜɨɟ ɭɫɥɨɜɢɟ ɦɨɠɟɬ ɨɤɚɡɚɬɶɫɹ ɩɟɪɟɜɵɩɨɥɧɟɧɧɵɦ ɢ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɨɡɦɨɠɧɚ ɟɟ ɝɟɧɟɪɚɰɢɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫ ɩɪɨɞɨɥɶɧɵɦɢ
ɦɨɞɚɦɢ. Ɋɚɡɭɦɟɟɬɫɹ, ɬɚɤɨɟ ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɜ ɚɤɬɢɜɧɵɯ ɫɪɟɞɚɯ
ɫ
ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɭɫɢɥɟɧɢɹ, ɤɚɤ ɷɬɨ ɢɦɟɟɬ ɦɟɫɬɨ,
ɧɚɩɪɢɦɟɪ ɜ ɬɜɟɪɞɨɬɟɥɶɧɵɯ ɢɥɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɨɜɵɯ ɥɚɡɟɪɚɯ. Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ, ɜɫɥɟɞɫɬɜɢɟ ɪɚɡɥɢɱɢɹ ɜ ɩɨɩɟɪɟɱɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɹɯ
ɩɨɥɟɣ ɩɪɨɞɨɥɶɧɵɯ ɢ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ ɜɨɡɦɨɠɧɚ ɢɯ ɨɞɧɨɜɪɟɦɟɧɧɚɹ
ɝɟɧɟɪɚɰɢɹ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɨɳɧɨɣ ɧɚɤɚɱɤɟ ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ,
ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɦ ɤɨɷɮɮɢɰɢɟɧɬɟ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ .
51
9.3.2. ɇɟɨɞɧɨɪɨɞɧɨɟ ɭɲɢɪɟɧɢɟ
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ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɜɫɥɟɞɫɬɜɢɟ ɬɨɝɨ, ɱɬɨ ɨɬɞɟɥɶɧɵɟ ɩɨɞɚɧɫɚɦɛɥɢ
ɱɚɫɬɢɰ ɢɦɟɸɬ ɪɚɡɥɢɱɧɵɟ ɱɚɫɬɨɬɵ ɩɟɪɟɯɨɞɚ, ɨɧɢ ɦɨɝɭɬ
ɜɡɚɢɦɨɞɟɣɫɬɜɨɜɚɬɶ ɬɨɥɶɤɨ ɫ ɬɟɦɢ ɨɩɬɢɱɟɫɤɢɦɢ ɤɨɥɟɛɚɧɢɹɦɢ, ɱɚɫɬɨɬɵ
ɤɨɬɨɪɵɯ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨ ɨɬɥɢɱɚɸɬɫɹ (ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɜ ɩɪɟɞɟɥɚɯ
ɲɢɪɢɧɵ ɥɨɪɟɧɰɟɜɨɣ ɥɢɧɢɢ) ɨɬ ɱɚɫɬɨɬ ɩɟɪɟɯɨɞɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ
ɩɨɞɚɧɫɚɦɛɥɟɣ. ȼ ɫɜɹɡɢ ɫ ɬɟɦ, ɱɬɨ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɩɪɨɞɨɥɶɧɵɦɢ
ɦɨɞɚɦɢ ɨɛɵɱɧɨ ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɟɜɨɫɯɨɞɢɬ ɲɢɪɢɧɭ
ɨɞɧɨɪɨɞɧɨ ɭɲɢɪɟɧɧɨɣ
ɥɢɧɢɢ ɩɨɞɚɧɫɚɦɛɥɹ, ɤɨɧɤɭɪɟɧɰɢɹ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ ɨɤɚɡɵɜɚɟɬɫɹ
ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɜɨɡɦɨɠɧɨɣ. ȼ ɪɚɜɧɨɣ ɫɬɟɩɟɧɢ ɩɨɞɨɛɧɚɹ ɫɢɬɭɚɰɢɹ ɦɨɠɟɬ
ɢɦɟɬɶ ɦɟɫɬɨ ɢ ɜ ɫɥɭɱɚɟ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɪɢ
ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɣ ɧɚɤɚɱɤɟ ɦɨɝɭɬ ɨɞɧɨɜɪɟɦɟɧɧɨ
ɫɭɳɟɫɬɜɨɜɚɬɶ ɚɜɬɨɤɨɥɟɛɚɧɢɹ ɤɚɤ ɧɚ ɱɚɫɬɨɬɚɯ ɧɟɫɤɨɥɶɤɢɯ
ɩɪɨɞɨɥɶɧɵɯ, ɬɚɤ ɢ ɧɟɫɤɨɥɶɤɢɯ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ. Ʉɨɥɢɱɟɫɬɜɨ ɢ
ɦɨɳɧɨɫɬɢ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɨɬɧɵɯ ɤɨɦɩɨɧɟɧɬ ɝɟɧɟɪɚɰɢɢ
ɨɩɪɟɞɟɥɹɸɬɫɹ
ɩɚɪɚɦɟɬɪɚɦɢ ɥɚɡɟɪɚ ɢ ɦɨɳɧɨɫɬɶɸ ɧɚɤɚɱɤɢ
ɉɪɢɛɥɢɡɢɬɟɥɶɧɵɣ ɜɢɞ ɫɩɟɤɬɪɚ ɝɟɧɟɪɚɰɢɢ ɥɚɡɟɪɚ ɫ ɧɟɨɞɧɨɪɨɞɧɨ
ɭɲɢɪɟɧɧɨɣ ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɟɣ ɧɚ ɧɟɫɤɨɥɶɤɢɯ ɩɪɨɞɨɥɶɧɵɯ ɢ
ɩɨɩɟɪɟɱɧɵɯ ɦɨɞɚɯ ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ. 9.5. Ƚɨɪɢɡɨɧɬɚɥɶɧɵɦɢ ɩɭɧɤɬɢɪɧɵɦɢ
ɥɢɧɢɹɦɢ ɧɚ ɷɬɨɦ ɪɢɫɭɧɤɟ ɨɬɦɟɱɟɧɵ ɩɨɪɨɝɨɜɵɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ
0
ɭɫɢɥɟɧɢɹ ɞɥɹ ɩɪɨɞɨɥɶɧɵɯ (
123
,,
(
ɩɨɪ ɩɨɪ ɩɨɪ
ɫɩɟɤɬɪɚɥɶɧɵɯ ɢɧɬɟɪɜɚɥɨɜ, ɜ ɩɪɟɞɟɥɚɯ ɤɨɬɨɪɵɯ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ
). ȼɟɪɬɢɤɚɥɶɧɵɦɢ ɩɭɧɤɬɢɪɧɵɦɢ ɥɢɧɢɹɦɢ ɨɬɦɟɱɟɧɵ ɤɪɚɹ
), ɢ ɬɪɟɯ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ
ɩɨɪ
ɩɪɟɜɵɲɚɟɬ ɟɝɨ ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɦɨɞ. Ɍɚɤɠɟ ɤɚɤ
ɢ ɜ ɩɪɢɜɟɞɟɧɧɨɦ ɜɵɲɟ ɩɪɢɦɟɪɟ ɫ ɨɞɧɨɪɨɞɧɵɦ ɭɲɢɪɟɧɢɟɦ, ɬɨɱɤɢ
ɩɟɪɟɫɟɱɟɧɢɹ ɷɬɢɯ ɩɭɧɤɬɢɪɧɵɯ ɥɢɧɢɣ ɫ ɤɨɧɬɭɪɨɦ ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɢ
ɨɩɪɟɞɟɥɹɸɬ ɬɨɬ ɫɩɟɤɬɪɚɥɶɧɵɣ ɢɧɬɟɪɜɚɥ, ɜ ɤɨɬɨɪɨɦ ɦɨɠɟɬ ɢɦɟɬɶ ɦɟɫɬɨ
ɝɟɧɟɪɚɰɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɦɨɞ.
