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Encyclopaedic Glossary
Bessel’s equation of zero order,
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d 2 y 1 dy
+ ⋅
+ k 2 y = 0 , is:
dx 2 x dx
y = A ⋅ J 0 (kx ) + B ⋅ Y0 (kx ) , where A and B are constants required to satisfy the boundary conditions, k
is a real constant {not imaginary or complex}, and Y0 (
)
is a Bessel function of the second kind. The
notation, name and value of Bessel functions of the second kind are remarkably inconsistent
between authors. The function can also be called a Neumann function and can be written as N 0 ( ) .
Since Y0 (0 + ) → − ∞ , boundary conditions often mean that the constant B is zero in the general
solution so that the Neumann function is not required.
d 2 y 1 dy
+ ⋅
− k 2 y = 0 , the general solution is y = A ⋅ I 0 (kx ) + B ⋅ K 0 (kx ) ,
dx 2 x dx
is the modified Bessel function of the first kind and K 0 ( ) is a modified Bessel function
For the differential equation
where I 0 (
)
of the second kind. Again it is found that K 0 (0 + ) → ∞ .
For an angle-modulated signal, VO = sin (ωc t + θ ⋅ sin[ω mt ]) , the carrier amplitude is
J 0 (θ ) and
amplitude of the first sideband pair (offset ±ω m from the carrier) is J 1 (θ ) . The first sidebands and
carrier are equal when J 0 (θ ) = J 1 (θ ) , θ = 1.4347 . When θ = 2.4048 , J 0 (θ ) = 0 , meaning that the
carrier is completely suppressed (zero).
BESSEL-THOMPSON FILTER 8 … is a set of maximally flat group delay filters (linear phase) for
use on time domain circuits. The rolloff in the frequency domain is not as fast as Butterworth,
Chebyshev and Elliptic filters, but those all overshoot in the time domain, whereas the BesselThompson does not. It originated in 1949, whereas the Butterworth originated in 1930.
BIFILAR WINDING … is a winding technique used to reduce inductance. Wrap a piece of wire
many times around a bobbin and you have made an inductor. Take the same length of wire folded
back on itself, making the go and return paths next to each other, and there is very little ‘loop area’
between the wires. Winding this doubled wire around a bobbin gives very little increase in
inductance. However, this method greatly increases the capacitance between the ends of the wire.
Regions of large potential difference between conductors give large amounts of self-capacitance.
For a low value resistor (say <10 Ω) the self-inductance will limit the high frequency response. For a
higher value of resistor (say >1 kΩ) the self-capacitance will limit the high frequency response. Thus
bifilar winding is only appropriate on low value resistors.
A better bifilar technique exists, however. Consider winding a wire as a clockwise helix around a
cylindrical former. If a second wire is now wound over the top of the first wire, but as an anticlockwise
helix, the magnetic fields will tend to cancel when the two wires are connected in parallel.
Furthermore, since the wires will be at the same potential where they cross, the self-capacitance will
be minimised. Using a flat former rather than a cylindrical former further improves the results and is
known as the Ayrton-Perry construction.
BIOT-SAVART LAW: The discovery by Oersted (also written as Örsted) in 1820 that a currentcarrying conductor influenced a magnetic compass needle opened up a whole new field of
exploration. Within the next few weeks the basis of the modern theories was discovered and
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published, Ampère, Biot & Savart being the key explorers.
Biot and Savart did their first experimental work using a magnetic compass needle and a long
straight wire. A long bar magnet was used to cancel out the effect of the earth’s magnetic field. The
actual measurement of the magnetic force was done using the period of oscillation of a small
8
W.E. Thomson, 'Delay Networks Having Maximally Flat Frequency Characteristics', in Proceedings of the
IEE, 96 part 3 (1949), pp. 487-490.
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R.A.R. Tricker, Early Electrodynamics, The First Law of Circulation (Pergamon, 1965).