Please Do Course Evaluation LC and RLC Circuits Oscillation Spring - PowerPoint PPT Presentation

Please Do Course Evaluation

LC and RLC Circuits

Oscillation ‐ Spring Potential energy  Kinetic energy 1 1 2 2 kx mv 2 2 k Conservation of energy: m 1 1   2 2 kx mv constant 2 2 1 1  2 2 kA or mv max 2 2 2 d  Equation of motion: m x - kx 2 dt k       Solution: x A sin ( t ) with m

Oscillation – LC circuit Electric energy  Magnetic energy 2 1 1 Q 2 LI 2 2 C Conservation of energy: C L 1 1 1   2 2 Q LI constant 2 C 2 1 1 1  2 2 Q or LI max max 2 C 2 2 dI 1 d Q 1      Kirchhoff’s rule : L Q Q 2 dt C C dt Solution: Solve the differential equation!

Similarity between Spring Oscillation and LC Oscillation I

Similarity between Spring Oscillation and LC Oscillation II k C m L Potential energy  Kinetic energy Electric energy  Magnetic energy 1 1 2 1 Q 1 2 2 kx mv 2 LI 2 2 2 C 2 Newton’s Law Kirchhoff’s rule: 2 d 2  dI 1 d Q 1 m x - kx      L Q Q 2 dt 2 dt C dt C 2 1 Q 1 2 Potential energy Electrical energy kx 2 C 2 1 1 2 mv Kinetic energy Magnetic energy 2 LI 2 2 1 Spring constant k 1/Capacitance C Mass m Inductance L Displacement x Charge Q dQ I  dx v  Velocity v Current I dt dt

Class 42 More LC Circuiturrrent

Oscillation – LC circuit Electric energy  Magnetic energy 2 1 1 Q 2 LI 2 2 C Conservation of energy: C L 1 1 1   2 2 Q LI constant 2 C 2 2 dI 1 d Q 1      Kirchhoff’s rule : L Q L Q 2 dt C C dt k       Solution: x A sin ( t ) with m 1 1        Q A sin ( t ) with LC LC

RLC circuit RLC circuit Damped Oscillation k m Friction = ‐ bv Equation of motion : Kirchhoff' s rule : 2 d d Q dI d       0 IR L (I Q ) m x - bv - kx (v x ) 2 C d t dt dt dt 2 d d Q 2 d d         L Q R Q 0 m x b x kx 0 2 dt C dt 2 dt dt

RLC circuit and Mechanical Oscillation RLC circuit Mechanical Q x I = dQ/dt v = dx/dt C 1/k R b L m Magnetic energy ½LI 2 Kinetic energy ½mv 2 Electrical energy ½(1/C)Q 2 Potential energy ½kx 2

RLC circuit RLC circuit Solution: R - t   2L Q(t) Q e cos 0 d 2   1 R      d   LC 2L   0 under damped 2    1 R      0 critically damped   LC 2L    0 over damped Kirchhoff' s rule : Q dI d     0 IR L (I Q ) C d t dt 2 d d Q     L Q R Q 0 2 dt C dt

Damping R - t    d real: under damped 2L Q(t) Q e cos 0 d  d = 0: critically damped  d imaginary: overdamped 2   1 R      d   LC 2L