6. 䝷䝥䝷䝇ኚ
(䝷䝥䝷䝇ኚ䛾⩦)
ᩍ⛉᭩䛻䛿┦ᙜ䛩䜛❶䛿䛺䛔
ྂไᚚ⌮ㄽ䛸⌧௦ไᚚ⌮ㄽ
z ⥺ᙧไᚚ⌮ㄽ䛻䛿⌧௦ไᚚ⌮ㄽ䛸ྂไᚚ⌮ㄽ䛜䛒䜛䚹
z 䛂⌧௦䛃䛂ྂ䛃䛸䛔䛖ゝ䛔᪉䛿䚸᭱ึ䛻ᥦ䛥䜜䛯ᮇ䛜㐪䛖䛛䜙䛷䛒䛳䛶䚸䛹
䛱䜙䛜ඃ䜜䛶䛔䜛䛸䛔䛖ヰ䛷䛿䛺䛔䚹
z 㠀⥺ᙧ䝅䝇䝔䝮䛾ሙྜ䛿⌧ᅾไᚚ⌮ㄽ䛧䛛䛺䛔䚹
ྂไᚚ⌮ㄽ ⌧௦ไᚚ⌮ㄽ
䝷䝥䝷䝇ኚ䛻ᇶ䛵䛟᪉ἲ ᚤศ᪉⛬ᘧ䛻ᇶ䛵䛟᪉ἲ
࿘Ἴᩘ㡿ᇦ 㛫㡿ᇦ
㏻ᖖ䚸ึᮇ್䜢0䛸䛧䛶⪃䛘䜛 ึᮇ್䜢㝧䛻⪃៖䛷䛝䜛
ྍไᚚ䞉ྍほ 䛺㒊ศ䛰䛡 ྍไᚚ䞉ྍほ 䛺㒊ศ䜒⪃៖ྍ⬟
≧ែ䛿⪃䛘䛺䛔(ฟຊ䝣䜱䞊䝗䝞䝑䜽) ≧ែ䝣䜱䞊䝗䝞䝑䜽䛜ᇶᮏ(ฟຊ䝣䜱䞊䝗
䝞䝑䜽䛾䛸䛝䛿䜸䝤䝄䞊䝞䜢⏝䛔䜛)
䝣䞊䝸䜶ኚ
z 㛫䛾㛵ᩘ: f(t) (–f < t < f ) ࿘Ἴᩘ䛾㛵ᩘ: F(Z) (–f <Z<f )
z 䝣䞊䝸䜶ኚ:
z ㏫䝣䞊䝸䜶ኚ:
z ඖ䛾㛵ᩘ
f(t) 䛜ᐇᩘ䛺䜙䜀䚸
z 䝣䞊䝸䜶ኚ䛾✚ศ䛜Ⓨᩓ䛧䛶䛧䜎䛖䛣䛸䛜䛒䜛䚹
(䝣䞊䝸䜶ኚᏑᅾ䛾䛯䜑䛾ᚲせ᮲௳) f(t) 䛜⤯ᑐྍ✚ศ
(䝣䞊䝸䜶ኚᏑᅾ䛾䛯䜑䛾༑ศ᮲௳) f(t) 䛜ྍ✚ศ
³
ff
dt
e
t
f
t
f
F(Z) F[ ()] ( ) jZ
t
³
ff Z Z
S
Z
F eZ
d
F
t
f - ( ) j t
2
1
)]
(
[
)
( 1
F
f
³
ff
f )(
t dt
f
³
ff
f t dt
2
)
(
)
(
)
( Z
F Z
F
⤯ᑐྍ✚ศ䛷䛺䛔䛺䜙䜀䚸
…
z ⤯ᑐྍ✚ศ䛾᮲௳ 䜢‶䛯䛥䛺䛔 䝣䞊䝸䜶ኚ䛷䛝䛺䛔
z ኚᩘ
t 䛜䇿㛫䇿䛺䜙䚸ᕤᏛⓗ䛻䛿㐣ཤ䛾ಙྕ䜢ᢅ䛖䛣䛸䛻䛭䜜䜋䛹ព䛿↓䛔䚹
f(t) = 0 (t d 0) 䛸௬ᐃ䛩䜛䚹
z 䛝䛔
t 䛻ᑐ䛧䛶ಙྕ䜢ῶ⾶䛥䛫䜛㡯䜢䛪䜛䚹
f
³
ff
f )(
t dt
⤯ᑐྍ✚ศ䛷䛺䛔ಙྕ䛻ᑐ䛧䛶䜒࿘Ἴᩘ㡿ᇦ䛻ኚ䛧䛯䛔
䛴䜎䜚䚸
䜢⪃䛘䜛䚹
c 䛜༑ศ䛝䛡䜜䜀䚸䜋䛸䜣䛹䛾ሙྜ䛻ኚ䛜Ꮡᅾ䛩䜛䚹
