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Файл:Расчет переходных процессов в электроэнергетических системах методом интеграла Дюамеля. Учебное пособие
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ɇɟɧɭɥɟɜɵɟ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɩɪɢ ɷɬɨɦ ɭɱɬɟɦ, ɜɵɩɨɥɧɢɜ ɪɚɫɱɟɬ ɫɨɫɬɚɜɥɹɸɳɟɣ
ɧɚɩɪɹɠɟɧɢɹ ɩɪɢ ɡɚɦɵɤɚɧɢɢ ɧɚɤɨɪɨɬɤɨ ɡɚɠɢɦɨɜ ɢɫɬɨɱɧɢɤɚ
u(ș), ɢɫɩɨɥɶɡɭɹ ɨɩɟɪɚɬɨɪɧɵɣ ɦɟɬɨɞ.
ɋɨɫɬɚɜɥɹɸɳɭɸ ɧɚɩɪɹɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚɣɞɟɦ ɩɭɬɟɦ
ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ
ᇱᇱ
ሺݐሻ
݀ݑ
ܴ
ܥ
ଵ
݀ݐ
+ ݑ
ᇱᇱ
ሺݐሻ
(2.17)
=0.
ɉɪɢɦɟɧɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ʌɚɩɥɚɫɚ, ɢɦɟɟɦ
ᇱᇱ
ܴଵܥݑ
ሺሻ
+ ݑ
ᇱᇱ
ሺሻ
= ܴଵܥܷ
,
(2.18)
ɨɬɤɭɞɚ
ܷ
ቀ+
1
ܴଵܥ
.
ቁ
(2.19)
ᇱᇱ
ሺሻ
ݑ
=
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɫɬɚɜɥɹɸɳɚɹ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɨɦ ɞɨ U
ɤɨɧɞɟɧɫɚɬɨɪɟ ɩɪɢ ɡɚɦɵɤɚɧɢɢ
0
ɧɚɤɨɪɨɬɤɨ ɡɚɠɢɦɨɜ ɢɫɬɨɱɧɢɤɚ ɧɚɩɪɹɠɟɧɢɹ u(ș) ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
ᇱᇱ
ሺɅሻ
ݑ
= ܷ݁
ିஔ
.
(2.20)
ɋɭɦɦɢɪɭɹ ɬɟɩɟɪɶ ɧɚɣɞɟɧɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɩɪɹɠɟɧɢɹ,
ɩɨɥɭɱɢɦ ɢɫɤɨɦɨɟ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ.
51
= ݑ
ܣ
ܣ
ܣ
ܣ
ܣ
ᇱ
ሺɅሻ
ሺɅሻ
ݑ
+ ݑ
ᇱᇱ
ሺɅሻ
,
(2.21)
ɢɥɢ
ሺɅሻ
ݑ
+
ሺ
ሾ
= ܷ
sin
ሺ
sin
Ʌଵ+ ɔሻെsin Ʌ
ሺ
Ʌ+ Ʌ
ሻ
െ
ଵ
ሺ
sin
Ʌ+ Ʌଵ+ ɔ
ିஔ
ሻ
݁
ଵ
൧+ ܷ݁
ିஔ
ሻ
+
,
(2.22)
ɢɥɢ, ɫ ɭɱɟɬɨɦ (2.16) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
ሺɅሻ
ݑ
= ܷ
ሾ
ሺ
sin
+
Ʌ+ Ʌ
sin
ሻ
െ
ଵ
ሺ
Ʌଵ+ ɔሻ݁
ሺ
sin
Ʌ+ Ʌଵ+ ɔ
ିஔ
൧.
ሻ
+
(2.23)
ɍɝɨɥ ș2 ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɱɤɨɣ ɩɟɪɟɫɟɱɟɧɢɹ ɤɪɢɜɨɣ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɫ ɩɨɥɭɜɨɥɧɨɣ ɢ ɩɨɷɬɨɦɭ ɦɨɠɟɬ ɛɵɬɶ
ɧɚɣɞɟɧ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ
ሺ
ሻ
Ʌ
ݑ
= ܷ
ଶ
ሺ
sin
Ʌଶ+ Ʌ
ሻ
,
ଵ
(2.24)
ɢɥɢ
ିఋ
ሾ
ሺ
sin
Ʌଵ+ ɔሻሿ݁
మ
=ܣsin
ሺ
Ʌଶ+ Ʌଵ+ ɔ
ሻ
(2.25)
.
