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Capacitance and RC Circuits
Capacitors are constructed by separating 2 sheets of conductor, which
is usually metallic, by a thin layer of insulating material. In parallel-plate
capacitor, the sheets are flat and parallel. The insulating material
between the plates, called a dielectric, can be air, Mylar, polyester,
polypropylene, mica, etc.
Stored charge in terms of voltage
q  Cv
The constant of proportionality is the capacitance C, which has units
of farads (F). Farads are equivalent to coulombs per volt.
Current in terms of voltage
dq d
i
 (Cv)
dt dt
Cdv
i
dt
Note: the current reference direction points into the
positive reference polarity.
Voltage in terms of current Suppose that we know the current i(t)
flowing through a capacitance C and we want to compute charge
and voltage.
t
q(t )   i (t )dt  q (t0 )
t0
q(t0 )
1 t
v(t )   i (t )dt 
C t0
C
1 t
v(t )   i (t )dt  v(t0 )
C t0
Stored Energy
Energy stored in the capacitance is given by
1 2
w(t )  Cv (t )
2
1
w(t )  v(t )q(t )
2
q 2 (t )
w(t ) 
2C
Capacitance of the parallel-plate capacitor
If the distance d between the plates is much smaller than both the
width and the length of the plates, the capacitance is approximately
C
A
d
In which  is the dielectric constant of the material between the plates.
For vacuum, the dielectric constant is   0  8.85x10-12 F/m
Inductance
An inductor is constructed by coiling a wire around some type of form.
Current flowing through the coil creates a magnetic field or flux that links
the coil. Frequently the coil form is composed of a magnetic material
such as iron or iron oxide that increases the magnetic flux for a given
current.
When the current changes in value, the resulting magnetic flux changes
according to Faraday’s law of electromagnetic induction,
time-varying magnetic flux linking a coil induces voltage across the coil.
For an ideal inductor, the voltage is proportional to the time rate of
change of the current. Furthermore, the polarity of the voltage is such as
to oppose the change in current. The constant of proportionality is called
inductance, L.
di
v(t )  L
dt
Current in terms of voltage
di
v(t )  L
dt
1
di  v(t )dt
L
i (t )
1 t
i (t0 )di  L t0 v(t )dt
1 t
i (t )   v(t )dt  i (t0 )
L t0
Stored Energy
1 2
w(t )  Li (t )
2