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RF Coils for MRI(VERY BASIC) THEORY AND (A FEW) PRACTICAL

CONSIDERATIONS

Outline RF Coil Theory

Model of an RF coil & Impedance Resonance Q Matching

Practical Considerations - Measurements Matching and Tuning

S11 Smith Chart

Measuring Q S21

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When you input a magnetic field, B1, at the Larmor frequency(an RF field generated by an RF coil ), you can tip themagnetization vector into the x-y plane, or transverse plane

Turn off the transmitter: The magnetization continues to precess atthe Larmor frequency in the transverse plane and this movingmagnetic field can be detected by a pickup coil. Transverse magnetization can be detected

D =90o

Longitudinal magnetization is not detectedwith the RF coil

RF excitation getting a detectable signal

A note on the Rotating FrameA frame of reference that isrotating at the Larmorfrequency. i.e. x and y axesare rotating at f0 and z=z.

Rotating frame

RF CoilsTHEORY In MRI and MRS, a radiofrequency (RF) coil, or resonator, is

designed to produce and/or detect a time-varyingmagnetic flux density, , at the Larmor frequency for ourmain field strength, f0= B0

Produce: Current in the coil produces . Themagnetization vector undergoes forced precession aboutthe transmitted in order to create a detectabletransverse component

Detect: As the magnetization vector precesses in thetransverse plane, it creates a time-varying magnetic fieldwhich induces a current in the coil according to FaradaysLaw of Induction, presenting a voltage at the terminals ofthe coil.

1B

1B

1B

RECIPROCITY

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Any RF coil produces magnetic flux in proportion tothe current

Biot-Savart Law

Energy is stored in the magnetic flux, obtained byintegrating B1 over all space

Inductance, L, is the ratio of the total stored energyto the current squared

We model an RF coil as an inductor able to store energy inits magnetic field created by the current running through it

RF CoilsTHEORY: Inductance, L

dvBWIWL

vm

m 212

121herew2

Inductance, L, and capacitance, C, describe a componentsability to store energy

Resistance, R, describes the energy dissipated in acomponent

The opposition the inductance presents to a change of currentis called inductive reactance, XL

The total impedance of the inductor contains a resistivecomponent

The opposition capacitance presents to a change of current iscalled capacitive reactance, Xc

The total impedance of the capacitor contains a resistivecomponent

Notes on impedance:

LX L

LLL jXRZ

CX C 1

CCC jXRZ

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We want max current able to flow in our coil in order to Generate the max B1 during transmit Have the max current (Icoil) induced during receive

RF CoilsTHEORY: Resonance

Max current => need minimum impedance [to flow of Icoil](purely real input impedance) (RESONANCE)

=> need to cancel the inductive reactance of the coil

Resonate a coil by adding a capacitor with value such that XC = -XL

Lcoil Icoil

Rcoil

Lcoil Icoil,RCR

Rcoil,tot

Impedance presented to Icoil :Zcoil = RL+jXL = Rcoil + jXL,coil

Zcoil,R = RL+RC+jXL+jXC= Rcoil,tot + jXL- jXL=Rcoil,tot

Add a known capacitor Lcoil is set

=> Coil will resonate at fres where Xc=-XL i.e. -1/C = - L

RF CoilsTHEORY: Resonance

fres

I coil

freq In practice, use a tunable capacitor to control the resonance frequency

Implications (howcan we~calculate CR?)

LfC

CfL

LCf

resR

rescoil

res

2

2

)2(1

)2(1

21

Lcoil IcoilCR

Rcoil,tot

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Q: quality factor

Higher Q indicates resistive losses are low andSignal-to-Noise Ratio (SNR) is high

RF CoilsTHEORY: Resonance & Coil Q

RLQ

QSNR1 R

BSNR

RI

LI

cycleperlostenergyinductorinstoredenergyQ

2

2

2122

More intuitive form more easily visualized:

fres

I coil

0.707f3dB

dB

res

ffQ

3

RF CoilsTHEORY: Resonance & Coil Q

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We have to get energy in and out of the coil: itneeds to be connected to the amplifier and/orpreamp.

Usually the transmitter and receivers are 50systems, so we want to impedance matchthe resonated coil to Z = 50 + j0 Ohms.

RF CoilsTHEORY: Matching

Lcoil IcoilCR

Rcoil,tot

ZIN

MatchingNetwork

There are many options for the matching network. The simplest is to make CRvariable and use it as part of the matching network.*i.e. What was CR is now variable, and called C1, the tuning capacitorTwo elements are required to match Rcoil+jXcoil to 50 + j0.

