Announcements: Today: RL, LC and RLC circuits and some maths

Transcription

1 Announcements: Today: RL, LC and RLC circuits and some maths

2 RL circuit (from last lecture) Kirchhoff loop: First order linear differential equation with general solution: i t = A + Be kt ε A + Be kt R LBke kt = 0 BR LBk = 0 k = R L A= ε/r i t = ε/r + Be R L t (general solution) i 0 = 0 B = ε R i t = ε R ε R e R L t

3 What is the current if I turn off the emf at t 0? A. B. C. D. i t = ε R e R L t i t = ε R e R L (t t 0) i t = ε R e R L (t t0) ε R e R L t i t = ε R e R L (t t0) (1 e R L t 0 )

4 RL circuit i t < t 0 = ε R ε R e R L t i t 0 = ε R ε R e R L t 0 = C t 0

5 RL circuit i t < t 0 = ε R ε R e R L t i t 0 = ε R ε R e R L t 0 = C i t > t 0 = ε R + Be R L t t 0

6 RL circuit i t < t 0 = ε R ε R e R L t i t 0 = ε R ε R e R L t 0 = C t 0 i t > t 0 = ε R + Be R L t = Be R L t 0

7 RL circuit i t < t 0 = ε R ε R e R L t i t 0 = ε R ε R e R L t 0 = C t 0 i t > t 0 = ε R + Be R L t = Be R L t 0 = Ce R L t 0 e R L t

8 circuit-construction-kit-ac_en

9 LC circuit

10 Courtesy of M. Devoret

11 2000 qubits!

12 LC circuit

13 LC circuit

14 LC circuit q C

15 LC circuit Kirchhoff loop: L di dt q C = 0

16 LC circuit Kirchhoff loop: L di dt q C = 0 I = dq dt

17 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt

18 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order (homogenous) linear differential equation

19 Survival guide on differential equations: Step 1: understand the equation and its notation Typical notations: dy dx = y d 2 y dx 2 = y dy dt = yሶ d 2 y dt 2 = yሷ dy dx = y x d 2 y dx 2 = 2 y x 2 dy dt = y t d 2 y dx 2 = 2 y x 2 One variable (x or t) y(x, t) x (partial) V(x, y, z, t) ; y Many variables (x,y,z,t) 2 y(x, t) x 2 ; 2 V(x, y, z, t) z 2 y(x, t) V(x, y, z, t) ; t t (total) (Most typical differential equations use only partial derivatives, e.g. divergence) dv(x, y, z, t) dy

20 Survival guide on differential equations: Step 2: How many variables? y x + 2 y x 2 = 0 y x + y t = 0 One variable (x) => ODE Many variables (x and t) => PDE

21 Survival guide on differential equations: Step 3: Is it linear? y x + 2 y x 2 = 0 y ay + b x + c 2 y + d = f(x) x2 Linear y x + 2 y x 2 + y x 2 = 0 Non-linear

22 Survival guide on differential equations: Step 4: What order? ay + b y x + c 2 y x 2 + d = 0 ay + b y x + c = 0 ay + b y x + c 2 y + d = f(x) x2 ay + b y x + c = f(x) a(x)y + b y x + c 2 y x 2 + d = 0 Second order a(x)y + b y x + c = 0 First order

23 Survival guide on differential equations: Step 5: Is it linear with constant coefficients? ay + b y x + c 2 y x 2 + d = 0 Yes if a,b,c,d are constants ay + b y x + c 2 y + d = f(x) No x2 a(x)y + b y x + c 2 y x 2 + d = 0 No

24 Survival guide on differential equations: Step 5: Is it linear with constant coefficients? ay + b y x + c 2 y x 2 + d = 0 a,b,c,d are constants y x = Second order Ae k 1x + Be k 2x + C or Ae k 1x + Be k 2x if d=0 (homogenous case) Examples: RLC circuit, mechanical oscillators, E&M waves, (All other PDEs or ODEs are usually more complicated) First order y x = A + Be kx or Be kx (homogenous case)

25 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae k 1t + Be k 2t

26 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae k 1t + Be k 2t LAk 1 2 e k 1t + LBk 2 2 e k 2t + 1 C Aek 1t + Be k 2t = 0

27 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae k1t + Be k 2t LAk 2 1 e k1t + LBk 2 2 e k2t + 1 C Aek1t + Be k 2t = 0 (LAk C A)ek1t +(LBk C B)ek2t = 0

28 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae k 1t + Be k 2t LAk 1 2 e k 1t + LBk 2 2 e k 2t + 1 C Aek 1t + Be k 2t = 0 (LAk C A)ek 1t +(LBk C B)ek 2t = 0 = 0 = 0

29 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae k 1t + Be k 2t LAk 1 2 e k 1t + LBk 2 2 e k 2t + 1 C Aek 1t + Be k 2t = 0 (LAk C A)ek 1t +(LBk C B)ek 2t = 0 = 0 = 0 Lk C = 0 k 1 = 1 LC = i 1 LC = iω imaginary

30 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae k 1t + Be k 2t LAk 1 2 e k 1t + LBk 2 2 e k 2t + 1 C Aek 1t + Be k 2t = 0 (LAk C A)ek 1t +(LBk C B)ek 2t = 0 = 0 = 0 Lk C = 0 k 1 = 1 LC = i 1 LC = iω imaginary and k 2 = iω

31 Who has seen e ix = cos x + i sin x? A. Yes B. No C. I vaguely remember

32 Survival guide on complex numbers: Step 1: understand the notation Notations: i = 1 (in most fields) j = 1 (in electrical engineering)

33 Survival guide on complex numbers: Step 2: basic properties Properties: i = 1 i 2 = 1 z = x + iy z 2 = x 2 y 2 + 2ixy Complex number Real part Imaginary part e ix = cos x + i sin x (Often called the Euler Formula) z = re iθ = r cos θ + i r sin θ One of the most important formulas in physics and engineering!

34 Challenge: i =?

35 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae k 1t + Be k 2t LAk 1 2 e k 1t + LBk 2 2 e k 2t + 1 C Aek 1t + Be k 2t = 0 (LAk C A)ek 1t +(LBk C B)ek 2t = 0 = 0 = 0 Lk C = 0 k 1 = 1 LC = i 1 LC = iω imaginary and k 2 = iω

36 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: q t = Ae kt + Be kt k = i 1 LC = iω

37 LC circuit Kirchhoff loop: L di dt q C = 0 L d2 q dt 2 + q C = 0 I = dq dt Second order homogenous linear differential equation with general solution: e ix = cos x + i sin x (Euler Formula) q t = Ae kt + Be kt q t = Q cos ωt + Φ k = i 1 LC = iω Q 2 = 4AB e 2iΦ = 4 B A

38 LC circuit General solution: q t = Q cos ωt + Φ 2 parameters depending on initial conditions (they are usually real numbers, even though the differential equation would also allow for complex solutions.)

39 LC circuit General solution: q t = Q cos ωt + Φ

40 LC circuit General solution: q t = Q cos ωt + Φ I t = ωq sin ωt + Φ

41 V General solution: q t = Q cos ωt + Φ I t = ωq sin ωt + Φ V t = Q cos ωt + Φ C LC circuit

42 V General solution: q t = Q cos ωt + Φ I t = ωq sin ωt + Φ V t = Q C cos ωt + Φ LC circuit

43 V General solution: q t = Q cos ωt + Φ I t = ωq sin ωt + Φ V t = Q C cos ωt + Φ LC circuit Total energy: E = 1 2 LI t 2 + q t 2 2C = Q2 2C ( 1 LC = ω) is constant

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