㐣Ώ⌧㇟
ᩓൢ‫ׅ‬ែ
• 䝇䜲䝑䝏䜢ධ䜜䛯┤ᚋ䛾㟁ᅽ䠈㟁ὶ䛾ኚ໬
• ᩘᏛⓗ䛻䛿ᚤศ᪉⛬ᘧ䛷グ㏙
• 䝷䝥䝷䝇ኚ᥮ἲ䜢⏝䛔䜛䛸ᐜ᫆䛻ゎ䛡䜛
– 䝦䝡䝃䜲䝗(1850-1925)䛜⊂Ꮫ䛷㛤Ⓨ䛧䛯₇⟬Ꮚἲ䜢
䝤䝻䝮䜴䜱䝑䝏䛜1916ᖺ䛻䝷䝥䝷䝇(1749-1827)ኚ᥮
䜢⏝䛔䛶ᩘᏛⓗ䛻᏶ᡂ䛥䛫䛯䜒䛾
– 䝣䞊䝸䜶䠄1768-1830䠅ኚ᥮ἲ䛾ᣑᙇ
ᇹᵓᇘᴾ‫ׅ‬ែእ‫܇‬ỉࣖሉᴾ
• 㟁Ẽᅇ㊰䠄㐣Ώ⌧㇟ㄽ䠅䠈⮬ືไᚚ䛾୺㢟
5-2 CR┤ิᅇ㊰
඘㟁㐣⛬䛾ゎ
(౛㢟5-1)
(౛㢟5-2~5)
5-1 ᢬ᢠᅇ㊰
᢬ᢠ䜢ὶ䜜䜛㟁ὶ䛾▐᫬್ i 䛿᢬ᢠ୧➃䛾㟁ᅽ▐᫬್ vR 䛻ẚ౛䠊
R
dq 1
+ q=E
dt C
(5-6)
ᚤศ=0 䛾䛸䛝䛾ᐃᖖゎ
୍⯡ゎ䛿
5-2 CR┤ิᅇ㊰
5-2-1 ᚤศ᪉⛬ᘧ䛸ゎἲ㻌㻌㻌䞉඘㟁㐣⛬
q = CE + k × e
䝁䞁䝕䞁䝃䛾▐᫬㟁Ⲵ䜢 q 䛸䛩䜜䜀
dq
i=
dt
q
C
dq 1
v R +vC = R
+ q=E
dt C
䜎䛯䠈㻌v㻌 R㻌 =
㻌㻌Ri
㻌䠈㻌v㻌 C㻌㻌=㻌㻌㻌䜘䜚
(5-6)
-
t
CR
䠍㝵䛾⥺ᙧᚤศ᪉⛬ᘧ
௵ពᐃᩘ䜢ྵ䜎䛺䛔ゎ
≉ゎ㻌䠇㻌⿵㛵ᩘ
ྠḟ᪉⛬ᘧ䛾୍⯡ゎ
䛣䛣䛻䠈k 䛿 t = 0 䛻䛚䛡䜛ึᮇ᮲௳䛷Ỵ䜎䜛௵ពᐃᩘ䠊
t = 0 䛷 q = 0 䛸䛩䜜䜀䠈k = -CE䠊䜘䛳䛶
t ö
æ
ç
÷
CR
q = CE 1 - e
ç
÷
è
ø
㟁ὶ i 䛸䠈䝁䞁䝕䞁䝃䛾㟁ᅽ vC 䜢ồ䜑䜛䛸
t
æ
t
㐣Ώゎ㻌
ö
q
dq E - CR
v C = = E ç1 - e CR ÷
i=
= e
,
ç
÷
C
(5-18)
dt R
è
ø
(5-22)
ᨺ㟁㐣⛬
᫬ᐃᩘ䠄䛨䛶䛔䛩䛖䠅
t
t
E E i = e CR = e t
R
R
t
(5-20)
(౛㢟 5-6)
v R +vC = R
= CR㻌䠖㻌᫬ᐃᩘ㻌
㻌㻌 ᣦᩘ㛵ᩘⓗኚ໬䛜㻌
㻌㻌 ึᮇ್䛾 1/e 䛻䛺䜛᫬㛫㻌
37 %
୍⯡ゎ䠖 q =
dq 1
+ q=0
dt C
(5-23)
t
CR
ke
t = 0 䛷䛾䝁䞁䝕䞁䝃䛾㟁Ⲵ䛿 CE 䛷䛒䜛䛛䜙 q = CEe
t
q
v C = = Ee CR
C
5-2-1 CR┤ิᅇ㊰
䛻䛚䛡䜛▴ᙧἼᛂ⟅
౛㢟 5-7
᫬ᐃᩘ㻌t
CR << T 2
౛㢟 5-8
CR << T 2
t
(5-26)
dq
E i=
= - e CR
dt
R
-
t
CR
(5-24)
(5-25)
QCUS䛻䜘䜛
䝅䝭䝳䝺䞊䝅䝵䞁
ᐃᩘಀᩘ n 㝵⥺ᙧ
ᚤศ᪉⛬ᘧ䛾ゎᵓ㐀
a0
dny
dt n
+ a1
d n -1 y
dt n -1
+ L + an -1
dy
+ an y = f (t )
dt
a0, a1, ... , an㻦 ᐃᩘ