ɂɬɚɤ, ɫ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥɵ (2.23) ɞɥɹ ɪɚɡɧɵɯ ɡɧɚɱɟɧɢɣ
ɬɟɤɭɳɟɝɨ ɭɝɥɚ ș ɦɨɠɟɬ ɛɵɬɶ ɜɵɱɢɫɥɟɧɨ ɧɚɩɪɹɠɟɧɢɟ uC(ș)
(ɪɢɫ. 2.2, ɛ).
2.2. Ʉɨɧɞɟɧɫɚɬɨɪɧɵɟ ɪɟɥɟ ɜɪɟɦɟɧɢ
ɞɥɹ ɭɫɬɪɨɣɫɬɜ ɚɜɬɨɦɚɬɢɤɢ ɢ ɪɟɥɟɣɧɨɣ ɡɚɳɢɬɵ
ȼ ɭɫɬɪɨɣɫɬɜɚɯ ɚɜɬɨɦɚɬɢɤɢ ɢ ɪɟɥɟɣɧɨɣ ɡɚɳɢɬɵ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɢ ɮɨɪɦɢɪɨɜɚɧɢɹ ɜɪɟɦɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɸɬɫɹ ɷɥɟɤɬɪɨɧɧɵɟ ɪɟɥɟ ɜɪɟɦɟɧɢ.
52
ɉɪɢɧɰɢɩɵ ɜɵɩɨɥɧɟɧɢɹ ɪɟɥɟ ɜɪɟɦɟɧɢ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɢɱɧɵ. Ⱦɥɹ ɭɫɬɪɨɣɫɬɜ ɪɟɥɟɣɧɨɣ ɡɚɳɢɬɵ ɨɫɨɛɟɧɧɨ ɰɟɥɟɫɨɨɛɪɚɡɧɵɦɢ ɹɜɥɹɸɬɫɹ ɪɟɥɟ, ɜ ɤɨɬɨɪɵɯ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɟɪɟɯɨɞɧɵɣ ɩɪɨɰɟɫɫ ɡɚɪɹɞɚ, ɪɚɡɪɹɞɚ ɢɥɢ ɩɟɪɟɡɚɪɹɞɚ ɤɨɧɞɟɧɫɚɬɨɪɚ ɜ
ɚɩɟɪɢɨɞɢɱɟɫɤɨɣ ɰɟɩɢ (ɤɨɧɞɟɧɫɚɬɨɪɧɵɟ ɪɟɥɟ ɜɪɟɦɟɧɢ), ɩɨɫɤɨɥɶɤɭ ɨɧɢ ɚɜɬɨɧɨɦɧɵ, ɝɨɬɨɜɵ ɤ ɞɟɣɫɬɜɢɸ ɞɚɠɟ ɩɨɫɥɟ
ɤɪɚɬɤɨɜɪɟɦɟɧɧɵɯ ɩɟɪɟɪɵɜɨɜ ɩɢɬɚɧɢɹ ɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ
ɜɵɫɨɤɢɦɢ ɩɨɤɚɡɚɬɟɥɹɦɢ.
ɉɨɤɚɠɟɦ
ɩɪɢɧɰɢɩ ɨɛɪɚɡɨɜɚɧɢɹ ɜɪɟɦɟɧɧɨɝɨ ɢɧɬɟɪɜɚɥɚ
ɩɪɢ ɡɚɪɹɞɟ, ɪɚɡɪɹɞɟ ɢ ɩɟɪɟɡɚɪɹɞɟ ɤɨɧɞɟɧɫɚɬɨɪɚ.
ȼ ɫɯɟɦɟ ɫ ɡɚɪɹɞɨɦ ɤɨɧɞɟɧɫɚɬɨɪɚ (ɪɢɫ. 2.3, a) ɤɥɸɱ K ɜ
ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɡɨɦɤɧɭɬ, ɚ ɧɚɩɪɹɠɟɧɢɟ ɧɚ
ɤɨɧɞɟɧɫɚɬɨɪɟ U
(0) = 0. ȼ ɦɨɦɟɧɬ ɡɚɦɵɤɚɧɢɹ ɤɥɸɱɚ ɤɨɧ-
ɋ
ɞɟɧɫɚɬɨɪ ɧɚɱɢɧɚɟɬ ɡɚɪɹɠɚɬɶɫɹ. ȿɫɥɢ ɩɚɪɚɥɥɟɥɶɧɨ ɤɨɧɞɟɧɫɚɬɨɪɭ ɜɤɥɸɱɢɬɶ ɩɨɪɨɝɨɜɨɟ ɭɫɬɪɨɣɫɬɜɨ, ɬɨ ɨɧɨ ɫɪɚɛɨɬɚɟɬ, ɤɨɝɞɚ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɞɨɫɬɢɝɧɟɬ ɩɨɪɨɝɚ ɫɪɚɛɚɬɵɜɚɧɢɹ u
(t) = Ucp (ɪɢɫ. 2.3, ɛ).
