Open access peer-reviewed chapter - ONLINE FIRST

# Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging

By Gabriele Barbaraci

Submitted: June 10th 2019Reviewed: July 11th 2019Published: March 13th 2020

DOI: 10.5772/intechopen.88561

## Abstract

A mathematical discussion is introduced to describe the receiver coil characterizing the NMRI system starting from a general shape of the conductor. A set of different inductance calculations have been introduced varying the shape of the conductor. The inductance calculation led to a general expression of the magnetic field of a single coil characterized by a rectangular shape. A dynamic model of the receiver coil has been developed to represent the natural frequencies that characterize the operational bandwidth. A nonstationary control strategy is implemented to make a real-time changing of the operational bandwidth. The frequency response of the coil generates the necessary conditional expression in order to let the peak of resonance move to a desired value of frequency.

### Keywords

• magnetic resonance
• inductance
• magnetic field
• transfer function

## 2. RF coils

In this section the RF coils are introduced. The RF coils have usually two tasks: produce the magnetic field pulse and receive the signal from the magnetization vector. As mentioned before, in this chapter, the receiving performance of the RF coil will be discussed. The RF coils must be able to provide high SNR and maximize the power of electrical current signal generated by the magnetic flow produced by the rotation of magnetization vector during the FID. The noise which they undergo is generated because of the inductance phenomena between one loop and another. The inductance is a function of current and magnetic flow linking the winding wire characterizing the coil. The cross section of wires can assume two different shapes, circular or square , and they will be discussed in this section by using mathematical analysis that allows a comparison between them.

### 2.1 Transmitter and receiver coils

The excitation of hydrogen nuclei placed inside the human body is produced by a magnetic field that magnetizes the nuclei along the direction of the magnetic field B0z. The field B0zis the magnetic field directed towards the direction feet to head of the patient, determining the magnetic polarization of the human body. The field B0zis produced by the uniform axial magnetic field and by the gradient coil. The transmitter coil produces a magnetic field called B1t, directed along the transversal direction to the magnetic field B0z, and is a function of time because of the impulse characteristic that has a function Sincωtwhere ω is the Larmor frequency. The resonance occurs because frequency ω of the impulse is the same of the oscillation frequency of hydrogen protons. To capture the oscillations, the receiver coil is introduced and characterized by several electrical parameters such as resistance, inductance, and capacitive components associated also to supplementary circuits providing current circulation that works as filter. The resistances are represented by copper wire or by the resistors; the inductance is produced by the mutual and auto effect of coils crossed by current. The capacitors are introduced in the receiver coil as common electrical components with a specific value according to the desired operative bandwidth. The most difficult part in designing a receiver coil is to model the inductance according to the shape of the coil. The inductance modeling and the knowledge of the operative bandwidth will lead to the dynamic mathematical model of the receiver coil from where the control system will be built up.

#### 2.1.1 Self-inductance of and finite length wire

The phenomenon of self-induction occurs every time there is the presence of a magnetic flux that is concatenated to the circuit where the current that generates that flow flows. This flow generates an EMF which opposes the causes that generated it causing alteration of voltage along the branches of the circuit. The case in which the wire is a conductor having a diameter 2rw will be analyzed to then study the case of a conductor with a rectangular amplitude cross section w. The hypothesis that is made in this study is that the current is uniformly distributed over the whole cross section of the conductor; this allows us to treat the problem as the one inherent to a filament as shown in Figure 1 .

In Figure 1 the wire has a finite length l where the surface for the calculation of the flow is given by one side equal to the length of the wire, while the other is of infinite value. Recall that inductance LΦis given by the ratio between the flow produced by the magnetic field produced by the current flowing in that length wire l and the current that generates that same field as described by (1):

LΦ=ΦBiBE1

To develop the expression of self-inductance, the expression of the magnetic field is developed in two variables describing the distance transverse to the wire and that along the wire itself as shown in Figure 2 .

From Biot-Savart’s law, it has:

dB=μ0I4πR2sinϑdzE2

The distance R is calculated from the wire element; the point can be represented as:

R=Zz2+r2E3

where

sinϑ=rRE4

Therefore, the magnetic field produced by a wire crossed by a certain current and having a total length equal to L in all points of space is given as:

B=μ0I4πL2L2rZz2+r23/2dzE5

By changing variables as Zz=λ,=dz, it has the integral (6):

B=μ0Ir4πZL2Z+L21λ2+r23/2E6

from where the expression of magnetic field:

BrZ=μ0I4πrZ+L/2r2+L/2+Z2ZL/2r2+L/2Z2uϑE7

This vector generates a flow through the surface shown in Figure 1 as:

ΦB=r=rwZ=l/2l/2BdZdrE8

that returns the expression in (9):

