Материал: Демирчян, Нейман, Коровкин, Чечурин. Теоретические основы электротехники.т.2.2003
I p
Z
p
D p
E p
Z
p
D p
E
p
I p
Z
1
22
11
12
22
2
2
( )
( )
( )
( )
( )
( )
( );
( )
=
+ -
= -
1
11
11
22
11
22
( )
( )
( )
( )
( )
( );
( )
( )
( )
p
D p
E p
Z p
D p
E
p
D p
Z p Z
p
+
=
-
[
( )] .
Z
p
12
2
Ðàññìîòðèì âèä ýòèõ ôóíêöèé, êîãäà
å
1
=
Å
0
=
const è
å
2
=
E
m
sin
w
t.
Ñîîòâåòñò-
âåííî,
E p
E
p
E p
E
p
m
1
0
2
2
2
( )
( )
.
=
=
+
è
w
w
Ïîäñòàâëÿÿ çíà÷åíèÿ
Z
11
(
p
),
Z
22
(
p
),
Z
12
(
p
), à òàêæå
E
11
(
p
) è
E
22
(
p
) äëÿ òîêà
I
1
(
p
), íàïðèìåð, ïîëó÷èì
I p
E G p
pH p
E
G p
p
H p
u
G
m
C
1
0 1
1
2
2
2
1
2
3
0
( )
( )
( )
( )
(
) ( )
( ) (
=
-
+
+
w
w
p
pH p
L i
G p
H p
L i
G p
H p
G
L
L
)
( )
( ) ( )
( )
( ) ( )
( )
(
1
1 1
4
1
3 3
5
1
0
0
+
+
+
=
p
p p
H p
)
(
) ( )
,
2
2
1
+ w
ãäå
H p
a p
a p
a p a
L L p
L r
r
L r
r
1
3
3
2
2
1
0
1 3
3
3
1
2
1
2
3
( )
[ (
)
(
)
=
+
+
+
=
+
+
+
+
]
(
)
;
p
r r
r
r r
L
L
C
p r
r
C
2
1
2
3
2 3
1
3
2
1
3
2
+
+
+
+
+
+
é
ë
ê
ù
û
ú +
+
G p
G p
pZ
p
r
r p
C
L p
G p
G p
1
4
22
2
3
2
3
2
2
3
1
( )
( )
( ) (
)
;
( )
( )
=
=
=
+
+
+
=
=
pZ
p
p r
pL
G p
pZ p
r p
C
12
3
3
5
2
2
2
1
( )
(
);
( )
( )
.
=
+
=
=
+
 ïîëó÷åííîì âûðàæåíèè
I
1
(
p
) ïåðâûå äâà ÷ëåíà îïðåäåëÿþò òîê â ïåðåõîäíîì
ïðîöåññå ïðè âêëþ÷åíèè öåïè ïîä äåéñòâèå ÝÄÑ
e
1
è
e
2
.
Ïîñëåäíèå òðè ÷ëåíà îï-
ðåäåëÿþò òîê ïåðåõîäíîãî ïðîöåññà, âîçíèêàþùåãî â öåïè çà ñ÷åò íåíóëåâûõ íà-
÷àëüíûõ çíà÷åíèé òîêîâ â êàòóøêàõ
L
1
è
L
3
è íàïðÿæåíèÿ íà êîíäåíñàòîðå
C
2
.
Åñëè äî ðàçìûêàíèÿ êëþ÷à òîê â êàòóøêå
L
3
îòñóòñòâîâàë, òî
i
3
L
(0)
=
0 è ÷ëåí, ñî-
äåðæàùèé ýòîò òîê, îòñóòñòâóåò. Èç ïîëó÷åííûõ âûðàæåíèé âûòåêàåò òàêæå âîç-
ìîæíîñòü ðàñ÷åòà ïåðåõîäíîãî ïðîöåññà íàëîæåíèåì ïåðåõîäíûõ ïðîöåññîâ, ðàñ-
ñ÷èòàííûõ îòäåëüíî îò êàæäîé ÝÄÑ, íà÷àëüíûõ òîêîâ è íàïðÿæåíèé.
