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Файл:Расчет переходных процессов в электроэнергетических системах методом интеграла Дюамеля. Учебное пособие
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Ɋɚɫɫɦɨɬɪɢɦ ɨɫɨɛɟɧɧɨɫɬɢ ɩɨɥɭɱɟɧɢɹ ɫɬɭɩɟɧɱɚɬɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɥɹ ɫɥɟɞɭɸɳɢɯ ɜɨɡɦɨɠɧɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɜɟɥɢɱɢɧ ɬɨɤɚ ɢ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɩɪɢɦɟɪɟ ɷɥɟɤɬɪɢɱɟɫɤɨɣ ɫɯɟɦɵ, ɩɪɟɞɫɬɚɜɥɟɧɧɨɣ ɧɚ ɪɢɫ. 1.12.
Ɋɢɫ. 1.12
Ɂɚɜɢɫɢɦɨɫɬɶ i
ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ u
ɋ
ȼ ɦɨɦɟɧɬ t = 0 ɤ ɰɟɩɢ ɩɪɢɥɨɠɟɧɨ ɟɞɢɧɢɱɧɨɟ ɫɬɭɩɟɧɱɚɬɨɟ
ɧɚɩɪɹɠɟɧɢɟ u{t} = H(t). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɧɬɟɝɪɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
௧
ܴ
+ ܴ
ܴଵ݅
ሺݐሻ
+
ଵ
1
ଶ
ܴ
ܥ
ଶ
න݅
ሺݐሻ
݀ݐ= ܪሺݐ
ሻ
.
(1.30)
Ⱦɚɧɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɩɟɪɚɰɢɨɧɧɨɟ
ܴଵ݅
+
ܴ
ሺሻ
ܴଵ+ ܴ
1
ଶ
ଶ
ܥ
ሺሻ
݅
=
1
,
(1.31)
21
ɨɬɤɭɞɚ ɢɡɨɛɪɚɠɟɧɢɟ ɞɥɹ ɬɨɤɚ ɜ ɜɟɬɜɢ ɫ ɟɦɤɨɫɬɶɸ
ሺሻ
1
=
݅
ܴ
ଵ
ή
+
1
,
(1.32)
1
ܴʬܥ
ɝɞɟ
ܴଵܴ
ܴଵ+ ܴ
ଶ
.
(1.33)
ଶ
=
ܴ
ʬ
Ɉɪɢɝɢɧɚɥ ɷɬɨɝɨ ɢɡɨɛɪɚɠɟɧɢɹ ɪɚɜɟɧ
ή݁
ି
௧
ோʬ
.
(1.34)
ሺݐሻ
݅
=
ܴ
ଵ
1
ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɡɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɤɨɝɞɚ ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɬɨɤ ɜ ɟɦɤɨɫɬɢ
ɰɟɩɢ
ή݁
ି
௧
ோʬ
ήܪሺݐ
ሻ
(1.35)
.
ሼݐሽ
݄
௨
=
ܴ
ଵ
1
Ɂɚɦɟɬɢɦ, ɱɬɨ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɜ
ɷɬɨɦ ɫɥɭɱɚɟɬ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «1/Ɉɦ».
Ɂɚɜɢɫɢɦɨɫɬɶ iɋ ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ i
1
ȼ ɦɨɦɟɧɬ t = 0 ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɬɨɤ ɜ ɜɢɞɟ ɟɞɢɧɢɱɧɨɣ
ɫɬɭɩɟɧɱɚɬɨɣ ɮɭɧɤɰɢɢ i1{t} = H(t). Ɍɨɝɞɚ ɩɨɥɭɱɚɟɦ ɢɧɬɟɝɪɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
22
௧
1
ሺݐሻ
݅
+
ܥ
ܴ
ଶ
න݅
ሺݐሻ
݀ݐ= ܪሺݐ
ሻ
.
(1.36)
ɋɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɟɦɭ ɨɩɟɪɚɰɢɨɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
ሺሻ
+
݅
ܴ
ଶ
ܥ
ሺሻ
݅
1
=
1
,
(1.37)
ɨɬɤɭɞɚ, ɪɟɲɢɜ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɤɚ ɜ ɜɟɬɜɢ ɫ ɟɦɤɨɫɬɶɸ, ɩɨɥɭɱɚɟɦ ɟɝɨ ɢɡɨɛɪɚɠɟɧɢɟ
1
ሺሻ
݅
=
+
1
ܴଶܥ
.
