Declaration of Conflict of Interest or Relationship Speaker Name: Greig Scott Consultant for Boston Scientific. Our lab also receives funding from General Electric Healthcare. I have no conflicts of interest to disclose with regard to the subject matter of this presentation
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Declaration of Conflict of Interest or Relationship
Speaker Name: Greig Scott
Consultant for Boston Scientific. Our lab also receives funding from General Electric Healthcare.
I have no conflicts of interest to disclose with regard to the subject matter of this presentation
Theory of RF Reciprocity
Greig Scott
Department of Electrical Engineering, Stanford,
PMRIL Stanford Electrical Engineering
Topics
• Intro to Reciprocity• Faraday Law & Reciprocity• Circular Polarization Mathematics• Equivalent Sources• Lorentz Reciprocity• Applications
What is Reciprocity?
Left hand polarized
Response to a source is unchanged when source and measurer swapped.
What is Reciprocity?
Left hand polarized
Response to a source is unchanged when source and measurer swapped.
Series Resistor Capacitor
)(ωI
+- Cj
IVω
=AjIdVωε
2=ε)(ωI
E1=J/σσd1
d2
AIJ /=
E2=J/jωε
Assume ejωt so fields match electric circuit convention
A
)(ωIAj
IdVωε
2=ε )(ωI
E1=J1=0σd1
d2 E2=i/jωεA
+-Cj
IVω
=
Response is unchanged when source & observer are swapped
Time Domain Convolution
∫ −= τττ dthitv )()()(
)(ωI +
-
)(1
)( ωω
ω IRCj
RV+
=
)()( ttI δ= )(th
Frequency Domain Output: Product of input & filter response.
Time Domain Output: convolution of input and impulse response.
To receive, a quadrature coil must create a field rotating opposite the direction of precession.
-90o
¼ cycle time delay
-90o
¼ cycle time delay
B M
-1
Transmit Receive
+
Constructing Polarized Fields
yyyxxxxy ththtΗ aa )cos()cos()( θωθω +++=
( ) ( ) −+ −++= aaΗ yxyx jy
jx
jy
jxxy ejhehejheh θθθθ
21
21
yx jaaa −=+yx jaaa +=−
yj
yxj
xxyyx eheh aaH θθ +=][)( tj
xyxy eetH ωHℜ= where
Time Domain:
Frequency Domain Phasor:
Circularly Polarized Phasor:
+ve/right hand -ve/ left hand
The Principle of Reciprocity in Signal Strength Calculations, D.I. Hoult, Concepts Magn. Reson. 12:173,2000.
yjHHm yx ⇒∗−ℑ ]2/)[(xjHHe yx ⇒∗−ℜ ]2/)[(
( )
( ) yxxyy
xyyxx
yx
hh
hh
jHHe
a
a
a
θθ
θθ
sincos21
sincos21
]2/)([
−
++
⇒−ℜ −
In Electromagnetics :
In MRI (Hoult):
Imaginary part gives y component
Extract x, y components from real part of complex vector
Reciprocity Tool Kit
• Impressed Electric Current Source• Impressed Magnetic Current Source• Impressed Magnetic Dipole• Lorentz Reciprocity Theorem• Rumsey Reaction Integral
Electric Current Source Ji
iJEjH ++=×∇ )( ωεσHjE ωµ=×∇−
Ji is an impressed current element independent of field
δz zoi rrIJ a)( −= δ
Induces E, H J=σED=εE
B=µH
E
H
Ji
∫∫∫ −=⋅=⋅=⋅ IVdzEIdvIdzEdvJEVV
i )(rδ+
-
EV
Can compute a voltage across Ji from a field E!
N Port Impedance Matrix
aiJ
biJ
ciJ
diJ
Wherever we impress a current, we create current source port
aI
bI
cIdI
+
-
+
-
+-
+ -
ZIV =
ba
V
ai
b VIdvJE −=⋅∫Electric fields integrated over this source reduce to a port voltage.
Magnetic Current Source Ki
EjH )( ωεσ +=×∇
iKHjE +=×∇− ωµ
Ki = impressed magnetic current independent of field
δz zooi rrMjK a)( −= δωµ
Induces E, HJ=σE
D=εEB=µH
E
H
Ki
A magnetic current element can be physically created by a time varying magnetic dipole.
Ki
ioi MjK ωµ~
Voltage Source Concept
biK
ciK
diK
Magnetic current looping a wire creates a voltage source port
aV
bV
cVdV
YVI =
ab
V
ai
b VIdvKH −=⋅∫A loop of K induces an emf in the wire like a transformer.
