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Файл:Расчет переходных процессов в электроэнергетических системах методом интеграла Дюамеля. Учебное пособие
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݂
݂
Ɋɢɫ. 1.7
ɉɪɨɢɡɜɟɞɟɧɢɟ ɟɞɢɧɢɱɧɨɣ ɮɭɧɤɰɢɢ ɧɚ ɥɸɛɭɸ ɝɥɚɞɤɭɸ
ɮɭɧɤɰɢɸ f (t) ɞɚɟɬ ɧɨɜɭɸ ɪɚɡɪɵɜɧɭɸ ɮɭɧɤɰɢɸ (ɪɢɫ. 1.8),
ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɭɫɥɨɜɢɸ
ሺݐሻήܪሺݐሻ
0 ˒˓ˋ ݐ<0,
= ൜
ሺݐሻ
˒˓ˋ ݐ>0,
(1.7)
ɬ. ɟ. ɫɨɜɩɚɞɚɸɳɭɸ ɫ f (t) ɩɪɢ ɜɫɟɯ t > 0 ɢ ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɭɸ ɧɭɥɸ ɩɪɢ t < 0. ɍɫɥɨɜɢɟ (1.7) ɨɡɧɚɱɚɟɬ ɭɫɟɱɟɧɢɟ ɮɭɧɤɰɢɢ f (t) ɫɥɟɜɚ.
Ɋɢɫ. 1.8
11
ɋ ɩɨɦɨɳɶɸ ɟɞɢɧɢɱɧɵɯ ɫɬɭɩɟɧɱɚɬɵɯ ɮɭɧɤɰɢɣ ɦɨɠɧɨ
݂
݂
݂
ɞɚɬɶ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɩɪɨɢɡɜɨɥɶɧɨɣ ɪɚɡɪɵɜɧɨɣ ɮɭɧɤɰɢɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɪɚɡɪɵɜɧɭɸ ɮɭɧɤɰɢɸ (ɪɢɫ. 1.9), ɤɨɬɨɪɚɹ ɞɨ
ɦɨɦɟɧɬɚ t = IJ ɫɨɜɩɚɞɚɟɬ ɫ ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɟɣ f
(t), ɚ ɩɨɫɥɟ
1
t = IJ – ɫ ɮɭɧɤɰɢɟɣ f2(t). ɉɪɢɦɟɧɹɹ ɟɞɢɧɢɱɧɵɟ ɮɭɧɤɰɢɢ, ɩɨɥɭɱɚɟɦ
ሼݐሽ
=
ሺݐሻήܪሺ
ଵ
ɒെݐ
ሻ
+
ሺݐሻήܪሺ
ଶ
ݐെɒ
ሻ
.
(1.8)
Ɋɢɫ. 1.9
Ɂɚɦɟɬɢɦ, ɱɬɨ ɮɭɧɤɰɢɹ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɜɵɪɚɠɟɧɢɹ (1.8) ɨɛɨɡɧɚɱɚɟɬɫɹ ɫɢɦɜɨɥɨɦ f {t} ɫ ɮɢɝɭɪɧɵɦɢ ɫɤɨɛɤɚɦɢ, ɱɬɨ ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɨɬɞɟɥɟɧɢɹ ɨɛɨɛɳɟɧɧɵɯ ɮɭɧɤɰɢɣ ɨɬ ɝɥɚɞɤɢɯ, ɞɥɹ
ɤɨɬɨɪɵɯ ɬɪɚɞɢɰɢɨɧɧɨ ɫɨɯɪɚɧɹɟɬɫɹ ɫɢɦɜɨɥ ɫ ɤɪɭɝɥɵɦɢ ɫɤɨɛɤɚɦɢ. ɂɫɤɥɸɱɟɧɢɟɦ ɹɜɥɹɸɬɫɹ ɬɨɥɶɤɨ ɟɞɢɧɢɱɧɚɹ ɢ ɞɟɥɶɬɚɮɭɧɤɰɢɢ.
