the capacitor: c the inductor: l relationships...
TRANSCRIPT
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Capacitors and Inductors
Capacitors and Inductors
The capacitor: CThe inductor: LRelationships between C and LRC/RL Circuits
Carruthers (ECE @ BU) EK307 Notes Spring 2020 138 / 275
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Capacitors and Inductors The capacitor: C
Capacitor
Scale is cm
The capacitance is C = ✏A/d, unit farad (F).
Carruthers (ECE @ BU) EK307 Notes Spring 2020 139 / 275
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Capacitors and Inductors The capacitor: C
Capacitor: Voltage/Current
q = Cv
i = Cdv
dt
v(t) = v(t0) +1
C
Z t
t0
i(⌧)d⌧
Carruthers (ECE @ BU) EK307 Notes Spring 2020 140 / 275
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Capacitors and Inductors The capacitor: C
Capacitor
typical values: µF to nF.
capacitor is an “open circuit to dc”: i.e. current is zero if voltageis a constant.
voltage on the capacitor must be continuous.
capacitor stores energy in the amount of w = 12Cv
2
when iv > 0, capacitor is storing energy.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 141 / 275
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Capacitors and Inductors The inductor: L
Inductor
The inductance is L = kN2, unit henry (H), where the constant k
depends on the construction of the inductor.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 142 / 275
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Capacitors and Inductors The inductor: L
Inductor: Voltage/Current
v = Ldi
dt
i(t) = i(t0) +1
L
Z t
t0
v(⌧)d⌧
Carruthers (ECE @ BU) EK307 Notes Spring 2020 143 / 275
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Capacitors and Inductors The inductor: L
Inductor
typical values: µH to H.
inductor is a “short circuit to dc”: i.e. voltage is zero if current isa constant.
current on the inductor must be continuous.
inductor stores energy in the amount of w = 12Li
2
when iv > 0, inductor is storing energy.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 144 / 275
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Capacitors and Inductors Relationships between C and L
Capacitors and Inductors: Duals
Any statement about a capacitor can be made about an inductor (andvice-versa) by:
swapping current and voltage
swapping C and L
swapping open and short circuit
swapping series and parallel
Capacitors store energy in an electric field.Inductors store energy in a magnetic field.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 145 / 275
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Capacitors and Inductors Relationships between C and L
Series and Parallel Combinations
Capacitors InductorsSeries 1
Ceq= 1
C1+ 1
C2+ · · · 1
CNLeq = L1 + L2 + · · ·LN
Parallel Ceq = C1 + C2 + · · ·CN1
Leq= 1
L1+ 1
L2+ · · · 1
LN
Memory aid:
Inductors combine like resistors; capacitors combine opposite toresistors
capacitors get capacitance through area; adding in parallelincreases area
inductors get inductance through turns; adding in series increasesnumber of turns.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 146 / 275
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Capacitors and Inductors Relationships between C and L
Example Problem 1
Find the energy stored in the capacitors, under dc conditions.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 147 / 275
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Capacitors and Inductors Relationships between C and L
Example Problem 2
Find the voltages across the capacitors.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 148 / 275
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Capacitors and Inductors Relationships between C and L
Example Problem 3
Determine iL, vC , and the energies stored in C and L.
Carruthers (ECE @ BU) EK307 Notes Spring 2020 149 / 275
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Capacitors and Inductors RC/RL Circuits
What does this circuit do? Integrator
v0(t) = � 1
RC
Z t
0vi(⌧)d⌧
Carruthers (ECE @ BU) EK307 Notes Spring 2020 150 / 275
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Capacitors and Inductors RC/RL Circuits
What does this circuit do? Di↵erentiator
v0(t) = �RCdvi
dt
Carruthers (ECE @ BU) EK307 Notes Spring 2020 151 / 275