Êàê âèäíî èç äàííîãî ïðèìåðà, îïåðàòîðíîå èçîáðàæåíèå òîêà ïðåäñòàâëÿåò
ñîáîé ðàöèîíàëüíóþ äðîáü, ãäå è ÷èñëèòåëü, è çíàìåíàòåëü ÿâëÿþòñÿ ïîëèíîìà-
ìè îïåðàòîðà
p.
 ñëåäóþùåì ïàðàãðàôå áóäåò ðàññìîòðåí ñïîñîá ïåðåõîäà îò
èçîáðàæåíèÿ ê îðèãèíàëó, â êîòîðîì èñïîëüçóþòñÿ íåêîòîðûå ñâîéñòâà ðàöèî-
íàëüíûõ äðîáåé.  íåêîòîðûõ ñëó÷àÿõ ýòè äðîáè ìîæíî ïðèâåñòè ê âèäó, ïðåä-
ñòàâëåííîìó â òàáëèöàõ, â êîòîðûõ äàíû ñîîòâåòñòâóþùèå èì îðèãèíàëû. Îä-
íàêî â ýòèõ òàáëèöàõ (íàïðèìåð,
Äèòêèí Â. À., Êóçíåöîâ Ï. È.
Ñïðàâî÷íèê ïî
îïåðàöèîííîìó èñ÷èñëåíèþ) òàêèå ôîðìóëû ïðèâåäåíû äëÿ ïîëèíîìîâ
H
1
(
p
)
îòíîñèòåëüíî íèçêîãî ïîðÿäêà.
Ãëàâà 10. Ðàñ÷åò ïåðåõîäíûõ ïðîöåññîâ îïåðàòîðíûì ìåòîäîì
103
10.5. Ïåðåõîä îò èçîáðàæåíèé ê îðèãèíàëó. Òåîðåìà ðàçëîæåíèÿ
Äëÿ íàõîæäåíèÿ îðèãèíàëà ïðåäñòàâèì èçîáðàæåíèå, ïîëó÷åííîå â âèäå ðàöèî-
íàëüíîé äðîáè, ïðîñòåéøèìè ñëàãàåìûìè, äëÿ êîòîðûõ èçâåñòíû îðèãèíàëû.
Ñ ýòîé öåëüþ âîñïîëüçóåìñÿ òåîðåìîé ðàçëîæåíèÿ. Ïóñòü èìååòñÿ èçîáðàæåíèå
â âèäå
X p
G p
H p
( )
( )
( )
,
=
ãäå
G
(
ð
) è
Í
(
ð
)
—
ïîëèíîìû îò
ð.
Çäåñü áóäåì ïðåäïîëàãàòü, ÷òî ñòåïåíü
m
ïîëè-
íîìà â ÷èñëèòåëå ìåíüøå ñòåïåíè
ï
ïîëèíîìà â çíàìåíàòåëå (
ò
<
ï
)
.
 äàëüíåé-
øåì, â ãë. 12, ñíèìåì ýòî îãðàíè÷åíèå è óâèäèì, ÷òî ïðè
ò
³
ï
ïîÿâëÿþòñÿ ÝÄÑ,
òîêè è íàïðÿæåíèÿ, èìåþùèå èìïóëüñíûé õàðàêòåð, ò. å. ïðèíèìàþùèå áåñêîíå÷íî
áîëüøèå çíà÷åíèÿ â òå÷åíèå áåñêîíå÷íî ìàëûõ èíòåðâàëîâ âðåìåíè. Íå ðàññìàòðè-
âàÿ çäåñü òàêèõ èìïóëüñíûõ ôóíêöèé, ÿâëÿþùèõñÿ, ïî ñóòè äåëà, ðåçóëüòàòîì
èäåàëèçàöèè ðåàëüíûõ ÝÄÑ, òîêîâ è íàïðÿæåíèé, áóäåì ïîëàãàòü
ò
<
ï.