Ⱦɟɥɶɬɚ-ɮɭɧɤɰɢɹ į(t), ɢɥɢ ɮɭɧɤɰɢɹ Ⱦɢɪɚɤɚ ɹɜɥɹɟɬɫɹ ɩɪɨɫɬɟɣɲɟɣ ɢɦɩɭɥɶɫɧɨɣ ɮɭɧɤɰɢɟɣ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɣ ɭɫɥɨɜɢɸ
12
0 ˒˓ˋ ݐ<0,
Ɋ
Ɂሺݐሻ= ൝
λ ˒˓ˋ ݐ=0,
(1.9)
0 ˒˓ˋ ݐ>0,
ɬ. ɟ. ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɣɫɹ ɬɟɦ, ɱɬɨ ɬɨɠɞɟɫɬɜɟɧɧɨ ɫɨɜɩɚɞɚɟɬ
ɫ ɧɭɥɟɦ ɧɚ ɜɫɟɣ ɨɫɢ ɜɪɟɦɟɧɢ ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɨɞɧɨɣ ɬɨɥɶɤɨ
ɬɨɱɤɢ t = 0, ɜ ɤɨɬɨɪɨɣ ɩɪɢɨɛɪɟɬɚɟɬ ɧɟɨɝɪɚɧɢɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ.
ɉɨɧɹɬɢɟ ɢɦɩɭɥɶɫɧɨɣ ɮɭɧɤɰɢɢ ɹɜɥɹɟɬɫɹ ɢɞɟɚɥɢɡɚɰɢɟɣ
ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ, ɫɨɫɪɟɞɨɬɨɱɟɧɧɨɣ ɝɥɚɜɧɵɦ ɨɛɪɚɡɨɦ ɜ
ɧɟɤɨɬɨɪɨɦ ɢɧɬɟɪɜɚɥɟ
ɢ ɫɯɨɞɹɳɟɣɫɹ ɤ ɧɭɥɸ ɜɧɟ ɟɝɨ, ɩɨɡɜɨɥɹɹ, ɬɟɦ ɫɚɦɵɦ, ɝɨɜɨɪɢɬɶ ɨ ɫɤɨɪɨɫɬɢ ɩɪɢɪɚɳɟɧɢɹ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ.
ɇɟɨɝɪɚɧɢɱɟɧɧɨɟ ɜɨɡɪɚɫɬɚɧɢɟ ɡɧɚɱɟɧɢɣ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ, ɫɨɫɪɟɞɨɬɨɱɟɧɧɨɣ ɜ ɢɧɬɟɪɜɚɥɟ, ɜɨɡɦɨɠɧɨ ɜ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɫ ɭɦɟɧɶɲɟɧɢɟɦ ɞɥɢɬɟɥɶɧɨɫɬɢ ɢɧɬɟɪɜɚɥɚ ɩɥɨɳɚɞɶ ɮɭɧɤɰɢɢ ɧɟ ɦɟɧɹɟɬɫɹ. ɉɪɢ ɷɬɨɦ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɨɬɧɨɲɟɧɢɟ
=
ᇱ
க
ሺݐሻ
.
(1.10)
ሺݐሻ
ɋ
க
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɢ ɭɦɟɧɶɲɟɧɢɢ ɢɧɬɟɪɜɚɥɚ –İ…+İ
ɤ ɧɭɥɸ (ɪɢɫ. 1.10) ɮɭɧɤɰɢɹ Ȟİ(t), ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɭɫɥɨɜɢɸ
0 ˒˓ˋ ݐ< െɂ,
ɋ
க
ሺݐሻ
=
1
൞
˒˓ˋ ݐ=|ɂ
2ɂ
(1.11)
|
,
0 ˒˓ˋ ݐ>+ɂ,
13
ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɢ ɜ ɩɪɟɞɟɥɟ ɪɚɜɧɚ
Ɋ
Ɋ
lim
க՜
க
ሺݐሻ
ɋ
= Ɂሺݐ
ሻ
.
(1.12)
Ɋɢɫ. 1.10
ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɩɪɟɞɟɥɶɧɨɝɨ ɩɟɪɟɯɨɞɚ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ į(t) ɜ ɬɨɱɤɟ t = 0 ɢɦɟɟɬ ɧɟɨɝɪɚɧɢɱɟɧɧɵɣ ɜɫɩɥɟɫɤ. ɉɨɷɬɨɦɭ ɟɟ ɜɟɥɢɱɢɧɚ ɦɨɠɟɬ ɛɵɬɶ ɨɰɟɧɟɧɚ ɫ ɩɨɦɨɳɶɸ ɢɧɬɟɝɪɚɥɚ ɩɨ ɜɪɟɦɟɧɢ, ɢɦɟɸɳɟɦ ɤɨɧɟɱɧɨɟ ɡɧɚɱɟɧɢɟ ɢ ɜɵɪɚɠɚɸɳɢɦɫɹ ɩɥɨɳɚɞɶɸ, ɡɚɤɥɸɱɟɧɧɨɣ ɦɟɠɞɭ ɮɭɧɤɰɢɟɣ ɢ ɨɫɶɸ
ɚɛɫɰɢɫɫ
ାக
නlim
ఌ՜
ିக
ሺݐሻ
ɋ
݀ݐ
க
ାக
= නlim
க՜
ିக
ᇱ
க
ሺݐሻ
݀ݐ
=lim
க՜
க
ሺݐሻ
.
(1.13)
14
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɟɟɦ
݂
݂
݂
݂
ஶ
ାக
නɁሺݐሻ݀ݐ
ିஶ
= නɁሺݐሻ݀ݐ
ିக
=1.
(1.14)
ɋ ɩɨɦɨɳɶɸ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ, ɢɥɢ ɟɞɢɧɢɱɧɨɣ ɢɦɩɭɥɶɫɧɨɣ
ɮɭɧɤɰɢɢ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɢɦɩɭɥɶɫɧɵɟ ɮɭɧɤɰɢɢ ɪɚɡɥɢɱɧɨɣ
ɚɦɩɥɢɬɭɞɵ.
Ɋɚɫɫɦɨɬɪɢɦ ɢɦɩɭɥɶɫɧɭɸ ɮɭɧɤɰɢɸ ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɭɸ ɧɭɥɸ ɩɪɢ ɜɫɟɯ t 0 ɫ ɚɦɩɥɢɬɭɞɨɣ Ⱥ. ɉɪɢɦɟɧɹɹ ɟɞɢɧɢɱɧɭɸ ɢɦɩɭɥɶɫɧɭɸ ɮɭɧɤɰɢɸ ɫ ɩɨɫɬɨɹɧɧɵɦ ɦɧɨɠɢɬɟɥɟɦ, ɩɨɥɭɱɚɟɦ
ஶ
න݃ሼݐሽ݀ݐ
ିஶ
ஶ
= නܣɁሺݐሻ݀ݐ
ିஶ
=ܣ.
(1.15)
ɉɪɢ ɩɪɨɢɡɜɟɞɟɧɢɢ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɧɚ ɥɸɛɭɸ ɝɥɚɞɤɭɸ
ɮɭɧɤɰɢɸ f (t) ɩɪɨɹɜɥɹɟɬɫɹ ɮɢɥɶɬɪɭɸɳɟɟ ɞɟɣɫɬɜɢɟ ɟɞɢɧɢɱɧɨɣ ɢɦɩɭɥɶɫɧɨɣ ɮɭɧɤɰɢɢ, ɨɩɪɟɞɟɥɹɟɦɨɟ ɬɟɦ, ɱɬɨ ɩɪɢ ɜɫɟɯ
t 0 ɞɟɥɶɬɚ-ɮɭɧɤɰɢɹ ɩɪɢɨɛɪɟɬɚɟɬ ɧɭɥɟɜɵɟ ɡɧɚɱɟɧɢɹ. Ɍɨɝɞɚ
ɢɦɟɟɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ
ሺݐሻήɁሺݐሻ
ሺ0ሻήɁሺݐሻ
=
,
(1.16)
ɢɥɢ
ሺݐሻήɁሺ
ݐെ߬
ሻ
=
ሺ߬ሻήɁሺ
ݐെ߬
ሻ
.