(1.38)
Ɉɪɢɝɢɧɚɥ ɷɬɨɝɨ ɬɨɤɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
௧
ି
ሺݐሻ
݅
= ݁
ோమ
.
(1.39)
Ɍɨɝɞɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫ ɪɟɚɤɰɢɟɣ ɜ ɜɢɞɟ ɬɨɤɚ ɜ ɟɦɤɨɫɬɢ ɰɟɩɢ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
௧
ሼݐሽ
݄
= ݁
ି
ோమ
ήܪሺݐ
ሻ
.
(1.40)
ɂɡ ɫɨɨɬɧɨɲɟɧɢɹ ɜɢɞɧɨ, ɱɬɨ ɞɚɧɧɚɹ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɟɫɬɶ ɛɟɡɪɚɡɦɟɪɧɚɹ ɜɟɥɢɱɢɧɚ.
23
Ɂɚɜɢɫɢɦɨɫɬɶ i2 ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ u
ȼ ɦɨɦɟɧɬ t = 0 ɤ ɰɟɩɢ ɩɪɢɥɨɠɟɧɨ ɟɞɢɧɢɱɧɨɟ ɫɬɭɩɟɧɱɚɬɨɟ
ɧɚɩɪɹɠɟɧɢɟ u{t} = H(t). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
ሺݐሻ
݀݅
ሺ
ܴ
ଵ
+ ܴ
ሻ
ሺݐሻ
݅
ଶ
+ ܴ
ଶ
ଵܴଶ
ܥ
ଶ
݀ݐ
= ܪሺݐ
.
ሻ
(1.41)
ɉɪɢɦɟɧɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ʌɚɩɥɚɫɚ, ɩɨɥɭɱɚɟɦ ɨɩɟɪɚɰɢɨɧɧɨɟ ɭɪɚɜɧɟɧɢɟ
1
ሺ
ܴଵ+ ܴ
ሻ
ሺሻ
݅
ଶ
+ ܴଵܴଶܥ݅
ଶ
ଶ
ሺሻ
=
,
(1.42)
ɚ ɞɥɹ ɢɡɨɛɪɚɠɟɧɢɹ ɬɨɤɚ ɜ ɜɟɬɜɢ ɫ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ R2, ɦɨɠɟɦ
ɧɚɩɢɫɚɬɶ
ሺሻ
1
݅
=
ଶ
ܴଵܴଶܥ
ή
ቀ+
1
ܴଵ+ ܴ
ܴଵܴଶܥ
.
(1.43)
ଶ
ቁ
Ɍɨɝɞɚ ɨɪɢɝɢɧɚɥ ɷɬɨɝɨ ɬɨɤɚ ɪɚɜɟɧ
ሺݐሻ
݅
=
ଶ
ܴଵ+ ܴ
ଶ
1 െ݁
൬
1
ି
௧
ோʬ
(1.44)
.
൰
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɫ ɪɟɚɤɰɢɟɣ ɜ
ɜɢɞɟ ɬɨɤɚ ɜ ɫɨɩɪɨɬɢɜɥɟɧɢɢ R2 ɰɟɩɢ ɜɵɪɚɠɚɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
24
ሼݐሽ
݄
௨
=
+ ܴ
ܴ
ଵ
൬
ଶ
1 െ݁
1
ି
௧
ோʬ
൰
ሻ
(1.45)
ܪሺݐ
.
ɂɡ ɜɵɪɚɠɟɧɢɹ ɫɥɟɞɭɟɬ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «1/Ɉɦ».
