MOGlabs DLC202, Manual, Page: 58 / 82

48 Appendix C. Modulation coils
where n is the number of turns per unit length and i the current. For
wire diameter 0.4 mm, n = 2500 m
− 1
, and the current requirement is
only 22 mA / MHz.
C.2 Coil impedance
However, driving an oscillating current through a coil is problematic
because the impedance grows with the frequency. The impedance
is given by
X
L
= ωL where ω is the radial frequency and L the
inductance. The inductance for a long solenoid is
L = µ
0
n
2
Al (C.2.3)
where A is the cross-section area of the coil ( πr
2
for a circular cross-
section) and l is the coil length. In practice, the inductance will be
less (e.g. see Wheeler [9]):
L
Wheeler =
N
2
r
2
228 r + 254 l
(mH) (C.2.4)
where N is the total number of turns, r is the coil radius in metres,
and l is the length in metres ( l > 0 . 8 r ). We have found that for
dimensions typical of coils wound around vapour cells, these two
formulae agree within a factor of two.
Note that the inductance increases with n
2
whereas the magnetic
field and hence modulation depth grows with n ; thus for our pur-
poses, we generally prefer small n and large currents. On the other
hand, the driving voltage requirement (the “back emf”) is given by
ε = −L
di
dt
ε
max =
Li
0
ω (C.2.5)
for a sinusoidal current of amplitude i
0
. The required output slew
rate is
dV /dt = −L
d
2
i
dt
2
Max ≡ Li
0
ω
2
. (C.2.6)