Mathematics
1,2
5. Energy Storage
Efficiency in an Inductor
The ability of
an inductor to store energy allows it to be used for many useful applications,
voltage converters being one example. Since real inductors are not ideal, it is
useful to know how the energy storage efficiency of an inductor varies as the
current builds toward its steady state value.
Consider the circuit
of fig 5.1. The resistance R represents the total circuit resistance. It can be
shown that the current and induced voltage obey exponential functions as given
by equations 5.1 and 5.2 respectively. Using these relationships we can determine
the efficiency of the inductor energy storage as a function of time.
Fig
5.1. Inductor storage circuit.
The energy stored
by the inductor is the time integral of the product of the current and induced
voltage. This is described by equation 5.3 
Performing the
integration leads to the relationship of Eqn 5.4.
Initial conditions
of i = 0 at t = 0 are used to establish the constant of integration for eqn 5.4.
In a similar vein
we can develop an equation that describes the energy supplied by the source. This
is simply the time integral of the supply power as shown in eqn 5.5.
The integration
yields eqn 5.6.
Again the constant
of integration is obtained from the initial conditions: i = 0 at t = 0.
The efficiency
is simply given by dividing eqn 5.4 by 5.6. We can plot the efficiency as shown
in fig 5.2.
Fig
5.2. Inductor storage efficiency.
The
behaviour is plotted against time units of .
This is done to illustrate that the behaviour is independent of specific values
of Vs, L, or R, when viewed in this way. For example, the storage efficiency will
always be 54.3 % after 1 time constant.
It
is interesting to note that in order to obtain a high storage efficiency, the
ON time of the voltage applied to the inductor must be only a small fraction of
a time constant.