Ɂɚɜɢɫɢɦɨɫɬɶ i
ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ i
2
1
ȼ ɦɨɦɟɧɬ t = 0 ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɬɨɤ ɜ ɜɢɞɟ ɟɞɢɧɢɱɧɨɣ
ɫɬɭɩɟɧɱɚɬɨɣ ɮɭɧɤɰɢɢ i1{t} = H(t). Ɇɨɠɟɦ ɧɚɩɢɫɚɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
ሺݐሻ
݀݅
݅
ଶ
ሺݐሻ
+ ܴଶܥ
ଶ
݀ݐ
= ܪሺݐ
. (1.46)
ሻ
Ⱦɚɧɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɩɟɪɚɰɢɨɧɧɨɟ
1
ሺሻ
݅
ଶ
+ ܴଶܥ݅
ሺሻ
=
,
(1.47)
ɨɬɤɭɞɚ ɢɡɨɛɪɚɠɟɧɢɟ ɞɥɹ ɬɨɤɚ ɜ ɜɟɬɜɢ ɫ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ R
ሺሻ
1
݅
=
ଶ
ܴଶܥ
ቀ+
1
1
ܴଶܥ
.
ቁ
2
(1.48)
Ɉɪɢɝɢɧɚɥ ɷɬɨɝɨ ɬɨɤɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
௧
ି
݅
ଶ
ሺݐሻ
=1െ݁
ோమ
.
(1.49)
25
Ɍɨɝɞɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫ ɪɟɚɤɰɢɟɣ ɜ ɜɢɞɟ ɬɨɤɚ ɜ ɫɨɩɪɨɬɢɜɥɟɧɢɢ R2 ɰɟɩɢ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
௧
ି
ோ
ሼݐሽ
݄
=
1 െ݁
൬
మ
ήܪሺݐ
൰
ሻ
.
(1.50)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɚɧɧɚɹ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɹɜɥɹɟɬɫɹ ɛɟɡɪɚɡɦɟɪɧɨɣ ɜɟɥɢɱɢɧɨɣ.
Ɂɚɜɢɫɢɦɨɫɬɶ uC ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ u
ȼ ɦɨɦɟɧɬ t = 0 ɤ ɰɟɩɢ ɩɪɢɥɨɠɟɧɨ ɟɞɢɧɢɱɧɨɟ ɫɬɭɩɟɧɱɚɬɨɟ
ɧɚɩɪɹɠɟɧɢɟ u{t} = H(t). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
ܴଵ+ ܴ
ଶ
ሺݐሻ
ݑ
ܴ
ଶ
+ ܴଵܥ
݀ݑ
݀ݐ
ሺݐሻ
= ݑሼݐ
ሽ
,
(1.51)
ɤɨɬɨɪɨɦɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɩɟɪɚɰɢɨɧɧɨɟ
ܴଵ+ ܴ
ଶ
ሺሻ
ݑ
ܴ
ଶ
+ ܴଵܥݑ
ሺሻ
1
=
, (1.52)
ɂɡɨɛɪɚɠɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɟɦɤɨɫɬɢ
ݑ
=
ܴଵܥ
ቀ+
ሺሻ
1
1
ܴଵ+ ܴ
ܴଵܴଶܥ
ଶ
ቁ
(1.53)
ɢ ɟɝɨ ɨɪɢɝɢɧɚɥ
ݑ
ሺݐሻ
=
ܴଵ+ ܴ
ܴ
ଶ
1 െ݁
൬
ଶ
26
ି
௧
ோʬ
.
൰
(1.54)
ɉɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫ ɪɟɚɤɰɢɟɣ ɜ ɜɢɞɟ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɟɦɤɨɫɬɢ ɰɟɩɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɜɵɪɚɠɚɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
ܴ
ሼݐሽ
݄
௨௨
=
ଶ
ܴଵ+ ܴ
൬
ଶ
1 െ݁
ି
௧
ோʬ
൰
ܪሺݐ
ሻ
.
(1.55)
ȼɢɞɢɦ, ɱɬɨ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɜ ɷɬɨɦ
ɫɥɭɱɚɟ ɧɟ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɢ.
Ɂɚɜɢɫɢɦɨɫɬɶ uC ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ i
1
ȼ ɦɨɦɟɧɬ t = 0 ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɬɨɤ ɜ ɜɢɞɟ ɟɞɢɧɢɱɧɨɣ
ɫɬɭɩɟɧɱɚɬɨɣ ɮɭɧɤɰɢɢ i1{t} = H(t). Ɍɨɝɞɚ ɩɨɥɭɱɚɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
1
ሺݐሻ
ݑ
+ ܥ
ܴ
ଶ
݀ݑ
݀ݐ
ሺݐሻ
= ݅
ሼݐሽ
.