Ïðåäïîëîæèì, êðîìå òîãî, ÷òî óðàâíåíèå
Í
(
ð
)
=
0 íå èìååò êðàòíûõ êîðíåé,
à òàêæå íå èìååò êîðíåé, ðàâíûõ êîðíÿì óðàâíåíèÿ
G
(
ð
)
=
0. Ïðè óêàçàííûõ
óñëîâèÿõ ðàöèîíàëüíóþ äðîáü ìîæíî ðàçëîæèòü íà ïðîñòåéøèå äðîáè:
G p
H p
A
p p
A
p p
A
p p
A
p p
n
n
k
k
k
n
( )
( )
,
=
-
+
-
+ +
-
=
-
=
å
1
1
2
2
1
K
ãäå
p
1
,
p
2
, ...,
ð
ï
—
êîðíè
Í
(
ð
)
.
Äëÿ îïðåäåëåíèÿ êîýôôèöèåíòîâ
A
k
ìîæíî âîñ-
ïîëüçîâàòüñÿ îäíèì èç ìíîãèõ ïðèåìîâ, èçâåñòíûõ èç àëãåáðû. Óìíîæèâ îáå
÷àñòè ðàâåíñòâà íà (
p – p
k
) è ïðèíÿâ
ð
=
ð
k
, ïîëó÷èì ñïðàâà
A
k
, à ñëåâà — íåîïðå-
äåëåííîñòü. Ðàñêðûâàÿ ýòó íåîïðåäåëåííîñòü, íàõîäèì
A
G p p p
H p
G p
p p
H p
G p
H
k
k
p p
k
p p
k
k
k
k
=
-
=
-
=
=
®
( )(
)
( )
( ) lim
( )
( )
'
( )
.
p
k
Òàêèì îáðàçîì,
X p
G p
H p
A
p p
G p
H p p p
k
k
k
k n
k
k
k
k n
( )
( )
( )
( )
( )
=
=
-
=
-
=
=
=
=
å
å
1
1
1
'
k
.
Òàê êàê
A
p p
A e
k
k
k
p t
k
-
Þ
, òî äëÿ èñêîìîé âåëè÷èíû
õ
(
t
) èìååì
x t
G p
H p
e
k
k
k
k n
p t
k
( )
( )
( )
.
=
=
=
å
'
1
Ýòî ðàâåíñòâî è íàçûâàþò ò å î ð å ì î é ð à ç ë î æ å í è ÿ.
 ÷àñòíîì ñëó÷àå, êîãäà îäèí èç êîðíåé ïîëèíîìà
Í
(
ð
), äîïóñòèì
p
1
, ðàâåí
íóëþ, òî
e
p t
k
=
1 è ñîîòâåòñòâóþùèé ÷ëåí â ðàçëîæåíèè îáðàùàåòñÿ â ïîñòîÿí-
íóþ âåëè÷èíó. Âûäåëÿÿ ýòîò ÷ëåí, íàïèøåì
X p
x t
G
H
G p
H p
e
k
k
k
k n
p t
k
( )
( )
( )
( )
( )
( )
.
Þ
=
+
=
=
å
0
0
2
'
'
Ïîëèíîì
Í
(
ð
) ìîæåò èìåòü êîðåíü, ëåæàùèé â íà÷àëå êîîðäèíàò (
p
1
=
0), êî-
ãäà â äàííîé öåïè èìåþòñÿ èñòî÷íèêè ïîñòîÿííîé ÝÄÑ (èëè èñòî÷íèêè ïîñòî-
104
×àñòü 2. Òåîðèÿ ëèíåéíûõ ýëåêòðè÷åñêèõ öåïåé
ÿííîãî òîêà). Âûäåëåííûé ïîñòîÿííûé ÷ëåí ïðåäñòàâëÿåò ñîáîé óñòàíîâèâøèå-
ñÿ òîê èëè íàïðÿæåíèå â öåïè.
