Energy stored in a magnetic field Energy Stored in an Inductor Energy - PowerPoint PPT Presentation

Energy stored in a magnetic field

Energy Stored in an Inductor Energy stored in an inductor: L 1 U  2 LI 2 dI   (Do not forget .) - L dt Energy density stored in an electric field: U 1   2 B u B   B 2 0

Capacitor and Inductor Capacitor C Inductor L Charge Q Current I E field B field Q V  d I   - L C d t Parallel plate capacitor (uniform E Solenoid (uniform B field)   A V field)      0 L nNA and B nI C and E 0 0 d d 1 1 1 1    2 2   U CV and u E 2 2 U LI and u B E E 0  B B 2 2 2 2 0

Class 40 RL Circuits

From Class 25 RC Circuits – Charging Charge At t=0, capacitance is uncharged and C Q=0 (initial condition). At t=0, switched is closed, if the capacitor has no charge, it behaves like a conductor and I=  /R.  R After the capacitor is completely charged, Q=C  ,  V C =  and  V R =0. I=0 and the capacitors behave like an insulator.

RL Circuits – Charging Current At t=0, inductor is uncharged and I=0 L (initial condition). At t=0, switched is closed, if the inductor has no current, it behaves  like an insulator (opposes changes) R and I=0. After the inductor is completely charged (with current), I=  /R,  V L = 0 and  V R =  . The inductor behaves like a conductor.

From Class 25 RC Circuits – Charging Charge q q d q        IR R C C d t C    CR dq (C - q) dt dq 1   Integration constant - dt  q - C CR t      n( q - C ) - K'  CR R t -     K' q - C Ke CR (K e ) t -     q C K e CR          At t 0, q 0 0 C K K - C t -     CR q C ( 1 e )   t t dq C - -    I e CR e CR d t CR R t -     V IR e CR R  V R +  V C =  t q -     CR V (1 - e ) C C

RL Circuits – Charging Current d I        L IR L dI I R dt dt d t L    L dI ( - IR)dt L dI   Integration constant dt  - IR L       n( - IR ) t K' R R R R      n( - IR ) t K' L L R R - t K'     L L - IR Ke (K e ) R - t     L IR Ke          At t 0, I 0 0 K K  R R - t - t        L L I R ( 1 e ) I ( 1 e ) R R - t      L V IR ( 1 e ) R R dI - t     L  V R +  V C =  V L e L dt

RC time constant From Class 25  =RC is known as the RC time constant. It indicates the response time (how fast you can charge up the capacitor) of the RC circuit.  t -  CR I e R   I  e 2.72 R -  1 e 0 . 37   - 1 I e R  ~ 0 . 37 R t t=RC t -    CR q C ( 1 e ) q   C  2 1.414    -1 q ( 1 e ) C  1 ~ 0 . 63 C  0 . 707 2 t Nothing to do with RC circuits t=RC

L/R time constant  =L/R is known as the time constant. It indicates the response time (how fast you can up a current) of the RC circuit.  R d I - t  L e d t L  dI   e 2.72 dt L -  1 e 0 . 37  dI  - 1 e dt L  ~ 0 . 37 L t t=L/R   R - t  I ( 1 e L ) R   I  R    - 1 2 1.414 I ( 1 e ) R  1 ~ 0 . 63  0 . 707 R 2 t t=L/R Nothing to do with RL circuits

From Class 25 RC Circuits – Discharging Charge q q d q      0 IR R 0 C C d t   CR dq - q dt dq 1   - dt Integration constant q CR t     n q - K' C R CR t -    K' q Ke CR (K e ) t -   CR q K e     At t 0, q Q Q K t -   CR q Qe t dq Q -    CR I e d t RC t Q -     CR V IR e R C  V R +  V C = 0 t q Q -    V e CR C C C