䛣䛣䛷䚸
c 䛸 jZ䜢䛔䛳䛧䜗䛻䜎䛸䜑䛶䚸
䛸᭩䛟䚹 ĺ䝷䝥䝷䝇ኚ)
³
f
0{ ()}
)
,
(
c e f t e dt
F Z ct jZ
t
³
f
0
)
(
)
(
)
(
c j f t e dt
F Z c jZ
t
䝷䝥䝷䝇ኚ
z (∦ഃ) 䝷䝥䝷䝇ኚ:
z ㏫䝷䝥䝷䝇ኚ:
z ኚᩘ
s 䛿」⣲ᩘ䛷䛒䜛䚹୍᪉䚸䝣䞊䝸䜶ኚ䛾䛸䛝䛾Z䛿ᐇᩘ䛷䛒䛳䛯䚹
z
s = c + jZ䛸䛚䛟䛺䜙䜀䚸
c = 0 䛾䛸䛝䛿䝣䞊䝸䜶ኚ䛻୍⮴䛩䜛䚹䛴䜎䜚䚸䝷䝥䝷
䝇ኚ䛿䝣䞊䝸䜶ኚ䛾ᣑᙇ䛷䛒䜛䚹
³
f
0 ( )
)]
(
[
)
(
s f t f t e dt
F st
L
³
f
o
c jp
jp
c
st
p j F s e ds
s
F
t
f ( )
2
1
lim
)]
(
[
)
( 1
S
L
䝷䝥䝷䝇ኚ䛾ᛶ㉁
(1) (
㔜せ
)
z ⥺ᙧᛶ
: L[af(t) + bg(t)] = aL[f(t)] + bL[g(t)]
z ᚤศබᘧ:
z ✚ศබᘧ:
z ㏫ኚ䛾ᚤศබᘧ:
z ㏫ኚ䛾✚ศබᘧ:
z ␚䜏㎸䜏✚ศ:
)
0
(
]
[
»¼
º
«¬
ª
s f f
dt
df
L
L
)]
(
[
1
)
(
0
f d s f t
t
L
L
«¬ª
³
W W
»¼º
ึᮇ್䛜0䛺䜙䚸䇿ᚤศ䇿 = “s 䜢䛡䜛䇿
”✚ศ䇿 = “(1 / s) 䜢䛡䜛䇿
)]
(
[
)]
(
[
)
(
)
(
]
[
0
f g t d f t g t
g
f L
t L L
L
«¬ª
³
W W W
»¼º
)]
(
[
)]
(
[ 1
1
F s t F s
c
L
L
)]
(
[
1
)
( 1
0
1
F s
t
dr
r
F
s
»¼
º
«¬
ª
³
L
L
䝷䝥䝷䝇ኚ䛾ᛶ㉁
(2)
z 㛫㍈䛾ఙ⦰:
z
s ㍈ୖ䛾ᖹ⾜⛣ື:
z 㛫㍈ୖ䛾ᖹ⾜⛣ື:
䛯䛰䛧䚸
u(t) 䛿䝦䞂䜱䝃䜲䝗䛾㝵ẁ㛵ᩘ䛷䚸
