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EE1000, Lecture 4
Capacitance, Inductance, Resonance
Capacitance
The symbol of a capacitor:
Examples:
Capacitors in parallel add.
Q: What about capacitors in series?
A:
Variable capacitor (increase or decrease capacitance): overlapping plates
Inductance
The symbol of an inductor:
Examples:
The cylindrical shape in the middle is called a solenoid.
Inductors in series add.
Q: What about inductors in parallel?
A:
Variable Inductor (increase or decrease inductance): ferrite slug moves into and out of the coil.
Capacitors
A capacitor's energy exists in its surrounding electric fields. It is proportional to the square of the
field strength, which is proportional to the charges on the plates. If we assume the plates carry
charges that are the same in magnitude, +q and -q, then the energy stored in the capacitor must
be proportional to q2. Let’s also assume that the voltage across the capacitor = V. For historical
reasons, we write the constant of proportionality as ½ C,
1 q2 1
EC 
 CV 2 …This gives us an important relationship: q  CV
2C 2
The constant C is a geometrical property of the capacitor, called its capacitance:
C
 r 0 A
d
,
where εr is the relative permittivity of the dielectric and ε0 = 8.854x10-12 F/m.
Inductors
Any current will create a magnetic field, so in fact every current-carrying wire in a circuit acts as
an inductor! All the loops' contribution to the magnetic field add together to make a stronger
field. Unlike capacitors and resistors, practical inductors are easy to make by hand. One can for
instance spool some wire around a short wooden dowel, put the spool inside a plastic aspirin
bottle with the leads hanging out, and fill the bottle with epoxy to make the whole thing rugged.
The energy density is proportional to the square of the magnetic field strength, which is in turn
proportional to the current flowing through the coiled wire, so the energy stored in the inductor
must be proportional to I2.
EL 
1 2
LI
2
The quantity L is called the inductance of the inductor:
L
0 N 2 A
l
,
Where, A=cross-sectional of the coil/solenoid, N =the number of turns (or loops of wire) in the
coil/solenoid and µ0 = 4π×10-7 Henrys per meter= permeability of free space.
Oscillations and RLC Circuit
The simplest type of resonator (the device that resonates or oscillates) is an RLC circuit.
A series RLC circuit:
A parallel RLC circuit:
Resonance effect
The resonance effect occurs when inductive and capacitive reactances are equal in magnitude.
The frequency at which this equality holds for the particular circuit is called the resonant
frequency. The resonant frequency of the LC circuit is
o 
1
LC
where L is the inductance in henries (H), and C is the capacitance in farads (F). The
angular frequency
has units of radians per second (rad/s).
The resonant frequency in units of Hertz (Hz) is
fo 
o
1

2 2 LC
LC circuits are often used as filters; the L/C ratio determines their "Q" and so selectivity. For a
series resonant circuit with a given resistance, the higher the inductance and the lower the
capacitance, the narrower the filter bandwidth. For a parallel resonant circuit, the opposite
applies.
At resonant frequency: series = short; parallel = open.
The most common application of an RLC circuit is tuning radio transmitters and receivers. For
example, when we tune a radio to a particular station, the RLC circuits are set at resonance for
that particular carrier frequency.
The Home-Made Inductor (Coil) and Home-made Variable Capacitor
Inductor
[HW on the last page…]
Variable Capacitor
****************************** Homework Assignment **********************************
*************************************************************************************
EE1000 HW #4

The capacitance of a parallel plate capacitor with area A and distance d between the plates is
C

Name:________________________________
 r 0 A
d
, where εr is the relative permittivity of the dielectric and ε0 = 8.854x10-12 Farads
per meter.
The inductance of an air-core solenoid of length l and cross sectional area A is given by
L
0 N 2 A
l
, where N is the number of turns (or loops of wire) in the solenoid and µ0 =
4π×10-7 Henrys per meter is the permeability of free space.

The resonant frequency is given by
fo 
1
2 LC
[Hz].
Problems: All units must be converted to m, m^2, F, H, etc. and then modified later for the final
answer.
1. Find the area (in square centimeters) of a 1.5x10-9 F parallel plate capacitor with a distance
between the plates of 0.002” (=5.08x10^-5 m). Assume a relative permittivity, εr = 2.5.
2. Find the inductance of a 5 cm (centi meter) diameter solenoid that is 10 cm (centi meter) in
length and has 156 turns. Assume an air core.
3. What is the resonant frequency of a 25 pF (pico Farads) capacitor connected to a 500 μH (micro
Henry) inductor?