ΦB=μ0I2πLlogeLrw+Lrw2+11+rwL2+rwLr=rwrE9

from where the auto-inductance is:

LΦ=μ02πLlogeLrw+Lrw2+11+rwL2+rwLE10

By using the property, it is defined as:

sinh1Lrw=logeLrw+Lrw2+1E11

Eq. (10) is defined as:

LΦ=μ02πLsinh1Lrw1+rwL2+rwLE12

Eq. (10) can be simplified in the hypothesis that Lrw:

LΦapprox.=μ02πLLoge2Lrw1E13

In case the wire has a rectangle cross section with wtas shown in Figure 3 ,

the expression of self-inductance is :

LΦ=μ02π1w2[Lw2loge(Lw+(Lw)2+1)+L2wloge(wL+(wL)2+1)+13(L3+w3)13(L2+w2)3/2]E14

that, when simplified according to the assumption wL, returns :

LΦapprox.=μ02πLLoge2Lw+12E15

In Figures 4 and 5 , a comparison between the exact expression of inductance and the approximated one, respectively, for round cross-section wire and rectangle is shown:

In the same figures, it is possible to see how the values of the characteristic cross section higher than 0.01 m of the approximated relation deviate from the exact one tending to reach a significant difference. All curves shown in Figures 4 and 5 are characterized also by a decreasing of inductance by increasing the characteristic dimension of the cross section. This happens because in the expression of inductance, the ratio is between the magnetic field flow and current, so by increasing the distance from the wire, the magnitude of the magnetic field decreases and the flow as well:

#### 2.1.2 Mutual inductance between two parallel finite length wires

For the calculation of the mutual inductance, it is assumed that the parallel wires have the same length and are aligned as shown in Figure 6 . Figure 6.Scheme to calculate the partial mutual inductance for two parallel finite length wires.

The calculation remains the same with the only difference that the integration interval changes as y=d+rw. Eq. (16) shows the general expression of the mutual inductance :

MΦ=μ02πLLogeLd+rw+Ld+rw2+11+d+rwL2+d+rwLE16

that based on the condition drwreturns the approximated expression:

MΦapprox.=μ02πLLogeLd+Ld2+11+dL2+dLE17

In Figure 7 , the variation of Eq. (2.16) as a function of d and rw has been shown. The inductance decreases with an increasing of the geometrical parameters characterizing the dimension of the cross section of wire and their mutual distance. In the same figure, the inductance as a function of wire’s ray shows a slower rate than the inductance as a function of the distance. In Figure 8 , the comparison between the exact and approximated expression of the inductances has been shown. For drwthe difference is significant. In the same figure, it is clear also that an increasing of d equal to d=10×rwand the two expressions tend to become coincident validating the assumption drw.

In case of rectangular cross-section wire with wt, the expression is shown in Eq. (18) obtained by imposing d=win Eq. (16) .

MΦlin=μ02πLLogeLd+w+Ld+w2+11+d+wL2+d+wLE18

In Figure 9 the pattern of Eq. (18) describing the inductance produced by a rectangular cross section wire has been shown. The inductance decreases by increasing the width of pads keeping the same fashion of the magnetic field pattern of a partial self-inductance shown in Figure 5 . Moreover, in Figure 10 a comparison between the round cross section and the circular one has been reported. In the same figure, it is clear how the rectangular cross section exhibits a higher value of the inductance for very low mutual distance between the pads. Again, in Figure 9 the two patterns tend to become coincident by increasing the mutual distance. This happens because for infinite distance the two wires can be represented as a filament no matter what their own shapes are.

In this section the self-inductance and mutual inductance of finite length wires crossed by a uniformly distributed current and characterized by different geometrical parameters have been calculated. In the next section, the partial mutual inductance for parallel wires located in different axial positions will be shown.

#### 2.1.3 Partial mutual inductance of two parallel offset wires

The calculation of the mutual inductance produced by two parallel wires like those ones shown in Figure 11 is reported as per Eq. (19) :

M˜Φ=μ04π[z2sinh1(z2d)z1sinh1(z1d)(z2m)sinh1(z2md)+(z1m)sinh1(z1md)z22+d2+z12+d2+(z2m)2+d2(z1m)2+d2]E19

In Figure 12 , the graphical representation of the inductance of two parallel wires located axially at a certain distance by s and orthogonally by d is shown. In the same figure, it is possible to see that in general the inductance decreases by increasing the geometrical distance since it depends on the magnetic field flow. However, the curves describing the inductance as a function of the distance d exhibit a decreasing rate that is higher by decreasing the axial distance between the wires which is more than the decreasing rate of the inductance as function of the axial distance by varying the axial distance along the orthogonal direction to the wire itself.