ଵ
(1.56)
ɋɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɟɦɭ ɨɩɟɪɚɰɢɨɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
1
ሺሻ
ݑ
ܴ
ଶ
+ ܥݑ
ሺሻ
1
=
, (1.57)
ɨɬɤɭɞɚ, ɪɟɲɢɜ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɟɦɤɨɫɬɢ, ɩɨɥɭɱɚɟɦ ɟɝɨ ɢɡɨɛɪɚɠɟɧɢɟ
ݑ
=
ܥ
ሺሻ
1
ቀ+
1
1
ܴଶܥ
.
(1.58)
ቁ
27
Ɉɪɢɝɢɧɚɥ ɷɬɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
௧
ି
ோ
మ
ሺݐሻ
ݑ
= ܴ
൬
ଶ
1 െ݁
(1.59)
.
൰
Ɍɨɝɞɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɤɨɝɞɚ
ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɟɦɤɨɫɬɢ ɰɟɩɢ,
ɩɪɢɧɢɦɚɟɬ ɜɢɞ
௧
ି
ሼݐሽ
௨
= ܴ
݄
൬
ଶ
1 െ݁
ோమ
൰
ܪሺݐ
ሻ
.
(1.60)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «Ɉɦ».
ȼɨɡɦɨɠɧɵɟ ɜɢɞɵ ɫɬɭɩɟɧɱɚɬɵɯ ɩɟɪɟɯɨɞɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɞɥɹ ɡɚɜɢɫɢɦɨɫɬɟɣ ɜɟɥɢɱɢɧ ɬɨɤɚ ɢ ɧɚɩɪɹɠɟɧɢɹ ɢ ɢɯ ɪɚɡɦɟɪɧɨɫɬɢ ɫɜɟɞɟɧɵ ɜ ɬɚɛɥ. 1.1.
Ɍɚɛɥɢɰɚ 1.1
ɋɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɚɹ
Ɍɢɩ
Ɍɢɩ ɪɟɚɤɰɢɢ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
ɜɨɡɞɟɣɫɬɜɢɹ
Ɉɛɨɡɧɚɱɟɧɢɟ Ɋɚɡɦɟɪɧɨɫɬɶ
u i hui{t} ɋɦ
u u huu{t} –
i i hii{t} –
i u hiu{t} Ɉɦ
28
ɉɪɢ ɜɨɡɞɟɣɫɬɜɢɢ ɜ ɜɢɞɟ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ į(t) ɧɚ ɜɯɨɞɟ ɧɟɜɨɡɛɭɠɞɟɧɧɨɣ ɰɟɩɢ ɟɟ ɜɵɯɨɞɧɚɹ ɜɟɥɢɱɢɧɚ ɛɭɞɟɬ ɢɡɦɟɧɹɬɶɫɹ
ɩɨ ɡɚɤɨɧɭ, ɜɵɪɚɠɚɟɦɨɦɭ ɟɞɢɧɢɱɧɨɣ ɢɦɩɭɥɶɫɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ k(t), ɹɜɥɹɸɳɟɣɫɹ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɩɭɥɶɫɧɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
k{t} ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɟɚɤɰɢɸ x{t} ɰɟɩɢ ɫ ɧɭɥɟɜɵɦɢ
ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɧɚ ɜɨɡɞɟɣɫɬɜɢɟ f{t
} ɜ ɜɢɞɟ ɞɟɥɶɬɚ-
ɮɭɧɤɰɢɢ (ɪɢɫ. 1.13).
Ɋɢɫ. 1.13
ɂɦɩɭɥɶɫɧɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ k{t} ɫɜɹɡɚɧɚ ɫ
ɟɞɢɧɢɱɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ h{t} ɡɚɜɢɫɢɦɨɫɬɶɸ
ᇱ
݇ሼݐሽ= ݄
ሼݐሽ
.
