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

*Gabriele Barbaraci*

## **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

## **1. Introduction**

For a NMRI system to work, a set of operations must be completed in a time order. The first operation is characterized by choosing the point of the human body that must be analyzed; determining the resonance magnetic field which must be associated to that point; the RF magnetic field pulsing sequence; phase encoding process; and data extraction processed by a digital microprocessor proving a high computational burden. In this chapter, one step characterizing the NMRI system that represents the most important physical effect involving a dynamic aspect which is the transmitter coil will be discussed. In the NMRI systems, it is strictly necessary to have a transmitter coil and a receiver coil, respectively, to excite the magnetization vector inside the human body and transform in electrical current signal the magnetic field variation coming from the magnetization vector. The modern NMRI exhibit a more compact design characterized by associating to one coil the task to generate the RF pulse and receive the signal. This is the reason why in the literature and in the real operating NMRI, the use of a term that refers to a coil performing both tasks called trans-receiver is very common. In this chapter, a model of receiver coil will be studied by focusing the attention in the receiving task because of the importance that is related to the image's quality. The receiver coils can be considered as a filter that allows to capture signal's frequency of interest. As a filter, the receiver coil can be a low-pass, high-pass, or band-pass filter. The challenge of all NMRI manufacturers and of most of the scientific researchers is to

improve the image's quality because of its importance in early-stage diagnosis and make visible the blood vessels. The improvement of image's quality can occur by eliminating the noise and at the same time having a wider spectrum. The images' quality can be also improved by increasing the signal-to-noise ratio (SNR) as a variable of a design which the receiver coil is related to. The SNR calculation in magnetic resonance imaging (MRI) coil arrays is a powerful tool in the development of new coil arrays. A proposed method describes a model that allows the calculation of the absolute SNR values of arbitrary coil arrays [1]. Another sophisticated strategy is to implement a filter that can estimate the current signal no matter what the noise is. This filter, called also estimator, is commonly used in the control system theory to capture those values of state's vector, whereas a sensor is of difficult implementation. The most common estimator provides a very accurate reconstruction of the missing signal or the signal affected by an amount of noise characterized by a certain variance. Typically, the *Kalman* filter and *Luenberger* observer can provide an excellent reconstruction of the signal [2]. The receiver coils might have different shapes and dimensions according to the specific part of the bod that must be analyzed. The shape can be also optimized to reduce the noise or to minimize the magnetic field coming from the RF pulse produced by the transmitter coil. The shape can be planar which is distributed as an array [1] or a solenoid shape [3, 4]. In high magnetic field pulse NMRI, the risk of inhomogeneity is very high; this leads to an alteration of the current signal carried out from the receiver coils. A classical finite difference time domain (FDTD) coupled with a transverse electromagnetic resonator provided the use of high magnetic field pulse in image extraction [5]. The dimension of a receiver coil may vary from order of magnitude like the NMRI system up to small dimension like a human finger. In some cases, micro-coils are used for NMR microscopy for the possibility to capture a signal not affected by noise and at the same time have a large bandwidth of mode signal [6–9]. A use of a simulation has characterized the design of NMRI system since the computers with strong computational capabilities have been built. The use of a simulator has also characterized the study of NMRI receiver coil and in some cases has constituted a valid tool to investigate on the coil performance over a range of different tuning, allowing the engineers in saving an enormous amount of time [10]. A good frequency resolution has been reached by using an optimization algorithm of strip-line chip [11]. This chip has been built up in a silicon substrate, and its geometry was modeled in respect of RF-homogeneity, sensitivity, and spectral resolution. This model allows to achieve a resolution of 0.7 *Hz*, leading to more visibility of all the harmonic during the DFT analysis. Another shape of receiver coils is the so-called birdcage coil [2, 3] that works also as a transmitter coil. This shape is characterized by a circular loop array having specific axial length and diameter approximately equal to the space where the patient is introduced. An interesting technique in improving the performance of a receiver coil is to use a sliding tuning of the ring characterizing the birdcage coil configuration. The sliding tuning allows to vary the capacity's values and so also the operative bandwidth [12]. The modern technique in NMRI receiver coil is the use of stacked resonator. This configuration is characterized by a multiple-connection of receiver coil one to another to cover the entire body's surface. This allows to maximize the magnetic energy the human body releases during the free induction decay (FID) [13, 14]. A method that develops an optimization of RF coil is based on the algorithm called method of moment that leads to a set of different geometry. This method allows to include in the design the desired bandwidth operation of the signal coming out from the human body. This is a very good strategy in designing RF coils, but it does not allow to vary in real time the bandwidth and so be adapted in several operational conditions [15, 16]. The value of current covers an extreme importance in designing RF coils since the power

of the signal and so the image quality depend on it. A free space time-harmonic electromagnetic *Green*'s functions and de-emphasized magnetic field pulse target fields are used to calculate the current density on the coil cylinder to avoid an overemphasis of some areas in the human body that do not allow to see in a clear way the details [17]. One of the most used techniques is to increase the value of uniform magnetic field since it provides a higher resonance frequency and so a higher energy transmitted from the body to the antennas. The modern values of NMRI system can reach a magnetic field equal to 7 T [18]. It has been demonstrated that a high value of magnetic field might not be a good solution for humans having an age up to 5 years old since the magnetic field can cause nerve contraction or temporary disease. This is the reason why a suitable RF coil design is the most accepted strategy particularly in those cases where it is necessary to increase the details without exceeding a certain value of magnetic field. The RF coils are typically shaped according to the surface where they must be located; for example, there can be one for the head, hand, and leg [19, 20]. An innovative method in the analysis of RF coils is represented by the application of the boundary element method with regularization technique. This method represents an effective approach for solving the electromagnetic forward problem and includes biplanar transverse gradient coils and RF phased array coils [21, 22]. With the consideration of the practical engineering requirements, physical constraints such as wire intervals are transformed into mathematical constraints and formulated into BEM equations. The examples demonstrate that the proposed method is efficient and flexible for the design of MRI coils with arbitrary geometries and engineering constraints. The modern NMRI system has also the possibility to work with low and high magnetic field values to make them more adapted to a wide range of clinical cases. In order to increase the quality of the images, there has to be an optimization to two different aspects or rather the transmission mode where RF coils must be able to produce a uniform magnetic field in the volume of interest so that the nuclei can be properly excited; the receiving mode, a SNR, is needed, and the coil must be able to collect the signal emitted by the nuclei with better sensitivity throughout the volume of interest. A strategy to increase the SNR has been proposed in [23] by

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

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

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 [24], and they will be discussed in this section by using mathe-

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 *B*

! 0*z*.

dropping the temperature up to 77 K.

**2.1 Transmitter and receiver coils**

matical analysis that allows a comparison between them.

