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MINISTÈRE DE L’ÉDUCATION

NATIONALE ET DE LA JEUNESSE

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‒ 2 ‒

j j² = 1

u(t) = U₀cos(!t+') u=U₀e^j!t =U₀e^j(!t+')

A^⇤ A

e

u

±q x

lame de quartz

´electrodes m´etalliques

fils de connexion e

(a)

e+x u

q q

(b)

EAE PHP 1

‒ 1 ‒ Tournez la page S.V.P. B

(EC) ( q

C +↵x =u

mx¨+δx˙ +kx+↵q = 0 (EC)

˙ x= dx

dt x¨= d²x

dt² C m δ k ↵

x

(E) v = ˙x

u = 0

k > C↵² (E)

!0 =

rk−C↵²

m Q= m!0

δ . u(t)

u(t) =

⇢ 0 t <0

E t >0

Q > 1/2 v(t)

v(t) =Ae^−t/⌧sin(!₀⁰t)

⌧ !₀⁰ Q !0

A !0 Q1

⌧q

⌧_q Q

Q

!

i

(EC) Z ! k ↵ C

δ m

Z_e

Z Cs Rs Ls Cp C ↵ m δ k

{R_s, C_s, L_s} Cp

(EC) ( q

C +↵x=u

m¨x+δx˙ +kx+↵q = 0 (EC)

˙ x= dx

dt x¨= d²x

dt² C m δ k ↵

x

(E) v = ˙x

u = 0

k > C↵² (E)

!0 =

rk−C↵²

m Q= m!0

δ . u(t)

u(t) =

⇢ 0 t <0

E t >0

Q >1/2 v(t)

v(t) =Ae^−t/⌧sin(!⁰₀t)

⌧ !₀⁰ Q !0

A !0 Q1

⌧q

⌧_q Q

Q

!

i

(EC) Z ! k ↵ C

δ m

Z_e

Z Cs Rs Ls Cp C ↵ m δ k

{Rs, C_s, L_s} Cp

branche dynamique (ou motionnelle, ou s´erie) Rs

Cp

L Ls

Cs

branche parall`ele (ou statique)

Q ⇡ 10⁵ Rs

Rs = 0 Z

Z = 1

j(Cp+Cs)! ⇥ 1−!²/!_s² 1−!²/!_p²

!_s !_p C_p C_s

Ls !s !0

!s !p

! =!_s

Cp ⇡ 10⁻¹² Cs ⇡ 10⁻¹⁴

!_p²−!²_s

!²_s

(!²_p−!²_s)/!_s² !s !p

Rs 6= 0 Q

!s !p Q = 10²

|Z| ! Q= 10⁵

(EC)

‒ 3 ‒ Tournez la page S.V.P.

C0 S

e "0

±q a

~p

O M

O OM �a M

V~p(M) = 1 4⇡"0

~p·−−!OM OM³

P~ P~

σp

⇢_p

⇢p

σp P~

⇢p =−divP~ σp =P~ ·~n

~n

M A

V(M)

σp ⇢p

χ

div ("0"rE) =~ ⇢libre

"r χ

C S e

"r

C S = 25 mm² e = 0,33 mm "r = 4,5

"0 = 8,85·10⁻¹²F·m⁻¹

C0 S

e "0

±q a

~p

O M

O OM �a M

V~p(M) = 1 4⇡"0

~p·−−!OM OM³

P~ P~

σp

⇢_p

⇢p

σp P~

⇢p =−divP~ σp =P~ ·~n

~n

M A

V(M)

σp ⇢p

χ

div ("0"rE) =~ ⇢libre

"r χ

C S e

"r

C S = 25 mm² e = 0,33 mm "r = 4,5

"0 = 8,85·10⁻¹²F·m⁻¹

SiO2

+ Si

F~

+ +

(a) (b)