z 䝷䝥䝷䝇ኚ
F(s) 䛜 s = s0䛷⤯ᑐ᮰䛩䜜䜀䚸
Re(s)t Re(s0
) 䛺䜛 s 䛾⠊ᅖ䛷
୍ᵝ᮰䛩䜛䚹䜒䛧䚸
F(s) (s = V+ Z
j) 䛜䚸V > V0䛷᮰䛧䚸V <V0䛷᮰䛧
䛺䛔䛾䛷䛒䜜䜀䚸V
0䜢᮰ᗙᶆ䛸䛔䛖䚹
z 䝷䝥䝷䝇ኚ
F(s) 䛿䚸䛭䛾᮰ᇦ䛷ṇ๎㛵ᩘ䛷䛒䜛䚹
)]
(
[
)
(
)
/
(
1
)]
(
[
F s a F s f t
a
at
f L
L 䛯䛰䛧䚸
)]
(
[
)
(
)]
(
[
)
(
s a e f t F s f t
F at
L
L
䛯䛰䛧䚸
)]
(
[
)]
(
)
(
[
f t aut a e as f t
L
L
¯
®
t
)
0
(
0
)
0
(
1
)
(
t
t
t
u
䛔䜝䛔䜝䛺㛵ᩘ䛻ᑐ䛩䜛䝷䝥䝷䝇ኚ
(1)
z 䜲䞁䝟䝹䝇㛵ᩘ= 䝕䜱䝷䝑䜽䛾䝕䝹䝍㛵ᩘ
✚ศ⠊ᅖ䛾➃䛻䝕䝹䝍㛵ᩘ䛾䜲䞁䝟䝹䝇䛜ධ䛳䛶䛧䜎䛔གྷ䛺䛾䛷䚸✚✚ศ⠊
ᅖෆ䛻䜲䞁䝟䝹䝇䛜ධ䜛䜒䛾䛸䛧䛶ᢅ䛔䚸䛭䛾䜲䞁䝟䝹䝇䛾⨨䜢0䛻㏆䛵䛡䛯
ᴟ㝈䜢⪃䛘䜛䚹
z ᐃᩘ= 䝦䞂䜱䝃䜲䝗䛾㝵ẁ㛵ᩘ
㈇䛾㛫䛾ಙྕ䛿0 䛸䛧䛶⪃䛘䜛䛾䛷䚸ᐃᩘ䛾䝷䝥䝷䝇ኚ䛿ᐇ䛿䝦䞂䜱䝃䜲
䝗䛾㝵ẁ㛵ᩘ䛾䝷䝥䝷䝇ኚ䜢⪃䛘䛶䛔䜛䛾䛻➼䛧䛔䚹
1
lim
)
(
lim
)]
(
[
0
0
0
o
f
o
³
sp
p
st
p t p e dt e
t G
G
L
s
a
s
a
dt
ae
a st
³
f
(0 1)
]
[
0
L
䛔䜝䛔䜝䛺㛵ᩘ䛻ᑐ䛩䜛䝷䝥䝷䝇ኚ
(2)
z 䜉䛝㛵ᩘ:
z ᣦᩘ㛵ᩘ:
z ᣦᩘ㛵ᩘ㽢䜉䛝㛵ᩘ䠖
1
1
1
]
1
[
1
)!
1
(
1
!
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
n
n
n
n
s
s
n
t
s
n
t
L
L
L
a
s
e
eat at
1] 1
[
]
[ L
L
1
)
(
1
!
1
»¼
º
«¬
ª
n
at
n
a
s
e
t
n
L
䛔䜝䛔䜝䛺㛵ᩘ䛻ᑐ䛩䜛䝷䝥䝷䝇ኚ
(3)
z ୕ゅ㛵ᩘ:
z ୕ゅ㛵ᩘ㽢ᣦᩘ㛵ᩘ:
z ୖグ䛾බᘧ䛻䛥䜙䛻㏫ኚ䛾ᚤศබᘧ䜢⤌䜏ྜ䜟䛫䜛䛸䚸୕ゅ㛵ᩘ䞉ᣦᩘ㛵
ᩘ䛻䜉䛝䜢䛡䛯ሙྜ䛾ᘧ䜒ᚓ䜙䜜䜛䚹
2
2
2
2 , [sin( )]
)]
[cos(
Z
Z
Z
Z
Z
t s
s
s
t L
L
2
2
2
2 , [ sin( )]
( )
)
(
)]
cos(
[
Z
Z
Z
Z
Z
a
s
t
e
a
s
a
s
t
eat L at
L
䝷䝥䝷䝇ኚ⾲
(
䛒䜛⛬ᗘᬯグ䛩䜉䛧
)
z 䝷䝥䝷䝇ኚ䛾୍ぴ⾲䛷䚸䝷䝥䝷䝇㏫ኚ䛻䜒䛘䜛䚹
z
[ὀព] f(t) = 0 (t < 0) 䛷䛒䜚䚸⾲䛾 f(t) 䛿 tt 0 䛻ᑐ䛩䜛䜒䛾䛷䛒䜛䚹
f(t) F(s) = L{f(t)}
G
(t) 1
1
eat
sin Z
t
cos Z
t
s
1
1
1
n
s
!
n
tn
a
s
1
2
2
Z
Z
s
2
2
Z
s
s
f(t) F(s) = L{f(t)}
eatsin Z
t
eatcos Zt
eatsinh bt
eatcosh bt
1
)
(
1
a n
s
2
2
)
( Z
Z
a
s
2
2
)
( Z
a
s
a
s
2
2
)
(
s a b
b
2
2
)
(
s a b
a
s
!
n
e
tn at
䝷䝥䝷䝇㏫ኚ䛾᪉ἲ
z ㏫ኚ䛾බᘧ(Bromwich ✚ศ)
z
F(s) 䛜᭷⌮ᘧ䛷䚸䛛䛴(ศᏊ䛾ḟᩘ)d (ศẕ䛾ḟᩘ)䛷䛒䜜䜀௨ୗ䛾᪉ἲ䛜
⡆༢䛷䛒䜛䚹
1. 䜎䛪䚸
F(s) 䜢㒊ศศᩘᒎ㛤䛩䜛䚹䛯䛸䛘䜀䚸
2. ྛ㡯ู䛻䝷䝥䝷䝇ኚ⾲䜢ඖ䛻㏫䝷䝥䝷䝇ኚ䛩䜛䚹
³
f
o
c jp
jp
c
st
p j F s e ds
s
F
t
f ( )
2
1
lim
)]
(
[
)
( 1
S
L
2
2
1
2
)
1
(
1
)
2
(
)
1
(
)
(
2 2
s s s s
s
s
s
F
t
t
t e e
te
s
s
s
2
2
1
2 2
2
2
1
2
)
1
(
1
»
¼
º
«
¬
ª
L
䝷䝥䝷䝇㏫ኚ䛾
z []
t
e
t
e
e
te
s
s
s
s
s
s
s
s
s
t
t
t
t cos sin
1
)
1
(
1
1
1
)
1
(
)
1
(
1
1
1
)
1
(
1
)
2
2
(
)
1
( 2 2 2
1
2
2
1
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
L
L
1. ศẕ䛾ᙧ䛿ീ䛷䛝䜛䚹
2. ศᏊ䛾ಀᩘ䜢ᮍᐃᐃᩘ䛸䛧䛶㏻ศ䛧䚸㡯ูẚ
㍑䛧䛶㐃❧᪉⛬ᘧ䜢స䜛䚹ĺศᏊ䛾ಀᩘỴᐃ
ึᮇ್ᐃ⌮
z 㛫㛵ᩘ
f(t) 䛾ึᮇ್ f(+0) 䜢䝷䝥䝷䝇ኚ䛛䜙ồ䜑䜛᪉ἲ䚹
z
[௬ᐃ1] ึᮇ್ f(+0) 䛜Ꮡᅾ䛩䜛 (ᴟ㝈䛜Ⓨᩓ䛧䛯䜚䛧䛺䛔)䚹
z
[௬ᐃ2] 㛵ᩘ f(t) 䛿ᣦᩘᆺ䚸䛴䜎䜚㐺ᙜ䛺 a, M 䛻ᑐ䛧䛶䚸|f(t)| d Meat䛸䛺䜛䛣䛸䚹
z
g(t) = (f(t) – f(+0))u(t) (u(t) 䛿䝦䞂䜱䝃䜲䝗䛾㝵ẁ㛵ᩘ) 䛜 t = 0 䛷㐃⥆䛷䛒䜛䛣
䛸䛻౫Ꮡ䛧䛶䛔䜛䚹䛣䛾௬ᐃ䜢‶䛯䛥䛺䛔ሙྜ䚸䛯䛸䛘䜀䝕䝹䝍㛵ᩘ䜢ྵ䜐
f(t)
= G
(t) + e–t䛾䜘䛖䛺ሙྜ䚸ୖグ䛾㛵ಀᘧ䛿ᡂ䜚❧䛯䛺䛔䚹
ึᮇ್ᐃ⌮:
ୖグ䛾௬ᐃ䜢‶䛯䛩䛺䜙䜀䚸
)]
(
[
)
(
),
(
lim
)
0
(
sF s F s f t
f
s L
f
o 䛯䛰䛧䚸
f
f
o
f
o
o
o
1
)
2
(
lim
)
(
lim
1
)
(
lim
lim
)
0
(
0
s
s
s
s
sF
e
t
f
s
s
t
t HH G H
᭱⤊್ᐃ⌮
(
㔜せ
)
z 㛫㛵ᩘ
f(t) 䛾᭱⤊್ f(+f) 䜢䝷䝥䝷䝇ኚ䛛䜙ồ䜑䜛᪉ἲ䚹
z
[௬ᐃ1] ௵ព䛾 T > 0 䛻ᑐ䛧䛶䚸f(t) 䛿༊㛫 [0, T] 䛷✚ศྍ⬟䚹
z
[௬ᐃ2] ᭱⤊್ f(+f) 䛜Ꮡᅾ䛩䜛 (ᴟ㝈䛜Ⓨᩓ䛧䛯䜚䚸ື䛜ṧ䛳䛯䜚䛧䛺䛔)䚹
z ᭱⤊್䛜Ⓨᩓ䛩䜛ሙྜ䚸䛯䛸䛘䜀䚸
f(t) = et䛾䜘䛖䛺ሙྜ䚸f(+f) = f 䛷䛒䜛䛜䚸
䛸䛺䜚䚸ୖグ䛾㛵ಀᘧ䛿ᡂ䜚❧䛯䛺䛔䚹
᭱⤊್ᐃ⌮:
ୖグ䛾௬ᐃ䜢‶䛯䛩䛺䜙䜀䚸
)]
(
[
)
(
),
(
lim
)
(
0
sF s F s f t
f
s L
f
o 䛯䛰䛧䚸
0
1
lim
)
(
lim
0
0 o
o
s
s
s
sF
s
s
䝷䝥䝷䝇ኚ䛷ᚤศ᪉⛬ᘧ䜢ゎ䛟
z ᚤศ䛾බᘧ 䜢䛳䛶䚸ᚤศ᪉⛬ᘧ䜢ゎ䛣䛖䚹
z ᚤศ䛾බᘧ䜢⧞䜚㏉䛧㐺⏝䛩䜛䛸䚸
z ⥺ᙧᚤศ᪉⛬ᘧ:
z ึᮇ᮲௳:
)
0
(
]
[
»¼
º
«¬
ª
s f f
dt
df
L
L
)
0
(
)
0
(
)
0
(
]
[ 2 1
1
1
»
¼
º
«
¬
ª
f
s
dt
df
s
dt
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Z(s) 䛻䛴䛔䛶ゎ䛔䛶
㏫䝷䝥䝷䝇ኚ
Ļ
z(t)䛜ồ䜎䜛
䝷䝥䝷䝇ኚ