ɋ
Ɋɢɫ. 2.3
53
ȼ ɫɯɟɦɟ ɫ ɪɚɡɪɹɞɨɦ (ɪɢɫ. 2.4, ɚ) ɤɨɧɞɟɧɫɚɬɨɪ ɩɟɪɜɨɧɚɱɚɥɶɧɨ ɡɚɪɹɠɟɧ ɞɨ ɧɚɩɪɹɠɟɧɢɹ Eɩ. ɉɪɢ ɩɟɪɟɤɥɸɱɟɧɢɢ
ɤɥɸɱɚ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɧɚɱɢɧɚɟɬ ɭɦɟɧɶɲɚɬɶɫɹ.
ȼ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɜɯɨɞɟ ɩɨɪɨɝɨɜɨɝɨ ɭɫɬɪɨɣɫɬɜɚ ɉɍ, ɪɟɚɝɢɪɭɸɳɟɝɨ ɧɚ ɩɨɧɢɠɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ, ɫɬɚɧɟɬ
ɦɟɧɶɲɟ ɭɫɬɚɧɨɜɥɟɧɧɨɝɨ ɭɪɨɜɧɹ, ɪɟɥɟ ɫɪɚɛɨɬɚɟɬ (ɪɢɫ. 2.4, ɛ).
Ɋɢɫ. 2.4
ȼ ɫɯɟɦɟ ɩɟɪɟɡɚɪɹɞɚ ɤɨɧɞɟɧɫɚɬɨɪɚ (ɪɢɫ. 2.5, a) ɩɨɫɥɟ ɩɟɪɟɤɥɸɱɟɧɢɹ ɤɥɸɱɚ K ɤɨɧɞɟɧɫɚɬɨɪ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɵɣ ɞɨ ɧɟɤɨɬɨɪɨɝɨ ɧɚɩɪɹɠɟɧɢɹ
+ Eɩ1, ɩɟɪɟɡɚɪɹɠɚɟɬɫɹ ɞɨ
ɧɚɩɪɹɠɟɧɢɹ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɝɨ ɡɧɚɤɚ – ȿɩ2. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ
(Eɩ1 ȿɩ2). ȼ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ
ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ Ucp, ɪɟɥɟ ɫɪɚɛɚɬɵɜɚɟɬ (ɪɢɫ. 2.5, ɛ).
Ɋɚɫɫɦɨɬɪɢɦ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ
ɡɚɪɹɠɟɧɧɨɦ ɞɨ uC(0) ɤɨɧɞɟɧɫɚɬɨɪɟ ɩɪɢ ɜɤɥɸɱɟɧɢɢ ɟɝɨ ɧɚ
ɩɪɨɢɡɜɨɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ u
(ɯ).
1
54
Ɋɢɫ. 2.5
Ɋɚɫɱɟɬ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɜɨɡɛɭɠɞɟɧɧɨɣ ɰɟɩɢ ɜɵɩɨɥɧɢɦ ɩɨ ɩɪɢɧɰɢɩɭ ɧɚɥɨɠɟɧɢɹ. ɋɧɚɱɚɥɚ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɥɢɧɟɣɧɭɸ ɷɥɟɤɬɪɢɱɟɫɤɭɸ ɰɟɩɶ ɩɚɫɫɢɜɧɨɣ ɫ ɧɭɥɟɜɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɬ. ɟ. ɭɱɬɟɦ ɬɨɥɶɤɨ ɜɤɥɸɱɚɟɦɨɟ ɧɚɩɪɹɠɟɧɢɟ u1(ɯ). Ɋɚɫɱɟɬ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɩɪɢ ɷɬɨɦ ɩɪɨɜɟɞɟɦ, ɢɫɩɨɥɶɡɭɹ ɢɧɬɟɝɪɚɥ Ⱦɸɚɦɟɥɹ. Ɂɚɬɟɦ ɭɱɬɟɦ ɬɨɥɶɤɨ ɢɫɬɨɱɧɢɤ ɚɤɬɢɜɧɨɝɨ ɞɜɭɯɩɨɥɸɫɧɢɤɚ, ɬ. ɟ. ɧɚɣɞɟɦ ɢɡɦɟɧɟɧɢɟ
ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ, ɡɚɪɹɠɟɧɧɨɦ ɞɨ uC(0), ɩɪɢ ɡɚɦɵɤɚɧɢɢ ɧɚɤɨɪɨɬɤɨ ɡɚɠɢɦɨɜ ɢɫɬɨɱɧɢɤɚ ɧɚɩɪɹɠɟɧɢɹ u1(ɯ).
