Capacitors and Inductors

Capacitors and Inductors

The capacitor: CThe inductor: LRelationships between C and LRC/RL Circuits

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Capacitors and Inductors The capacitor: C

Capacitor

Scale is cm

The capacitance is C = ✏A/d, unit farad (F).

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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⌧

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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.

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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.

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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⌧

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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.

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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.

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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.

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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.

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Capacitors and Inductors Relationships between C and L

Example Problem 2

Find the voltages across the capacitors.

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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.

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Capacitors and Inductors RC/RL Circuits

What does this circuit do? Integrator

v0(t) = � 1

RC

Z t

0vi(⌧)d⌧

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Capacitors and Inductors RC/RL Circuits

What does this circuit do? Di↵erentiator

v0(t) = �RCdvi

dt

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