**2. RF coils**

**13**

#### *Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

of the signal and so the image quality depend on it. A free space time-harmonic electromagnetic *Green*'s functions and de-emphasized magnetic field pulse target fields are used to calculate the current density on the coil cylinder to avoid an overemphasis of some areas in the human body that do not allow to see in a clear way the details [17]. One of the most used techniques is to increase the value of uniform magnetic field since it provides a higher resonance frequency and so a higher energy transmitted from the body to the antennas. The modern values of NMRI system can reach a magnetic field equal to 7 T [18]. It has been demonstrated that a high value of magnetic field might not be a good solution for humans having an age up to 5 years old since the magnetic field can cause nerve contraction or temporary disease. This is the reason why a suitable RF coil design is the most accepted strategy particularly in those cases where it is necessary to increase the details without exceeding a certain value of magnetic field. The RF coils are typically shaped according to the surface where they must be located; for example, there can be one for the head, hand, and leg [19, 20]. An innovative method in the analysis of RF coils is represented by the application of the boundary element method with regularization technique. This method represents an effective approach for solving the electromagnetic forward problem and includes biplanar transverse gradient coils and RF phased array coils [21, 22]. With the consideration of the practical engineering requirements, physical constraints such as wire intervals are transformed into mathematical constraints and formulated into BEM equations. The examples demonstrate that the proposed method is efficient and flexible for the design of MRI coils with arbitrary geometries and engineering constraints. The modern NMRI system has also the possibility to work with low and high magnetic field values to make them more adapted to a wide range of clinical cases. In order to increase the quality of the images, there has to be an optimization to two different aspects or rather the transmission mode where RF coils must be able to produce a uniform magnetic field in the volume of interest so that the nuclei can be properly excited; the receiving mode, a SNR, is needed, and the coil must be able to collect the signal emitted by the nuclei with better sensitivity throughout the volume of interest. A strategy to increase the SNR has been proposed in [23] by dropping the temperature up to 77 K.

## **2. RF coils**

improve the image's quality because of its importance in early-stage diagnosis and make visible the blood vessels. The improvement of image's quality can occur by eliminating the noise and at the same time having a wider spectrum. The images' quality can be also improved by increasing the signal-to-noise ratio (SNR) as a variable of a design which the receiver coil is related to. The SNR calculation in magnetic resonance imaging (MRI) coil arrays is a powerful tool in the development of new coil arrays. A proposed method describes a model that allows the calculation of the absolute SNR values of arbitrary coil arrays [1]. Another sophisticated strategy is to implement a filter that can estimate the current signal no matter what the noise is. This filter, called also estimator, is commonly used in the control system theory to capture those values of state's vector, whereas a sensor is of difficult implementation. The most common estimator provides a very accurate reconstruction of the missing signal or the signal affected by an amount of noise characterized by a certain variance. Typically, the *Kalman* filter and *Luenberger* observer can provide an excellent reconstruction of the signal [2]. The receiver coils might have different shapes and dimensions according to the specific part of the bod that must be analyzed. The shape can be also optimized to reduce the noise or to minimize the magnetic field coming from the RF pulse produced by the transmitter coil. The shape can be planar which is distributed as an array [1] or a solenoid shape [3, 4]. In high magnetic field pulse NMRI, the risk of inhomogeneity is very high; this leads to an alteration of the current signal carried out from the receiver coils. A classical finite difference time domain (FDTD) coupled with a transverse electromagnetic resonator provided the use of high magnetic field pulse in image extraction [5]. The dimension of a receiver coil may vary from order of magnitude like the NMRI system up to small dimension like a human finger. In some cases, micro-coils are used for NMR microscopy for the possibility to capture a signal not affected by noise and at the same time have a large bandwidth of mode signal [6–9]. A use of a simulation has characterized the design of NMRI system since the computers with strong computational capabilities have been built. The use of a simulator has also characterized the study of NMRI receiver coil and in some cases has constituted a valid tool to investigate on the coil performance over a range of different tuning, allowing the engineers in saving an enormous amount of time [10]. A good frequency resolution has been reached by using an optimization algorithm of strip-line chip [11]. This chip has been built up in a silicon substrate, and its geometry was modeled in respect of RF-homogeneity, sensitivity, and spectral resolution. This model allows to achieve a resolution of 0.7 *Hz*, leading to more visibility of all the harmonic during the DFT analysis. Another shape of receiver coils is the so-called birdcage coil [2, 3] that works also as a transmitter coil. This shape is characterized by a circular loop array having specific axial length and diameter approximately equal to the space where the patient is introduced. An interesting technique in improving the performance of a receiver coil is to use a sliding tuning of the ring characterizing the birdcage coil configuration. The sliding tuning allows to vary the capacity's values and so also the operative bandwidth [12]. The modern technique in NMRI receiver coil is the use of stacked resonator. This configuration is characterized by a multiple-connection of receiver coil one to another to cover the entire body's surface. This allows to maximize the magnetic energy the human body releases during the free induction decay (FID) [13, 14]. A method that develops an optimization of RF coil is based on the algorithm called method of moment that leads to a set of different geometry. This method allows to include in the design the desired bandwidth operation of the signal coming out from the human body. This is a very good strategy in designing RF coils, but it does not allow to vary in real time the bandwidth and so be adapted in several operational conditions [15, 16]. The value of current covers an extreme importance in designing RF coils since the power

*Nuclear Magnetic Resonance*

**12**

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 [24], 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 *B* ! 0*z*. The field *B* ! <sup>0</sup>*<sup>z</sup>* is the magnetic field directed towards the direction feet to head of the patient, determining the magnetic polarization of the human body. The field *B* ! <sup>0</sup>*<sup>z</sup>* is produced by the uniform axial magnetic field and by the gradient coil. The transmitter coil produces a magnetic field called *B* ! <sup>1</sup>ð Þ*t* , directed along the transversal direction to the magnetic field *B* ! <sup>0</sup>*z*, and is a function of time because of the impulse characteristic that has a function *Sinc*ð Þ *ωt* where *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 2*rw* 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 *L*<sup>Φ</sup> is 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):

$$L\_{\Phi} = \frac{\Phi\left(\overrightarrow{B}\right)}{i\_{\overrightarrow{B}}}\tag{1}$$

*R* ¼

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

where

**Figure 2.**

**15**

**Figure 1.**

*Self-inductance case study for a finite length wire.*

*Scheme for magnetic field B.*

q

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *Z* � *z*

sin ð Þ¼ *<sup>ϑ</sup> <sup>r</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup>

*<sup>R</sup>* (4)

(3)

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:

$$dB = \frac{\mu\_0 I}{4\pi R^2} \sin\left(\theta\right) dz\tag{2}$$

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

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

$$R = \sqrt{\left(Z - z\right)^2 + r^2} \tag{3}$$

where

The field *B*

!

*Nuclear Magnetic Resonance*

the magnetic field *B*

system will be built up.

coil produces a magnetic field called *B*

!

*2.1.1 Self-inductance of and finite length wire*

inherent to a filament as shown in **Figure 1**.

that along the wire itself as shown in **Figure 2**.

From *Biot-Savart*'s law, it has:

represented as:

**14**

<sup>0</sup>*<sup>z</sup>* is the magnetic field directed towards the direction feet to head of the

<sup>0</sup>*z*, and is a function of time because of the impulse characteristic

<sup>1</sup>ð Þ*t* , directed along the transversal direction to

! <sup>0</sup>*<sup>z</sup>* is

(1)

patient, determining the magnetic polarization of the human body. The field *B*

!

produced by the uniform axial magnetic field and by the gradient coil. The transmitter

that has a function *Sinc*ð Þ *ωt* where *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

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 2*rw* 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

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 *L*<sup>Φ</sup> is given by the ratio between the flow produced by the magnetic field produced by the current flowing in that length wire

> Φ *B* !

> > *i B* !