+
-
+
-
+-
+ -
aiK
wire
Magnetic dipole toroid K
Lorentz Reciprocity Theorem
ai
aa JEjH ++=×∇ )( ωεσai
aa KHjE +=×∇− ωµ
bi
bb JEjH ++=×∇ )( ωεσbi
bb KHjE +=×∇− ωµ
aiK
aiJ
Exp. A: electric current source Jia,
magnet current source Kia
biK
biJ
Exp. B: electric current source Jib,
magnet current source Kib
0)(∫ =⋅×−×S
abba ndSHEHE
Reaction in Reciprocity
ai
aa JEjH ++=×∇ )( ωεσai
aa KHjE +=×∇− ωµ
bi
bb JEjH ++=×∇ )( ωεσbi
bb KHjE +=×∇− ωµ
aiK
aiJ
Exp. A: electric current source Jia,
magnet current source Kia
biK
biJ
Exp. B: electric current source Jib,
magnet current source Kib
∫ ∫ ⋅−⋅=⋅−⋅V V
ai
bai
bbi
abi
a dvKHJEdvKHJE )()(
Beware! σ, ε, µ are actually tensors and must be symmetric.
Reaction: <a,b> <b,a>
NMR Reciprocity CaseExp. A: electric current filament Ji
a Exp. B: magnetic current Kib=jωµoM
∫ ∫ ⋅−=⋅=−V V
bio
aai
bab dvMjHdvJEIV ωµcoil voltage
+ stuff0
aiJ
dtdMK
bi
obi µ=Unit current
I(ω) bV
+
- dtdj ↔ω
If symmetric σ, ε, µ
Vesselle et al, IEEE Trans. Biomed. Eng. 42, 497,1995
Ibrahim, T., Magn. Reson. Med. 54, 677, 2005
aaai EHJ ,→ bbb
io HEMj ,→ωµ
NMR Reciprocity CaseExp. A: electric current filament Ji
a Exp. B: magnetic current Kib=jωµoM
dvMjHIV bio
V
aab ωµ⋅−=− ∫
aaai EHJ ,→ bbb
io HEMj ,→ωµ
aiJ
dtdMK
bi
obi µ=Unit current
I(ω) bV
+
-( ) ++−−++ ⋅+=⋅ aaa mHHMH aab
ia
dvmHI
jVVa
ob+−∫=
ωµ2
Field rotating opposite precession determines sensitivity
( ) 0=⋅ ++++ aa mH
Time Domain Reciprocity
∫ ⋅−=t
io d
ddMtHtV
0
)()()( ττ
τµτ
)(tI δ=
Let a unit current impulse δ(t) generate the time varying field Hxy(t).
The receive signal is convolution of the field impulse response with the time varying magnetization
)(tV dtdM
oµ)(tH xy+
-
Reciprocal Media
HB µ=
=
z
y
x
zzyzxz
yzyyxy
xzxyxx
z
y
x
HHH
BBB
µµµµµµµµµ
=
z
y
x
z
y
x
z
y
x
HHH
BBB
µµ
µ
000000
scalar
x
y
z
x
y
z
x
y
zPrincipal axes align x,y,z Principal axes arbitrary
Reciprocal Media have symmetric material property tensors. Reciprocity is satisfied if material is reciprocal.
T][][ µµ = 0][][ =⋅−⋅ abba HHHH µµIf then
Circuit Reciprocity
)(1 ωi
+- )(2 ωv
+=
2
1
2
1
11
11
ii
CjCj
CjCjR
vv
ωω
ωω
2221
1211
zzzz
1i 2i
2v1v
2112 zz =In reciprocal circuits, Zmn = Znm for n!=m
When Can Reciprocity Fail?
Left hand polarized Left hand
polarized
ionosphereq-
Earth magnetic field
Cyclotron motion
Hmm? What if I’ve got charged ions moving in a magnetic field? Or electron spin acting like a gyroscope, or NMR?
Gyrotropic Media
−=
z
y
x
oz
y
x
HHH
jj
BBB
µµκκµ
0000
Material tensors not symmetric so reciprocity is not satisfied.
Foundations for Microwave Engineering, Robert E Collin
Bloch equation describes electron motion in ferrites!
Phenomena & Applications
• Surface coil sensitivity asymmetry• Guide wire artifact patterns• Reversed Polarization• RF Current Density Imaging• Electric Properties Tomography
Transmit & Receive Asymmetry
Reciprocity & Gyrotropism in magnetic resonance transduction, James Tropp, Phys. Rev. A, 74, 062103, fig 3, 2006
Different Excitation & Reception Distributions with a Single Loop Transmit-Receive Surface Coil near a head-sized spherical phantom at 300 MHz, C.M. Collins et al, MRM, 47, 1026, fig 2, 2002
3T7T
Guidewire Artifacts
15° flip Ross Venook
Transmit and Receive coupling of a guidewire to a body coil creates two distinct null locations