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Young Won Lim05/07/2015
Capacitor and Inductor
Young Won Lim05/07/2015
Copyright (c) 2015 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
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Capacitor 3 Young Won Lim05/07/2015
Energy
+
−
+
−
+
−
+
−
+
−−
+
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 vL = 0Φ
Capacitor 4 Young Won Lim05/07/2015
Newton's First Law of Motion
tendency to try to maintain voltage at a constant level
capacitor resists changes in voltage drop
when vc is increased
the capacitor resists the increase
by drawing current ic
from the source of the voltage change
in opposition to the change
tendency to try to maintain current at a constant level
inductor resists changes in current flow
when iL is increased
the inductor resists the increase
by dropping voltage vL
from the source of the current change
in opposition to the change
Capacitor 5 Young Won Lim05/07/2015
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
Capacitor 6 Young Won Lim05/07/2015
Newton's First Law of Motion
when vc is decreased
the capacitor resists the decrease
by supplying current ic
to the source of the voltage change
in opposition to the change
when iL is decreased
the inductor resists the decrease
by producing voltage vL
to the source of the current change
in opposition to the change
Capacitor 7 Young Won Lim05/07/2015
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
Capacitor 8 Young Won Lim05/07/2015
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 +
−
Capacitor 9 Young Won Lim05/07/2015
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
Capacitor 10 Young Won Lim05/07/2015
Fast Change
Δ v
+
−
+
−
+
−
+
−−
+
Q = C⋅V Φ = L⋅I
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
Capacitor 11 Young Won Lim05/07/2015
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 vC
dtv L = L⋅
d iLdt
Capacitor 12 Young Won Lim05/07/2015
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
Q = C⋅V Φ = L⋅I
dqdt
= C⋅d vdt
dϕ
dt= L⋅
d idt
iC = C⋅d vC
dtv L = L⋅
d iLdt
Capacitor 13 Young Won Lim05/07/2015
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
Capacitor 14 Young Won Lim05/07/2015
Some (i,v) signal pair examples
vC
dd t
dd t
iC
vC
iC
iL
vL
iL
vL
Capacitor 15 Young Won Lim05/07/2015
Everchanging signal pairs
decreasing decreasing decreasingincreasing increasing increasing
negative negative negativepositive positive positive
dd t
vC
iC
iL
vL
Capacitor 16 Young Won Lim05/07/2015
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
Capacitor 17 Young Won Lim05/07/2015
Sinusoidal voltage and current
iC = C⋅d vC
dtv 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
Capacitor 18 Young Won Lim05/07/2015
Leading and Lagging Current
iC = C⋅d vC
dtv 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
Capacitor 19 Young Won Lim05/07/2015
Ohm's Law
vC
iC v L
iL
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
Capacitor 20 Young Won Lim05/07/2015
Phasor and Ohm's Law
V C
IC
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
I L
V L
− jωC
jω L
Capacitor 21 Young Won Lim05/07/2015
Phase Lags and Leads
Young Won Lim05/07/2015
References
[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003
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