Projet 1

Régimes(transitoires(d’un(circuit(RC(série(
( (
!"#$%&#'"#("#)*"+,"*#)*-."&#/0&1-2#"3&#'"#+-'45,3"*#2%+4*,6%"+"2&#57#(17*8"#"&#57#'4(17*8"#'9%2#
(-2'"237&"%*# '723# %2# (,*(%,&# RC#34*,":# 6%"# 2-%3# 7;-23# 4&%',4# '723# 5"# (17),&*"# <=# '%# (-%*3# '"#
)103,6%">#
1. #Etude#analytique##
#?2#4&%',"#57#*4)-23"#'9%2#(,*(%,&#RC#34*,"#@#%2# (*42"7%#'"#&"23,-2:#"2#4&7$5,3372&#@#&-%&#,23&72&#57#
&"23,-2#𝑢!𝑡#7%A#$-*2"3#'%#(-2'"237&"%*>#
#
#
1.1. #Etude#de#la#charge#du#condensateur#
B723#("&&"#)7*&,":#-2#4&%',"#57#*4)-23"#'%#(,*(%,&#RC#@#%2#4(1"5-2#'"#&"23,-2:#(9"3&C@C',*"#)-%*#D#
0≤𝑡≤
𝑇
2#
#
E>EF#B4&"*+,2"*#5"3#(-2',&,-23#,2,&,75"3#'%#(,*(%,&>#
E>GF# B4&"*+,2"*# 5946%7&,-2# ',HH4*"2&,"55"# *48,3372&# 594;-5%&,-2# '"# 57# &"23,-2# 𝑢!𝑡#7%A# $-*2"3# '%#
(-2'"237&"%*#"&#"2#'4'%,*"#57#(-23&72&"#'"#&"+)3!𝜏#(7*7(&4*,3&,6%"#'9%2#(,*(%,&#𝑅𝐶#34*,">#
E>=F#B4&"*+,2"*#59"A)*"33,-2#'"#57#&"23,-2#𝑢!𝑡#7%A#$-*2"3#'%#(-2'"237&"%*>#
E>IF#J%"55"#"3&#57#;75"%*#'"#𝑢!𝑡#)-%*#𝑡=𝜏!K#𝑡=5𝜏!K#L-++"2&"*#5"3#*43%5&7&3>#
E>MF# N*7("*# 59755%*"# '"# 57# &"23,-2# 𝑢!𝑡#7%A# $-*2"3# '%# (-2'"237&"%*#)-%*# 0≤𝑡≤𝑇/2># ?2# H"*7#
7))7*7O&*"#57#(-23&72&"#'"#&"+)3#𝜏#3%*#5"#8*7)1,6%">#
1.2. #Etude#de#la#décharge#du#condensateur#
B723#("&&"#)7*&,":#-2#4&%',"#57#*4)-23"#'%#(,*(%,&#RC#"2#*48,+"#5,$*":#(9"3&C@C',*"#)-%*#D#
𝑇
2≤𝑡≤𝑇#
E>PF#B4&"*+,2"*#5"3#(-2',&,-23#,2,&,75"3#'%#(,*(%,&>#
E>QF# B4&"*+,2"*# 5946%7&,-2# ',HH4*"2&,"55"# *48,3372&# 594;-5%&,-2# '"# 57# &"23,-2# 𝑢!𝑡#7%A# $-*2"3# '%#
(-2'"237&"%*>#
E>RF#B4&"*+,2"*#59"A)*"33,-2#'"#57#&"23,-2#𝑢!𝑡#7%A#$-*2"3#'%#(-2'"237&"%*#𝑇/2≤𝑡≤𝑇>#
E>SF#N*7("*#59755%*"#'"#57#&"23,-2#𝑢!𝑡#7%A#$-*2"3#'%#(-2'"237&"%*>#?2#H"*7#7))7*7O&*"#57#(-23&72&"#
'"#&"+)3#𝜏#3%*#5"#8*7)1,6%">#