O

(c) +

+ +

+

+ +

F~ =~0

F~

P~p

e S

±σp

σp >0

σ D~ D~ = −σ~n D~ ="0E~ +P~

e S F~ = F ~ux

e+x |x| ⌧ e P~

↵ P~ ="0χ ~E+"↵x

e ~ux

"="0"r χ

‒ 5 ‒ Tournez la page S.V.P.

m´etal quartz

σ

~n

x e+x

F~

quartz

( q) (q)

u

e x

E~ x

q

q

C +↵x=u (CEM1)

q

F =kx+↵q (CEM2) (CEM₁) (CEM₂)

q= 0 (CEM2)

k

x m¨x+δx˙ +kx+↵q = 0

m δ

(EC)

P´electrique =Pfrottements+ d

dt(Ecin´etique+E´elastique+Econdensateur+Eem)

m´etal quartz

σ

~n

x e+x

F~

quartz

( q) (q)

u

e x

E~ x

q

q

C +↵x=u (CEM1)

q

F =kx+↵q (CEM2) (CEM₁) (CEM₂)

q= 0 (CEM2)

k

x m¨x+δx˙ +kx+↵q = 0

m δ

(EC)

P´electrique =Pfrottements+ d

dt(Ecin´etique+E´elastique+Econdensateur+Eem)

Eem

R1

R2

R3

+ ie

A

M ve

ve ie AM

R⁰ R₁ R₂ R₃

AM RLC R3

R⁰ R3

R₃

!0 R3c

R3

|Z| RLC

!=!0

‒ 7 ‒ Tournez la page S.V.P.

R

L

C

R1

R2

R3

+ i

RLC

RLC !

! =!₀

R3 =R3c R3

RLC

R3 1,0 k⌦

R1= 3,9 k⌦

R2= 560 ⌦ R3= 1,0 k⌦

+ Z

vs

vs

f₀ = 32768 Q = 22 ·10³

Rs Ls Cs Q

Q = 1 Rs

rLs

Cs

R

L

C

R1

R2

R3

+ i

RLC

RLC !

! =!₀

R3 =R3c R3

RLC

R3 1,0 k⌦

R1= 3,9 k⌦

R2= 560 ⌦ R3= 1,0 k⌦

+ Z

vs

vs

f₀ = 32768 Q = 22·10³

Rs Ls Cs Q

Q = 1 Rs

rLs

Cs

f0 = 32768 32768 = 2¹⁵

f₀

D K K

CLOCK K

f₀ = 32768

K K

K

K K = D

CLOCK

fr´equencef

t K

D=K 0 1 0 1 0 1

t t

‒ 9 ‒ Tournez la page S.V.P.

P

D

✓R

P

X Y x y

h(x, y) h(x, y) = C

λf Z +1

−1

Z +1

−1

(X, Y)e^{−2j⇡(xXf+yYf)}dXdY,

C λ

f

(X, Y) P

i(X, Y)

i(0,0) = 0

P

D

✓R

P

X Y x y

h(x, y) h(x, y) = C

λf Z +1

−1

Z +1

−1

(X, Y)e^{−2j⇡(xXf+yYf)}dXdY,

C λ

f

(X, Y) P

i(X, Y)

i(0,0) = 0

(X, Y)

⌧(X, Y) 0

h(x, y)

u=−X/λf v =−Y /λf h(x, y)

u v

H(x, y)

H(x, y) = |h(x, y)|². H(u, v)ˆ H(u, v) =ˆ F[H](u, v)

u v x y

⌧(X, Y)

H(u, v) =ˆ K Z ₊₁

−1

Z ₊₁

−1

⌧(X, Y)⌧(X−λf u, Y −λf v)dXdY, K =|C 0λf|²

D ⌧(X, Y)

⌧(X, Y) =

(1 p

X²+Y² < D/2,

0 .

H(u,ˆ 0) |u| > uc H(u,ˆ 0)

uc D λ f

Hˆn(u,0) = ˆH(u,0)/K

Hˆn(u,0) = 2

⇡ 0

@arccos|u| uc − |u|

uc

s 1−

✓u uc

◆21 A Hˆn(0,0)

Hˆn(u,0) u/uc

‒ 11 ‒ Tournez la page S.V.P.