The combination of square roots and inverse hyperbolic sine function produces a double concave surface asymptotically tending to the infinity:

#### 2.1.4 Mutual inductance of wires run by current and lying on the same line

The system shown in this paragraph is shown in Figure 13 where both wires of finite length crossed by a generic amount lay coaxially:

The mutual inductance generated by such a configuration is described by Eq. (20) :

MΦ=μ04π[(l+s+m)sinh1l+s+mrw(m+s)sinh1m+srw+(l+s)sinh1l+srw+ssinh1srw(l+s+m)2+rw2]E20

that has been shown in Figure 14 where the inductance as a function of the axial distance between the edges of two wires has been reported:

In Figure 14 the inductance exhibits a decreasing of the inductance by increasing the distance s between two coaxial wires. This is the same concept that has been shown in the previous figures as demonstration that in case of the presence of current in a specific conductor pattern, there will always be an influence between those patterns coupled to a decreasing of inductance by increasing the distance between two conductors. These patterns can be considered as partial, characterized by an insulated wire for a specific length, or they can be coils constituting a closed loop. In this last case, it is necessary to consider the single contribution coming from each side of the closed loop in terms of auto-inductance and mutual inductance.

The receiving coil links the signal coming from the magnetization vector that rotates in space. This rotation generates an electromotive force that in the presence of an electrical resistance produces the electrical current. The current represents a signal characterized by a certain frequency content that processed through the discrete Fourier transform allows the images’ generation. The circuit elements must also be able to store the electrical energy produced by a signal that in general can be decomposable in sinusoidal functions. For this reason, the receiving coils are designed in different configurations according to the frequency band of interest. It is for this founded reason the receiving coils are designed in three different possible ways: low-pass coils, high-pass coils, or band-pass coils. Two of the most applied configurations are the so-called birdcage coil and the phased array coils as shown, respectively, in Figures 15 and 16 .

In Figure 15 the coils are characterized by two rings joined by columns and rings, and columns are made of copper material. According to the desired frequency, the capacitors are in different positions of coils as shown in Figure 15 . Figure 16 shows the phase array configuration which is the Cartesian representation of birdcage coils with a small difference, or rather, there is an overlap of the circuits to reduce the electromagnetic coupling between the nearest coils , but for what concerns the filtering capabilities, the capacitors’ placement reflects the birdcage coil according to the high-pass, low-pass, and pass-band filtering. The phased array structure is what will be considered to develop the experimental validation of results. In the next section, we will present first the analysis of a single loop as a derivation of the study developed in this section.

### 3.1 Single loop coil

In this section we determine the inductance of the rectangular loop shown in Figure 17 , whose length is l and width is w. The conductors of the loop are rectangular flat having a width 2rw . We assume that the current I is uniformly distributed across the cross section of the wires, so that with regard to computing the magnetic field from it, the current can be considered to be concentrated in a filament on the axes of those wires. For isolated direct currents (dc) not in proximity to other currents, the current is, in fact, uniformly distributed over the wire cross section. However, for a current that is near other currents, the current in the wire will not be distributed uniformly over the wire cross section. Nearby currents will cause the current to be concentrated on the side of the nearest wire, a phenomenon known as the proximity effect. Proximity effect is usually not pronounced unless the two currents are within about four radii of each other (i.e., one wire will just fit between the two) . To determine the total flux through that loop, we determine the flux through the loop caused by the current of each wire separately and then add the four fluxes:

Lloop=i=14ΦiI=i=14iByzdydzIE21

where Byzis the magnetic field expression Eq. (7) that is Eq. (22) for the vertical wire as per Figure 16 :

Byz=μ0I4πyz+l/2y2+l/2+z2zl/2y2+l/2z2uϑE22

The integral of vertical wires that is developed according to the integration edges as per Eqs. (23) and (24) is:

Φvlwrw=Φ1lwrw+Φ3lwrw=2z=rwl/2l/2rwy=rwwrwByzdydzE23
Φhlwrw=Φ2lwrw+Φ4lwrw=2z=rww/2w/2rwy=rwlrwByzdydzE24

where Φ1lwrw=Φ3lwrwand Φ2lwrw=Φ4lwrw.

According to the integral table  and dividing the total magnetic flow for the current I, the inductance for the entire loop is carried out as:

L(l,,w,,rw)loop=Φv(l,,w,,rw)+Φh(l,,w,,rw)I=μ0π[(rwl)sinh1lrwwrw+(rww)sinh1wrwlrw+(lrw)sinh1lrwrw+(wrw)sinh1wrwrw+rwsinh1rwwrw+rwsinh1rwlrw+2(lrw)2+(wrw)22(wrw)2+(rw)22(lrw)2+(rw)22rwln(1+2)+22rw]E25

In Figure 18 the inductance produced by a closed loop varying its dimension has been shown.