୍⯡ゎ㻌䛿䈈㻌
㟁ᅽ
㻌
㻌㻌㻌ྑ㎶ = 0 䛸䛧䛯ྠḟ᪉⛬ᘧ䛾୍⯡ゎ㻌
㻌㻌㻌㻌㻌㻌㻌㻌㻔 n ಶ䛾௵ពᐃᩘ䜢ྵ䜐㻕㻌䞉䞉䞉㻌 ⿵㛵ᩘ㻌
㻌㻌㻌ྑ㎶ = f(t) 䜎䛷⪃䛘䛯᪉⛬ᘧ䛾㻌
㻌㻌㻌㻌㻌㻌㻌㻌 ௵ពᐃᩘ䜢ྵ䜎䛺䛔ゎ㻌㻌㻌䞉䞉䞉㻌 ≉㻌ゎ㻌
㻌
㻌㻌䛾࿴䛷⾲䛥䜜䜛䠊㻌
㟁ὶ
5-3 LR┤ิᅇ㊰
㐣Ώゎ䛸ᐃᖖゎ
䕿㻌᫬㛫䜢⊂❧ኚᩘ䛸䛩䜛⥺ᙧᚤศ᪉⛬ᘧ䛷䛿䈈㻌
㻌
㻌㻌㻌㻌⿵㛵ᩘ䠖㻌㐣Ώ⌧㇟䜢⾲䛩ゎ䠄㐣Ώゎ䠅㻌
㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌n ಶ䛾௵ពᐃᩘ䛾ලయⓗ䛺್䛿 n ಶ䛾㻌
㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌ึᮇ᮲௳䛷Ỵ䜎䜛䠊㻌
㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌㻌Ᏻᐃ䛺ᅇ㊰䛷䛿 t → ∞ 䛷㻌㻜㻌䛸䛺䜛䠊㻌
㻌
ᚤศ䛾㡯
㻌㻌㻌㻌≉㻌ゎ䠖㻌ᐃᖖ≧ែ䜢⾲䛩ゎ䠄ᐃᖖゎ䠅㻌
䜢䝊䝻䛻
㻌
㻌㻌 ୍⯡ゎ㻌䠙㻌⿵㛵ᩘ㻌䠇㻌≉㻌ゎ㻌
(౛㢟5-9䡚11)
di
+ Ri = E
dt
t
R
- tö
E æç
E æç
÷
t
L
=
i=
1- e
1- e
÷ Rç
Rç
è
ø
è
v L +v R = L
L
t = 䠖㻌᫬ᐃᩘ㻌
R
(5-27)
ö
÷
÷
ø
(5-28)
QCUS䛻䜘䜛
䝅䝭䝳䝺䞊䝅䝵䞁
㟁ὶ
㟁ᅽ
⿵㛵ᩘ
5-4 LCR┤ิᅇ㊰
v R + v L + v C = Ri + L
i=
dq
,
dt
dq
d 2q 1
R
+ L 2 + q = E 䜘䜚
dt
C
dt
di d 2 q
=
dt dt 2
di q
+ =E
dt C
䜢౑䛳䛶ᩚ⌮
d 2 q R dq 1
E
+
+
q=
2
L dt LC
L
dt
䛣䛾ᐃᩘಀᩘ䛾㻞㝵⥺ᙧᚤศ᪉⛬ᘧ䛾ゎ䛿㻌
d 2q
dt 2
R dq
L dt
(5-32)
1
LC
ྠḟ᪉⛬ᘧ㻌㻌㻌㻌㻌 +㻌㻌㻌㻌㻌㻌+㻌㻌㻌㻌q㻌 =㻌 0㻌㻌䛾ゎ䠄⿵㛵ᩘ䠅䛸
E
d 2 q R dq 1
q
+
+
=
䛾≉ゎ䛾࿴䛷୚䛘䜙䜜䜛䠊
L
dt 2 L dt LC
d 2 q R dq 1
+
+
q=0
dt 2 L dt LC
R
1
m2 + m +
=0
L
LC
m=(5-33)
䛻ᑐ䛩䜛ḟ䛾≉ᛶ᪉⛬ᘧ
䛾㻞᰿䛻䜘䜚䠈Ỵᐃ䛥䜜䜛䠊㻌
L
1 æç
R ± R2 - 4
ç
C
2L è
ö
÷
÷
ø
qC = c1em1t + c2e m2t
⿵㛵ᩘ 䛿䠈䠄㼕䠅㻘䠄㼕㼕䠅㻘䠄㼕㼕㼕䠅䛻ᑐᛂ䛧䛶㻟✀㢮㻌
䠄i䠅 m 䛜2ᐇ᰿㻌㻌䠖㻌 R2 䠚 4L/C
䠄ii䠅 m 䛜㔜᰿㻌㻌㻌䠖㻌 R2 䠙 4L/C
