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

Please Do Course Evaluation

LC and RLC Circuits

Oscillation ‐ Spring

mv 2 1

2

m k

Potential energy  Kinetic energy

kx 2 1

2 2 max 2 2 2

mv 2 1

• r

kA 2 1 constant mv 2 1 kx 2 1   

Conservation of energy: Equation of motion:

kx

• x

dt d m

2 2



Solution:

m k with ) t ( sin A x      

Oscillation – LC circuit

C Q 2 1

2

Electric energy  Magnetic energy

LI 2 1

2

LI 2 1

• r

Q C 1 2 1 constant LI 2 1 Q C 1 2 1

2 max 2 max 2 2

  

Conservation of energy: Kirchhoff’s rule :

Q C 1 dt Q d Q C 1 dt dI L

2 2

    

Solution: Solve the differential equation!

C L

Similarity between Spring Oscillation and LC Oscillation I

Similarity between Spring Oscillation and LC Oscillation II

m k C L

Potential energy  Kinetic energy Electric energy  Magnetic energy

C Q 2 1

2

LI 2 1

2

kx 2 1

2

mv 2 1

2

kx

• x

dt d m

2 2

 Q C 1 dt Q d Q C 1 dt dI L

2 2

    

Kirchhoff’s rule: Newton’s Law

Potential energy Kinetic energy Spring constant k Mass m Displacement x Velocity v

kx 2 1

2

mv 2 1

2

dt dx v 

Electrical energy Magnetic energy 1/Capacitance Inductance L Charge Q Current I

C Q 2 1

2

LI 2 1

2

C 1 dt dQ I 

Class 42 More LC Circuiturrrent

Oscillation – LC circuit

C Q 2 1

2

Electric energy  Magnetic energy

LI 2 1

2

constant LI 2 1 Q C 1 2 1

2 2

 

Conservation of energy: Kirchhoff’s rule :

Q C 1 dt Q d L Q C 1 dt dI L

2 2

    

Solution:

C L

m k with ) t ( sin A x       LC 1 LC 1 with ) t ( sin A Q       

RLC circuit

C Q Q dt d R Q dt d L ) Q dt d (I t d dI L IR C Q : rule s Kirchhoff'

2 2

       

RLC circuit Damped Oscillation

m k

kx x dt d b x dt d m ) x dt d (v kx

• bv
• x

dt d m : motion

• f

Equation

2 2 2 2

     

Friction = ‐bv

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 ½LI2 Kinetic energy ½mv2 Electrical energy ½(1/C)Q2 Potential energy ½kx2

RLC circuit

C Q Q dt d R Q dt d L ) Q dt d (I t d dI L IR C Q : rule s Kirchhoff'

2 2

       

RLC circuit Solution:

                        damped

• ver

damped critically damped under 2L R LC 1 2L R LC 1 cos e Q Q(t)

2 2 d d t 2L R

• 



Damping

2 d d t 2L R

• 2L

R LC 1 cos e Q Q(t)           

d real: under damped d = 0: critically damped d imaginary: overdamped