Ɋɚɫɱɟɬ ɧɚɩɪɹɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɜɵɩɨɥɧɢɦ, ɧɚɩɪɢɦɟɪ, ɨɩɟɪɚɬɨɪɧɵɦ ɦɟɬɨɞɨɦ. ɋɭɦɦɢɪɭɹ ɧɚɣɞɟɧɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ
ɧɚɩɪɹɠɟɧɢɹ, ɩɨɥɭɱɢɦ ɢɫɤɨɦɨɟ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ
ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɨɦ ɤɨɧɞɟɧɫɚɬɨɪɟ.
ɂɫɩɨɥɶɡɭɹ ɢɧɬɟɝɪɚɥ Ⱦɸɚɦɟɥɹ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ
ɧɟɡɚɪɹɠɟɧɧɨɦ ɤɨɧɞɟɧɫɚɬɨɪɟ ɩɪɢ ɜɤɥɸɱɟɧɢɢ ɟɝɨ ɧɚ ɧɚɩɪɹɠɟɧɢɟ u
(ɯ) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ
1
55
௧
ሺɒሻ
݀ݑ
ᇱ
ሺݐሻ
ݑ
= ݑ
ሺ0ሻ ݄ሺݐሻ
ଵ
+ න
ଵ
݄ሺݐെɒሻ݀ɒ
(2.26)
,
݀ɒ
(0) – ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɜɯɨɞɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɰɟɩɢ ɜ ɦɨ-
ɝɞɟ u
1
ɦɟɧɬ t = 0; h(t), h(t – IJ) – ɩɟɪɟɯɨɞɧɵɟ ɮɭɧɤɰɢɢ ɩɪɢ ɟɞɢɧɢɱɧɨɦ ɫɤɚɱɤɟ ɧɚɩɪɹɠɟɧɢɹ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɰɟɩɢ.
ɇɚ ɜɯɨɞɟ ɧɟɜɨɡɛɭɠɞɟɧɧɨɣ ɰɟɩɢ ɞɟɣɫɬɜɭɟɬ ɩɨɫɬɨɹɧɧɨɟ
ɧɚɩɪɹɠɟɧɢɟ
ሺɒሻ
ݑ
ଵ
= ݑ
ଵ
ሺ0ሻ
= ܿ݊ݏݐ
,
(2.27)
ɨɬɤɭɞɚ ɢɦɟɟɦ
ሺɒሻ
݀ݑ
ଵ
(2.28)
=0.
݀ɒ
ɇɚɣɞɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɪɢ
ɟɞɢɧɢɱɧɨɦ ɫɤɚɱɤɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɜɯɨɞɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ
ɰɟɩɢ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
݀ݑ
ݑ+ ܴʬܥ
= ܪሺݐ
(2.29)
ሻ
,
݀ݐ
ɤɨɬɨɪɨɦɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɩɟɪɚɰɢɨɧɧɨɟ
1
ሺሻ
ݑ
+ ܴʬܥݑ
ሺሻ
=
.
(2.30)
Ɂɞɟɫɶ Rɗ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɷɤɜɢɜɚɥɟɧɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɰɟɩɢ.
56
ɂɡɨɛɪɚɠɟɧɢɟ ɩɨ Ʌɚɩɥɚɫɭ ɞɥɹ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɩɪɢɦɟɬ ɜɢɞ
ቀ+
1
1
ܴʬܥ
,
ቁ
(2.31)
ݑ
=
ܥ
ܴ
ʬ
ሺሻ
1
ɨɬɤɭɞɚ
௧
ሺݐሻ
ݑ
=1െ݁
ି
(2.32)
ோʬ
.
Ɍɨɝɞɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɬɭɩɟɧɱɚɬɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫ ɪɟɚɤɰɢɟɣ ɜ ɜɢɞɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
௧
ି
ሼݐሽ
݄
௨௨
=
1 െ݁
൬
ோʬ
൰
ήܪሺݐ
.
ሻ
(2.33)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɮɭɧɤɰɢɸ, ɜɵɪɚɠɚɸɳɭɸ ɫɬɭɩɟɧɱɚɬɭɸ
ɩɟɪɟɯɨɞɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɜ (2.26) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ
௧
݄ሺݐሻ=1െ݁
ି
(2.34)
த
ʬ
,
ɝɞɟ
ɒʬ= ܴʬܥ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɧɚɩɪɹɠɟɧɢɹ ɧɚ
ɧɟɡɚɪɹɠɟɧɧɨɦ ɤɨɧɞɟɧɫɚɬɨɪɟ
௧
ᇱ
ሺݐሻ
ݑ
= ݑ
ଵ
ሺ0ሻ
57
൬
1 െ݁
ି
த
ʬ
(2.36)
.
൰
(2.35)
Ɂɚɦɵɤɚɹ ɧɚɤɨɪɨɬɤɨ ɡɚɠɢɦɵ ɢɫɬɨɱɧɢɤɚ ɧɚɩɪɹɠɟɧɢɹ u1(ɯ)
ɧɚɣɞɟɦ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɨɦ ɞɨ u
(0) ɤɨɧɞɟɧɫɚɬɨɪɟ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶ-
C
ɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
ᇱᇱ
݀ݑ
ݑ
ᇱᇱ
+ ܴʬܥ
݀ݐ
=0
.
(2.37)
ɉɪɢɦɟɧɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ʌɚɩɥɚɫɚ, ɩɨɥɭɱɢɦ
ᇱᇱ
ሺሻ
ݑ
+ ܴʬܥݑ
ᇱᇱ
ሺሻ
= ܴʬܥݑ
ሺ0ሻ
,
(2.38)
ɨɬɤɭɞɚ
ሺ0ሻ
ݑ
ቀ+
1
ܴʬܥ
.
(2.39)
ቁ
ᇱᇱ
ሺሻ
ݑ
=
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɨɦ ɤɨɧɞɟɧɫɚɬɨɪɟ ɩɪɢ ɡɚɦɵɤɚɧɢɢ ɧɚɤɨɪɨɬɤɨ
ɡɚɠɢɦɨɜ ɢɫɬɨɱɧɢɤɚ ɧɚɩɪɹɠɟɧɢɹ u1(ɯ) ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
௧
ᇱᇱ
ሺݐሻ
ݑ
= ݑ
ሺ0ሻ
ି
த
ʬ
݁
.
(2.40)
ɋɥɨɠɢɜ ɬɟɩɟɪɶ ɧɚɣɞɟɧɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɩɪɹɠɟɧɢɹ,
ɩɨɥɭɱɢɦ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ
= ݑ
1 െ݁
൬
ᇱ
ሺݐሻ
ି
ሺݐሻ
ݑ
ሺ0ሻ
= ݑ
ଵ
த
௧
ʬ
+ ݑ
+ ݑ
൰
ᇱᇱ
ሺݐሻ
ሺ0ሻ
=
௧
ି
த
ʬ
݁
.
(2.41)
58
ɉɪɢɦɟɦ ɜ (2.41) uC(0) = ȿɩ1 = Uɧ ɧɚɩɪɹɠɟɧɢɟɦ ɧɚ ɤɨɧɞɟɧɫɚɬɨɪɟ ɞɨ ɧɚɱɚɥɚ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɚ u1(0) = ȿɩ2 = Uɤ –
ɩɨɫɥɟ ɟɝɨ ɨɤɨɧɱɚɧɢɹ. Ɍɨɝɞɚ
௧
ି
ሺݐሻ
ݑ
= ܷ
ˍ
+
ሺ
ܷːെܷ
ˍ
(2.42)
த
ʬ
ሻ
݁
.
