Page 108 - Maxwell House

88 Chapter 2

Clearly, the rotating magnetic fields () alternate continuously the orientation forcing the
±
total magnetic field vector to rotate thereby endlessly regaining the magnetic moment
precession. As a result, this total rotating vector almost literally drags along all magnetic
moments thereby generating the forced precession as Figure 2.7.1c illustrates. In general, such
forced precession occurs with frequency of dragging force, i.e. ≠ , and continues as long
0
as () is applied. If so, the additional rotating component of magnetic moment () of the
± ±
same frequency is generated in ferrite and its total magnetization becomes () = +
Σ
().
±
2.7.4 Permeability of Fully Magnetized Ferrite
With the harmonic component () in operation, the total torque forcefully rotating N spins
±
is () = () x () where () = () + (). Newton’s law dictates that the
Σ
Σ

0
Σ
Σ
Σ
deviation in time of total magnetization vector () should be proportional to the ratio ≅
Σ
/ and torque force, i.e.

= () x () (2.93)
Σ ()
Σ Σ
The frequency domain transition yields
()/ = () � (2.94)
±
Σ
() x () = x ( − ) + () x ()
±
±
Σ
±

0

0
±
Σ
Here = . The last term () x () is the product of two harmonic values and
±

±

contains, evidently, the frequency component like (cos ) . It reflects the fact that the spin
2
precessing is an inharmonious phenomenon, in principle. If so, such nonlinearity can be used
deliberately for highly efficient frequency doubling at level of power where nonlinear elements
as diodes cannot survive. Magnetized ferrites nonlinearity also opens way for design multiple
active elements like power limiters, modulators, etc. but this subject is far beyond the scope of
this book.
So assume that the harmonic components are relatively weak, i.e. |()| ≪ , |()| ≪
0

and the term | () x ()| can be omitted. In such linear approach, the expression (2.94)
±
±
could be written as
= x ( − ) (2.95)
±

0
±
0

±
The frequency note in (2.95) is implied but omitted as obvious. Using (2.92) written as a
phasor () = � ± � we can find that x = ∓ and the same
±

0
±
±
0
for . Therefore,
±
0
= − and = + = �1 − � (2.96)
±
±
±
0
0
±
±
±
0 ∓ 0 ∓
The expression (2.96) proves the fact mentioned before that the interaction EM wave with bias
ferrite medium depends on how the wave magnetic field is polarized. The frequency dependable
coefficient relating and should be taken as the polarization dependable permeability
±
±

= �1 − � (2.97)
0
±
0 ∓
We derived all expressions ignoring the fact that the alternating magnetic field that supports the
forced spin precessing should transfer part of its energy to spinning electrons at any frequency.