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The electric power associated with an element is:
where
is the current going through it and
the voltage across it.
Energy is power integrated over time:
- Energy Dissipation: For a resistor with
and
, the energy dissipated in time period
is:
This energy is converted irreversibly from electrical energy to heat
which cannot be converted back to electrical energy. The rate of
dissipation is:
When the current and voltage are both constant
and
,
we have
.
Example: When
and
, the energy dissipated in time period
is
Some useful trigonometric identities:
- Energy Storage:
- Capacitor
Here we have assumed
and
. We see that capacitance
is a measure of the capacitor's ability to store energy in electric
field (separated charge in terms of voltage
). In particular, under zero
initial condition
and
, the potential energy
stored in the capacitor is proportional to
and voltage
squared:
Example: When
,
, the energy dissipated in
period
is
This results indicates that there is no energy dissipated over the complete
period
, as in the first and third
the energy is stored in the capacitor
(equivalent to a battery being charged), but in the 2nd and 4th
the energy is
released from the capacitor again (equivalent to a battery delivering power).
- Inductor
Here we have assumed
and
. We see that inductance
is a measure of the inductor's ability to store energy in magnetic
field (moving charge in terms of current
). In particular, under zero
initial condition
and
, the potential energy
stored in the inductor is proportional to
and the current
squared:
Example: When
,
, the energy dissipated in
time period
is
Similarly, for a sinusoidal current of period
, no energy will be dissipated
by the inductor during the complete period
. In both of the cases of capacitor
and inductor, the energy is converted into potential energy stored in either
the electric or magnetic field of the element, instead of being dissipated
(converted to heat).
Comparison with mechanical systems:
The work of a mechanical system does is
where
is force and
is displacement.
- Potential energy: A spring can be described by Hooke's law
where
is the stiffness, or
where
is the
compliance. The potential energy stored in the spring is:
i.e., the compliance
is a measure of the spring's ability to
store potential energy (the less stiff, the more potential energy can
be stored in the spring with the same force).
- Kinetic energy: A mass moving at a velocity
has kinetic
energy
But as
, the kinetic energy becomes:
i.e., the mass
of a body is a measure of the body's ability to
store kinetic energy (the more mass, the more kinetic energy can be
stored in the body with the same velocity).
Examples:
- A mechanical system composed of a spring
and a mass
can be
described by:
where
is the horizontal displacement of the mass. The
homogeneous solution of the differential equation can be found to be
where
is the compliance. When the displacement is zero
, the velocity
is maximal. This
corresponds to the fact that when the potential energy stored in
the spring is zero, the kinetic energy is maximal. On the other hand,
when the displacement is maximal
, the
velocity
is zero. This corresponds
to the fact that when the potential energy stored in the spring is
maximal, the kinetic energy is zero.
- An electrical system composed of a capacitor
and an inductor
can be described by:
where
is the voltage across both components, and
and
are currents through
and
, respectively. As
,
i.e.
, we have
The homogeneous solution of the differential equation can be found to be
When the energy stored in the capacitor is zero
,
the energy stored in the inductor is maximal
; but when the energy stored in the
capacitor is maximal
, the energy stored
in the inductor is zero
.
In both cases, the energy in the system is converted back and forth
between different forms (potential vs. kinetic, electrical vs. magnetic),
while the total amount is always reserved.
However, when a dash-pot (causing friction proportional to speed
)
is added (in parallel to the spring) in the mechanical system, and a
resistor is added in series with the electrical circuit, the energy is
dissipated (converted to heat) in both systems:
The corresponding solution of the DEs will be decaying sinusoidal,
indicating the dissipation of the energy in the system.
Next: Sinusoidal Variables
Up: Chapter 1: Basic Quantities
Previous: Linear Elements and Experimental
Ruye Wang
2009-02-02