Biomedical Engineering Reference
In-Depth Information
z
z
B
0
B
0
υ
L
t = 0
M
0
t > 0
M
0
φ
y
y
B
1
x
x
(a)
(b)
FIGURE 16.33
(a) Initial alignment of magnetization with the static magnetic field
B
0
. (b) Tipping of the net
magnetization from the
z
-axis through an angle
f
by the application of the rf
B
1
field aligned along the
x
-axis
causes
M
0
to spin about the
z
-axis at the Larmor frequency.
0
. One reason for using this frame is that the trajectories of these spinning, preces-
sing magnetic vectors are complicated, and simplified methods are helpful in visualizing
them. Another reason is that a bundle of frequencies is usually involved in these measure-
ments, and it is easier to track their movements relative to the reference frequency
such as
x
v
L
.
Rotating Frame
The approach is called a rotating frame, a coordinate frame of reference that rotates with
the magnetization vector whirling at a Larmor frequency,
, in contrast to a reference point
in a fixed Cartesian coordinate system. This methodology is similar to a stroboscopic view-
point: To make a rotating object appear still, a stroboscopic light flashes in synchronism with
each revolution of the object. This light frequency serves as a reference so if the revolving
object rotates faster or slower, its deviation from this reference frequency is easily seen. Simi-
larly, a frequency rotating clockwise relative to
v
L
in a rotating frame is called a positive fre-
quency, and a frequency rotating counterclockwise is called a negative frequency.
Recall case 3 from Section 16.3.2, a description of a configuration where a signal was
detected from a permanent magnet spinning around the
v
L
z
-axis. The resulting detected sig-
nal was a sinusoidal signal with a frequency
. Instead of a permanent magnet, consider a
time-varying magnetization vector spinning at a Larmor frequency about the
o
z
-axis in the
x
-
y
plane at a position initially aligned with the
x
-axis at time
t
:
B
x
ð
t
Þ¼
xB
1
cos
o
t
ð
16
:
59
Þ
To find out what this magnetic field vector would look like in a rotating frame, a coordi-
nate transformation is applied to obtain the rotated components of the field in the new
frame, which are part of a new vector
B
0
1
with components
B
x
0
¼
cos
ðo
t
Þ
B
x
sin
ðo
t
Þ
B
y
ð
16
:
60a
Þ
B
y
0
¼
sin
ðo
t
Þ
B
x
þ
cos
ðo
t
Þ
B
y
ð
16
:
60b
Þ
which with the substitution of Eq. (16.58) becomes
B
x
0
¼
B
1
cos
2
o
t
ð
16
:
60c
Þ
B
y
0
¼
B
1
cos
o
t
sin
o
t
ð
16
:
60d
Þ
