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Young Won Lim05/18/2017

Capacitor and Inductor


Copyright (c) 2015 – 2017 Young W. Lim.

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Capacitor 3 Young Won Lim05/18/2017

Energy Storage

+

−

+

−

+

−

+

−

+

−−

+

Final

energy stored in electric field

+

−

Final

energy stored in magnetic field

potential energy of accumulated electrons

kinetic energy of moving electrons

f (v c) f (iL)

Q Φ

Q = CV N Φ = LI

W =12CV 2

W =12L I2

Φ = BA


Newton's First Law of Motion

tendency to try to maintain voltage at a constant level

a capacitor resists changes in voltage drop vc

tendency to try to maintain current at a constant level

an inductor resists changes in current flow iL

iC

iC

iL

iL

vC

vC

vL

vL

f (v c) f (iL)potential energy kinetic energy


Inertia

when vc is increasedthe capacitor resists the increase

by drawing current ic

from the source of the voltage change

in opposition to the change

when iL is increasedthe inductor resists the increase

by dropping voltage vL

from the source of the current change


Δ v s

Δ iC

Δ is

Δ vC = 0 → Δ v s

Δ iL = 0 → Δ is

0+∞

0+∞

+

−

Δ v L

sc oc

oc sc

temporarily short circuit temporarily open circuit


Resisting changes

resists the change of voltage

increasing vc =

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 iL =

increasing magnetic field : opposing emf → back current acting as reducing the current :resists the change of current

v s

Δ v s

is

Δ is

tries to cancel

+

−

−Δ v s

Δ v s tries to cancel Δ is

−Δ is +

−


Opposition to the change

temporarily short circuit temporarily open circuit

Δ is

Δ v s change at t = 0 Δ is change at t = 0

+

−

Δ vC = 0

Δ v s

−Δ v s −Δ is

Δ iL = 0

Δ isΔ v s

by drawing current Δ iC by inducing voltage Δ vL

+

−


Opposition to the change

Δ v s change at t = 0

Δ v s

−Δ v s

Δ is change at t = 0

initial t = 0

Δ v s

initial t = 0

final t = ∞

Δ v s

final t = ∞

Δ is Δ is Δ is

Δ vC = 0 Δ vC = Δ v s

Δ iC peak Δ iC = 0

Δ iL = 0 Δ iL = Δ is

Δ v L peak Δ v L = 0

sc

oc

sc

oc

−Δ is


Charging

+

−

+

−

+

−

+

−

+

−−

+

v s is

+

−

+

−

+

−

+

−−

+

iC = 0

v s

Δ v s++

− −

is

Δ is

Δ iC → 0

+

−

Δ v L → 0

vL = 0

power load power load


Discharging

+

−

+

−

+

−

+

−

+

−−

+

v s is

+

−

+

− −

+

iC = 0

v s

Δ v s

is

Δ is

Δ iC → 0

+

−

Δ v L → 0

vL = 0

power source power source


Dissipating Energy

when vc is decreasedthe capacitor resists the decrease

by supplying current ic

to the source of the voltage change


when iL is decreasedthe inductor resists the decrease

by producing voltage vL

to the source of the current change


Δ v s

Δ iC

Δ is

+

−

Δ v L

Δ vC = 0

Δ vC → Δ v s

Δ iL = 0

Δ iL → Δ is

Δ iC


To store more energy

● static nature ● a function of V

to store more energyv

c must be increased →

increasing charges on both sides produces electron movement :a way of resisting change of V

● dynamic nature● a function of I

to store more energyiL must be increased →

increasing magnetic flux induces emf to prohibit electron movement : a way of resisting change of I

+

−

+

−

+

−

+

−−

+

Q = C⋅V Φ = L⋅I

more charges

more magnetic flux


Fast Change

Δ v

+

−

+

−

+

−

+

−−

+


more electron imbalance

more moving electrons

fast change Δ qfast change

Δ vΔ t

large ΔqΔ t

∝ ilarge

Δ ifast change Δϕfast change

Δ iΔ t

largeΔϕ

Δ t∝ elarge


Ohm's Law

whenever voltage vC

changes

there is a flowing current ic

the magnitude of the current is proportional to the rate of change

whenever current IL changes

there is an induced voltage vL

the magnitude of the voltage is proportional to the rate of change

iC = C⋅d vCdt

v L = L⋅d iLdt


Stored Energy

resists to the change of voltage

to increase vc,

energy should betransferred to capacitors by the flowing current i

c

The stored energy is a function of voltage

resists to the change of current

to increase iL,

energy should betransferred to the inductors by the induced voltage v

L

The stored energy is a function of current


dqdt

= C⋅d vdt

dϕ

dt= L⋅

d idt

iC = C⋅d vCdt

v L = L⋅d iLdt


Short term effects

iC → 0

vC → v + Δ v

vL → 0

iL → i + Δ i

the short term effect: a way of resisting

the long term result: voltage change

the short term effect: a way of resisting

the long term result: current change

+

−

+

−

+

−

+

−−

+

v s

Δ v s++

− −

is

Δ is+

−

vs is


Some (i,v) signal pair examples

vC

dd t

dd t

iC

vC

iC

iL

vL

iL

vL


Everchanging signal pairs

decreasing decreasing decreasingincreasing increasing increasing

negative negative negativepositive positive positive

dd t

vC

iC

iL

vL


Current & Voltage Changes

constant v constant id vd t

= 0 i = 0d id t

= 0

increasing v increasing id vd t

> 0 i > 0d id t

> 0

+ −

+ −

v > 0

v = 0

decreasing v increasing id vd t

< 0 i < 0d id t

< 0

− +

v < 0


Sinusoidal voltage and current

iC = C⋅d vCdt

v L = L⋅d iLdt

vC = A cos(ω t ) iL = A sin (ω t)

iC =−ωC A sin(ω t )

vL = ωL A cos(ω t)= ωC A cos(ω t + π/2)

vC

iC=

1ωC

cos(ω t)cos(ω t+π/2)

vL

iL= ω L

cos(ω t)cos(ω t−π/2)

= A cos (ω t−π/2)

V C = A 0 I L = A −π/2

IC = ωC A π/2 V L = ω L A 0

ZL =V L

I L

= ω L +π/2ZC =V C

IC

=1

ωC−π/2

= jω L=− jωC

=1

jωC


Leading and Lagging Current

iC = C⋅d vCdt

v L = L⋅d iLdt

vC = cos(2 π t ) iL = sin (2π t)

iC =−π sin (2π t)

vL = π cos(2π t)= πcos (2π t + π/2)

= cos(2π t−π/2)

vC

iC v L

iL

C=0.5 L=0.5


Ohm's Law

vC

iC v L

iL

V C = A 0 I L = A −π/2


ZL =V L

I L

= ω L +π/2ZC =V C

IC

=1

ωC−π/2

= jω L=− jωC

=1

jωC


Phasor and Ohm's Law

V C

IC

V C = A 0 I L = A −π/2


ZL =V L

I L

= ω L +π/2ZC =V C

IC

=1

ωC−π/2

= jω L=− jωC

=1

jωC

I L

V L

− jωC

jω L


Phase Lags and Leads


References

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

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