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɫɪɚɛɚɬɵɜɚɧɢɟ ɪɟɥɟ ɩɪɨɢɫɯɨɞɢɬ ɜ ɦɨɦɟɧɬ,
ɤɨɝɞɚ uɋ(t) = Uɫɪ, ɩɨɥɭɱɚɟɦ
௧
ܷ˔˓= ܷ
ˍ
+
ሺ
ܷːെܷ
ି
(2.43)
த
ʬ
ሻ
݁
ˍ
,
ɨɬɤɭɞɚ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ
ݐ˔˓= ɒ
ln
ʬ
ܷ˔˓െܷ
ܷːെܷ
ˍ
. (2.44)
ˍ
Ɍɨɝɞɚ ɞɥɹ ɫɯɟɦɵ ɫ ɡɚɪɹɞɨɦ ɤɨɧɞɟɧɫɚɬɨɪɚ, ɩɪɢɧɹɜ ɜ ɜɵɪɚɠɟɧɢɢ (2.44) Uɧ = 0; Uɤ = ȿɩ2 = ȿɩ ɢ IJɗ = IJ = Rɋ, ɩɨɥɭɱɢɦ
ܧ
ݐ
˔˓ˊ
= ɒln
˒
ܧ˒െܷ
˔˓
.
(2.45)
Ⱦɥɹ ɫɯɟɦɵ ɫ ɪɚɡɪɹɞɨɦ ɤɨɧɞɟɧɫɚɬɨɪɚ ɩɪɢ Uɧ = ȿɩ ɢ IJɗ = IJ
ɛɭɞɟɦ ɢɦɟɬɶ
ܧ
ݐ
˔˓˓
= ɒln
˒
. (2.46)
ܷ
˔˓
59
Ⱦɥɹ ɫɯɟɦɵ ɫ ɩɟɪɟɡɚɪɹɞɨɦ ɤɨɧɞɟɧɫɚɬɨɪɚ, ɭɱɢɬɵɜɚɹ, ɱɬɨ Uɧ
ɢ Uɤ ɢɦɟɸɬ ɪɚɡɧɵɟ ɡɧɚɤɢ (Uɧ = Eɩ1 ɢ Uɤ = – Eɩ2) ɢ IJɗ = IJ,
ɩɨɥɭɱɢɦ
ݐ
˔˓˒
ܷ˒ଶ+ ܷ
= ɒln
ܧ˒ଵ+ ܧ
˒ଶ
˔˓
.
(2.47)
2.3. Ɉɬɤɥɸɱɟɧɢɟ ɜɵɤɥɸɱɚɬɟɥɟɦ ɰɟɩɟɣ
ɩɨɫɬɨɹɧɧɨɝɨ ɬɨɤɚ
ɉɪɢ ɪɟɲɟɧɢɢ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɬɹɝɨɜɨɝɨ ɷɥɟɤɬɪɨɫɧɚɛɠɟɧɢɹ ɧɟɪɟɞɤɨ ɜɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɚɧɚɥɢɡɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɩɟɪɟɯɨɞɧɵɯ ɩɪɨɰɟɫɫɨɜ ɩɪɢ ɨɬɤɥɸɱɟɧɢɢ ɰɟɩɟɣ
ɩɨɫɬɨɹɧɧɨɝɨ ɬɨɤɚ ɜɵɤɥɸɱɚɬɟɥɟɦ.
ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɨɩɵɬɧɵɯ ɞɚɧɧɵɯ ɞɥɹ ɚɧɚɥɢɡɚ ɩɪɨɰɟɫɫɚ
ɨɬɤɥɸɱɟɧɢɹ ɷɥɟɤɬɪɢɱɟɫɤɨɣ ɰɟɩɢ ɩɪɢɛɟɝɚɸɬ ɤ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦɭ ɦɨɞɟɥɢɪɨɜɚɧɢɸ. ɋɯɟɦɚ ɬɚɤɨɣ ɰɟɩɢ ɞɥɹ ɬɹɝɨɜɨɣ ɫɟɬɢ
ɩɨɫɬɨɹɧɧɨɝɨ ɬɨɤɚ ɢɦɟɟɬ ɜɢɞ, ɩɪɢɜɟɞɟɧɧɵɣ ɧɚ ɪɢɫ. 2.6, ɜɫɟ
ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɣ ɥɢɧɟɣɧɵɟ.
Ɋɢɫ. 2.6
60
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