Digital Signal Processing Reference
In-Depth Information
−
E
0
M
0
cos
(α)
sin
(θ)
K
=
M
0
sin
(α)
sin
(φ)
B
0
−
B
0
E
0
e
j
(φ
−
θ)
⇒
K
=
tan
(α)
−
E
0
M
0
cos
(α)
e
j
(
−
γ
B
0
t
+
θ)
−
B
0
E
0
e
j
(φ
−
θ)
⇒
M
xy
(
t
)
=
tan
(α)
B
0
e
j
(
−
γ
B
0
t
+
φ)
⇒
M
xy
(
t
)
=
M
0
sin
(α)
Note that, the transverse component is not changed due to the external field
E
(
t
)
.
What we achieved is that the transverse magnetic moment
M
xy
(
t
)
due to the external
field
E
is in phase as that of the transvere component obtained using the static
magnetic field
B
0
. The resultant transverse magnetic moment due to
B
0
and
E
(
t
)
(
t
)
is
given as
e
j
(
−
γ
B
0
t
+
φ)
M
xy
(
t
)
=
2
M
0
sin
(α)
(2.22)
E
The effect of the external magnetic moment
(
t
)
on the
z
-component of the magnetic
M
(
)
moment
t
is obtained by solving (
2.19
) as shown below.
dM
z
(
)
t
=
γ(
M
x
(
)
E
y
(
)
−
M
y
(
)
E
x
(
))
t
t
t
t
dt
=
γ
(α)(
(
−
γ
+
θ)
(
−
γ
+
φ)
M
0
E
0
sin
sin
B
0
t
cos
B
0
t
−
(
−
γ
+
θ)
(
−
γ
+
φ))
cos
B
0
t
sin
B
0
t
=
γ
M
0
sin
(α)
E
0
sin
(
−
γ
B
0
t
+
θ
−
(
−
γ
B
0
t
+
φ)
Thus
dM
z
(
t
)
=
γ
M
0
sin
(α)
E
0
sin
(θ
−
φ)
(2.23)
dt
⇒
M
z
(
t
)
=
γ
M
0
sin
(α)
E
0
sin
(θ
−
φ)
t
+
M
z
(
0
)
(2.24)
Applying the initial condition
M
z
(
0
)
=
M
0
cos
(α)
in (
2.24
), we get
M
z
(
t
)
=
γ
M
0
sin
(α)
E
0
sin
(θ
−
φ)
t
+
M
0
cos
(θ)
. Recall that the
z
-component due to
B
0
is
M
0
cos
(α)
and hence resultant
z
-component is obtained as follows
M
z
(
t
)
=
γ
M
0
sin
(α)
E
0
sin
(θ
−
φ)
t
+
2
M
0
cos
(α)
(2.25)
It is also noted that the resultant
z
-component with external field
B
1
(instead of
B
0
) is given as
M
z
(
t
)
=
γ
M
0
sin
(α)
E
0
sin
(
−
γ(
B
1
−
B
0
)
+
θ
−
φ)
t
+
2
M
0
cos
(α)
(2.26)
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