Capacitor and Inductor

Capacitor and Inductor

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Energy Storage Final Final Q Φ energy stored in electric field energy stored in magnetic field potential energy of accumulated electrons f (v c ) kinetic energy of f (i L ) moving electrons Q = CV N Φ = LI Φ = BA W = 1 2 CV 2 W = 1 2 L I 2 Capacitor 3

Newton's First Law of Motion tendency to try to maintain voltage at a constant level tendency to try to maintain current at a constant level a capacitor resists changes in voltage drop v c an inductor resists changes in current flow i L v C i L v C i L potential energy f (v c ) kinetic energy f (i L ) Capacitor 4

Inertia when v c is increased the capacitor resists the increase when i L is increased the inductor resists the increase by drawing current i c from the source of the voltage change in opposition to the change Δ by dropping voltage from the source of the current change in opposition to the change oc sc 0 Δ i L = 0 Δ i s Δ v s 0 Δ v C = 0 Δ v s sc oc Δ i s Δ temporarily short circuit temporarily open circuit Capacitor 5

Resisting changes resists the change of voltage increasing v c = increasing electric field : opposing current back voltage drop acting as reducing the voltage drop : resists the change of voltage resists the change of current increasing i L = increasing magnetic field : opposing emf back current acting as reducing the current : resists the change of current Δ v s Δ i s Δ i s v s Δ v s is tries to cancel Δ v s tries to cancel Δ i s Capacitor 6

Opposition to the change Δ v s change at t = 0 Δ i s change at t = 0 Δ i L = 0 Δ v s Δ v s Δ i Δ v C = 0 s Δ i s temporarily short circuit temporarily open circuit by drawing current Δ by inducing voltage Δ Δ v s Δ i s Capacitor 7

Opposition to the change Δ v s change at t = 0 initial t = 0 final t = Δ v s Δ v s Δ v C = 0 sc Δ v s Δ v C = Δ v s Δ v s Δ peak Δ = 0 oc Δ i s change at t = 0 initial t = 0 final t = Δ i s Δ i s Δ i L = 0 oc Δ i s Δ i L = Δi s Δ i s Δ peak Δ = 0 sc Capacitor 8

Charging = 0 = 0 v s i s Δ 0 Δ 0 Δ v s Δ i s v s i s power load power load Capacitor 9

Discharging = 0 = 0 v s i s Δ 0 Δ 0 Δ v s v s Δ i s is power source power source Capacitor 10

Dissipating Energy when v c is decreased the capacitor resists the decrease when i L is decreased the inductor resists the decrease by supplying current i c to the source of the voltage change in opposition to the change by producing voltage to the source of the current change in opposition to the change Δ i L = 0 Δ Δ i L Δ i s Δ v s Δ v C = 0 Δ i s Δ Δ v C Δ v s Δ Capacitor 11

To store more energy static nature a function of V to store more energy v c must be increased increasing charges on both sides produces electron movement : a way of resisting change of V Q = C V dynamic nature a function of I to store more energy i L must be increased increasing magnetic flux induces emf to prohibit electron movement : a way of resisting change of I Φ = L I more charges more magnetic flux Capacitor 12

Fast Change Q = C V Φ = L I more electron imbalance more moving electrons fast change Δ v fast change Δ q fast change Δ i fast change Δ ϕ large Δ v Δ t large Δ q Δ t i large Δi Δ t large Δ ϕ Δ t e Capacitor 13

Ohm's Law whenever voltage v C changes there is a flowing current i c the magnitude of the current is proportional to the rate of change whenever current I L changes there is an induced voltage the magnitude of the voltage is proportional to the rate of change = C d v C dt = L d i L dt Capacitor 14

Stored Energy resists to the change of voltage to increase v c, energy should be transferred to capacitors by the flowing current i c The stored energy is a function of voltage Q = C V resists to the change of current to increase i L, energy should be transferred to the inductors by the induced voltage The stored energy is a function of current Φ = L I dq dt = C d v dt d ϕ dt = L d i dt = C d v C dt = L d i L dt Capacitor 15

Short term effects Δ v s Δ i s v s i s v s i s the short term effect: a way of resisting 0 the long term result: voltage change v C v Δ v the short term effect: a way of resisting 0 the long term result: current change i L i Δ i Capacitor 16

Some (i,v) signal pair examples d d t v C i L d d t v C i L Capacitor 17

Everchanging signal pairs decreasing increasing decreasing increasing decreasing increasing d d t v C i L negative positive negative positive negative positive Capacitor 18

Current & Voltage Changes constant v d v d t = 0 i = 0 constant i d i d t = 0 v = 0 increasing v d v d t > 0 i > 0 increasing i d i d t > 0 v > 0 decreasing v d v d t < 0 i < 0 increasing i d i d t < 0 v < 0 Capacitor 19

Sinusoidal voltage and current = C d v C dt = L d i L dt v C = A cos(ωt ) i L = A sin (ω t) =ω C A sin(ωt ) = ω C A cos(ωt π/2) = A cos(ω tπ/2) = ω L A cos(ω t) v C = 1 ωc cos(ω t) cos(ω tπ/2) i L = ω L cos(ω t) cos(ω tπ/2) V C = A 0 I L = A π/2 I C = ωc A π/2 V L = ω L A 0 Z C = V C I C = 1 ωc π/2 = j ωc = 1 j ωc Z L = V L I L = ω L π/2 = j ω L Capacitor 20

Leading and Lagging Current = C d v C dt = L d i L dt v C = cos(2 πt ) i L = sin(2π t) =πsin (2 π t) = cos(2 πtπ/2) = πcos(2 π t π/2) = π cos(2 π t) C=0.5 L=0.5 v C i L Capacitor 21

Ohm's Law V C = A 0 I L = A π/2 I C = ωc A π/2 V L = ω L A 0 Z C = V C I C = 1 ωc π/2 = j ωc = 1 j ωc Z L = V L I L = ω L π/2 = j ω L v C i L Capacitor 22

Phasor and Ohm's Law V C = A 0 I L = A π/2 I C = ωc A π/2 V L = ω L A 0 Z C = V C I C = 1 ωc π/2 = j ωc = 1 j ωc Z L = V L I L = ω L π/2 = j ω L I C j ω C V L V C j ω L I L Capacitor 23

Phase Lags and Leads Capacitor 24

References [1] http://en.wikipedia.org/ [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003