Ʉɚɤ ɜɢɞɧɨ ɢɡ ɪɢɫɭɧɤɚ, ɦɨɳɧɨɫɬɢ
ɝɟɧɟɪɚɰɢɢ ɧɚ ɱɚɫɬɨɬɚɯ ɪɚɡɧɵɯ ɩɪɨɞɨɥɶɧɵɯ ɢ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ ɦɨɝɭɬ ɛɵɬɶ
ɫɭɳɟɫɬɜɟɧɧɨ ɪɚɡɧɵɦɢ, ɩɨɫɤɨɥɶɤɭ ɭ ɪɚɡɧɵɯ ɦɨɞ ɪɚɡɥɢɱɧɚɹ ɜɟɥɢɱɢɧɚ
ɩɪɟɜɵɲɟɧɢɹ ɧɚɞ ɩɨɪɨɝɨɦ ɝɟɧɟɪɚɰɢɢ. ɉɨɞ ɩɪɟɜɵɲɟɧɢɟɦ ɧɚɞ ɩɨɪɨɝɨɦ ɜ
ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɩɨɧɢɦɚɟɬɫɹ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɡɧɚɱɟɧɢɟɦ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɭɫɢɥɟɧɢɹ ɧɚ ɱɚɫɬɨɬɟ ɦɨɞɵ ɫ ɧɨɦɟɪɨɦ m ɢ ɩɨɪɨɝɨɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ
ɭɫɢɥɟɧɢɹ ɞɥɹ
ɷɬɨɣ ɠɟ ɦɨɞɵ, ɨɬɧɟɫɟɧɧɚɹ ɤ ɩɨɪɨɝɨɜɨɦɭ ɡɧɚɱɟɧɢɸ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ
m
q
ɩɨɪ
ɩɨɪ
.
m
()
ɉɪɢ ɷɬɨɦ, ɨɱɟɜɢɞɧɨ, ɱɟɦ ɞɚɥɶɲɟ ɨɬ ɰɟɧɬɪɚ ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɢ
ɧɚɯɨɞɢɬɫɹ ɬɚ ɢɥɢ ɢɧɚɹ ɦɨɞɚ, ɬɟɦ ɦɟɧɶɲɟ ɟɟ ɦɨɳɧɨɫɬɶ.
52
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Z
Z
3
ɩɩɪɨ
2
ɩɨɪ
1
ɨɨɪɪɩ
0
m 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 01
q-2
0 1 2 3 0 1 2 0 1
q-1 q q+1
P
en
q
q+1q-1
Ɋɢɫ. 9.5.
ɂɥɥɸɫɬɪɚɰɢɹ ɫɥɭɱɚɹ ɨɞɧɨɜɪɟɦɟɧɧɨɣ ɝɟɧɟɪɚɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɪɨɞɨɥɶɧɵɯ
ɢ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ ɜ ɥɚɡɟɪɟ ɫ ɧɟɨɞɧɨɪɨɞɧɨ ɭɲɢɪɟɧɧɨɣ ɫɩɟɤɬɪɚɥɶɧɨɣ
0
ɥɢɧɢɟɣ:
1
- ɩɨɪɨɝɨɜɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɞɥɹ ɩɟɪɜɨɣ ɩɨɩɟɪɟɱɧɨɣ ɦɨɞɵ,
ɩɨɪ
2
- ɩɨɪɨɝɨɜɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɞɥɹ ɜɬɨɪɨɣ ɩɨɩɟɪɟɱɧɨɣ ɦɨɞɵ,
ɩɨɪ
3
- ɩɨɪɨɝɨɜɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɞɥɹ ɬɪɟɬɶɟɣ ɩɨɩɟɪɟɱɧɨɣ ɦɨɞɵ.
ɩɨɪ
- ɩɨɪɨɝɨɜɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɞɥɹ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ,
ɩɨɪ
9.4. ɋɟɥɟɤɰɢɹ ɝɟɧɟɪɢɪɭɸɳɢɯ ɦɨɞ ɜ ɥɚɡɟɪɚɯ
ȼ ɪɹɞɟ ɩɪɚɤɬɢɱɟɫɤɢɯ ɩɪɢɦɟɧɟɧɢɣ ɥɚɡɟɪɨɜ ɧɟɨɛɯɨɞɢɦɨ ɢɦɟɬɶ
ɬɨɥɶɤɨ ɨɞɧɭ ɱɚɫɬɨɬɧɭɸ ɤɨɦɩɨɧɟɧɬɭ ɜ ɫɩɟɤɬɪɟ ɢɡɥɭɱɟɧɢɹ. ɋɥɭɱɚɣ ɬɚɤɨɣ
ɝɟɧɟɪɚɰɢɢ ɨɛɵɱɧɨ ɧɚɡɵɜɚɸɬ ɨɞɧɨɱɚɫɬɨɬɧɵɦ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɬɚɤɨɣ
ɪɟɠɢɦ ɝɟɧɟɪɚɰɢɢ ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɟɟ ɩɪɟɞɩɨɱɬɢɬɟɥɶɧɵɦ ɜ ɫɥɭɱɚɟ
ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɩɨɥɭɱɢɬɶ ɧɚɢɜɵɫɲɭɸ ɦɨɧɨɯɪɨɦɚɬɢɱɧɨɫɬɶ ɥɚɡɟɪɧɨɝɨ
ɢɡɥɭɱɟɧɢɹ. Ⱦɥɹ ɪɟɚɥɢɡɚɰɢɢ ɷɬɨɝɨ ɪɟɠɢɦɚ ɝɟɧɟɪɚɰɢɢ ɦɨɝɭɬ ɛɵɬɶ
ɢɫɩɨɥɶɡɨɜɚɧɵ ɪɚɡɥɢɱɧɵɟ ɦɟɬɨɞɵ ɫɟɥɟɤɰɢɢ ɨɞɧɨɣ ɩɪɨɞɨɥɶɧɨɣ
ɝɟɧɟɪɢɪɭɸɳɟɣ
ɦɨɞɵ. Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɢɡ ɧɢɯ. Ɂɚɦɟɬɢɦ, ɱɬɨ
ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɩɨɞɨɛɧɨɣ ɫɟɥɟɤɰɢɢ ɦɨɞ ɱɚɳɟ ɜɨɡɧɢɤɚɟɬ ɜ ɫɥɭɱɚɟ
53
ɧɟɩɪɟɪɵɜɧɵɯ ɝɚɡɨɜɵɯ ɥɚɡɟɪɨɜ, ɫɬɟɩɟɧɶ ɦɨɧɨɯɪɨɦɚɬɢɱɧɨɫɬɢ ɤɨɬɨɪɵɯ
'
d
' '
ɦɨɠɟɬ ɛɵɬɶ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ, ɩɨɬɟɧɰɢɚɥɶɧɨ, ɧɚɢɜɵɫɲɟɣ.
ɇɚɢɛɨɥɟɟ ɩɪɨɫɬɨɣ ɫɩɨɫɨɛ ɩɨɥɭɱɢɬɶ ɝɟɧɟɪɚɰɢɸ ɧɚ ɨɞɧɨɣ
ɩɪɨɞɨɥɶɧɨɣ ɦɨɞɟ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɜ ɰɟɧɬɪɟ ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɢ, ɫɨɫɬɨɢɬ
ɜ ɩɨɞɛɨɪɟ ɬɚɤɨɣ ɦɨɳɧɨɫɬɢ ɧɚɤɚɱɤɢ, ɱɬɨ ɩɪɟɜɵɲɟɧɢɟ ɧɚɞ ɩɨɪɨɝɨɦ
ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɬɨɥɶɤɨ ɞɥɹ ɷɬɨɣ ɰɟɧɬɪɚɥɶɧɨɣ ɦɨɞɵ. ɇɟɞɨɫɬɚɬɤɨɦ ɬɚɤɨɝɨ
ɦɟɬɨɞɚ ɫɟɥɟɤɰɢɢ ɹɜɥɹɟɬɫɹ ɧɟɢɡɛɟɠɧɨɟ ɫɧɢɠɟɧɢɟ ɦɨɳɧɨɫɬɢ
ɝɟɧɟɪɚɰɢɢ
ɫɟɥɟɤɬɢɪɭɟɦɨɣ ɦɨɞɵ ɢɡ-ɡɚ ɨɞɧɨɜɪɟɦɟɧɧɨɝɨ ɭɦɟɧɶɲɟɧɢɹ ɩɪɟɜɵɲɟɧɢɹ
ɧɚɞ ɩɨɪɨɝɨɦ ɞɥɹ ɧɟɟ.
Ⱦɪɭɝɨɣ ɦɟɬɨɞ ɩɪɟɞɩɨɥɚɝɚɟɬ ɭɦɟɧɶɲɟɧɢɟ ɞɥɢɧɵ ɥɚɡɟɪɚ. Ʉɚɤ
ɢɡɜɟɫɬɧɨ, ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɩɪɨɞɨɥɶɧɵɦɢ ɦɨɞɚɦɢ ɨɛɪɚɬɧɨ
ɫ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɞɥɢɧɟ ɪɟɡɨɧɚɬɨɪɚ, ɚ ɢɦɟɧɧɨ,
ɨɛɪɚɡɨɦ, ɭɦɟɧɶɲɟɧɢɟ ɞɥɢɧɵ ɪɟɡɨɧɚɬɨɪɚ ɩɪɢɜɟɞɟɬ ɤ ɭɜɟɥɢɱɟɧɢɸ
f
. Ɍɚɤɢɦ
q
2
Ln
ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɩɪɨɞɨɥɶɧɵɦɢ ɦɨɞɚɦɢ. Ⱦɥɹ ɡɚɞɚɧɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɭɫɢɥɟɧɢɹ ɫɪɟɞɵ ɜɫɟɝɞɚ ɦɨɠɧɨ ɩɨɞɨɛɪɚɬɶ ɬɚɤɭɸ ɞɥɢɧɭ ɪɟɡɨɧɚɬɨɪɚ, ɩɪɢ
ɤɨɬɨɪɨɣ ɞɥɹ ɛɨɤɨɜɵɯ ɦɨɞ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɛɭɞɟɬ ɦɟɧɶɲɟ
ɩɨɪɨɝɨɜɨɝɨ. Ɋɚɡɭɦɟɟɬɫɹ, ɩɪɢ ɭɦɟɧɶɲɟɧɢɢ ɞɥɢɧɵ ɪɟɡɨɧɚɬɨɪɚ ɬɚɤɠɟ
ɞɨɥɠɧɚ ɧɟɫɤɨɥɶɤɨ ɭɦɟɧɶɲɢɬɶɫɹ ɦɨɳɧɨɫɬɶ ɝɟɧɟɪɚɰɢɢ ɧɚ ɰɟɧɬɪɚɥɶɧɨɣ
ɦɨɞɟ, ɩɨɫɤɨɥɶɤɭ ɭɦɟɧɶɲɚɟɬɫɹ ɨɛɴɟɦ
ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ. ɋɞɟɥɚɟɦ ɨɰɟɧɤɭ
ɞɥɢɧɵ ɪɟɡɨɧɚɬɨɪɚ ɝɟɥɢɣ-ɧɟɨɧɨɜɨɝɨ ɥɚɡɟɪɚ, ɩɪɢ ɤɨɬɨɪɨɣ ɜɨɡɦɨɠɧɚ
ɝɟɧɟɪɚɰɢɹ ɬɨɥɶɤɨ ɨɞɧɨɣ ɩɪɨɞɨɥɶɧɨɣ ɦɨɞɵ. ɇɚ ɱɚɫɬɨɬɟ ɥɚɡɟɪɧɨɝɨ
ɩɟɪɟɯɨɞɚ
ɪɚɜɧɚ
ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ ɫɨɫɬɚɜɥɹɟɬ
0,6328 ɦɤɦ ɲɢɪɢɧɚ ɞɨɩɥɟɪɨɜɫɤɨɣ ɥɢɧɢɢ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ
1500 ɆȽɰ. Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɩɨɪɨɝɨɜɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɞɥɹ
0,5 ɨɬ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɜ
ɰɟɧɬɪɟ ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɢ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɬɨɥɶɤɨ ɨɞɧɚ ɦɨɞɚ ɞɨɥɠɧɚ
ɧɚɯɨɞɢɬɶɫɹ ɜ ɫɩɟɤɬɪɚɥɶɧɨɦ ɢɧɬɟɪɜɚɥɟ, ɪɚɜɧɨɦ ɲɢɪɢɧɟ ɫɩɟɤɬɪɚɥɶɧɨɣ
ɥɢɧɢɢ. ȿɫɥɢ ɱɚɫɬɨɬɚ ɷɬɨɣ ɦɨɞɵ ɫɨɜɩɚɞɚɟɬ ɫ ɰɟɧɬɪɨɦ ɫɩɟɤɬɪɚɥɶɧɨɣ
ɥɢɧɢɢ, ɬɨ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɩɪɨɞɨɥɶɧɵɦɢ ɦɨɞɚɦɢ ɞɨɥɠɧɨ
ɛɵɬɶ ɩɨɪɹɞɤɚ
750 ɆȽɰ ɢɥɢ ɛɨɥɟɟ. Ɉɬɫɸɞɚ ɩɨɥɭɱɚɟɦ ɦɚɤɫɢɦɚɥɶɧɭɸ
ɞɥɢɧɭ ɪɟɡɨɧɚɬɨɪɚ ɥɚɡɟɪɚ
cc
L ɫɦ
2
ff
ȼɨɡɦɨɠɟɧ ɛɨɥɟɟ ɫɥɨɠɧɵɣ ɫɩɨɫɨɛ ɫɟɥɟɤɰɢɢ, ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ
qD
20
.
ɡɚɤɥɸɱɚɸɳɢɣɫɹ ɜ ɬɨɦ, ɱɬɨ ɜɧɭɬɪɶ ɥɚɡɟɪɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ ɦɟɠɞɭ ɚɤɬɢɜɧɨɣ
ɫɪɟɞɨɣ ɢ ɨɞɧɢɦ ɢɡ ɟɝɨ ɨɬɪɚɠɚɬɟɥɟɣ ɩɨɦɟɳɚɟɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ
ɡɟɪɤɚɥɨ ɫ ɡɚɞɚɧɧɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɨɬɪɚɠɟɧɢɹ (ɫɦ. ɪɢɫ. 9.6). ȼ
ɪɟɡɭɥɶɬɚɬɟ ɪɟɡɨɧɚɬɨɪ ɥɚɡɟɪɚ ɫɨɫɬɨɢɬ ɤɚɤ ɛɵ ɢɡ ɞɜɭɯ ɫɜɹɡɚɧɧɵɯ ɞɪɭɝ ɫ
ɞɪɭɝɨɦ ɪɟɡɨɧɚɬɨɪɨɜ, ɢɦɟɸɳɢɯ ɪɚɡɧɵɟ ɞɥɢɧɵ. Ɉɬɧɨɲɟɧɢɟ ɪɚɫɫɬɨɹɧɢɣ
ɦɟɠɞɭ ɞɨɩɨɥɧɢɬɟɥɶɧɵɦ ɡɟɪɤɚɥɨɦ ɢ ɞɜɭɦɹ ɞɪɭɝɢɦɢ ɨɬɪɚɠɚɬɟɥɹɦɢ
ɪɟɡɨɧɚɬɨɪɚ (L
) ɞɨɥɠɧɨ ɫɬɪɨɝɨ ɪɚɜɧɹɬɶɫɹ ɰɟɥɨɦɭ ɱɢɫɥɭ.
1/L2
54
'
/
'
Z
/
'Z
Z
Z
Z
Ⱥɤɬɢɜɧɚɹ ɫɪɟɞɚ
N>0
L
1
L
2
L
Ɋɢɫ.9.6.
ɋɯɟɦɚ ɤɨɧɫɬɪɭɤɰɢɢ ɥɚɡɟɪɚ, ɜ ɤɨɬɨɪɨɦ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɫɟɥɟɤɰɢɹ
ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ ɫ ɩɨɦɨɳɶɸ ɫɥɨɠɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ, ɫɨɫɬɨɹɳɟɝɨ ɢɡ ɞɜɭɯ
ɫɜɹɡɚɧɧɵɯ ɪɟɡɨɧɚɬɨɪɨɜ.
ɉɪɢɦɟɪ ɮɨɪɦɢɪɨɜɚɧɢɹ ɫɩɟɤɬɪɚ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ ɬɚɤɨɝɨ ɫɥɨɠɧɨɝɨ,
ɫɨɫɬɚɜɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ.9.7. ȼɵɫɨɬɚ ɱɚɫɬɨɬɧɵɯ ɤɨɦɩɨɧɟɧɬ
ɧɚ ɪɢɫɭɧɤɟ ɨɡɧɚɱɚɟɬ ɞɨɛɪɨɬɧɨɫɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɦɨɞ.
q
/
q
ɚ)
ɛ)
/
//
q00
q00
ɜ)
q00
Ɋɢɫ.9.7.
Ɏɨɪɦɢɪɨɜɚɧɢɟ ɫɩɟɤɬɪɚ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ ɥɚɡɟɪɚ ɫɨ ɫɥɨɠɧɵɦ
ɪɟɡɨɧɚɬɨɪɨɦ ɩɪɢ ɨɬɧɨɲɟɧɢɢ ɞɥɢɧ L
ɪɟɡɨɧɚɬɨɪɚ ɥɚɡɟɪɚ ɫ ɞɥɢɧɨɣ L
, ɛ) ɫɩɟɤɬɪ ɦɨɞ ɪɟɡɨɧɚɬɨɪɚ ɫ ɞɥɢɧɨɣ L2, ɜ)
1
=3: ɚ) ɫɩɟɤɬɪ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ
1/L2
ɫɩɟɤɬɪ ɦɨɞ ɫɥɨɠɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ ɫ ɞɥɢɧɨɣ L.
Ʉɚɤ ɞɨɥɠɧɨ ɛɵɬɶ ɹɫɧɨ ɢɡ ɷɬɨɝɨ ɪɢɫɭɧɤɚ, ɫɩɟɤɬɪ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ
ɬɚɤɨɝɨ ɥɚɡɟɪɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ ɨɤɚɡɵɜɚɟɬɫɹ ɛɨɥɟɟ ɪɚɡɪɟɠɟɧɧɵɦ.
Ⱦɨɛɪɨɬɧɨɫɬɶ ɦɨɞɵ ɫɨɫɬɚɜɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ ɛɭɞɟɬ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɣ
ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɱɚɫɬɨɬɵ ɦɨɞ ɪɟɡɨɧɚɬɨɪɨɜ ɫ ɞɥɢɧɚɦɢ L
ɢ L
1
ɫɨɜɩɚɞɚɸɬ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɢɡɥɭɱɟɧɢɟ, ɜɵɯɨɞɹɳɟɟ ɢɡ ɨɞɧɨɝɨ
ɪɟɡɨɧɚɬɨɪɚ, ɛɭɞɟɬ ɨɫɥɚɛɥɹɬɶɫɹ ɩɪɢ ɩɪɨɯɨɞɟ ɱɟɪɟɡ ɞɪɭɝɨɣ ɪɟɡɨɧɚɬɨɪ,
ɩɨɫɤɨɥɶɤɭ ɨɧ ɧɟ ɧɚɫɬɪɨɟɧ ɧɚ ɱɚɫɬɨɬɭ ɷɬɨɣ ɦɨɞɵ ɢ ɧɚɨɛɨɪɨɬ. ȼ
ɩɪɢɜɟɞɟɧɧɨɦ ɜɵɲɟ ɩɪɢɦɟɪɟ ɛɨɥɶɲɭɸ ɞɨɛɪɨɬɧɨɫɬɶ ɢɦɟɟɬ ɬɨɥɶɤɨ ɤɚɠɞɚɹ
ɱɟɬɜɟɪɬɚɹ ɩɪɨɞɨɥɶɧɚɹ ɦɨɞɚ ɬɚɤɨɝɨ ɫɨɫɬɚɜɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ. Ɉɱɟɜɢɞɧɨ,
2
55
ɱɬɨ ɱɟɦ ɛɨɥɶɲɟ ɨɬɧɨɲɟɧɢɟ ɞɥɢɧ L
O
, ɬɟɦ ɪɟɠɟ ɛɭɞɟɬ ɫɩɟɤɬɪ ɬɚɤɨɝɨ
1/L2
ɪɟɡɨɧɚɬɨɪɚ. ɇɟɞɨɫɬɚɬɨɤ ɬɚɤɨɝɨ ɦɟɬɨɞɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɧɟɨɛɯɨɞɢɦɨ ɫ
ɜɟɫɶɦɚ ɜɵɫɨɤɨɣ ɬɨɱɧɨɫɬɶɸ ɭɫɬɚɧɨɜɢɬɶ ɢ ɩɨɞɞɟɪɠɢɜɚɬɶ ɨɬɧɨɲɟɧɢɟ ɷɬɢɯ
ɞɥɢɧ. Ⱦɨɩɭɫɬɢɦɵɟ ɨɬɤɥɨɧɟɧɢɹ ɞɥɢɧ ɨɬ ɡɚɞɚɧɧɨɝɨ ɨɬɧɨɲɟɧɢɹ
ɨɩɪɟɞɟɥɹɸɬɫɹ ɲɢɪɢɧɨɣ ɪɟɡɨɧɚɧɫɧɵɯ ɤɪɢɜɵɯ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ, ɢ,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, - ɞɨɛɪɨɬɧɨɫɬɶɸ ɷɬɢɯ ɦɨɞ. ȼ ɥɸɛɨɦ ɫɥɭɱɚɟ ɨɬɤɥɨɧɟɧɢɟ
ɨɬ ɡɚɞɚɧɧɨɣ ɞɥɢɧɵ ɤɚɠɞɨɝɨ ɪɟɡɨɧɚɬɨɪɚ
ɞɨɥɠɧɚ ɛɵɬɶ ɧɚɦɧɨɝɨ ɦɟɧɶɲɟ
/2.
ȿɳɟ ɨɞɢɧ ɦɟɬɨɞ ɫɨɫɬɨɢɬ ɜ ɩɨɜɵɲɟɧɢɢ ɩɨɪɨɝɨɜɨɣ ɦɨɳɧɨɫɬɢ
ɧɚɤɚɱɤɢ ɩɭɬɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɭɦɟɧɶɲɟɧɢɹ ɞɨɛɪɨɬɧɨɫɬɢ ɨɬɞɟɥɶɧɵɯ
ɦɨɞ ɪɟɡɨɧɚɬɨɪɚ ɬɟɦ ɢɥɢ ɢɧɵɦ ɫɩɨɫɨɛɨɦ. Ɉɞɢɧ ɢɡ ɬɚɤɢɯ ɫɩɨɫɨɛɨɜ
ɩɨɧɢɠɟɧɢɹ ɞɨɛɪɨɬɧɨɫɬɢ ɫɨɫɬɨɢɬ ɜ ɭɫɬɚɧɨɜɤɟ ɜɧɭɬɪɶ ɪɟɡɨɧɚɬɨɪɚ
ɞɢɚɮɪɚɝɦɵ, ɨɝɪɚɧɢɱɢɜɚɸɳɟɣ ɚɩɟɪɬɭɪɭ ɪɟɡɨɧɚɬɨɪɚ. ȼ ɛɨɥɟɟ ɩɪɨɫɬɨɦ
ɜɚɪɢɚɧɬɟ ɦɨɠɧɨ ɩɪɨɫɬɨ ɨɝɪɚɧɢɱɢɬɶ ɩɨɩɟɪɟɱɧɵɣ ɪɚɡɦɟɪ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ,
ɧɚɩɪɢɦɟɪ, - ɞɢɚɦɟɬɪ ɚɤɬɢɜɧɨɝɨ ɷɥɟɦɟɧɬɚ
ɜ ɬɜɟɪɞɨɬɟɥɶɧɨɦ ɥɚɡɟɪɟ ɢɥɢ
ɜɧɭɬɪɟɧɧɢɣ ɞɢɚɦɟɬɪ ɝɚɡɨɜɨɣ ɪɚɡɪɹɞɧɨɣ ɬɪɭɛɤɢ ɜ ɝɚɡɨɜɨɦ ɥɚɡɟɪɟ. ɗɬɨɬ
ɦɟɬɨɞ ɧɚɢɛɨɥɟɟ ɷɮɮɟɤɬɢɜɟɧ ɩɪɢ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɩɨɞɚɜɥɟɧɢɹ ɝɟɧɟɪɚɰɢɢ
ɧɚ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞɚɯ, ɩɨɫɤɨɥɶɤɭ ɢɯ ɞɨɛɪɨɬɧɨɫɬɶ ɫɢɥɶɧɟɟ ɡɚɜɢɫɢɬ ɨɬ
ɞɢɚɦɟɬɪɚ ɚɩɟɪɬɭɪɵ ɪɟɡɨɧɚɬɨɪɚ, ɨɫɨɛɟɧɧɨ ɜ ɫɥɭɱɚɟ ɩɪɢɦɟɧɟɧɢɹ
ɫɮɟɪɢɱɟɫɤɢɯ ɨɬɪɚɠɚɬɟɥɟɣ. ɇɟɞɨɫɬɚɬɨɤ ɬɚɤɨɝɨ ɦɟɬɨɞɚ ɫɟɥɟɤɰɢɢ ɬɚɤɠɟ
ɫɜɹɡɚɧ ɫ ɧɟɤɨɬɨɪɵɦ ɭɦɟɧɶɲɟɧɢɟɦ ɦɨɳɧɨɫɬɢ ɫɟɥɟɤɬɢɪɭɟɦɨɣ
ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ
.
ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɩɪɢ ɨɬɤɥɨɧɟɧɢɢ ɨɬɪɚɠɚɬɟɥɟɣ ɪɟɡɨɧɚɬɨɪɚ ɨɬ
ɫɬɪɨɝɨɣ ɩɚɪɚɥɥɟɥɶɧɨɫɬɢ ɞɨɛɪɨɬɧɨɫɬɶ ɩɪɨɞɨɥɶɧɵɯ ɦɨɞ (ɤɨɬɨɪɚɹ ɧɚɦɧɨɝɨ
ɜɵɲɟ ɞɨɛɪɨɬɧɨɫɬɢ ɛɥɢɠɚɣɲɢɯ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ) ɫɢɥɶɧɟɟ ɡɚɜɢɫɢɬ ɨɬ
ɫɬɟɩɟɧɢ ɪɚɡɴɸɫɬɢɪɨɜɤɢ, ɧɟɠɟɥɢ ɞɨɛɪɨɬɧɨɫɬɶ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ. ȼ ɫɜɹɡɢ ɫ
ɷɬɢɦ ɩɪɢ ɧɟɤɨɬɨɪɨɣ ɪɚɡɴɸɫɬɢɪɨɜɤɟ ɡɟɪɤɚɥ ɪɟɡɨɧɚɬɨɪɚ ɩɨɪɨɝ ɝɟɧɟɪɚɰɢɢ
ɞɥɹ ɩɪɨɞɨɥɶɧɨɣ ɦɨɞɵ ɦɨɠɟɬ ɨɤɚɡɚɬɶɫɹ ɜɵɲɟ ɬɚɤɨɝɨ ɠɟ ɩɨɪɨɝɚ ɞɥɹ
ɨɞɧɨɣ ɢɡ ɩɨɩɟɪɟɱɧɵɯ ɦɨɞ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɷɬɨɝɨ ɦɨɠɧɨ ɩɭɬɟɦ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɪɚɡɴɸɫɬɢɪɨɜɤɢ ɪɟɡɨɧɚɬɨɪɚ ɩɨɞɚɜɢɬɶ ɝɟɧɟɪɚɰɢɸ ɧɚ
ɩɪɨɞɨɥɶɧɨɣ ɦɨɞɟ ɢ ɩɨɥɭɱɢɬɶ ɝɟɧɟɪɚɰɢɸ ɧɚ ɨɞɧɨɣ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ
ɩɨɩɟɪɟɱɧɵɯ ɦɨɞɚɯ.
9.5. ɋɟɥɟɤɰɢɹ ɝɟɧɟɪɢɪɭɸɳɢɯ ɥɚɡɟɪɧɵɯ ɩɟɪɟɯɨɞɨɜ
ȼ ɪɹɞɟ ɫɥɭɱɚɟɜ ɚɤɬɢɜɧɚɹ ɫɪɟɞɚ ɥɚɡɟɪɚ ɨɛɥɚɞɚɟɬ ɧɟɫɤɨɥɶɤɢɦɢ
ɩɟɪɟɯɨɞɚɦɢ, ɧɚ ɤɨɬɨɪɵɯ ɜɨɡɦɨɠɧɚ ɨɞɧɨɜɪɟɦɟɧɧɚɹ ɝɟɧɟɪɚɰɢɹ ɫ ɪɚɡɧɵɦɢ
ɱɚɫɬɨɬɚɦɢ. ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɦɨɠɧɨ ɭɤɚɡɚɬɶ ɝɟɥɢɣ-ɧɟɨɧɨɜɵɣ ɥɚɡɟɪ,
ɢɦɟɸɳɢɣ ɬɪɢ ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɦɵɯ ɥɚɡɟɪɧɵɯ ɩɟɪɟɯɨɞɚ ɫ
ɞɥɢɧɚɦɢ ɜɨɥɧ:
ɩɟɪɟɯɨɞɨɜ, ɥɟɠɚɳɢɯ ɜ ɫɩɟɤɬɪɚɥɶɧɨɦ ɢɧɬɟɪɜɚɥɟ
0,6328 ɦɤɦ, 1,1523 ɦɤɦ ɢ 3,39 ɦɤɦ ɢ ɟɳɟ ɰɟɥɵɣ ɪɹɞ
1…1,2 ɦɤɦ.
ȼ ɫɥɭɱɚɟ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɩɨɞɚɜɥɟɧɢɹ ɝɟɧɟɪɚɰɢɢ ɧɚ ɬɨɦ ɢɥɢ ɢɧɨɦ
ɩɟɪɟɯɨɞɟ ɜ ɪɟɡɨɧɚɬɨɪ ɥɚɡɟɪɚ ɦɨɠɟɬ ɛɵɬɶ ɜɧɟɫɟɧ ɩɨɝɥɨɬɢɬɟɥɶ ɧɚ ɱɚɫɬɨɬɟ
ɷɬɨɝɨ ɩɟɪɟɯɨɞɚ. ɇɚɩɪɢɦɟɪ, ɜ ɝɟɥɢɣ ɧɟɨɧɨɜɨɦ ɥɚɡɟɪɟ ɥɟɝɤɨ ɭɞɚɟɬɫɹ
ɩɨɞɚɜɢɬɶ ɝɟɧɟɪɚɰɢɸ ɧɚ ɞɥɢɧɟ ɜɨɥɧɵ
3,39 ɦɤɦ ɢɫɩɨɥɶɡɭɹ ɜ ɤɚɱɟɫɬɜɟ
56
ɨɤɨɲɟɤ ɧɚ ɬɨɪɰɚɯ ɝɚɡɨɪɚɡɪɹɞɧɨɣ ɬɪɭɛɤɢ ɨɛɵɱɧɨɟ ɫɬɟɤɥɨ, ɩɨɫɤɨɥɶɤɭ ɨɧɨ
Z
Z
Z
Z
ZmlZ
q
ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɩɪɨɩɭɫɤɚɟɬ ɢɡɥɭɱɟɧɢɟ ɫ ɬɚɤɨɣ ɞɥɢɧɨɣ ɜɨɥɧɵ, ɜ ɨɬɥɢɱɢɟ
ɨɬ ɤɜɚɪɰɟɜɨɝɨ ɫɬɟɤɥɚ, ɩɪɨɡɪɚɱɧɨɝɨ ɜ ɛɥɢɠɧɟɦ ɂɄ ɞɢɚɩɚɡɨɧɟ ɞɥɢɧ ɜɨɥɧ.
Ⱦɢɷɥɟɤɬɪɢɱɟɫɤɢɟ ɦɧɨɝɨɫɥɨɣɧɵɟ ɡɟɪɤɚɥɚ ɬɚɤɠɟ ɦɨɝɭɬ ɜɨ ɦɧɨɝɢɯ
ɫɥɭɱɚɹɯ ɜɵɩɨɥɧɹɬɶ ɪɨɥɶ ɱɚɫɬɨɬɧɨ-ɢɡɛɢɪɚɬɟɥɶɧɨɝɨ ɷɥɟɦɟɧɬɚ,
ɜɵɞɟɥɹɸɳɟɝɨ ɬɨɬ ɢɥɢ ɢɧɨɣ ɥɚɡɟɪɧɵɣ ɩɟɪɟɯɨɞ.
ɇɚɤɨɧɟɰ, ɜ ɬɨɦ
ɫɥɭɱɚɟ, ɤɨɝɞɚ ɷɬɨɝɨ ɧɟɞɨɫɬɚɬɨɱɧɨ, ɜɧɭɬɪɶ
ɪɟɡɨɧɚɬɨɪɚ ɦɨɠɟɬ ɛɵɬɶ ɩɨɦɟɳɟɧɚ ɞɢɮɪɚɤɰɢɨɧɧɚɹ ɪɟɲɟɬɤɚ (ɫɦ. ɪɢɫ.
9.7), ɩɨɡɜɨɥɹɸɳɚɹ ɨɛɟɫɩɟɱɢɬɶ ɧɭɠɧɨɟ ɱɚɫɬɨɬɧɨɟ ɪɚɡɪɟɲɟɧɢɟ ɛɥɢɡɤɨ
ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɫɩɟɤɬɪɚɥɶɧɵɯ ɥɢɧɢɣ. ɉɭɬɟɦ ɩɨɜɨɪɨɬɚ ɪɟɲɟɬɤɢ ɦɨɠɧɨ
ɢɡɛɢɪɚɬɟɥɶɧɨ ɧɚɫɬɪɚɢɜɚɬɶ ɥɚɡɟɪ ɧɚ ɬɭ ɢɥɢ ɢɧɭɸ ɞɥɢɧɭ ɜɨɥɧɵ.
nk
s
Ɋɢɫ. 9.7.
ɋɯɟɦɚ ɜɵɞɟɥɟɧɢɹ ɨɞɧɨɝɨ ɥɚɡɟɪɧɨɝɨ ɩɟɪɟɯɨɞɚ (ɫ ɱɚɫɬɨɬɨɣ
ɧɟɫɤɨɥɶɤɢɯ ɞɪɭɝɢɯ (
,
) ɩɪɢ ɩɨɦɨɳɢ ɞɢɮɪɚɤɰɢɨɧɧɨɣ ɪɟɲɟɬɤɢ,
qs
ɩɨɦɟɳɟɧɧɨɣ ɜɧɭɬɪɶ ɥɚɡɟɪɧɨɝɨ ɪɟɡɨɧɚɬɨɪɚ.
ml
) ɢɡ
nk
Ʌɟɤɰɢɹ ʋ10.
ɗɎɎȿɄɌ ɇȺɋɕɓȿɇɂə ȼ ɅȺɁȿɊȺɏ
Ʉɚɤ ɨɬɦɟɱɚɥɨɫɶ ɪɚɧɟɟ ɜ ɥɟɤɰɢɢ ʋ 8, ɷɮɮɟɤɬ ɧɚɫɵɳɟɧɢɹ ɜ
ɚɤɬɢɜɧɵɯ ɫɪɟɞɚɯ ɥɚɡɟɪɨɜ ɢɝɪɚɟɬ ɨɫɧɨɜɧɭɸ ɪɨɥɶ ɩɪɢ ɭɫɬɚɧɨɜɥɟɧɢɢ
ɫɬɚɰɢɨɧɚɪɧɨɣ ɝɟɧɟɪɚɰɢɢ, ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɩɪɢɜɨɞɹ ɤ ɨɝɪɚɧɢɱɟɧɢɸ
ɜɟɥɢɱɢɧɵ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ. Ɉɞɧɚɤɨ ɷɬɢɦ ɧɟ
ɢɫɱɟɪɩɵɜɚɸɬɫɹ ɜɨɡɦɨɠɧɨɫɬɢ ɩɪɨɹɜɥɟɧɢɹ ɷɮɮɟɤɬɚ ɧɚɫɵɳɟɧɢɹ ɜ
ɥɚɡɟɪɚɯ. ȼ ɱɚɫɬɧɨɫɬɢ, ɷɮɮɟɤɬ ɧɚɫɵɳɟɧɢɹ ɩɨɡɜɨɥɹɟɬ ɩɪɢɞɚɬɶ ɥɚɡɟɪɚɦ
ɧɨɜɵɟ ɫɜɨɣɫɬɜɚ, ɤɚɤ ɷɬɨ
ɢɦɟɟɬ ɦɟɫɬɨ ɩɪɢ ɩɨɦɟɳɟɧɢɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ
ɩɨɝɥɨɬɢɬɟɥɹ ɜɧɭɬɪɶ ɪɟɡɨɧɚɬɨɪɚ ɥɚɡɟɪɚ. ɗɮɮɟɤɬ ɧɚɫɵɳɟɧɢɹ ɬɚɤɠɟ ɥɟɠɢɬ
ɜ ɨɫɧɨɜɟ ɬɚɤɢɯ ɹɜɥɟɧɢɣ, ɤɚɤ ɡɚɯɜɚɬɵɜɚɧɢɟ ɱɚɫɬɨɬɵ ɢɡɥɭɱɟɧɢɹ
ɥɚɡɟɪɚ, ɥɷɦɛɨɜɫɤɢɣ ɩɪɨɜɚɥ ɢ ɬ.ɞ.
10.1. ɅȺɁȿɊ ɋ ɇȺɋɕɓȺɘɓɂɆɋə ɉɈȽɅɈɌɂɌȿɅȿɆ ȼ ɊȿɁɈɇȺɌɈɊȿ
Ʉɨɧɫɬɪɭɤɰɢɹ ɬɚɤɨɝɨ ɥɚɡɟɪɚ ɢ ɭɩɪɨɳɟɧɧɵɟ ɫɯɟɦɵ ɭɪɨɜɧɟɣ
ɭɫɢɥɢɜɚɸɳɟɣ ɢ ɩɨɝɥɨɳɚɸɳɟɣ ɫɪɟɞ ɞɥɹ ɷɬɨɝɨ ɫɥɭɱɚɹ ɢɡɨɛɪɚɠɟɧɵ ɧɚ
ɪɢɫ.10.1.
57
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ɚ
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G
G
V
V
H
U U U U U
ɍɫɢɥɢɬ. ɫɪɟɞɚ
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L
1
ɉɨɝɥ.cɪɟɞ
N2<0
L
2
k
1
k
2
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n
1
n
2
L
Ɋɢɫ.10.1.
ɋɯɟɦɚ ɥɚɡɟɪɚ ɫ ɧɚɫɵɳɚɸɳɢɦɫɹ ɩɨɝɥɨɬɢɬɟɥɟɦ ɜɧɭɬɪɢ ɪɟɡɨɧɚɬɨɪɚ ɢ
ɫɯɟɦɵ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ ɚɤɬɢɜɧɨɣ ɢ ɩɨɝɥɨɳɚɸɳɟɣ ɫɪɟɞ.
ȼ ɤɚɱɟɫɬɜɟ ɩɨɝɥɨɬɢɬɟɥɹ ɢɫɩɨɥɶɡɭɟɬɫɹ ɫɪɟɞɚ ɫ ɪɚɡɪɟɲɟɧɧɵɦ
ɞɢɩɨɥɶɧɵɦ ɩɟɪɟɯɨɞɨɦ. ȼ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɧɟɨɛɯɨɞɢɦɨ ɩɨɥɭɱɢɬɶ
ɛɨɥɶɲɨɟ ɩɨɝɥɨɳɟɧɢɟ, ɫɥɟɞɭɟɬ ɜɵɛɢɪɚɬɶ ɜ ɤɚɱɟɫɬɜɟ ɧɢɠɧɟɝɨ ɭɪɨɜɧɹ
ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ. ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɢɩɚ ɥɚɡɟɪɚ ɪɨɥɶ ɩɨɝɥɨɬɢɬɟɥɟɣ
ɦɨɝɭɬ ɜɵɩɨɥɧɹɬɶ ɪɚɡɥɢɱɧɵɟ ɜɟɳɟɫɬɜɚ:
1. ȼ ɬɜɟɪɞɨɬɟɥɶɧɵɯ ɥɚɡɟɪɚɯ – ɪɚɫɬɜɨɪɵ ɨɪɝɚɧɢɱɟɫɤɢɯ ɤɪɚɫɢɬɟɥɟɣ
ɢɥɢ ɫɬɟɤɥɚ, ɜ ɤɨɬɨɪɵɟ ɜɜɟɞɟɧɵ ɤɪɚɫɹɳɢɟ ɞɨɛɚɜɤɢ,
2. ȼ
ɝɚɡɨɜɵɯ ɥɚɡɟɪɚɯ – ɤɸɜɟɬɵ ɫɨ ɫɩɟɰɢɚɥɶɧɨ ɩɨɞɨɛɪɚɧɧɵɦ ɝɚɡɨɦ,
ɭ ɤɨɬɨɪɨɝɨ ɢɦɟɟɬɫɹ ɩɟɪɟɯɨɞ, ɫɨɜɩɚɞɚɸɳɢɣ ɩɨ ɱɚɫɬɨɬɟ ɫ
ɱɚɫɬɨɬɨɣ ɥɚɡɟɪɧɨɝɨ ɩɟɪɟɯɨɞɚ,
3. ȼ ɩɨɥɭɩɪɨɜɨɞɧɢɤɨɜɨɦ ɢɧɠɟɤɰɢɨɧɧɨɦ ɥɚɡɟɪɟ – ɭɱɚɫɬɨɤ
p-n
ɩɟɪɟɯɨɞɚ, ɱɟɪɟɡ ɤɨɬɨɪɵɣ ɧɟ ɩɪɨɩɭɫɤɚɟɬɫɹ ɬɨɤ ɧɚɤɚɱɤɢ.
10.1.1. ɋɌȺɐɂɈɇȺɊɇȺə ȽȿɇȿɊȺɐɂə ɅȺɁȿɊȺ ɋ ɇȺɋɕɓȺɘɓɂɆɋə
ɉɈȽɅɈɌɂɌȿɅȿɆ ȼ ɊȿɁɈɇȺɌɈɊȿ
ȼɜɟɞɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ ɡɚɩɨɥɧɟɧɢɹ ɪɟɡɨɧɚɬɨɪɚ ɚɤɬɢɜɧɨɣ (1) ɢ
=L
ɩɨɝɥɨɳɚɸɳɟɣ (2) ɫɪɟɞɚɦɢ:
ɩɪɟɥɨɦɥɟɧɢɹ ɫɪɟɞ: n
ɩɪɨɫɬɨɬɵ ɫɱɢɬɚɟɦ, ɱɬɨ ɪɟɡɨɧɚɬɨɪ ɩɨɥɧɨɫɬɶɸ ɡɚɩɨɥɧɟɧ ɚɤɬɢɜɧɨɣ ɢ
ɩɨɝɥɨɳɚɸɳɟɣ ɫɪɟɞɚɦɢ: L
, n2, ɫɟɱɟɧɢɹ ɩɟɪɟɯɨɞɚ ɜ ɫɪɟɞɚɯ:
1
=L. ɉɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɨɩɬɢɱɟɫɤɨɝɨ ɩɨɥɹ ɜ
1+L2
/L ɢ
1
L2/L. ɉɭɫɬɶ ɩɨɤɚɡɚɬɟɥɢ
ɢ
1
. Ɋɚɞɢ
2
ɤɚɠɞɨɣ ɢɡ ɫɪɟɞ ɬɨɝɞɚ ɛɭɞɭɬ:
2
E
00
2
22
nn
, , .
ɝɞɟ
1102 20 0
Ɂɚɩɢɲɟɦ ɫɤɨɪɨɫɬɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɬɚɤɨɝɨ ɥɚɡɟɪɚ, ɩɨ ɚɧɚɥɨɝɢɢ ɫ
ɭɪɚɜɧɟɧɢɹɦɢ (8.11), (8.14), ɨɞɧɚɤɨ, ɫ ɭɱɟɬɨɦ ɜ ɧɢɯ ɧɚɥɢɱɢɹ ɞɜɭɯ ɫɪɟɞ.
ɉɪɨɢɧɬɟɝɪɢɪɭɟɦ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨ ɨɛɴɟɦɭ ɪɟɡɨɧɚɬɨɪɚ, ɚɧɚɥɨɝɢɱɧɨ
ɬɨɦɭ, ɤɚɤ ɷɬɨ ɛɵɥɨ ɫɞɟɥɚɧɨ ɜ ɩɭɧɤɬɟ 8.5. ȼɜɟɞɟɦ ɡɚɬɟɦ ɜ ɭɪɚɜɧɟɧɢɹ
ɤɨɷɮɮɢɰɢɟɧɬɵ ɡɚɩɨɥɧɟɧɢɹ ɪɟɡɨɧɚɬɨɪɚ ɨɛɟɢɦɢ ɫɪɟɞɚɦɢ, ɤɜɚɞɪɚɬ
58
ɷɮɮɟɤɬɢɜɧɨɝɨ ɩɨɤɚɡɚɬɟɥɹ ɩɪɟɥɨɦɥɟɧɢɹ (
U
ª º
G U U
¬ ¼
Q Q
Q
ª º
U U
W V G ' W V G '
« »
W
¬ ¼
' ' '
V ' U
Z
' ' '
V ' U
Z
!
!
U0 '
'
Q Q
V
V
J W V J W V D D
Z Z
! !
' '
J G ' J G ' ' '
D U D U
'
J1G1'
'
D U D U
U
U
U
ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɩɨ ɨɛɴɟɦɭ ɪɟɡɨɧɚɬɨɪɚ (
22 22 2
eff 2 1 1 2 c eff 0
ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɭɸ ɫɢɫɬɟɦɭ ɫɤɨɪɨɫɬɧɵɯ ɭɪɚɜɧɟɧɢɣ:
d
dt
dN N N nc
dt T
dN N N nc
dt T
Ȼɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɬɚɰɢɨɧɚɪɧɵɣ ɫɥɭɱɚɣ ɝɟɧɟɪɚɰɢɢ. Ɍɨɝɞɚ
cc
nn
12
cNc NP
11 1 2 2 2 v
22
nn
eff eff
111 1
222 2
e
-
1
1
e
-
1
2
2 ,
2 .
2
n
nk eff
2
n
nk eff
ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɜɟɥɢɱɢɧ
ɪɚɜɧɵɦɢ ɧɭɥɸ. ɉɪɟɧɟɛɪɟɠɟɦ ɬɚɤɠɟ ɦɚɥɵɦ ɜɤɥɚɞɨɦ ɫɩɨɧɬɚɧɧɨɝɨ
ɢɡɥɭɱɟɧɢɹ ɜ ɷɧɟɪɝɢɸ ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ. ȼɜɟɞɟɦ ɫɥɟɞɭɸɳɢɟ
2
n ), ɚ ɬɚɤɠɟ - ɭɫɪɟɞɧɟɧɧɨɟ
eff
):
ɫ
, nn nn n
N
11c
N
22c
N
1
.
sp
1,
(10.1)
N
ɧɟɨɛɯɨɞɢɦɨ ɩɨɥɨɠɢɬɶ
2
ɨɛɨɡɧɚɱɟɧɢɹ:
nn ncTncT
12 11112122
1221 2
22 2 2
nn n n
ȼ ɪɟɡɭɥɶɬɚɬɟ ɫɢɫɬɟɦɚ (10.1) ɩɪɢɦɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
ɉɨɞɫɬɚɜɢɜ ɜɵɪɚɠɟɧɢɹ ɞɥɹ
eff eff nk eff nk eff
11 1 2 2 2 1 2
ɜɜɟɞɹ ɨɛɨɡɧɚɱɟɧɢɹ
Ɂɞɟɫɶ j
, , 2 , 2
cc
10
NN N N
j
=
1
11
ɟɫɬɶ ɜɟɥɢɱɢɧɚ ɨɬɪɢɰɚɬɟɥɶɧɚɹ.
2
, , . (10.2)
'
j
2 J2G2
N
e
N
ң
jj
12
1c 2 c
() ()
ee
NN
12
11
1c 2 c
N
ɜ ɩɟɪɜɨɟ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ ɢ
1
2
e
N
, ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɟɟ:
2
10
. (10.3)
ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ.
ȼɨ-ɩɟɪɜɵɯ, ɧɚɩɨɦɧɢɦ, ɱɬɨ (10.3) ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɞɜɚ ɪɚɡɧɵɯ
ɫɨɫɬɨɹɧɢɹ ɥɚɡɟɪɚ. ɉɟɪɜɨɟ ɫɨɫɬɨɹɧɢɟ ɩɪɢ
ɜɵɩɨɥɧɟɧɢɸ ɭɫɥɨɜɢɹ ɫɚɦɨɜɨɡɛɭɠɞɟɧɢɹ:
>>0 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɬɚɰɢɨɧɚɪɧɨɣ ɝɟɧɟɪɚɰɢɢ: ɫɭɦɦɚ ɭɫɢɥɟɧɢɹ ɢ
ɫ
ɩɨɝɥɨɳɟɧɢɹ ɜ ɨɛɟɢɯ ɫɪɟɞɚɯ ɥɚɡɟɪɚ ɩɪɢ ɷɬɨɦ ɫ ɭɱɟɬɨɦ ɷɮɮɟɤɬɚ
ɧɚɫɵɳɟɧɢɹ ɞɨɥɠɧɚ ɛɵɬɶ ɬɚɤɨɜɚ, ɱɬɨɛɵ ɩɨɥɧɨɫɬɶɸ ɤɨɦɩɟɧɫɢɪɨɜɚɥɢɫɶ
=0 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɨɱɧɨɦɭ
ɫ
j
=1. ȼɬɨɪɨɟ ɫɨɫɬɨɹɧɢɟ - ɩɪɢ
1+j2
ɩɨɬɟɪɢ ɜ ɪɟɡɨɧɚɬɨɪɟ.
ɉɨɫɦɨɬɪɢɦ ɬɟɩɟɪɶ, ɤɚɤ ɜɟɞɭɬ ɫɟɛɹ ɩɟɪɜɵɟ ɞɜɚ ɫɥɚɝɚɟɦɵɯ ɜ
ɜɵɪɚɠɟɧɢɢ (10.3) ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɜɟɥɢɱɢɧɵ
.
ɫ
.
59
10.1.2. Ɇɹɝɤɢɣ ɪɟɠɢɦ ɫɚɦɨɜɨɡɛɭɠɞɟɧɢɹ ɤɨɥɟɛɚɧɢɣ
D
D
J J
U
D
U
U
D U
U
U
U
U
ɫ
U
ɫ
U
U
U
U
U
U
U
D
D
V
V
U
U
Ɋɚɫɫɦɨɬɪɢɦ ɫɧɚɱɚɥɚ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɩɚɪɚɦɟɬɪɵ ɫɪɟɞ ɫɨɜɩɚɞɚɸɬ:
=
, (ɱɟɦɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ: (Ɍ1)1=(Ɍ1)2,
1
2
ɩɪɟɜɨɫɯɨɞɢɬ ɩɨɝɥɨɳɟɧɢɟ ɜ ɫɪɟɞɟ 2 ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ ɩɥɨɬɧɨɫɬɢ
ɷɧɟɪɝɢɢ. ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ:
ɜɟɥɢɱɢɧɵ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɫɢɥɟɧɢɹ (
(
g
<0) ɫɪɟɞ 1 ɢ 2 ɫ ɭɱɟɬɨɦ ɷɮɮɟɤɬɚ ɧɚɫɵɳɟɧɢɹ. ɉɪɢ ɡɧɚɱɟɧɢɢ
2
g
>1. ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ ɡɚɜɢɫɢɦɨɫɬɢ g1 ɢ g2 ɢ ɢɯ ɫɭɦɦɵ
1+g2
g
) ɨɬ
(
1 +g2
ɢɦɟɸɬ ɜɢɞ, ɩɨɤɚɡɚɧɧɵɣ ɧɚ ɪɢɫ.10.2.
ɫ
g1(
) g2(
)
1
g
(
)=
1
0
1
g
(
)
1
ɫ
), ɚ ɭɫɢɥɟɧɢɟ ɜ ɫɪɟɞɟ 1
1
j
1
g
(
)=
,
2
10
0
1
g
>0) ɢ ɩɨɝɥɨɳɟɧɢɹ
1
g1(
)+g2(
)
ɫ
ɫ
j
2
20
=0 ɛɭɞɟɬ
0
. ɗɬɢ
st
g2(
ɫ
)
ɫ
Ɋɢɫ.10.2
Ɂɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɫɢɥɟɧɢɹ ɢ ɩɨɝɥɨɳɟɧɢɹ ɨɬ ɩɥɨɬɧɨɫɬɢ
ɷɧɟɪɝɢɢ ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ ɩɪɢ ɦɹɝɤɨɦ ɪɟɠɢɦɟ ɝɟɧɟɪɚɰɢɢ.
Ʉɚɤ ɜɢɞɧɨ ɢɡ ɷɬɢɯ ɝɪɚɮɢɤɨɜ, ɩɨɪɨɝɨɜɨɟ ɭɫɥɨɜɢɟ ɜ ɥɚɡɟɪɟ
ɜɵɩɨɥɧɹɟɬɫɹ ɫ ɩɪɟɜɵɲɟɧɢɟɦ ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ 0>
ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɢ ɫɤɚɱɤɨɨɛɪɚɡɧɨɦ ɜɤɥɸɱɟɧɢɢ ɧɚɤɚɱɤɢ ɝɟɧɟɪɚɰɢɹ ɧɚɱɧɟɬ
ɪɚɡɜɢɜɚɬɶɫɹ ɨɬ ɭɪɨɜɧɹ ɫɩɨɧɬɚɧɧɵɯ ɲɭɦɨɜ ɢ ɞɨɪɚɫɬɟɬ ɞɨ ɫɜɨɟɝɨ
>
. ɗɬɨ
0ɫ
st
ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɡɧɚɱɟɧɢɹ. ȼ ɪɚɞɢɨɬɟɯɧɢɤɟ ɬɚɤɨɣ ɪɟɠɢɦ ɪɚɛɨɬɵ
ɝɟɧɟɪɚɬɨɪɚ ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ
ɦɹɝɤɢɦ.
10.1.3. ɀɟɫɬɤɢɣ ɪɟɠɢɦ ɫɚɦɨɜɨɡɛɭɠɞɟɧɢɹ ɤɨɥɟɛɚɧɢɣ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɫɥɭɱɚɣ, ɤɨɝɞɚ
Ɍ
<(Ɍ1)2,
(
1)1
ɩɨɝɥɨɳɟɧɢɟɦ ɜ ɫɪɟɞɟ 2. ɉɪɢ ɬɚɤɨɦ ɜɵɛɨɪɟ ɩɚɪɚɦɟɬɪɨɜ ɷɮɮɟɤɬ
ɧɚɫɵɳɟɧɢɹ ɞɨɥɠɟɧ ɫɢɥɶɧɟɟ ɩɪɨɹɜɥɹɬɶɫɹ ɜ ɫɪɟɞɟ 2, ɜɫɥɟɞɫɬɜɢɟ ɱɟɝɨ ɩɪɢ
<
), ɚ ɭɫɢɥɟɧɢɟ ɜ ɫɪɟɞɟ 1 ɩɪɢ
1
2
<
(ɱɟɦɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ:
1
2
=0 ɫɨɢɡɦɟɪɢɦɨ ɫ
ɫ
ɜɨɡɪɚɫɬɚɧɢɢ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ ɤɨɷɮɮɢɰɢɟɧɬ
ɩɨɝɥɨɳɟɧɢɹ ɛɭɞɟɬ ɭɦɟɧɶɲɚɬɶɫɹ ɛɵɫɬɪɟɟ ɱɟɦ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ.
g
Ƚɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ
ɪɢɫ.10.3.
ɢ g2 ɨɬ
1
ɞɥɹ ɷɬɨɝɨ ɫɥɭɱɚɹ ɩɪɢɜɟɞɟɧɵ ɧɚ
0
60
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