(1.17)
15
ɋɨɨɬɧɨɲɟɧɢɹ (1.1), (1.9) ɢ (1.10) ɩɨɡɜɨɥɹɸɬ ɭɫɬɚɧɨɜɢɬɶ
݂
݂
݂
݂
݂
݂
݂
݂
݂
݂
݂
݂
݂
ɫɜɹɡɶ ɦɟɠɞɭ ɟɞɢɧɢɱɧɨɣ ɮɭɧɤɰɢɟɣ ɢ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɟɣ
Ɂሺݐ
݀
ሻ
ܪሺݐሻ݀ݐ= ܪ
=
ᇱ
ሺݐሻ
,
݀ݐ
ɢɥɢ
௧
(1.18)
නɁሺݐሻ݀ݐ
ିஶ
= ܪሺݐ
ሻ
.
(1.19)
ɂɧɬɟɝɪɚɥ ɜ ɩɨɫɥɟɞɧɟɦ ɫɨɨɬɧɨɲɟɧɢɢ ɦɟɧɹɟɬɫɹ ɫ ɢɡɦɟɧɟɧɢɟɦ t, ɚ ɢɦɟɧɧɨ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ t < 0 ɢ ɪɚɜɟɧ ɟɞɢɧɢɰɟ ɩɪɢ t > 0, ɬ. ɟ. ɨɧ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɜɟɪɯɧɟɝɨ ɩɪɟɞɟɥɚ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ.
ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɩɪɨɢɡɜɟɞɟɧɢɹ ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ f (t) ɧɚ
ɟɞɢɧɢɱɧɭɸ ɮɭɧɤɰɢɸ H(t) ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɝɥɚɫɧɨ ɢɡɜɟɫɬɧɵɦ
ɩɪɚɜɢɥɚɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɩɪɨɢɡɜɟɞɟɧɢɣ ɮɭɧɤɰɢɣ ɢ ɫɨɨɬɧɨɲɟɧɢɹ (1.16), ɬ. ɟ.
ᇱ
ሾ݂ሺݐሻήܪሺݐሻሿ
=
ᇱ
ᇱ
ሺݐሻήܪሺݐሻ
=
ሺݐሻήܪሺݐሻ
ሺ0ሻήɁሺݐሻ
+
ሺݐሻήɁሺݐሻ
+
=
.
(1.20)
Ɋɚɫɫɦɨɬɪɢɦ ɪɟɡɭɥɶɬɚɬ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɪɚɡɪɵɜɧɨɣ ɮɭɧɤɰɢɢ (1.8), ɪɚɫɱɟɬɧɚɹ ɮɨɪɦɭɥɚ ɤɨɬɨɪɨɝɨ
ɢɦɟɟɬ ɜɢɞ
ᇱ
ሼݐሽ
+
=
ᇱ
ሺݐሻήܪሺ
=
ଵ
ᇱ
ሺݐሻήܪሺ
ଶ
ᇱ
ሺݐሻήܪሺ
ଵ
ሾ
+
ଶ
ሺɒሻ
ɒെݐ
ݐെɒ
ɒെݐ
െ
ሻ
ሻ
+
ሻ
+
ሺɒሻሿήɁሺ
ଵ
ሺɒሻήɁሺ
+
ଵ
ሺɒሻήɁሺ
ଶ
ᇱ
ሺݐሻήܪሺ
ଶ
ݐെɒ
ݐെɒ
ɒെݐ
ݐെɒ
ሻ
.
ሻ
+
ሻ
=
ሻ
+
(1.21)
16
ȼ ɩɨɥɭɱɟɧɧɨɦ ɜɵɪɚɠɟɧɢɢ ɩɪɨɢɡɜɨɞɧɚɹ f '(t) ɫɨɜɩɚɞɚɟɬ ɫ
݂
݂
݂
݂
݂
݂
݂
݂
݂
ɨɛɵɱɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɝɥɚɞɤɢɯ ɮɭɧɤɰɢɣ f
(t) ɢ f2(t) ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ, ɝɞɟ ɩɨɫɥɟɞɧɢɟ ɫɭɳɟɫɬɜɭɸɬ,
1
ɬ. ɟ. ɩɪɢ ɜɫɟɯ t 0. ȼ ɬɨɱɤɟ ɪɚɡɪɵɜɚ t = IJ ɩɪɨɢɡɜɨɞɧɚɹ ɜɵɪɚɠɚɟɬɫɹ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɟɣ ɫ ɚɦɩɥɢɬɭɞɨɣ
ሺɒሻ
ο
=
ଶ
ሺɒሻ
െ
ଵ
ሺɒሻ
,
(1.22)
ɪɚɜɧɨɣ ɜɟɥɢɱɢɧɟ ɫɤɚɱɤɚ ɢɥɢ ɩɪɢɪɚɳɟɧɢɸ ɮɭɧɤɰɢɢ ɜ ɷɬɨɣ
ɬɨɱɤɟ.