䠄iii䠅 m 䛜2」⣲᰿䠖㻌 R2 䠘 4L/C
≉㻌ゎ㻌
䛣䛾ሙྜ䛾≉ゎ䛿
q p = CE
5-4-2 R2 = 4L/C 䛾
ሙྜ䠄⮫⏺ไື䠅
5-4-1 R2 > 4L/C 䛾
ሙྜ䠄㐣ไື䠅
≉ᛶ᪉⛬ᘧ䛿㈇䛾┦␗䛺䜛2ᐇ᰿ m1 = -p1, m2 = -p2 䜢
䜒䛴䠊
p1 =
L
1 æç
R - R2 - 4
ç
C
2L è
æ
ö
÷, p2 = 1 ç R + R 2 - 4 L
÷
C
2 L çè
ø
ö
÷
÷
ø
୍⯡ゎ䛿 ⿵㛵ᩘ + ≉ ゎ䜘䜚
q = c1e - p1t + c2e - p2t + CE
i=
c1䠈c2䠖 ௵ពᐃᩘ
dq
= -c1 p1e - p1t - c2 p2 e - p2t
dt
q = (c1t + c2 )e -a t + CE
i=
dq
= -(c1t + c2 )ae -a t + c1e -a t
dt
௵ពᐃᩘ c1䠈c2 䛾ලయ್䛿㻌
t = 0 䛻䛚䛡䜛 q䠈dq/dt 䛾್㻌
䛛䜙Ỵ䜑䜙䜜䜛䠊㻌
(5-36)
௵ពᐃᩘ c1䠈c2 䛾ලయ್䛿
t = 0 䛻䛚䛡䜛 q䠈dq/dt 䛾್
䛛䜙Ỵ䜑䜙䜜䜛䠊
R
≉ᛶ᪉⛬ᘧ䛿㔜᰿㻌 -㻌 a
㻌㻌=㻌-㻌㻌㻌䜢䜒䛴䠊㻌
2L
୍⯡ゎ䛿 ⿵㛵ᩘ + ≉ ゎ䜘䜚
c1䠈c2䠖 ௵ពᐃᩘ
(5-40)
q 䛸㻌i 䛾≉ᛶ䛿䠈㐣
ไື䛾ሙྜ䛸㢮ఝ㻌
R2㻌䠚㻌4L㻛C㻌䛾ሙྜ䛾㟁Ⲵ䛸㟁ὶ㻌
5-4-3 R2 < 4L/C 䛾
ሙྜ䠄୙㊊ไື䠅
≉ᛶ᪉⛬ᘧ䛿」⣲᰿ -a + jb, -a - jb 䜢䜒䛴䠊㻌
R
1
R2
a=
, b=
LC 4 L2
2L
୍⯡ゎ䛿㻌⿵㛵ᩘ + ≉ ゎ㻌䜘䜚㻌
a1䠈a2䠖 ௵ពᐃᩘ
q = e -a t (a1 cos bt + a2 sin bt ) + CE
dq
= e -a t [(- aa1 + ba2 )cos bt - (aa2 + ba1 )sin bt ]
i=
(5-44)
dt
௵ពᐃᩘ a1䠈a2 䛾ලయ್䛿
t = 0 䛻䛚䛡䜛 q䠈dq/dt 䛾್
䛛䜙Ỵ䜑䜙䜜䜛䠊
QCUS䛻䜘䜛
䝅䝭䝳䝺䞊䝅䝵䞁
(3 ´10 )
>4
(2 ´10 )
=4
(1´10 )
<4
3 2
3 2
3 2
R2㻌䠘㻌4L㻛C㻌䛾ሙྜ䛾㟁Ⲵ䛸㟁ὶ㻌
5 ´10 -3
0.1´10 -6
20 ´10 -3
0.02 ´10 -6
100 ´10 -3
0.01´10 -6
R2 > 4L/C 䛾ሙྜ
䠄㐣ไື䠅
R2 = 4L/C 䛾ሙྜ
䠄⮫⏺ไື䠅
R = 3 kW, L = 5 mH, C = 0.1 mF
㟁ᅽ
R = 2 kW, L = 20 mH, C = 0.02 mF
㟁ᅽ
㟁ὶ
㟁ὶ
R2 < 4L/C 䛾ሙྜ
䠄୙㊊ไື䠅
R = 1 kW, L = 100 mH, C = 0.01 mF