Åñëè
Í
(
ð
) èìååò ïàðó ñîïðÿæåííûõ ÷èñòî ìíèìûõ êîðíåé, ëåæàùèõ íà îñè
ìíèìûõ
p
1
=
j
w
,
p
2
=
–
j
w
, òî ìîæíî çàïèñàòü
X p
x t
G j
H j
e
G j
H
j
e
G p
j t
j t
k
( )
( )
( )
( )
(
)
(
)
( )
Þ
=
+
-
-
+
-
w
w
w
w
w
w
'
'
H p
e
k
k
k n
p t
k
'
( )
.
=
=
å
3
Ïîëèíîì
Í
(
ð
) ìîæåò èìåòü ïàðó ÷èñòî ìíèìûõ ñîïðÿæåííûõ êîðíåé â ñëó-
÷àå, åñëè ðàññìàòðèâàåòñÿ ïåðåõîäíûé ïðîöåññ ïðè íàëè÷èè â öåïè èñòî÷íèêîâ
ñèíóñîèäàëüíûõ ÝÄÑ èëè èñòî÷íèêîâ ñèíóñîèäàëüíûõ òîêîâ. Äâà ïåðâûõ âûäå-
ëåííûõ ÷ëåíà îïðåäåëÿþò ñèíóñîèäàëüíûé òîê èëè íàïðÿæåíèå óñòàíîâèâøåãî-
ñÿ ðåæèìà.
Äëÿ èëëþñòðàöèè ïðèìåíåíèÿ òåîðåìû ðàçëîæåíèÿ ðàññìîòðèì íåêîòîðûå
ïðèìåðû.
 êà÷åñòâå ïåðâîãî ïðèìåðà ðåøèì çàäà÷ó î ðàçðÿäå êîíäåíñàòîðà íà öåïü
(
r
,
L
), ðàññìîòðåííóþ ðàíåå êëàññè÷åñêèì ìåòîäîì (ñì. § 9.8). Ïóñòü íà÷àëüíîå
íàïðÿæåíèå êîíäåíñàòîðà
u
C
(0)
=
U
0
, a íà÷àëüíîå çíà÷åíèå òîêà â êàòóøêå
i
(0)
=
0. Îïåðàòîðíîå âûðàæåíèå òîêà èìååò âèä
I p
U p
Li
p
u
Z p
U p
r pL
pC
U L
p
C
( )
( )
( )
( )
( )
( )
=
+
-
=
-
+
+
=
=
-
0
1
0
1
0
0
2
+
+
=
-
+
+
r
L
p
LC
U L
p
p
1
2
0
2
0
2
( )
,
d
w
òàê êàê
U
(
p
)
=
0. Ñëåäîâàòåëüíî, â äàííîì ñëó÷àå
G
(
ð
)
=
–
U
0
/
L
è
Í
(
ð
)
=
p
2
+
+ 2
d
p
+ w
0
2
. Êîðíè óðàâíåíèÿ
Í
(
ð
)
=
0 áóäóò
p
1,2
=
- ±
d
d - w
2
0
2
. Òàê êàê
H p
p
G p
H p
U L
p
'
'
( )
,
( )
( )
(
.
=
+
= -
+
2
2
2
0
d
d)
òî
Ïîëüçóÿñü òåîðåìîé ðàçëîæåíèÿ, ïîëó÷àåì
i t
U
L p
e
U
L p
e
U
L
e
p t
p t
( )
(
(
(
=
-
+
+
-
+
=
-
0
1
0
2
0
2
0
2
2
2
2
1
2
d)
d)
d - w
p t
p t
e
1
2
-
).
Ïóñòü êîðíè
p
1
è
p
2
ðàâíû äðóã äðóãó:
p
1
=
p
2
=
–
d
è
d = w
0
, ò. å. ïîëèíîì
Í
(
ð
)
èìååò êðàòíûå êîðíè. Ïðåäïîëîæèâ ñíà÷àëà, ÷òî
p
1
¹
p
2
, ïîëó÷èì òîëüêî ÷òî
íàéäåííîå ðåøåíèå, îáðàùàþùååñÿ ïðè
p
1
=
p
2
â íåîïðåäåëåííîñòü. Ðàñêðûâàÿ
ýòó íåîïðåäåëåííîñòü ïðè
p
1
®
p
2
, ïîëó÷èì, êàê ýòî áûëî ñäåëàíî â § 9.8, èñêîìîå
ðåøåíèå â âèäå
i
U
L
te
t
= -
-
0
d
.