In the same figure, the inductance increases by increasing the area enclosed by the loop. So, the higher is the surface area, the higher is the induction phenomena the magnetization vector generates. An important information is carried out by the frequency response (FR) characterizing the loop that can be considered as an LC circuit having the voltage expression as in Eq. (26):

Vω=jωLlwrwloopjωCxIω=jω2LlwrwloopCx1ωCxIωE26

In Eq. (25) the capacitor is the variable that allows the shifting of peak of resonance (pr) in the FR diagram.

#### 3.1.1 Frequency tuning and magnitude variation

The fact that we are introducing a single loop means there is only one frequency of resonance once the capacitor value is chosen. The value of the resonance ωres=1/LlwrwloopCxthat will produce the singularity in the magnitude diagram in a bode plot according to the transfer function (tf) is described in Eq. (27):

IωVωdB=20LogωCxω2LlwrwloopCx1E27

The FR is carried out as shown in Figure 18 for a loop having a square shape with l=w=101mand rw=3.5×103mvarying the capacity in a certain value and neglecting the resistance effect. In Figure 18 it can be seen that an increasing of capacitive value corresponds to a decreasing of frequency of resonance (fr) value and an increasing of the magnitude at that frequency. The range of frequency where the Bode plot has been shown goes from 10 MHz to 10GHz with a useful signal amplification between 50 MHz and 120 MHz. According to the same figure, the frequency of resonance is always the frontier where on its right the curve is decreasing, while on the left the answer increases. It can be seen that by differentiating the tf in Eq. (27), the condition of these two regions separated by the value ωres=1/LlwrwloopCxis carried out. Eq. (27) establishes the effect of the capacitor is double, or rather, it varies the resonance frequency; it moves up and down the pr, decreasing the power of the receiving signal at high frequency.

In Figure 19 , we can consider the FR referred to a capacitive value of C0=55pFin order to perform some analysis that allows to understand better the dynamic of a single receiver coil. A desired value of fr called ωrthat is found to be at the right of the fr corresponding to the capacitive C 0 that we call ω0=40MHzsatisfies the condition that ωr>ω0. In this case we write the system with C0=1/Lloopω02, Cr=1/Lloopωr2, and Cr=C0+ΔC0rcarrying out the relation as:

ΔC0r=1Lloopω02ωr2ω02ωr2E28

Eq. (27) establishes that in order to reach the desired frequency ωr>ω0, the capacitive value has to decrease since ΔC0r<0. A decreasing of capacitor value implies a decreasing of the magnitude as per Eq. (27); this effect can be seen also by considering the curve describing the pr varying the frequency in Eq. (29) obtained after a differentiation of the tf and substituting the capacitive value as a function of frequency in Eq. (27) and shown in a dashed line joining all the pr in Figure 18 :

Ŵω=12LloopωE29

The manipulation of Eq. (29) with the same methodology that led to Eq. (28) returns the expression shown in Eq. (30):

ΔŴor=12Lloopω0ωrω0ωr<0E30

Eq. (30) establishes that for ωr>ω0the magnitude of the FR decreases or rather Ŵrωr<Ŵ0ω0.

The case where ωr<ω0produces opposite sign in the differences ΔC0rand ΔŴor, but the relation between the ΔC0rand ΔŴorremains the same as in Eq. (31):

ΔŴorΔC0r=12ω0ωrω0+ωrdBFE31

One has the infinitesimal differentiation for ΔC0r0from where it has ωrω0

dŴordC0r=14ωrdBFE32

The receiver coil behaves as a low-pass filter since it tends to amplify the signal having a low frequency in a certain operating frequency established by the possible values that the capacitor might have.

## 4. Conclusions

In this chapter, a meticulous study on the basic design of a receiver coil characterized by a single turning coil has been developed. The study has started by introducing the technique used in calculating the inductance produced by different shapes of wire crossed by a constant value of current. The inductance is a fundamental parameter that affects the performance in static and dynamic conditions, revealing how the intensity of magnetic field produced by the coil may interfere, through the mutual inductance to the other branches of the same coil. The frequency response of a given single coil loop corresponding to a value of capacitor is characterized by a single peak of resonance that shifts varying the capacitor tuned for a nominal frequency ω 0. The frequency resonance shifts are due to the capacitive effect and have as second consequence a variation of the peak of resonance. The variation of resonance frequency is a desired effect the author wants to reach in order to capture a signal having that frequency, while a numerical strategy has been introduced in order to quantify the variation of the magnitude that will affect the output signal produced by the coil.

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Gabriele Barbaraci (March 13th 2020). Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging [Online First], IntechOpen, DOI: 10.5772/intechopen.88561. Available from: