Snapshot 1: a shorted five-turn coil is placed upright beside a third-derivative Gaussian pulse field
Snapshot 2: a five-turn coil is connected to a
detection circuit for the field of a third-derivative Gaussian
Snapshot 3: an increased-turn (30) coil resonates with the capacitor of a
The Gaussian function
and its derivatives (i.e.
, ⋯) are often used to represent temporal changes. Assuming the applied magnetic field
to be one of those waveforms, the transient current and voltage induced in a circular coil are analyzed. In all of those cases, the maximum is normalized to 1 A/m, and the origin is shifted to
, for example, as
. According to Faraday's law, the electromotive force (EMF) in a circular coil is given by
being the radius and number of turns and
being the angle between field and coil axis. Assuming a circular section of coil of diameter
, the coil's inductance is approximated by
. Then the transfer function of current or voltage in the detection circuit can be obtained. Let
be the indicial response of the system; the actual response can be calculated by Duhamel's integral
can be obtained by a Laplace transform.
The coil current produces a magnetic field that is superimposed on the original field. The additional magnetic field from the ring current can be calculated in terms of Bessel functions. The magnetic field pattern is readily obtained as a function of time.
Among the four circuits considered, the detected current increases in this order: open circuit,
, and shorted conditions in general. The detected voltages vary in the opposite order. However, in the detection circuit of
, a large current and voltage can occur due to resonance between the coil inductance and the capacitor in the detection circuit, as shown in Snapshot 3.
 J. D. Jackson, Classical Electrodynamic
s, New York: John Wiley & Sons, 1998.
 A. L. Shenkman, Transient Analysis of Electric Power Circuit Handbook
, New York: Springer, 2005.