(1.61)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɚɧɧɚɹ ɫɜɹɡɶ ɨɛɭɫɥɨɜɥɟɧɚ ɬɟɦ, ɱɬɨ ɢɦɩɭɥɶɫɧɚɹ ɢ ɫɬɭɩɟɧɱɚɬɚɹ ɩɟɪɟɯɨɞɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɪɟɚɤɰɢɢ ɥɢɧɟɣɧɨɣ ɰɟɩɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ ɜɨɡɞɟɣɫɬɜɢɹ H(t) ɢ į(t), ɤɨɬɨɪɵɟ ɫɨɝɥɚɫɧɨ (1.18), ɫɜɹɡɚɧɵ ɬɚɤɨɣ
ɠɟ ɡɚɜɢɫɢɦɨɫɬɶɸ, ɬ. ɟ. į(t) ɟɫɬɶ ɩɪɨɢɡɜɨɞɧɚɹ ɨɬ H(t).
Ɋɚɫɤɪɵɜɚɹ (1.61) ɫ ɭɱɟɬɨɦ ɫɨɨɬɧɨɲɟɧɢɹ (1.29), ɩɨɥɭɱɚɟɦ
ᇱ
ሽ=ሾ݄ሺݐሻܪሺݐሻሿ
݇ሼݐ
= ݄
ᇱ
ሺݐሻܪሺݐሻ
+ ݄
ሺ0ሻɁሺݐሻ
(1.62)
,
29
ɢɥɢ
݇ሼݐሽ= ݇ሺݐሻܪሺݐሻ+ ݄
ሺ0ሻɁሺݐሻ
,
(1.63)
ɝɞɟ k(t) = h'(t) – ɝɥɚɞɤɚɹ ɮɭɧɤɰɢɹ, ɜɵɪɚɠɚɸɳɚɹ ɢɦɩɭɥɶɫɧɭɸ
ɩɟɪɟɯɨɞɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ; h(0)į(t) – ɢɦɩɭɥɶɫɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɷɬɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɩɭɥɶɫɧɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
ɫɨɞɟɪɠɢɬ ɝɥɚɞɤɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ, ɹɜɥɹɸɳɭɸɫɹ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ ɫɬɭɩɟɧɱɚɬɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ,
ɚ ɬɚɤɠɟ ɦɨɠɟɬ ɢɦɟɬɶ ɢɦɩɭɥɶɫɧɭɸ ɤɨɦɩɨɧɟɧɬɭ.
ɂɦɩɭɥɶɫɧɚɹ ɩɟɪɟɯɨɞɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ, ɤɚɤ ɢ ɫɬɭɩɟɧɱɚɬɚɹ, ɢɦɟɟɬ ɨɩɪɟɞɟɥɟɧɧɭɸ ɪɚɡɦɟɪɧɨɫɬɶ, ɡɚɜɢɫɹɳɭɸ ɨɬ ɮɢɡɢɱɟɫɤɨɣ ɩɪɢɪɨɞɵ ɜɨɡɞɟɣɫɬɜɢɹ ɢ ɪɟɚɤɰɢɢ.
ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɪɚɧɟɟ ɷɥɟɤɬɪɢɱɟɫɤɨɣ
ɫɯɟɦɟ ɧɚ ɪɢɫ. 1.12 ɧɚɣɞɟɦ ɢɦɩɭɥɶɫɧɵɟ ɩɟɪɟɯɨɞɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɥɹ ɫɥɟɞɭɸɳɢɯ ɜɨɡɦɨɠɧɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɜɟɥɢɱɢɧ
ɬɨɤɚ ɢ ɧɚɩɪɹɠɟɧɢɹ.
Ɂɚɜɢɫɢɦɨɫɬɶ i
ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ u
ɋ
– ɤɨɝɞɚ ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɬɨɤ ɜ ɟɦɤɨɫɬɢ ɰɟɩɢ, ɚ
ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɢɱɧɵɣ ɢɦɩɭɥɶɫ ɧɚɩɪɹɠɟɧɢɹ
ଵ
Ɂሺݐ
ᇱ
=
(1.64)
ሻ
.
= ݄
ᇱ
ሼݐሽ
௨
1
ή݁
ሼݐሽ
݇
௨
= െ
=
ି
ோʬ
ܴ
௧
1
ଵ
ήܪሺݐ
ܴଵܴʬܥ
ή݁
ି
ோʬ
ሻ
௧
+
ήܪሺݐሻ൨
1
ܴ
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «ɋɦ/ɫɟɤ».
30
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