ᶵᲔ䛸䛾䜰䝘䝻䝆䞊
ຊ-㟁ᅽἲ
• 䝞䝛 k
• 䝎䞁䝟䞊 r
• ㉁㔞 m
m
㟁ᅽ
㟁ὶ
L
d 2x
dt 2
d 2q
dt
2
+r
я㻌 1/C
я㻌 R
я㻌 L
dx
+ kx = F (t )
dt
+R
dq 1
+ q=E
dt C
k
m
F(t)
r
䝷䝥䝷䝇ኚ᥮䛻䜘䜛
ゎἲ
• CR┤ิᅇ㊰
• LR┤ิᅇ㊰
1
q=E
䝇䝔䝑䝥㛵ᩘ
C
1
Þ Ri (t ) +
i (t )dt = Eu (t )
C
1
1 I ( s) E
E
E/s
=
\ I (s) =
=
RI ( s ) +
1
R s+ 1
s
C s
R+
CR
sC
u(t) 䛿༢఩
Ri +
ò
t
E i (t ) = e CR u (t )
R
䝷䝥䝷䝇ኚ᥮䛻䜘䜛
ゎἲ
㏫䝷䝥䝷䝇ኚ᥮
di
+ Ri = E
dt
d
Þ L i (t ) + Ri (t ) = Eu (t )
dt
L
æ
1
E
E çç 1
E
LsI ( s ) + RI ( s ) =
\ I (s) =
=
s ( sL + R ) R ç s s + R
s
ç
L
è
t
ö
E æç
L
u (t ) - e / R u (t ) ÷
i (t ) =
÷
Rç
ø
è
➨5❶ ₇⩦ၥ㢟 2
䝁䞁䝕䞁䝃䛜㟁Ⲵ Q0 (< CE) 䜢䜒䛳䛶䛔䜛䛸䛝䠈䝇䜲䝑䝏㻿䜢㛢䛨
䛶䛛䜙䛾䝁䞁䝕䞁䝃䛾㟁Ⲵ q 䛸ᅇ㊰䜢ὶ䜜䜛㟁ὶ i 䜢ồ䜑䜘䠊㻌
v R +vC = R
dq 1
+ q=E
dt C
䛣䛾ᚤศ᪉⛬ᘧ䛾୍⯡ゎ䛿
q = CE + k × e
k = Q0 - CE
t = 0 䛷 q = Q0 䜘䜚䠈
\ q = CE + (Q0 - CE ) e
i=
dq E - Q0 / C
e
=
dt
R
-
-
t
CR
t
CR
ö
÷
÷
÷
÷
ø
-
t
CR
t ö
t
æ
= CEç1 - e CR ÷ + Q0 e CR
ç
÷
è
ø
➨5❶ ₇⩦ၥ㢟 3
䝇䜲䝑䝏㻿䜢㛢䛨䛶䝁䞁䝕䞁䝃䜢᏶඲䛻඘㟁䛧䛶䛚䛝䠈ḟ䛻㻿䜢㛤䛟䠊㻌
㻿䜢㛤䛔䛶䛛䜙䛾⤒㐣᫬㛫 t 䛸䝁䞁䝕䞁䝃୧➃㟁ᅽ vC 䛾㛵ಀ䜢ồ㻌
䜑䜘䠊䛯䛰䛧䠈rS << R 䛸䛩䜛䠊㻌
䝇䜲䝑䝏S䜢㛢䛨䛶ᐃᖖ≧ែ䛻㐩䛧䛯䛸䛝䠈
䡎S << R 䜘䜚 vC Ҹ E䠊
䜘䛳䛶䠈䝁䞁䝕䞁䝃䛻⵳✚䛧䛶䛔䜛㟁Ⲵ䛿 CE䠊
䝇䜲䝑䝏䜢㛤䛔䛯䛒䛸䛿
q
v C = = Ri
C
䝁䞁䝕䞁䝃䛾㟁Ⲵ q 䛾᫬㛫ኚ໬䛜ὶ䜜ฟ䛩㟁ὶ䛻䛺䜛䛛䜙
dq
= -i
dt
\
q
dq
= -R
C
dt
t = 0 䛷 q = CE 䜢⪃៖䛩䜛䛸䚸ゎ䛿
Қ
v C = Ee
-
t
CR
э
dq 1
+
q=0
dt t CR
q = CEe
-
CR
୧㎶䛾⮬↛ᑐᩘ䜢䛸䛳䛶䠈ᩚ⌮䛩䜛䛸
t = CR ln
E
vC