 áîëåå îáùåì ñëó÷àå, êîãäà îäèí èç êîðíåé, äîïóñòèì
p
1
, ïîëèíîìà
Í
(
ð
) ñòå-
ïåíè
n
èìååò êðàòíîñòü
q
, ðàöèîíàëüíóþ äðîáü ìîæíî ðàçëîæèòü íà ïðîñòåéøèå
â âèäå
Ãëàâà 10. Ðàñ÷åò ïåðåõîäíûõ ïðîöåññîâ îïåðàòîðíûì ìåòîäîì
105
G p
H p
G p
p p
H p
A
p p
A
p p
q
q
q
( )
( )
( )
(
)
( )
(
)
(
)
=
-
=
-
+
-
+
-
1
1
11
1
12
1
1
é
ë
ê
+
-
+
+
-
ù
û
ú + - + + - +
- +
K
K
K
K
A
p p
A
p p
A
p p
A
p p
s
q s
q
k
k
1
1
1
1
1
2
2
(
)
K
+
-
A
p p
n
n
,
ãäå
A
s
d
dp
p p G p
H p
A
G
s
s
s
q
p p
k
1
1
1
1
1
1
1
=
-
-
é
ë
ê
ù
û
ú
=
-
-
=
(
) !
(
) ( )
( )
;
( )
( )
( )
(
)
( )
.
p
H p
G p
p
p
H p
k
k
k
k
q
k
=
-
¢
1
1
Îðèãèíàë ôóíêöèè
A
p p
s
q s
1
1
1
(
)
-
- +
èìååò âèä
( )
A
p p
A
q s
t
e
s
q s
s
q s
p t
1
1
1
1
1
(
)
(
) !
.
-
Þ
-
- +
-
Îðèãèíàë æå ôóíêöèè
A
p p
k
k
-
ðàâåí
À
k
e
p t
k
.
Ðåøèì ñ ïîìîùüþ òåîðåìû ðàçëîæåíèÿ çàäà÷ó î âêëþ÷åíèè öåïè (
r
,
L
) ïîä
ñèíóñîèäàëüíîå íàïðÿæåíèå
è
=
U
m
sin (
w
t
+
y
u
) ïðè óñëîâèè
i
(0)
=
0, ðàññìîò-
ðåííóþ ðàíåå êëàññè÷åñêèì ìåòîäîì (ñì. § 9.5). Èçîáðàæåíèå òîêà â öåïè ïîëó-
÷èì, ïðèíèìàÿ âî âíèìàíèå, ÷òî
Z
(
p
)
=
r + pL
, à èçîáðàæåíèå ñèíóñîèäàëüíîé
ôóíêöèè èìååò âèä
u t
j
U e e
U e
e
j
U
p j
U
p
m
j
j t
m
j
j t
m
m
u
u
( )
(
)
&
*
=
-
Þ
-
-
+
-
-
1
2
1
2
y
w
y
w
w
j
U p
w
æ
è
ç
ç
ç
ö
ø
÷
÷
÷
=
( ).
Îïåðàòîðíîå èçîáðàæåíèå òîêà â öåïè îïðåäåëèòñÿ èç âûðàæåíèÿ
I p
U p
Z p
jL
U
p j
p r L
U
p j
p r L
m
m
( )
( )
( )
&
(
(
) (
(
)
*
=
=
-
+
-
+
+
é
1
2
w)
w)
ë
ê
ê
ù
û
ú
ú
.