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u/uc

0.0 0.2 0.4 0.6 0.8 1.0

ˆHn(u,0)

∆L

e = 0,3

∆T = 1

∆L e T ∆T n T

(n 1)

⇢

λ =

0,5µ T = 300

n = 1,0002678 ∆L

n ∆L

λ/4 λ= 0,5µ

H(x, y, t)

H(u, v, t) =ˆ K Z ₊₁

−1

Z ₊₁

−1

⌧(X, Y)⌧(X λf u, Y λf v)A(X, Y, t)A^⇤(X λf u, Y λf v, t)dXdY,

⌧(X, Y) A(X, Y, t)

A^⇤(X λf u, Y λf v, t)

A(X, Y, t) =Am(X, Y, t) exp[jφ(X, Y, t)] Am φ(X, Y, t) A(X, Y, t)

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u/uc

0.0 0.2 0.4 0.6 0.8 1.0

ˆHn(u,0)

∆L

e = 0,3

∆T = 1

∆L e T ∆T n T

(n 1)

⇢

λ =

0,5µ T = 300

n= 1,0002678 ∆L

n ∆L

λ/4 λ= 0,5µ

H(x, y, t)

H(u, v, t) =ˆ K Z ₊₁

−1

Z ₊₁

−1

⌧(X, Y)⌧(X λf u, Y λf v)A(X, Y, t)A^⇤(X λf u, Y λf v, t)dXdY,

⌧(X, Y) A(X, Y, t)

A^⇤(X λf u, Y λf v, t)

A(X, Y, t) =Am(X, Y, t) exp[jφ(X, Y, t)] Am φ(X, Y, t) A(X, Y, t)

Am

Am(X, Y, t)

Am

H(u, v, t)ˆ ⌧(X, Y) φ

✓R

D r₀

D r0

D = 8,0

λ = 0,50µm r0 = 10 cm

φc(X, Y, t) A(X, Y, t) A_c(X, Y, t) = A(X, Y, t) exp [ jφ_c(X, Y, t)]

φc

L

A A⁰

L A A⁰ A

L A F

A

‒ 13 ‒ Tournez la page S.V.P.

↵

⇡/2 < ↵ < ⇡/2

↵

↵

⇡/2 < ↵ < ⇡/2

↵

front de l’onde incidente

une microlentille un capteur CCD f⁰

(a)

matrice de microlentilles

matrice CCD (b)

D= 2 N = 200

r0

N D r0

r0

‒ 15 ‒ Tournez la page S.V.P.

j j² =−1

A^⇤ A

f F[f]

F[f](u, v) = Z +1

−1

Z +1

−1

f(x, y)e−2j⇡(ux+vy)dxdy

f(x, y) =F⁻¹[F[f]](x, y) = Z +1

−1

Z +1

−1 F[f](u, v)e^2j⇡(ux+vy)dudv

F[f(−x,−y)] =F⁻¹[f(x, y)] (1) F[f(ax, by)](u, v) = 1

|ab|F[f(x, y)]⇣u a,v

b

⌘ (2) F[f^⇤](u, v) =F^⇤[f](−u,−v) (3)

f⇤g (f⇤g)(x, y) =

Z +1

−1

Z +1

−1

f(x−x⁰, y−y⁰)g(x⁰, y⁰) dx⁰dy⁰

F[f ⇤g] =F[f]⇥ F[g] (4) F[f ⇥g] =F[f]⇤ F[g] (5)

⇥

ZZ

�

S(V)

A~·−!

dS = ZZZ

V

A~dV I

C

A~·−! d` =

ZZ

S(C)

−!A~·−!

dS

−−! ✓ 1 r

◆

=− 1 r²~ur

(U ~A) =U A~+A~· −−!

U U A~