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.

)(tI

Fourier transform

pair

)(ωH

BMdt

dM×= γ

zBoγω −=

1H, 31P, 19F: γ>0, negative (left hand) angular velocity

17O, electrons : γ< 0, positive (right hand) angular velocity

Spin Precession

Faraday Law & Reciprocity

∫ ⋅−=V

dvdtdemf MB

dtddlEemf Φ

−=⋅= ∫

I = 1

Hoult & Richards, J Magn. Reson, 24:71, 1976for vector potential proof: Haacke et al, Magnetic Resonance Imaging: Physical Principles and Sequence Design, pg 97-99, 1999

M

B

Quasistatic approximation of B. EMF imply loop contour integral.

Real Coils aren’t Simple Loops

Which loop do we use for contour integration?

How do I include matching capacitors, inductors etc?

Circular Polarizationyx jaaa −=+ yx jaaa +=−

yx

tjj

tteee

aaa

)sin()cos(][

φωφω

ωφ

+++=ℜ +

yx

tjj

tteee

aaa

)sin()cos(][

φωφω

ωφ

+−+=ℜ −

0=⋅ ∗+ −

aa 2=⋅−+ aa0=⋅

−− aa

y

x

y

φ

KEY PROPERTIES

xφ

Right hand Left hand

0=⋅ ++ aa

Quadrature Field

Chen, Hoult & Sank: Quadrature Detection Coils…J. Magn. Reson., 54:324,1983

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.

T][][ µµ ≠ abba HHHH ][][ µµ ⋅≠⋅If then

−=

z

y

x

z

tt

tt

z

y

x

EEE

jj

DDD

εεηηε

0000

Magnetized plasma (electrons undergo cyclotron motion

Magnetized ferrite

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

experiment simulation

Reversed RF PolarizationForward Polarization Reversed Polarization

• MR signal is created only by one circular polarization• This has been exploited for wireless catheter tracking

Haydar Celik et al., MRM 58:1224, 2007.

Pacemaker Lead

Forward Polarization Reversed Polarization

Linear polarized fields generated by conducting structures

Electric Properties Tomography

U Katscher et al, Proc ISMRM 14, 3035, 2006

U Katscher et al, Proc ISMRM 15, 1774, 2007

σ

raw

ε ε

qoq HjjH )(2 ωεσωµ +=∇++ +=∇ HjjH o )(2 ωεσωµ

−− +=∇ HjjH o )(2 ωεσωµ

CHALLENGE: Independently measure H+ and H-

q=x,y,z

Summary

• Lorentz Reciprocity Theorem central to deriving NMR signal detection.

• Impressed current dipole creates H.• Magnetic dipole induces V at current

dipole location.• Rotating phasor a+ “reacts” with a-• Time domain convolution requires

time reversal of one field.

Problems

• I need final freq domain formula showing H+ and m-, to go with time domain convolution, and show H- dot m- gives 0.

• Want time domain to show as picture the applied impulse and then the voltage response.

• See if can make picture of B- and B+ (just as in my paper!)

• Photocopy the Lorentz proof from a textbook.• Can I get a dyadic green’s function plot?• Add impedance definitions, power definitions &

computation

NMR Reciprocity Case

ai

aaa J

dtdDJH ++=×∇

dtdBE

aa =×∇−

dtdDJH

bbb +=×∇

dtdM

dtdBE

bi

o

bb µ+=×∇−

Exp. A: electric current filament Jia Exp. B: magnetic current Ki

b=jωµoM

∫ ∫ ⋅−=⋅V V

bi

oaa

ib dv

dtdMHdvJE µ)(

coil voltage

+ stuff0

Warning: Not time domain YET! Assume ωjdtd

=

aiJ

dtdMK

bi

obi µ=Unit current

I(ω) bV

+

-

Reciprocity Theorems

• Response to a source is unchanged when source and measurer swapped

• Relate response at one source due to a second source to the response at the second source due to the first source.

PMRIL Stanford Electrical Engineering

What is Reciprocity?

Left hand polarized Left hand

polarized

ionosphereq-

Earth magnetic field

Cyclotron motion

Response to a source is unchanged when source and measurer swapped.

Hmm? What if I’ve got charged ions moving in a magnetic field? Or electron spin acting like a gyroscope, or NMR?

Time Domain Interpretation

yx tttm aa )sin()cos()( ωω −=−

yx tttH aa )sin()cos()( ωω −=−

yx tttH aa )sin()cos()( ωω +=+

0=⋅−−−− aa Hm 0)()()(

0

→−⋅= −−∫ τττ dtHmtyt

0≠⋅ ++−− aa Hm )cos()()()(0

tdtHmtyt

ωτττ →−⋅= +−∫

Time convolution is zero for same sense of rotation.

Time convolution yields cos(ωt) for opposing rotation.

Ignored “Stuff”

[ ] dvdt

dBHdt

dBHdvJEJEdvdt

dDEdt

dDEa

bb

aabbaa

bb

a ∫∫∫

⋅−⋅+⋅−⋅+

⋅−⋅

( )abba EEEEj ][][ εεω ⋅−⋅ ( )abba EEEE ][][ σσ ⋅−⋅ ( )abba HHHHj ][][ µµω ⋅−⋅

ED ][ε= EJ ][σ= HB ][µ=

If [ε], [σ], [µ] are symmetric tensors, all terms are zero!

0 0 0

Reciprocal Media have symmetric material property tensors.

NMR Reciprocity CaseExp. A: electric current filament Ji

a Exp. B: magnetic current Kib=jωµoM

∫ ∫ ⋅−=⋅=−V V

bio

aai

bab dvMjHdvJEIV ωµcoil voltage

+ stuff0

ai

aa JEjH ++=×∇ )( ωεσaa HjE ωµ=×∇−

bb EjH )( ωεσ +=×∇bio

bb MjHjE ωµωµ +=×∇−

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

Rotating Frame Components

Scott et al, Trans. Med. Imag. 14, 515, 1995