Ⱦɥɹ ɨɩɟɪɚɰɢɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɪɚɡɪɵɜɧɵɯ ɮɭɧɤɰɢɣ ɫɨɯɪɚɧɹɸɬɫɹ ɢɡɜɟɫɬɧɵɟ ɩɪɚɜɢɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɝɥɚɞɤɢɯ ɮɭɧɤɰɢɣ, ɩɪɟɞɭɫɦɚɬɪɢɜɚɸɳɢɟ ɪɚɡɛɢɟɧɢɟ ɢɧɬɟɝɪɚɥɚ ɧɚ ɫɭɦɦɭ ɢɧɬɟɝɪɚɥɨɜ ɟɫɥɢ ɮɭɧɤɰɢɹ ɬɟɪɩɢɬ ɪɚɡɪɵɜɵ.
Ⱦɥɹ ɫɥɭɱɚɹ ɪɚɡɪɵɜɧɨɣ ɮɭɧɤɰɢɢ (1.8), ɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɪɚɡɛɢɜɚɟɬɫɹ ɧɚ ɞɜɚ ɢɧɬɟɝɪɚɥɚ, ɬ. ɤ. ɬɨɱɤɚ ɪɚɡɪɵɜɚ IJ
ɪɚɫɩɨɥɨɠɟɧɚ ɜɧɭɬɪɢ
ɢɧɬɟɪɜɚɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
ఛ
න
ሼݐሽ
݀ݐ
= න
ଵ
ሼݐሽ
݀ݐ
+ න
ఛ
ሼݐሽ
݀ݐ
ଶ
.
(1.23)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɫɨɜɩɚɞɟɧɢɟ ɨɞɧɨɝɨ ɢɡ ɩɪɟɞɟɥɨɜ ɫ ɬɨɱɤɨɣ
ɪɚɡɪɵɜɚ ɬɨɥɶɤɨ ɭɫɬɪɚɧɹɟɬ ɨɞɢɧ ɢɡ ɢɧɬɟɝɪɚɥɨɜ, ɧɟ ɢɡɦɟɧɹɹ
ɧɚɩɢɫɚɧɢɹ ɞɪɭɝɨɝɨ.
ɂɧɬɟɝɪɢɪɭɹ ɢɦɩɭɥɶɫɧɭɸ ɮɭɧɤɰɢɢ, ɩɨɥɭɱɢɦ ɫɨɨɬɧɨɲɟɧɢɟ
ሺݐሻήɁሺ
න
=
ݐെɒሻ݀ݐ
ሺɒሻήሾܪሺ
=
ܾെɒሻെܪሺܽെɒ
17
ሺɒሻ
නɁሺݐെɒሻ݀ݐ
ሻሿ
=
(1.24)
.
ȿɫɥɢ ɬɨɱɤɚ IJ ɪɚɫɩɨɥɨɠɟɧɚ ɜɧɭɬɪɢ ɢɧɬɟɪɜɚɥɚ a…b, ɬɨ
݂
݂
݂
ɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ (1.24) ɪɚɜɟɧ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɟ
න
ሺݐሻήɁሺ
ݐെɒሻ݀ݐ
=
(1.25)
ሺɒሻ
.
Ɂɞɟɫɶ ɩɪɨɹɜɥɹɟɬɫɹ ɮɢɥɶɬɪɭɸɳɟɟ ɞɟɣɫɬɜɢɟ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ f (t). Ⱦɚɧɧɵɣ ɢɧɬɟɝɪɚɥ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɟɫɥɢ ɬɨɱɤɚ IJ ɪɚɫɩɨɥɨɠɟɧɚ ɜɧɟ ɷɬɨɝɨ
ɢɧɬɟɪɜɚɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ.