The first element, C1, the tuning cap, transforms Rcoil to 50 ohms.The second element, C2, the matching cap, cancels the remaining reactance

Note: C1 is typically very close to CR, which should be within the range of the variable capacitor C1.

RF CoilsTHEORY: Matching

IcoilMatchingNetwork

C1

C2

Lcoil

Rcoil,tot

Z2=Zin = 50+j0 Z1= 50+jX Zcoil= Rcoil+jXcoil

C1In practice

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There are many options for the matching network. The simplest is to make CR variable anduse it as part of the matching network.Two elements are required to match Rcoil+jXcoil to 50 + j0.

The first element, C1, the tuning cap, transforms Rcoil to 50 ohms.The second element, C2, the matching cap, cancels the remaining reactance

Note: C1 is typically very close to CR, which should be within the range of the variablecapacitor C1.

RF CoilsTHEORY: Matching

Impedance curves with CR:Larmor frequency = Resonant frequency

Impedance curves with C1:Larmor frequency Resonant frequency butR = 50 at Larmor frequency, with remaining reactance

f0=fR f0fR but Z=50+jX

In practice, we can determine C1 and C2 by:1) Finding CR as above, (see calculations on slide 8)2) Choosing a suitable variable cap, C1, encompassing CR3) Finding a suitable C2 by trial and error- guided by experience

Or, we can mathematically solve for C1 and C2 giventhe coil impedance

1) Solve for C1 from 50 = Re{C1 || (Rcoil + jLcoil)} (analytically [next slide],graphically, or iteratively)

2) Then, 1/(jC2) = -jX, where X is the reactive part of Z1 after step 1

RF CoilsTHEORY: Matching

Icoil,RMatchingNetwork

C1

C2

Lcoil

Rcoil

Z2=Zin = 50+j0 Z1= 50+jX

Note that there are two values of C1 thatresult in Re{Z1} = 50. We choose the onethat gives a positive value for X, simplybecause we dont like to use inductors in thematching network. (Why?)

Zcoil= Rcoil+jXcoil

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Analytical Solution for C1 (step 1 on previous slide):

RF CoilsTHEORY: Matching

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)-R(150

:50toequalpartrealset the0,Rwhencasefor thethen.algebra...

)(1

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1

Rc

LbRLa

wherea

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Z1

http://www.fritz.dellsperger.net/

Downloads

Smith V2.03

RF CoilsTHEORY: Matching (& tuning)

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Outline RF Coil Theory

Model of an RF coil & Impedance Resonating Q Matching

Practical Considerations - Measurements Matching and Tuning

S11 Smith Chart

Measuring Q S21

Practical considerations These are RF phenomena cannot use voltmeter for

measurements; need NETWORK ANALYZER, or spectrum analyzer

Relevant Network Analyzer measurements Matching and tuning:

using S11 and the Smith Chart Measuring Q: using S21

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Matching/Tuning: S11 S11 : the input port voltage reflection coefficient

S11= 20 log () or S11= 10 log (Pref/Pinc)

S-Parameter, S11 Configuration on the NA, single port measurement Set your marker at your desired resonance frequency. Then --

incident

reflected

VV

When a coil is matched and tuned,this point will be minimized on thenetwork analyzer log mag/phasedisplay option.

Note S11 value. Need this to be-20dB or better

Matching makes the dip deeperand tuning moves the dip infrequency

Matching/Tuning: S11 & the Smith Chart

The Smith Chart provides a graphical aid to visualizeimpedances

The Smith Chart display option for the S11 configuration onthe NA provides a nice visual tool for matching and tuning

When a coil is matched andtuned, the marker you set at theresonance frequency will be atthis point on the Smith Chart,representing Z=50+j0

Matching makes the circle biggerand smaller. Tuning moves themarker around the circle.

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Measuring Q: S21 S21 : the voltage transmission coefficient

S21= 20 log ()

S-Parameter, S21 Configuration on the NA,two port measurement

Measuring Q: Coil RESONATED (not match/tuned) Connect one pickup loop to port 1 of the NA and

one to port 2 The peak transmission (S21 in dB) between the

two pickup probes will be at the coil resonance Recall Q=fres/f3dB S21 is a power measurement thus the

bandwidth 3dB down from the peak can bemeasured directly from the curve

incident

dtransmitte

VV

Measuring Q: S21

NETWORK ANALYZER

Port 2

Port 1

S21 display

Notes:

It is important thatthe probes do notcouple to one anotherwithout the deviceunder test present

Also important thatthe probes do notperturb the resonancecircuit of the deviceunder test

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VNA 2180 demo

Referenceshttp://submissions.mirasmart.com/ismrm2013/proceedings/isv7/main.htm