Ïðèìåíèâ òåîðåìó ðàçëîæåíèÿ, ïîëó÷èì
I p
jL
U
j
r L p j
U
j
r L p r L
U
j
r
m
m
m
( )
&
&
*
=
+
-
+
- -
+
é
ë
ê
ê
-
- +
1
2
1
1
w
w
w
w
L p j
U
j
r L p r L
j
U
r j L
e
U
r
m
m
j t
m
1
1
1
2
+
-
-
-
+
ù
û
ú
ú
Þ
+
é
ë
ê
ê
-
w
w
w
w
*
&
&
+
-
-
+
-
ù
û
ú
ú
-
-
-
j L
e
U
r j L
e
U
r j L
e
r
L
t
m
j t
m
r
L
t
w
w
w
w
*
*
.
106
×àñòü 2. Òåîðèÿ ëèíåéíûõ ýëåêòðè÷åñêèõ öåïåé
Ãðóïïèðóÿ ïåðâûé ÷ëåí ñ òðåòüèì è âòîðîé ñ ÷åòâåðòûì è èìåÿ â âèäó, ÷òî
j =
arctg
w
L
r
, íàõîäèì
i t
U
r
L
t
e
m
u
u
r
L
t
( )
sin(
) sin(
)
.
=
+
+
-
-
-
é
ë
ê
ê
ù
û
ú
ú
-
2
2 2
w
w y j
y j
 êà÷åñòâå ïðèìåðà îïðåäåëåíèÿ ïåðåõîäíîãî òîêà â ðàç-
âåòâëåííîé öåïè ðàññìîòðèì òàêîâîå ïðè âêëþ÷åíèè ïîä
ïîñòîÿííîå íàïðÿæåíèå
U
0
öåïè, ïðèâåäåííîé íà ðèñ. 10.3,
ïðè íóëåâûõ íà÷àëüíûõ óñëîâèÿõ. Îïåðàòîðíîå ñîïðîòèâ-
ëåíèå òàêîé öåïè
Z p
pL
r
pC
r
pC
p rLC pL r
prC
( )
( )
.
=
+
+
=
+
+
+
1
1
1
2
Èçîáðàæåíèå ïðèëîæåííîãî íàïðÿæåíèÿ
U
0
Þ
U
0
/
p.
Èçîáðàæåíèå òîêà ïðåäñòàâèòñÿ â âèäå
I p
U p
Z p
U
prC
p p rLC pL r
U
rLC
prC
p p
( )
( )
(
)
(
)
(
)
=
=
+
+
+
=
+
0
0
2
0
1
1
2
0
2
0
2
1
1
1
2
+
+
æ
è
ç
ö
ø
÷
=
+
+
+
rC
p
LC
U
rLC
prC
p p
p
(
)
(
)
.
d
w
Òàêèì îáðàçîì,
G p
U
rLC
prC
( )
(
);
=
+
0
1
H p
p
p
p H p
p
p
( )
;
( )
;
=
+
+
=
+
+
3
2
0
2
2
0
2
2
3
4
d
w
d
w
'
I p
G p
H p
A
p p
A
p p
A
p p
( )
( )
( )
;
=
=
-
+
-
+
-
1
1
2
2
3
3
p
p
p
1
2
2
0
2
3
2
0
2
0
=
= - +
= - -
;
;
;
d
d - w
d
d - w
A
G
H
U
r
A
G p
H p
U
rLC
p rC
p
1
0
2
2
2
0
2
2
0
0
1
2
=
=
=
=
+
×
( )
( )
;
( )
( )
(
)
(
'
'
p
2
+ d
)
;
A
G p
H p
U
rLC
p rC
p
p
3
3
3
0
3
3
3
1
2
=
=
+
×
+
( )
( )
(
)
(
)
.
'
d
Òîãäà
i t
U
r
U
rLC
p rC
p
e
U
rLC
p rC
p t
( )
(
)
(
)
=
+
+
×
-
-
+
0
0
2
2
2
0
2
0
3
1
2
1
2
d
w
p
e
p t
3
2
0
2
2
3
×
-
d
w
.
Ãëàâà 10. Ðàñ÷åò ïåðåõîäíûõ ïðîöåññîâ îïåðàòîðíûì ìåòîäîì
107
Ðèñ. 10.3