<sup>4</sup>*πR*<sup>2</sup> sin ð Þ *<sup>ϑ</sup> dz* (2)

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

*l* and the current that generates that same field as described by (1):

*L*<sup>Φ</sup> ¼

*dB* <sup>¼</sup> *<sup>μ</sup>*0*<sup>I</sup>*

The distance *R* is calculated from the wire element; the point can be

$$\sin\left(\theta\right) = \frac{r}{R} \tag{4}$$

**Figure 1.** *Self-inductance case study for a finite length wire.*

**Figure 2.** *Scheme for magnetic field B.*

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:

$$\left|\overrightarrow{B}\right| = \frac{\mu\_0 I}{4\pi} \int\_{-\frac{L}{2}}^{\frac{L}{2}} \frac{r}{\left[\left(Z - z\right)^2 + r^2\right]^{3/2}} dz \tag{5}$$

By changing variables as *Z* � *z* ¼ *λ*, *dλ* ¼ �*dz*, it has the integral (6):

$$\left|\overrightarrow{B}\right| = \frac{\mu\_0 I r}{4\pi} \int\_{Z-\frac{1}{2}}^{Z+\frac{1}{2}} \frac{1}{\left(\lambda^2 + r^2\right)^{3/2}} d\lambda \tag{6}$$

In case the wire has a rectangle cross section with *w* ≫ *t* as shown in **Figure 3**,

þ 1

<sup>2</sup>*<sup>π</sup> L Loge*

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:

For the calculation of the mutual inductance, it is assumed that the parallel wires

The calculation remains the same with the only difference that the integration interval changes as *y* ¼ ½ Þ *d* þ *rw*, ∞ . Eq. (16) shows the general expression of the

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L d* þ *rw* � �<sup>2</sup>

þ 1

1 A �

1 <sup>A</sup> <sup>þ</sup> *<sup>L</sup>*<sup>2</sup>

> þ 1 3

> > 2*L w* þ 1 2

*w Loge*

*w L* þ

*<sup>L</sup>*<sup>3</sup> <sup>þ</sup> *<sup>w</sup>*<sup>3</sup> � � � <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *w L* � �<sup>2</sup>

*<sup>L</sup>*<sup>2</sup> <sup>þ</sup> *<sup>w</sup>*<sup>2</sup> � �<sup>3</sup>*=*<sup>2</sup>

! r

3

� � (15)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*d* þ *rw L* � �<sup>2</sup>

þ

*d* þ *rw L*

3 5

(16)

1 þ

s

þ 1

(14)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L w* � �<sup>2</sup>

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

that, when simplified according to the assumption *w* ≪ *L*, returns [3]:

*<sup>L</sup>*Φ*approx:* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

*2.1.2 Mutual inductance between two parallel finite length wires*

have the same length and are aligned as shown in **Figure 6**.

þ

*L d* þ *rw*

@

0 s

the expression of self-inductance is [24]:

*L w* þ

@

0 s

*Lw*<sup>2</sup> *Loge*

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

*<sup>L</sup>*<sup>Φ</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> 2*π* 1 *w*<sup>2</sup>

*Rectangular cross-section wire.*

**Figure 3.**

mutual inductance [24]:

<sup>2</sup>*<sup>π</sup> L Loge*

2 4

*<sup>M</sup>*<sup>Φ</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

**17**

from where the expression of magnetic field:

$$\overrightarrow{B}(r,Z) = \frac{\mu\_0 I}{4\pi r} \left( \frac{Z + L/2}{\sqrt{r^2 + \left(L/2 + Z\right)^2}} - \frac{Z - L/2}{\sqrt{r^2 + \left(L/2 - Z\right)^2}} \right) u\_\theta \tag{7}$$

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

$$\Phi\left(\overrightarrow{B}\right) = \int\_{r=r\_w}^{\infty} \int\_{Z=-l/2}^{l/2} \overrightarrow{B} \cdot dZ \, dr \tag{8}$$

that returns the expression in (9):

$$\Phi\left(\overrightarrow{B}\right) = \frac{\mu\_0 I}{2\pi} L \left[ \log\_e \left( \frac{L}{r\_w} + \sqrt{\left(\frac{L}{r\_w}\right)^2 + 1} \right) - \sqrt{1 + \left(\frac{r\_w}{L}\right)^2} + \frac{r\_w}{L} \right]\_{r = r\_w}^{r \to \infty} \tag{9}$$

from where the auto-inductance is:

$$L\_{\Phi} = \frac{\mu\_0}{2\pi} L \left[ \log\_{\epsilon} \left( \frac{L}{r\_w} + \sqrt{\left(\frac{L}{r\_w}\right)^2 + 1} \right) - \sqrt{1 + \left(\frac{r\_w}{L}\right)^2} + \frac{r\_w}{L} \right] \tag{10}$$

By using the property, it is defined as:

$$\sinh^{-1}\left(\frac{L}{r\_w}\right) = \log\_e\left(\frac{L}{r\_w} + \sqrt{\left(\frac{L}{r\_w}\right)^2 + 1}\right) \tag{11}$$

Eq. (10) is defined as:

$$L\_{\Phi} = \frac{\mu\_0}{2\pi} L \left[ \sinh^{-1} \left( \frac{L}{r\_w} \right) - \sqrt{1 + \left( \frac{r\_w}{L} \right)^2} + \frac{r\_w}{L} \right] \tag{12}$$

Eq. (10) can be simplified in the hypothesis that *L* ≫ *rw*:

$$L\_{\Phi\text{approx.}} = \frac{\mu\_0}{2\pi} L \left( \text{Log}\_{\epsilon} \frac{2L}{r\_w} - 1 \right) \tag{13}$$

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

#### **Figure 3.** *Rectangular cross-section wire.*

Therefore, the magnetic field produced by a wire crossed by a certain current

ð Þ *Z* � *z*

*r*

<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup>

1

<sup>q</sup> � *<sup>Z</sup>* � *<sup>L</sup>=*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

h i3*=*<sup>2</sup> *dz* (5)

*<sup>λ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup> � �3*=*<sup>2</sup> *<sup>d</sup><sup>λ</sup>* (6)

1

CA

*dZ dr* (8)

3 5

3

*r*!∞

(9)

(12)

(13)

*r*¼*rw*

5 (10)

A (11)

*u<sup>ϑ</sup>* (7)

*<sup>r</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>L</sup>=*<sup>2</sup> � *<sup>Z</sup>* <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L rw* � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ

*rw L* � �<sup>2</sup>

þ 1

þ *rw L*

1

*rw L* � �<sup>2</sup>

*rw L* � �<sup>2</sup> þ *rw L*

þ *rw L*

and having a total length equal to *L* in all points of space is given as:

ð*L* 2 �*L* 2

By changing variables as *Z* � *z* ¼ *λ*, *dλ* ¼ �*dz*, it has the integral (6):

ð*Z*þ*<sup>L</sup>* 2 *<sup>Z</sup>*�*<sup>L</sup>* 2

*<sup>Z</sup>* <sup>þ</sup> *<sup>L</sup>=*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>r</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>L</sup>=*<sup>2</sup> <sup>þ</sup> *<sup>Z</sup>* <sup>2</sup>

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

*r*¼*rw*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L rw* � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L rw* � �<sup>2</sup>

¼ log *<sup>e</sup>*

þ 1

*L rw* þ

�

<sup>2</sup>*<sup>π</sup> L Loge*

r

2*L rw* � 1 � �

" #

@

*rw* � �

0 s

1 A �

ð*<sup>l</sup>=*<sup>2</sup>

*Z*¼�*l=*2 *B* !

þ 1

1 A �

r

r

*B* ! � � � � � � <sup>¼</sup> *<sup>μ</sup>*0*<sup>I</sup>* 4*π*

> *B* ! � � � � � � <sup>¼</sup> *<sup>μ</sup>*0*Ir* 4*π*

from where the expression of magnetic field:

0

B@

Φ *B* � �! ¼ ð<sup>∞</sup>

*L rw* þ

*L rw* þ

@

sinh �<sup>1</sup> *<sup>L</sup>*

*rw* � �

<sup>2</sup>*<sup>π</sup> <sup>L</sup>* sinh �<sup>1</sup> *<sup>L</sup>*

*<sup>L</sup>*Φ*approx:* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

Eq. (10) can be simplified in the hypothesis that *L* ≫ *rw*:

0 s

@

0 s

4*πr*

*B* !

*Nuclear Magnetic Resonance*

Φ *B* � �! ð Þ¼ *<sup>r</sup>*, *<sup>Z</sup> <sup>μ</sup>*0*<sup>I</sup>*

that returns the expression in (9):

<sup>2</sup>*<sup>π</sup> <sup>L</sup>* log *<sup>e</sup>*

from where the auto-inductance is:

<sup>2</sup>*<sup>π</sup> <sup>L</sup>* log *<sup>e</sup>*

By using the property, it is defined as:

*<sup>L</sup>*<sup>Φ</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

2 4

2 4

<sup>¼</sup> *<sup>μ</sup>*0*<sup>I</sup>*

*<sup>L</sup>*<sup>Φ</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

Eq. (10) is defined as:

**16**

In case the wire has a rectangle cross section with *w* ≫ *t* as shown in **Figure 3**, the expression of self-inductance is [24]:

$$L\_{\Phi} = \frac{\mu\_0}{2\pi} \frac{1}{w^2} \left[ Lw^2 \text{Log}\_{\epsilon} \left( \frac{L}{w} + \sqrt{\left(\frac{L}{w}\right)^2 + 1} \right) + L^2 w \text{Log}\_{\epsilon} \left( \frac{w}{L} + \sqrt{\left(\frac{w}{L}\right)^2 + 1} \right) \right] \tag{14}$$

$$+ \frac{1}{3} \left( L^3 + w^3 \right) - \frac{1}{3} \left( L^2 + w^2 \right)^{3/2} \tag{14}$$

that, when simplified according to the assumption *w* ≪ *L*, returns [3]:

$$L\_{\Phiappy.} = \frac{\mu\_0}{2\pi} L \left( L \text{og}\_{\epsilon} \frac{2L}{w} + \frac{1}{2} \right) \tag{15}$$

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**.

The calculation remains the same with the only difference that the integration interval changes as *y* ¼ ½ Þ *d* þ *rw*, ∞ . Eq. (16) shows the general expression of the mutual inductance [24]:

$$M\_{\Phi} = \frac{\mu\_0}{2\pi} L \left[ \text{Log}\_{\epsilon} \left( \frac{L}{d + r\_w} + \sqrt{\left(\frac{L}{d + r\_w}\right)^2 + 1} \right) - \sqrt{1 + \left(\frac{d + r\_w}{L}\right)^2} + \frac{d + r\_w}{L} \right] \tag{16}$$

**Figure 4.** *Partial self-inductance comparison for round wire cross section.*

that based on the condition *d* ≫ *rw* returns the approximated expression:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L d* � �<sup>2</sup>

þ 1

1 A � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*d L* � �<sup>2</sup>

þ *d L*

3

5 (17)

1 þ

s

*<sup>M</sup>*Φ*approx:* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

**Figure 6.**

**Figure 7.**

**19**

<sup>2</sup>*<sup>π</sup> L Loge*

2 4

*Partial mutual inductance for round cross-section wires.*

*L d* þ

*Scheme to calculate the partial mutual inductance for two parallel finite length wires.*

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

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

@

0 s

**Figure 5.** *Partial self-inductance for rectangle wire cross section.*

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

**Figure 6.** *Scheme to calculate the partial mutual inductance for two parallel finite length wires.*

**Figure 7.** *Partial mutual inductance for round cross-section wires.*

that based on the condition *d* ≫ *rw* returns the approximated expression:

$$M\_{\Phi\text{approx.}} = \frac{\mu\_0}{2\pi} L \left[ \text{Log}\_{\epsilon} \left( \frac{L}{d} + \sqrt{\left(\frac{L}{d}\right)^2 + 1} \right) - \sqrt{1 + \left(\frac{d}{L}\right)^2} + \frac{d}{L} \right] \tag{17}$$

**Figure 4.**

*Nuclear Magnetic Resonance*

**Figure 5.**

**18**

*Partial self-inductance for rectangle wire cross section.*

*Partial self-inductance comparison for round wire cross section.*

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 *d* ffi *rw* the difference is significant. In the same figure, it is clear also that an increasing of *d* equal to *d* ¼ 10 � *rw* and the two expressions tend to become coincident validating

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

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

In case of rectangular cross-section wire with *w* ≫ *t*, the expression is shown in

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L d* þ *w* � �<sup>2</sup>

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

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

þ 1

1 A � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*d* þ *w L* � �<sup>2</sup>

þ

*d* þ *w L*

3 5

(18)

1 þ

s

Eq. (18) obtained by imposing *d* ¼ *w* in Eq. (16) [24].

*L d* þ *w* þ

@

0 s

the assumption *d* ≫ *rw*.

*Partial mutual inductance comparison.*

**Figure 10.**

*<sup>M</sup>*Φ*lin* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

be shown.

**21**

<sup>2</sup>*<sup>π</sup> L Loge*

no matter what their own shapes are.

2 4

**Figure 8.** *Partial mutual inductance comparison.*

**Figure 9.** *Partial mutual inductance for rectangular cross-section wires.*

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

**Figure 10.** *Partial mutual inductance comparison.*

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

**Figure 9.**

**20**

**Figure 8.**

*Partial mutual inductance comparison.*

*Nuclear Magnetic Resonance*

*Partial mutual inductance for rectangular cross-section wires.*

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 *d* ffi *rw* the difference is significant. In the same figure, it is clear also that an increasing of *d* equal to *d* ¼ 10 � *rw* and the two expressions tend to become coincident validating the assumption *d* ≫ *rw*.

In case of rectangular cross-section wire with *w* ≫ *t*, the expression is shown in Eq. (18) obtained by imposing *d* ¼ *w* in Eq. (16) [24].

$$M\_{\Phi\text{lin}} = \frac{\mu\_0}{2\pi} L \left[ \text{Log}\_\epsilon \left( \frac{L}{d+w} + \sqrt{\left(\frac{L}{d+w}\right)^2 + 1} \right) - \sqrt{1 + \left(\frac{d+w}{L}\right)^2} + \frac{d+w}{L} \right] \tag{18}$$

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.

*<sup>M</sup>*<sup>~</sup> <sup>Φ</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> 4*π*

Eq. (20) [24]:

*<sup>M</sup>*<sup>Φ</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> 4*π*

**Figure 12.**

**23**

*Partial mutual inductance for two parallel wires.*

*<sup>z</sup>*<sup>2</sup> sinh �<sup>1</sup> *<sup>z</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

þ

q

*d*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *z*<sup>2</sup> � *m*

<sup>þ</sup> ð Þ *<sup>z</sup>*<sup>1</sup> � *<sup>m</sup>* sinh �<sup>1</sup> *<sup>z</sup>*<sup>1</sup> � *<sup>m</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup>

double concave surface asymptotically tending to the infinity:

ð Þ *<sup>l</sup>* <sup>þ</sup> *<sup>s</sup>* <sup>þ</sup> *<sup>m</sup>* sinh �<sup>1</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>s</sup>* <sup>þ</sup> *<sup>m</sup>*

*rw*

finite length crossed by a generic amount lay coaxially:

þð Þ *<sup>l</sup>* <sup>þ</sup> *<sup>s</sup>* sinh �<sup>1</sup> *<sup>l</sup>* <sup>þ</sup> *<sup>s</sup>*

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

� � � *<sup>z</sup>*<sup>1</sup> sinh �<sup>1</sup> *<sup>z</sup>*<sup>1</sup>

*d* � � �

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

q

�

*d*

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *z*<sup>1</sup> � *m*

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

The system shown in this paragraph is shown in **Figure 13** where both wires of

The mutual inductance generated by such a configuration is described by

*rw*

<sup>þ</sup> *<sup>s</sup>* sinh �<sup>1</sup> *<sup>s</sup>*

*rw* �

� � � ð Þ *<sup>z</sup>*<sup>2</sup> � *<sup>m</sup>* sinh �<sup>1</sup> *<sup>z</sup>*<sup>2</sup> � *<sup>m</sup>*

þ

q

� ð Þ *<sup>m</sup>* <sup>þ</sup> *<sup>s</sup>* sinh �<sup>1</sup> *<sup>m</sup>* <sup>þ</sup> *<sup>s</sup>*

q

*rw* �

<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup> *w*

(20)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *l* þ *s* þ *m*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *z*2 <sup>2</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup>

*d* � �

(19)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *z*2 <sup>1</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup>

**Figure 11.** *Physical system for mutual inductance calculation.*

#### *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) [24]:

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

$$\tilde{M}\_{\Phi} = \frac{\mu\_0}{4\pi} \left[ + \left( z\_1 - m \right) \sinh^{-1} \left( \frac{\mathbb{Z}\_1}{d} \right) - \left( z\_2 - m \right) \sinh^{-1} \left( \frac{\mathbb{Z}\_2 - m}{d} \right) \right]$$

$$\tilde{M}\_{\Phi} = \frac{\mu\_0}{4\pi} \left[ + \left( z\_1 - m \right) \sinh^{-1} \left( \frac{\mathbb{Z}\_1 - m}{d} \right) - \sqrt{z\_2^2 + d^2} + \sqrt{z\_1^2 + d^2} \right.$$

$$\left[ + \sqrt{\left( z\_2 - m \right)^2 + d^2} - \sqrt{\left( z\_1 - m \right)^2 + d^2} \right.$$

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) [24]:

$$M\_{\Phi} = \frac{\mu\_0}{4\pi} \left[ \begin{array}{c} (l+s+m)\sinh^{-1}\frac{l+s+m}{r\_w} - (m+s)\sinh^{-1}\frac{m+s}{r\_w} \\\\ + (l+s)\sinh^{-1}\frac{l+s}{r\_w} + s\sinh^{-1}\frac{s}{r\_w} - \sqrt{\left(l+s+m\right)^2 + r\_w^2} \end{array} \right] \tag{20}$$

**Figure 12.** *Partial mutual inductance for two parallel wires.*

*2.1.3 Partial mutual inductance of two parallel offset wires*

*Physical system for mutual inductance calculation.*

*Nuclear Magnetic Resonance*

**Figure 11.**

**22**

those ones shown in **Figure 11** is reported as per Eq. (19) [24]:

The calculation of the mutual inductance produced by two parallel wires like

**3. Receiver coils**

*Partial mutual inductance for two parallel wires.*

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

**Figure 14.**

**Figure 15.**

**25**

*Birdcage coil. (a) Low-pass, (b) high-pass, (c) band-pass.*

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

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

**Figure 13.** *Physical system for mutual inductance calculation.*

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.

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

**Figure 14.** *Partial mutual inductance for two parallel wires.*

## **3. Receiver coils**

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

**Figure 15.** *Birdcage coil. (a) Low-pass, (b) high-pass, (c) band-pass.*

that has been shown in **Figure 14** where the inductance as a function of the axial

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

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

distance between the edges of two wires has been reported:

*Physical system for mutual inductance calculation.*

*Nuclear Magnetic Resonance*

mutual inductance.

**24**

**Figure 13.**

between two conductors. These patterns can be considered as partial,

configurations are the so-called birdcage coil and the phased array coils as shown, respectively, in **Figures 15** and **16**.

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) [24]. 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

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

P<sup>4</sup> *i*¼1 ÐÐ *i B* !

*<sup>z</sup>* <sup>þ</sup> *<sup>l</sup>=*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *y*<sup>2</sup> þ ð Þ *l=*2 þ *z*

ð Þ *y*, *z* is the magnetic field expression Eq. (7) that is Eq. (22) for the

2 <sup>q</sup> � *<sup>z</sup>* � *<sup>l</sup>=*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The integral of vertical wires that is developed according to the integration edges

According to the integral table [26] and dividing the total magnetic flow for the

*w* � *rw*

*w* � *rw*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>* � *rw* <sup>2</sup> <sup>þ</sup> ð Þ *rw* <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>* � *rw* <sup>2</sup> <sup>þ</sup> ð Þ *<sup>w</sup>* � *rw* <sup>2</sup>

*rw*

In **Figure 18** the inductance produced by a closed loop varying its dimension has

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

<sup>þ</sup> *rw* sinh �<sup>1</sup> *rw*

� 2

q

ð*<sup>l</sup>=*2�*rw*

*z*¼*rw*�*l=*2

*z*¼*rw*�*w=*2

ð*<sup>w</sup>=*2�*rw*

ð Þ *y*, *z dydz*

*y*<sup>2</sup> þ ð Þ *l=*2 � *z*

ð*<sup>w</sup>*�*rw y*¼*rw B* !

ð*<sup>l</sup>*�*rw y*¼*rw B* !

<sup>þ</sup> ð Þ *rw* � *<sup>w</sup>* sinh �<sup>1</sup> *<sup>w</sup>* � *rw*

*l* � *rw*

q

� <sup>2</sup>*rw* ln 1 <sup>þ</sup> ffiffi

<sup>þ</sup> ð Þ *<sup>w</sup>* � *rw* sinh �<sup>1</sup> *<sup>w</sup>* � *rw*

*l* � *rw*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>w</sup>* � *rw* <sup>2</sup> <sup>þ</sup> ð Þ *rw* <sup>2</sup>

2 � � <sup>p</sup> <sup>þ</sup> <sup>2</sup> ffiffi

*rw*

2 <sup>p</sup> *rw*

*<sup>I</sup>* (21)

2

1

CA

*u<sup>ϑ</sup>* (22)

ð Þ *y*, *z dydz* (23)

ð Þ *y*, *z dydz* (24)

(25)

*Lloop* ¼

*μ*0*I* 4*πy*

Φ*v*ð Þ¼ *l*, *w*,*rw* Φ1ð Þþ *l*, *w*,*rw* Φ3ð Þ¼ *l*, *w*,*rw* 2

Φ*h*ð Þ¼ *l*, *w*,*rw* Φ2ð Þþ *l*, *w*,*rw* Φ4ð Þ¼ *l*, *w*,*rw* 2

*L l*ð Þ , *<sup>w</sup>*,*rw loop* <sup>¼</sup> <sup>Φ</sup>*v*ð Þþ *<sup>l</sup>*, *<sup>w</sup>*,*rw* <sup>Φ</sup>*h*ð Þ *<sup>l</sup>*, *<sup>w</sup>*,*rw*

þ 2

� 2

q

q

¼ *μ*0 *π*

been shown.