ȿɫɥɢ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɜɩɚɞɟɧɢɟ ɨɞɧɨɝɨ ɢɡ ɩɪɟɞɟɥɨɜ ɢɧɬɟɝɪɚɥɚ (1.24) ɫ ɦɨɦɟɧɬɨɦ t = IJ, ɩɪɟɞɟɥ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢɫɤɭɫɫɬɜɟɧɧɨ ɩɟɪɟɦɟɳɚɟɬɫɹ ɜɩɪɚɜɨ ɥɢɛɨ ɜɥɟɜɨ ɫ ɬɟɦ, ɱɬɨɛɵ, ɪɭɤɨɜɨɞɫɬɜɭɹɫɶ ɨɫɨɛɟɧɧɨɫɬɹɦɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɢ,
ɜɤɥɸɱɢɬɶ ɥɢɛɨ ɢɫɤɥɸɱɢɬɶ ɢɦɩɭɥɶɫɧɭɸ ɮɭɧɤɰɢɸ ɢɡ ɪɚɫɫɦɨɬɪɟɧɢɹ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɧɟɧɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɚɩɥɚɫɚ ɤ
ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɜ ɮɨɪɦɟ
ஶ
ି௧
ሺݐሻ
න
ήe
݀ݐ
ஶ
= නɁሺݐሻήe
ି௧
݀ݐ
.
(1.26)
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɬɚɤɚɹ ɡɚɩɢɫɶ ɧɟ ɢɦɟɟɬ ɫɦɵɫɥɚ, ɢɛɨ ɧɢɠɧɢɣ
ɩɪɟɞɟɥ ɜɵɛɪɚɧ ɜ ɦɨɦɟɧɬ t = 0, ɤɨɝɞɚ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɧɟɨɝɪɚɧɢɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ. ɉɨɷɬɨɦɭ, ɟɫɥɢ ɭɱɢɬɵɜɚɬɶ ɢɡɦɟɧɟɧɢɹ
ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ ɜ ɦɨɦɟɧɬ t = 0, ɞɥɹ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ
ɫɩɪɚɜɟɞɥɢɜɨ ɫɨɨɬɧɨɲɟɧɢɟ
18
݂
ஶ
݂
ି௧
ሺݐሻ
න
ష
ήe
݀ݐ
ஶ
= නɁሺݐሻ݀ݐ
ష
=1.
(1.27)
ȿɫɥɢ ɠɟ ɢɡɦɟɧɟɧɢɟ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ ɜ ɦɨɦɟɧɬ t = 0
ɧɟ ɭɱɢɬɵɜɚɬɶ, ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ʌɚɩɥɚɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
ஶ
ି௧
ሺݐሻ
ήe
න
శ
݀ݐ
ஶ
= නɁሺݐሻ݀ݐ
శ
=0.
(1.28)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ (1.27) ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɮɨɪɦɭɥɢɪɭɸɬɫɹ ɞɥɹ ɦɨɦɟɧɬɚ, ɩɪɟɞɲɟɫɬɜɭɸɳɟɝɨ ɜɨɡɧɢɤɧɨɜɟɧɢɸ
ɢɦɩɭɥɶɫɚ, ɚ ɜ ɫɥɭɱɚɟ (1.28) – ɞɥɹ ɦɨɦɟɧɬɚ, ɫɥɟɞɭɸɳɟɝɨ ɡɚ
ɢɫɱɟɡɧɨɜɟɧɢɟɦ ɢɦɩɭɥɶɫɚ.