2(((((((((((((Projet(Python(1(:(Régime(transitoire(d’un(circuit(RC#série(
1.3. #Charges#et#décharges#successives##
/-%*#;,3%75,3"*#5"3#(17*8"3#"&#'4(17*8"3#3%(("33,;"3#'%#(-2'"237&"%*:#-2#75,+"2&"#5"#(,*(%,&#RC#7;"(#
%2#TUV#'45,;*72&#%2"#&"23,-2#(*42"7%#'"#H*46%"2("#𝑓>##
E>EWF# X# 6%"55"# (-2',&,-2# 3%*# 5"3# ;75"%*3# '"# 𝑓,𝑅!et!𝐶#;,3%75,3"*7C&C-2# 5"3# (17*8"3# "&# '4(17*8"3#
complètes#'%#(-2'"237&"%*#K#
E>EEF#N*7("*#3%*# %2#+Y+"# 8*7)1,6%":# 59755%*"# '"# 57# &"23,-2#(*42"7%# 𝑒(𝑡)#'"# H*46%"2("#𝑓#"&# '"# 57#
&"23,-2#𝑢!𝑡#7%A#$-*2"3#'%#(-2'"237&"%*#5-*36%"#57#(-2',&,-2#)*4(4'"2&"#"3&#;4*,H,4">#
2. Modélisation#numérique#
?2#3-%17,&"#,(,#+-'45,3"*#2%+4*,6%"+"2&#5"3#(17*8"3#"&#'4(17*8"3#3%(("33,;"3#'9%2#(,*(%,&#𝑅𝐶#34*,":#
75,+"2&4#)7*#%2"#&"23,-2#(*42"7%#'"#H*46%"2("#𝑓>#
/-%*#5"3#+-'45,37&,-23:#-2#)*"2'*7#5"3#;75"%*3#2%+4*,6%"3#3%,;72&"3#D#
• #𝑅=1!kΩ#
• 𝐶=100!n𝐹#
• 𝑓=500!Hz#
• 𝐸=5!V#
G>EF#B723#594',&"%*:#,+)-*&"*#5"3#+-'%5"3#3(,"2&,H,6%"3#24("337,*"3#)-%*#5"#)*-$5Z+"#D#
C#2%+)0#
C#+7&)5-&5,$>)0)5-&#
C#3(,)0>,2&"8*7&"#[)-%*#57#*43-5%&,-2#'"3#46%7&,-23#',HH4*"2&,"55"3#'%#)*"+,"*#-*'*"F>#
G>GF#B4(57*"*#"&#7HH"(&"*#5"3#;7*,7$5"3#3%,;72&"3#D#
C#*43,3&72("#[2-&4"#\F#"&#(7)7(,&4#[2-&4"#LF#'%#(,*(%,&#
C#H*46%"2("#[2-&4"#HF#"&#7+)5,&%'"#[2-&4"#<F#'"#57#&"23,-2#(*42"7%#
C#(-2',&,-2#,2,&,75"#)-%*#57#&"23,-2#7%A#$-*2"3#'%#(-2'"237&"%*#"2#(17*8"#[2-&4"#%E]WF#
C#(-2',&,-2#,2,&,75"#)-%*#57#&"23,-2#7%A#$-*2"3#'%#(-2'"237&"%*#"2#'4(17*8"#[2-&4"#%G]WF#
2.1. #Modélisation#numérique#de#la#charge#du#condensateur#
G>=F#B4H,2,*#%2#;"(&"%*#[2-&4#&EF#(-2&"272&#EWWW#)-,2&3#"&#'-2&#5"3#;75"%*3#3-2&#(-+)*,3"3#"2&*"#
𝑇=0#"&#𝑡=𝑇/2>##
G>IF#B4H,2,*#%2"#H-2(&,-2#[7))"54"#(17*8"F#'4H,2,3372&#5946%7&,-2#',HH4*"2&,"55"#*48,3372&#57#(17*8"#
'%#(-2'"237&"%*>#
G>MF# \43-%'*"# 5946%7&,-2# ',HH4*"2&,"55"# *48,3372&#57# (17*8"# '%# (-2'"237&"%*>#?2# 2-&"*7# %E# 5"#
;"(&"%*#*43%5&72&>#
G>PF#N*7("*#594;-5%&,-2#'"#57#&"23,-2#7%A#$-*2"3#'%#(-2'"237&"%*#7%#(-%*3#'"#57#(17*8">#
2.2. #Modélisation#numérique#de#la#décharge#du#condensateur#
G>QF# X# 57# 3%,&"# '%# )*-8*7++"# )*4(4'"2&:#'4H,2,*#%2# 2-%;"7%# ;"(&"%*# [2-&4# &GF# (-2&"272&# EWWW#
)-,2&3#"&#'-2&#5"3#;75"%*3#3-2&#(-+)*,3"3#"2&*"#𝑇=𝑇/2#"&#𝑡=𝑇>##
G>RF# B4H,2,*# %2"# H-2(&,-2# [7))"54"# '"(17*8"F# '4H,2,3372&# 5946%7&,-2# ',HH4*"2&,"55"# *48,3372&# 57#
'4(17*8"#'%#(-2'"237&"%*>#
G>SF# \43-%'*"# 5946%7&,-2# ',HH4*"2&,"55"# *48,3372&#57# '4(17*8"# '%# (-2'"237&"%*>#?2# 2-&"*7# %G# 5"#
;"(&"%*#*43%5&72&>#
G>EWF#N*7("*#594;-5%&,-2#'"#57#&"23,-2#7%A#$-*2"3#'%#(-2'"237&"%*#7%#(-%*3#'"#57#'4(17*8">#

Projet(Python(1(:(Régime(transitoire(d’un(circuit(RC#série((((((((((((((3(
2.3. #Charge#et#décharge#successives#
G>EEF# <2# %&,5,372&# 57# fonction- np.concatenate#['-2&# -2# (1"*(1"*7# 57# H-2(&,-2# "&# 5"# )*,2(,)"#
'9%&,5,37&,-2#3%*#5"#3,&"#'"#^%+)0F:#'4H,2,*#%2#2-%;"7%#;"(&"%*#[2-&4#&F#6%,#(-2(7&Z2"#5"3#;"(&"%*3#
&E#"&#&G>#
G>EGF#<2#%&,5,372&#57#fonction-np.concatenate:#'4H,2,*#%2#2-%;"7%#;"(&"%*#[2-&4#%F#6%,#(-2(7&Z2"#
5"3#;"(&"%*3#%E#"&#%G>#
G>E=F# N*7("*# 594;-5%&,-2# '"# 57# &"23,-2# 7%A# $-*2"3# '%# (-2'"237&"%*# 7%# (-%*3# '"# 57# 346%"2("#
(17*8"C'4(17*8">#
G>EIF# J%"3&,-2# $-2%3#D# 3%*# 5"# 8*7)1,6%"# )*4(4'"2&:# 3%)"*)-3"*# @# 57# &"23,-2# 7%A# $-*2"3# '%#
(-2'"237&"%*:#57#&"23,-2#'975,+"2&7&,-2#(*42"7%>#
#