**27**

where Φ1ð Þ¼ *l*, *w*,*rw* Φ3ð Þ *l*, *w*,*rw* and Φ2ð Þ¼ *l*, *w*,*rw* Φ4ð Þ *l*, *w*,*rw* .

current *I*, the inductance for the entire loop is carried out as:

*I*

<sup>þ</sup> *rw* sinh �<sup>1</sup> *rw*

ð Þ *rw* � *<sup>l</sup>* sinh �<sup>1</sup> *<sup>l</sup>* � *rw*

<sup>þ</sup> ð Þ *<sup>l</sup>* � *rw* sinh �<sup>1</sup> *<sup>l</sup>* � *rw*

0

B@

P<sup>4</sup> *<sup>i</sup>*¼1Φ*<sup>i</sup> I* ¼

four fluxes:

where *B* !

vertical wire as per **Figure 16**:

*B* !

as per Eqs. (23) and (24) is:

ð Þ¼ *y*, *z*

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

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 [25], 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 2*rw*. 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

**Figure 16.** *Phased array coil: band-pass.*

**Figure 17.** *Single loop coil.*

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

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) [24]. 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:

$$L\_{loop} = \frac{\sum\_{i=1}^{4} \Phi\_i}{I} = \frac{\sum\_{i=1}^{4} \iint \overrightarrow{B}(y, z) dydz}{I} \tag{21}$$

where *B* ! ð Þ *y*, *z* is the magnetic field expression Eq. (7) that is Eq. (22) for the vertical wire as per **Figure 16**:

$$\overrightarrow{B}(\mathbf{y},\mathbf{z}) = \frac{\mu\_0 I}{4\pi\mathfrak{y}} \left( \frac{\mathbf{z} + \mathbf{l}/2}{\sqrt{\mathbf{y}^2 + \left(\mathbf{l}/2 + \mathbf{z}\right)^2}} - \frac{\mathbf{z} - \mathbf{l}/2}{\sqrt{\mathbf{y}^2 + \left(\mathbf{l}/2 - \mathbf{z}\right)^2}} \right) \mathbf{u}\_\theta \tag{22}$$

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

$$\Phi\_v(l, w, r\_w) = \Phi\_1(l, w, r\_w) + \Phi\_3(l, w, r\_w) = 2 \int\_{x = r\_w - l/2}^{l/2 - r\_w} \int\_{y = r\_w}^{w - r\_w} \overline{B}(y, z) dy dz \tag{23}$$

$$\Phi\_h(l, w, r\_w) = \Phi\_2(l, w, r\_w) + \Phi\_4(l, w, r\_w) = 2 \int\_{x = r\_w - w/2}^{w/2 - r\_w} \prod\_{y = r\_w}^{l - r\_w} \overrightarrow{B}(y, z) dy dz \tag{24}$$

where Φ1ð Þ¼ *l*, *w*,*rw* Φ3ð Þ *l*, *w*,*rw* and Φ2ð Þ¼ *l*, *w*,*rw* Φ4ð Þ *l*, *w*,*rw* .

According to the integral table [26] and dividing the total magnetic flow for the current *I*, the inductance for the entire loop is carried out as:

$$\begin{aligned} L(l, w, r\_w)\_{loop} &= \frac{\Phi\_{\boldsymbol{\nu}}(l, \boldsymbol{w}, r\_w) + \Phi\_{\boldsymbol{h}}(l, \boldsymbol{w}, r\_w)}{I} \\ &= \frac{\boldsymbol{l}}{\pi} \begin{bmatrix} (r\_w - l) \sinh^{-1} \frac{l - r\_w}{w - r\_w} + (r\_w - w) \sinh^{-1} \frac{w - r\_w}{l - r\_w} \\ + (l - r\_w) \sinh^{-1} \frac{l - r\_w}{r\_w} + (w - r\_w) \sinh^{-1} \frac{w - r\_w}{r\_w} \\ + r\_w \sinh^{-1} \frac{r\_w}{w - r\_w} + r\_w \sinh^{-1} \frac{r\_w}{l - r\_w} \\ + 2\sqrt{(l - r\_w)^2 + (w - r\_w)^2} - 2\sqrt{(w - r\_w)^2 + (r\_w)^2} \\ - 2\sqrt{(l - r\_w)^2 + (r\_w)^2} - 2r\_w \ln\left(1 + \sqrt{2}\right) + 2\sqrt{2}r\_w \end{bmatrix} \tag{25}$$

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

configurations are the so-called birdcage coil and the phased array coils as shown,

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 [25], 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

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 2*rw*. 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

loop as a derivation of the study developed in this section.

respectively, in **Figures 15** and **16**.

*Nuclear Magnetic Resonance*

**3.1 Single loop coil**

**Figure 17.** *Single loop coil.*

**26**

**Figure 16.**

*Phased array coil: band-pass.*

**Figure 18.** *Inductance of a rectangular lop varying the side dimension.*

frequency response (FR) characterizing the loop that can be considered as an LC circuit having the voltage expression as in Eq. (26):

$$V(o) = \left[ joL(l, w, r\_w)\_{loop} - \frac{j}{o\epsilon \mathbf{C}\_x} \right] I(o) = j \left[ \frac{o^2 L(l, w, r\_w)\_{loop} \mathbf{C}\_x - 1}{o\epsilon \mathbf{C}\_x} \right] I(o) \tag{26}$$

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* ¼ 1*=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L l*ð Þ , *w*,*rw loopCx* q that will produce the singularity in the magnitude diagram in a bode plot according to the transfer function (*tf*) is described in Eq. (27):

$$\left| \frac{I(o)}{V(o)} \right|\_{dB} = 20Log \left| \frac{oC\_{\rm x}}{o^2 L(l, w, r\_w)\_{loop} C\_{\rm x} - 1} \right| \tag{27}$$

differentiating the *tf* in Eq. (27), the condition of these two regions separated by the

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

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.

*<sup>r</sup>*, and *Cr* ¼ *C*<sup>0</sup> þ Δ*C*0*<sup>r</sup>* carrying out the relation as:

<sup>Δ</sup>*W*^ *or* <sup>¼</sup> <sup>1</sup>

<sup>Δ</sup>*C*0*<sup>r</sup>* <sup>¼</sup> <sup>1</sup>

*Lloop*

Eq. (27) establishes that in order to reach the desired frequency *ω<sup>r</sup>* > *ω*0, the capacitive value has to decrease since Δ*C*0*<sup>r</sup>* < 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**:

*<sup>W</sup>*^ ð Þ¼ *<sup>ω</sup>* <sup>1</sup>

2*Lloop*

The manipulation of Eq. (29) with the same methodology that led to Eq. (28)

*ω*<sup>0</sup> � *ω<sup>r</sup> ω*0*ω<sup>r</sup>*

In **Figure 19**, we can consider the FR referred to a capacitive value of *C*<sup>0</sup> ¼ 55pF in order to perform some analysis that allows to understand better the dynamic of a single receiver coil. A desired value of *fr* called *ω<sup>r</sup>* that is found to be at the right of the *fr* corresponding to the capacitive *C*<sup>0</sup> that we call *ω*<sup>0</sup> ¼ 40MHz satisfies the condition that *<sup>ω</sup><sup>r</sup>* <sup>&</sup>gt;*ω*0. In this case we write the system with *<sup>C</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*=Lloopω*<sup>2</sup>

> *ω*2 <sup>0</sup> � *<sup>ω</sup>*<sup>2</sup> *r*

*ω*2 0*ω*<sup>2</sup> *r*

is carried out. Eq. (27) establishes the effect of the

� � (28)

<sup>2</sup>*Lloop<sup>ω</sup>* (29)

� � <sup>&</sup>lt;<sup>0</sup> (30)

0, *Cr* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L l*ð Þ , *w*,*rw loopCx*

*Frequency response of a single coil with variable capacitor.*

*DOI: http://dx.doi.org/10.5772/intechopen.88561*

returns the expression shown in Eq. (30):

value *ωres* ¼ 1*=*

**Figure 19.**

1*=Lloopω*<sup>2</sup>

**29**

q

The FR is carried out as shown in **Figure 18** for a loop having a square shape with *<sup>l</sup>* <sup>¼</sup> *<sup>w</sup>* <sup>¼</sup> <sup>10</sup>�<sup>1</sup> m and *rw* <sup>¼</sup> <sup>3</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>3</sup> m varying 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

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

**Figure 19.** *Frequency response of a single coil with variable capacitor.*

differentiating the *tf* in Eq. (27), the condition of these two regions separated by the value *ωres* ¼ 1*=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L l*ð Þ , *w*,*rw loopCx* q is 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 *C*<sup>0</sup> ¼ 55pF in order to perform some analysis that allows to understand better the dynamic of a single receiver coil. A desired value of *fr* called *ω<sup>r</sup>* that is found to be at the right of the *fr* corresponding to the capacitive *C*<sup>0</sup> that we call *ω*<sup>0</sup> ¼ 40MHz satisfies the condition that *<sup>ω</sup><sup>r</sup>* <sup>&</sup>gt;*ω*0. In this case we write the system with *<sup>C</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*=Lloopω*<sup>2</sup> 0, *Cr* ¼ 1*=Lloopω*<sup>2</sup> *<sup>r</sup>*, and *Cr* ¼ *C*<sup>0</sup> þ Δ*C*0*<sup>r</sup>* carrying out the relation as:

$$
\Delta C\_{0r} = \frac{1}{L\_{loop}} \left( \frac{\alpha\_0^2 - \alpha\_r^2}{\alpha\_0^2 \alpha\_r^2} \right) \tag{28}
$$

Eq. (27) establishes that in order to reach the desired frequency *ω<sup>r</sup>* > *ω*0, the capacitive value has to decrease since Δ*C*0*<sup>r</sup>* < 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**:

$$
\hat{W}(\alpha) = \frac{1}{2L\_{loop}\alpha} \tag{29}
$$

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

$$
\Delta\hat{W}\_{or} = \frac{\mathbf{1}}{2L\_{loop}} \left(\frac{\alpha\_0 - \alpha\_r}{\alpha\_0 \alpha\_r}\right) < \mathbf{0} \tag{30}
$$

frequency response (FR) characterizing the loop that can be considered as an LC

*I*ð Þ¼ *ω j*

In Eq. (25) the capacitor is the variable that allows the shifting of peak of

The fact that we are introducing a single loop means there is only one frequency

that will produce the singularity in the magnitude diagram in a

� � � � �

m varying the capacity in a certain value and

*ω*<sup>2</sup>*L l*ð Þ , *w*,*rw loopCx* � 1

of resonance once the capacitor value is chosen. The value of the resonance *ωres* ¼

<sup>¼</sup> <sup>20</sup>*Log <sup>ω</sup>Cx*

The FR is carried out as shown in **Figure 18** for a loop having a square shape with

� � � � �

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

bode plot according to the transfer function (*tf*) is described in Eq. (27):

*<sup>ω</sup>*<sup>2</sup>*L l*ð Þ , *<sup>w</sup>*,*rw loopCx* � <sup>1</sup> *ωCx* " #

*I*ð Þ *ω* (26)

(27)

circuit having the voltage expression as in Eq. (26):

� �

*Inductance of a rectangular lop varying the side dimension.*

*3.1.1 Frequency tuning and magnitude variation*

*I*ð Þ *ω V*ð Þ *ω*

m and *rw* <sup>¼</sup> <sup>3</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>3</sup>

� � � � *dB*

� � � � *ωCx*

*<sup>V</sup>*ð Þ¼ *<sup>ω</sup> <sup>j</sup>ωL l*ð Þ , *<sup>w</sup>*,*rw loop* � *<sup>j</sup>*

resonance (*pr*) in the FR diagram.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *L l*ð Þ , *w*,*rw loopCx*

1*=*

**28**

q

**Figure 18.**

*Nuclear Magnetic Resonance*

*<sup>l</sup>* <sup>¼</sup> *<sup>w</sup>* <sup>¼</sup> <sup>10</sup>�<sup>1</sup>

Eq. (30) establishes that for *ω<sup>r</sup>* > *ω*<sup>0</sup> the magnitude of the FR decreases or rather *<sup>W</sup>*^ *<sup>r</sup>*ð Þ *<sup>ω</sup><sup>r</sup>* <sup>&</sup>lt;*W*^ <sup>0</sup>ð Þ *<sup>ω</sup>*<sup>0</sup> .

The case where *ω<sup>r</sup>* <*ω*<sup>0</sup> produces opposite sign in the differences Δ*C*0*<sup>r</sup>* and Δ*W*^ *or*, but the relation between the Δ*C*0*<sup>r</sup>* and Δ*W*^ *or* remains the same as in Eq. (31):

$$\frac{\Delta\hat{W}\_{or}}{\Delta C\_{0r}} = \frac{1}{2} \frac{o\nu\_0 o\nu\_r}{o\nu\_0 + o\nu\_r} \left[\frac{\text{dB}}{\text{F}}\right] \tag{31}$$

**References**

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*DOI: http://dx.doi.org/10.5772/intechopen.88561*

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

[9] Watzlaw J, Glöggler S, Blümich B, et al. Stacked planar micro coils for single-sided NMR applications. Journal of Magnetic Resonance. 2013;**230**: 176-185. DOI: 10.1016/j.jmr.2013.02.013

[10] Kozlov M, Turner R. Fast MRI coil analysis based on 3-D electromagnetic and RF circuit co-simulation. Journal of Magnetic Resonance. 2009;**200**:147-152.

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jmr.2012.05.016

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[12] Qian C, Masad IS, Rosenberg JT, et al. A volume birdcage coil with an adjustable sliding tuner ring for neuroimaging in high field vertical magnets: Ex and in vivo applications at 21.1 T. Journal of Magnetic Resonance. 2012;**221**:110-116. DOI: 10.1016/j.

[13] Georget E, Luong M, Vignaud A, et al. Stacked magnetic resonators for MRI RF coils decoupling. Journal of Magnetic Resonance. 2017;**275**:11-18. DOI: 10.1016/j.jmr.2016.11.012

[14] Ohliger MA, Sodickson DK. An introduction to coil array design for parallel MRI. NMR in Biomedicine. 2006;**19**:300-315. DOI: 10.1002/

[15] Rogovich A, Monorchio A, Nepa P, et al. Design of magnetic resonance imaging (MRI) RF coils by using the method of moments. In: IEEE Antennas and Propagation Society Symposium. 2004. DOI: 10.1109/APS.2004.1329829

[16] Stara R, Fontana N, Alecci M, et al. RF coil design for low and high field MRI: numerical methods and

measurements. In: IEEE Nuclear Science

[11] Bart J, Janssen JWG, van Bentum PJM, et al. Optimization of stripline-based microfluidic chips for high-resolution NMR. Journal of Magnetic Resonance. 2009;**201**:175-185.

[2] Barbaraci G, D'Ippolito F. An estimator algorithm for the rotation time of magnetization vector in nuclear

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Florida: CRC Press; 1999. ISBN 9780849396939- CAT# 9693

[3] Jin J. Electromagnetic Analysis and Design in Magnetic Resonance Imaging.

[4] Idziak S, Haeberlen U. Design and construction of a high homogeneity rf coil for solid-state multiple-pulse NMR. Journal of Magnetic Resonance. 1982;**50**:

[5] Ibrahim TS, Lee R, Baertlein BA, et al. Effect of RF coil excitation on field inhomogeneity at ultra high fields: A field optimized TEM resonator.