1.3. ɋɬɭɩɟɧɱɚɬɚɹ ɢ ɢɦɩɭɥɶɫɧɚɹ
ɩɟɪɟɯɨɞɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ
ɉɪɢ ɟɞɢɧɢɱɧɨɦ ɫɬɭɩɟɧɱɚɬɨɦ ɜɨɡɞɟɣɫɬɜɢɢ H(t) ɧɚ ɜɯɨɞɟ
ɧɟɜɨɡɛɭɠɞɟɧɧɨɣ ɰɟɩɢ ɟɟ ɜɵɯɨɞɧɚɹ ɜɟɥɢɱɢɧɚ ɛɭɞɟɬ ɢɡɦɟɧɹɬɶɫɹ ɩɨ ɡɚɤɨɧɭ, ɜɵɪɚɠɚɟɦɨɦɭ ɟɞɢɧɢɱɧɨɣ ɫɬɭɩɟɧɱɚɬɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ h{t}, ɹɜɥɹɸɳɟɣɫɹ ɮɭɧɤɰɢɟɣ ɜɪɟ-
ɦɟɧɢ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
h{t} ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɟɚɤɰɢɸ x{t} ɰɟɩɢ ɫ ɧɭɥɟɜɵɦɢ
ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɧɚ ɜɨɡɞɟɣɫɬɜɢɟ f{t} ɜ ɜɢɞɟ ɟɞɢɧɢɱɧɨɣ ɮɭɧɤɰɢɢ (ɪɢɫ. 1.11).
19
Ɋɢɫ. 1.11
ɋɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ
ɜɢɞɟ
݄ሼݐሽ= ݄ሺݐሻܪሺݐ
ሻ
,
(1.29)
ɝɞɟ h(t) – ɝɥɚɞɤɚɹ ɮɭɧɤɰɢɹ, ɜɵɪɚɠɚɸɳɚɹ ɫɬɭɩɟɧɱɚɬɭɸ ɩɟɪɟɯɨɞɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɞɚɧɧɨɦ ɫɨɨɬɧɨɲɟɧɢɢ ɦɧɨɠɢɬɟɥɶ ɜ ɜɢɞɟ ɟɞɢɧɢɱɧɨɣ ɮɭɧɤɰɢɢ ɭɫɬɪɚɧɹɟɬ ɡɧɚɱɟɧɢɹ ɪɟɚɤɰɢɢ ɩɪɢ t < 0, ɤɨɝɞɚ ɟɟ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɧɟɜɨɡɦɨɠɧɨ,
ɬ. ɤ. ɪɟɚɤɰɢɹ
ɧɟ ɦɨɠɟɬ ɜɨɡɧɢɤɧɭɬɶ ɞɨ ɩɨɹɜɥɟɧɢɹ ɜɨɡɞɟɣ-
ɫɬɜɢɹ.
Ɉɛɵɱɧɨ ɡɚɩɢɫɶ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɞɜɭɦɹ ɢɧɞɟɤɫɚɦɢ, ɨɛɨɡɧɚɱɚɸɳɢɦɢ ɬɢɩ ɜɨɡɞɟɣɫɬɜɢɹ ɢ
ɬɢɩ ɪɟɚɤɰɢɢ. ȿɫɥɢ, ɧɚɩɪɢɦɟɪ, ɧɚ ɜɯɨɞɟ ɧɟɜɨɡɛɭɠɞɟɧɧɨɝɨ ɱɟɬɵɪɟɯɩɨɥɸɫɧɢɤɚ ɞɟɣɫɬɜɭɟɬ ɧɚɩɪɹɠɟɧɢɟ ɜ ɜɢɞɟ ɟɞɢɧɢɱɧɨɣ
ɫɬɭɩɟɧɱɚɬɨɣ ɮɭɧɤɰɢɢ, ɬ. ɟ. ɜ ɦɨɦɟɧɬ t = 0 ɜɤɥɸɱɚɟɬɫɹ ɩɨɫɬɨ-
ɹɧɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɜ 1 ȼ, ɬɨ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɬɨɤɚ ɧɚ ɟɝɨ ɜɵɯɨɞɟ ɜɵɪɚɠɚɟɬ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ hui{t}.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
ɢɦɟɟɬ ɨɩɪɟɞɟɥɟɧɧɭɸ ɪɚɡɦɟɪɧɨɫɬɶ, ɡɚɜɢɫɹɳɭɸ ɨɬ ɮɢɡɢɱɟɫɤɨɣ ɩɪɢɪɨɞɵ ɜɨɡɞɟɣɫɬɜɢɹ ɢ ɪɟɚɤɰɢɢ.
20
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