Electronique : régime sinusoïdal forcé (PCSI)

u= 0
r R2= 70 Ω
R3= 2.5 kΩ
R1
u e
R1
R1= 1.5 kΩ r
L
r R1
R1= 1 kΩR2= 42 kΩ
R3= 100 Ω C= 5 nF L r
UR1=R1
R1+Zeq e1 Zeq =1 R2+jCω UR1=R1
R1+R2
1+jR2Cω
e=R1+jR1R2Cω
R1+R2+jR1R2Cω
ULr =r+jLω
R3+r+jLω e
r=R1R3
R2L=R1R3C

E(t) = E0sin ωt R0
Z=R+iLω U(t)
I(t)
PZ
Rm
Eeff = 220 V R0= 10 Ω ω
2π= 50 Hz
L= 0.1H
R0
m
I(t) = I0sin(ωt +ϕI)
(t) = (t)Z+ (t)R0⇔E0ei(ωt+π/2) =I0ei(ωt+π/2+ϕI)(R0+R+iLω)
I0eiϕI=E0
R+R0+iLω =E0R+R0−iωL
(R+R0)2+L2ω2
I0=E0
√(R+R0)2+L2ω2Ieff =I0
√2cos ϕI=
R+R0
√(R+R0)2+L2ω2sin ϕI=−ωL √(R+R0)2+L2ω2
P=hI(t)U(t)i
=1
TˆT
0
I0sin(ωt +ϕI)U0sin(ωt +ϕU)
=I0U0
1
2Tˆ(cos (ϕU−ϕI)−cos (2ωt +ϕU+ϕI))
=I0U0
2cos (ϕU−ϕI)
U0eiϕU=I0eiϕIU0=I0√R2+L2ω2cos (ϕU−ϕI) = R
√R2+L2ω2sin (ϕU−ϕI) =
Lω
√R2+L2ω2P=RI2
0
2=RE2
ef f
(R+R0)2+L2ω2
L= 0 P=R
(R+R0)2E2
eff dP
dR =(R+R0)2−2R(R+R0)
(R+R0)4= 0 ⇔R+R0−2R= 0 ⇔R=R0
P=E2
ef f
4R0= 1.2 MW
P0=RE2
ef f
(R+R0)2+L2ω2P0
P=(R+R0)2
(R+R0)2+L2ω2E2
eff ≤1dP0
dR =
0⇔1
(R+R0)2+L2ω2−2R(R+R0)
((R+R0)2+L2ω2)2= 0 (R+R0)2+L2ω2−2R(R+R0) = R2+R2
0+
2RR0+L2ω2−2R2−2RR0= 0 R=pL2ω2+R2
0= 33Ω

C L
R
E(t)
E(t)(0t < 0
E0t≥0t > 0
UC
UCE
E(t)(0t < 0
E0cos ωt t ≥0t > 0
UC
UC(t) = U0
Ccos (ωt +ϕC)
UC
UCE
ω U0
C
Z1, Z2Z3Z
0 +∞Gi=|ui|
|ue|, i ∈[[1,3]].
ω→0G1→R1R0
1
R1+R0
1
1
R1R0
1
R1+R0
1
+R0
2+R0 3
G2→R0
2
R1R0
1
R1+R0
1
+R0
2+R0 3
G3→R0
3
R1R0
1
R1+R0
1
+R0
2+R0 3
ω→+∞G1→R1R0
1
R1+R0
1
1
R1R0
1
R1+R0
1
+R2+R0
3
G2→R2
R1R0
1
R1+R0
1
+R2+R0
3
G3→R0
3
R1R0
1
R1+R0
1
+R2+R0
3
ω→ω1G1→R0
1
Z(ω)
ω→ω3G3→R3R0
3
R3+R0
3
1
Z(ω)

ue
ue=U0cos (ωt)
H(jω) = us
ue
us
T
vm=Vmcos (ωmt)
vm→vmvp+vpvpvp=Vpcos (ωpt)ωpωm
1Uei Usi
•H1(jω) = us1
ue1
G
1+ j
Qω ω0−ω0 ω
•Q1
Q1ω0=ωpUs1
Ue2> Us2
r Us2Us2
Us2
τ=RC
Us2
Us3Us2

VNUsUe
UsVNUe
H(jω) = Us
Ue
R=R0HdB(x)x=ω
ω0
ω0=1
RC
E(t) =
+∞
P
n=0
E04
π
1
2n+1 cos (2n+ 1) 2π Tt