Magnetic Resonance Imaging. 2001;**19**:

[6] Peck TL, Magin RL, Lauterbur PC. Design and analysis of microcoils for

Magnetic Resonance, Series B. 1995;**108**:

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jmr.2008.12.020

**31**

1077546318791608

281-288

1339-1347

114-124

One has the infinitesimal differentiation for Δ*C*0*<sup>r</sup>* ! 0 from where it has *ω<sup>r</sup>* ffi *ω*<sup>0</sup>

$$\frac{d\hat{W}\_{or}}{dC\_{0r}} = \frac{1}{4}\alpha\_r \left[\frac{\text{dB}}{\text{F}}\right] \tag{32}$$

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.

## **Author details**

Gabriele Barbaraci University of Palermo, Palermo, Italy

\*Address all correspondence to: gabriele.barbaraci@yorku.ca

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Control Flow Strategy in a Receiver Coil for Nuclear Magnetic Resonance for Imaging DOI: http://dx.doi.org/10.5772/intechopen.88561*

## **References**

Eq. (30) establishes that for *ω<sup>r</sup>* > *ω*<sup>0</sup> the magnitude of the FR decreases or rather

The case where *ω<sup>r</sup>* <*ω*<sup>0</sup> produces opposite sign in the differences Δ*C*0*<sup>r</sup>* and Δ*W*^ *or*,

*ω*0*ω<sup>r</sup> ω*<sup>0</sup> þ *ω<sup>r</sup>*

dB F 

dB F  (31)

(32)

but the relation between the Δ*C*0*<sup>r</sup>* and Δ*W*^ *or* remains the same as in Eq. (31):

¼ 1 2

*dW*^ *or dC*0*<sup>r</sup>*

One has the infinitesimal differentiation for Δ*C*0*<sup>r</sup>* ! 0 from where it has

¼ 1 4 *ωr*

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

In this chapter, a meticulous study on the basic design of a receiver coil charac-

terized 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

Δ*W*^ *or* Δ*C*0*<sup>r</sup>*

*<sup>W</sup>*^ *<sup>r</sup>*ð Þ *<sup>ω</sup><sup>r</sup>* <sup>&</sup>lt;*W*^ <sup>0</sup>ð Þ *<sup>ω</sup>*<sup>0</sup> .

*Nuclear Magnetic Resonance*

values that the capacitor might have.

output signal produced by the coil.

University of Palermo, Palermo, Italy

provided the original work is properly cited.

\*Address all correspondence to: gabriele.barbaraci@yorku.ca

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Author details**

Gabriele Barbaraci

**30**

*ω<sup>r</sup>* ffi *ω*<sup>0</sup>

**4. Conclusions**

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[15] Rogovich A, Monorchio A, Nepa P, et al. Design of magnetic resonance imaging (MRI) RF coils by using the method of moments. In: IEEE Antennas and Propagation Society Symposium. 2004. DOI: 10.1109/APS.2004.1329829

[16] Stara R, Fontana N, Alecci M, et al. RF coil design for low and high field MRI: numerical methods and measurements. In: IEEE Nuclear Science Symposium Conference Record. 2011. DOI: 10.1109/NSSMIC.2011.6152634

[17] Xu B, Wei Q, Liu F. An inverse methodology for high-frequency RF coil design for MRI with de-emphasized B1 fields. IEEE Transactions on Biomedical Engineering. 2005;**52**(9). DOI: 10.1109/ TBME.2005.851514

[18] Abraham R, Ibrahim TS. Humanbody coil design for magnetic resonance imaging at 7 Tesla. In: IEEE Antennas and Propagation Society International Symposium. 2005. DOI: 10.1109/ APS.2005.1551696

[19] Li BK, Xu B, Hui HT, et al. A new approach for magnetic resonance RF head coil design. In: IEEE Engineering in Medicine and Biology 27th Annual Conference. 2005. DOI: 10.1109/ IEMBS.2005.1615624

[20] Morey AM, Bhujade S, Bhuiya T. Design and development of surface coil for 1.5T MRI scanner. In: International Conference on Smart Technologies and Management for Computing, Communication, Controls, Energy and Materials (ICSTM). 2015. DOI: 10.1109/ ICSTM.2015.7225428

[21] Shou G, Xia L, Liu F, et al. MRI coil design using boundary-element method with regularization technique: A numerical calculation study. IEEE Transactions on Magnetics. 2010;**46**(4)

[22] Basari ASH, Rahardjo ET, Zulkifli FY. Eight-channel phased array RF coils design for 3T parallel MRI system. In: IEEE 4th Asia-Pacific Conference on Antennas and Propagation (APCAP). 2015. DOI: 10.1109/APCAP.2015.7374269

[23] Wei S, Yang W. A high-temperature superconducting RF coil design for low field MRI. In: Asia-Pacific International Symposium on Electromagnetic Compatibility (APEMC). 2016. DOI: 10.1109/APEMC.2016.7522954

[24] Paul CR. Inductance-Loop and Partial. John Wiley & Sons, Inc.; 2010. ISBN: 978-0-470-46188-4

[25] Brown RW, Cheng YN, Haacke EM, et al. Magnetic Resonance Imaging: Physical Principles and Sequence Design. John-Wiley & Sons, Inc.; 2014

[26] Dwight HB. Tables of Integrals and Other Mathematical Data. 4th ed. New York: Macmillan; 1961

**33**

**Chapter 3**

**Abstract**

tile usage of 2D 1

**1. Introduction**

bio-methane.

Facile NMR Relaxation Sensor

Degradation Products during

The chemical and morphological composition of animal biowaste is known to limit the efficiency of methane production by bacterial anaerobic digestion (AD). To better understand these material limitations, we studied degradative changes in cattle manure's organic complex components chemical and morphological composition during its AD to methane. This was achieved using low field 1

relaxation times domain (TD) spectral mapping combined with T1 (spin-lattice) and T2 (spin-spin) TD of cattle manure biomass (CM) peaks assignment, starting from samples of initial freshly collected CM biomass sample followed by several time points sampling during 21 days cycle of the AD process. A T1-T2 relaxation TD graph giving a stable reproducible pattern of 12 peaks was generated, and assigned to different domains, whose changes during AD could be observed. These 12 peaks were assigned to TDs of crystalline nano-aggregated complexes of different degrees of crystallinity with low porosity and low hydration rate and a morphological group of amorphous domains with increased pore size, density, and higher hydration. In agreement with models of elementary cellulose fibrils, these domains were designated as three layers of cellulose consisting of interior, subsurface, and surface. The most amorphous TD volume showed good correlation with biogas production and could serve as an indicator for digestibility and cellulose conversion to a glucose intermediate during the AD process. This study demonstrated the facile and versa-

systems, with the potential for improving CM biomass conversion efficiency into

To meet the needs of a growing world population, fossil fuels limited supply and global warming by greenhouse gas emissions, economically competitive biofuels with neutral greenhouse effects are being developed such as bacterial anaerobic digestion (AD) of biomass into methane [1, 2]. Although extensive studies have been carried out to maximize AD methane production, the major limitation is still

**Keywords:** anaerobic digestion (AD), biogas, lignocellulose, 1

time domain (TD), cellulose crystallinity, biofilm

H NMR T1-T2 sensorial technology in studying complex biowaste

H NMR

H LF-NMR relaxation,

for Monitoring of Biomass

Conversion to Biogas

*Wiesman Zeev and Linder Charles*

## **Chapter 3**

Symposium Conference Record. 2011. DOI: 10.1109/NSSMIC.2011.6152634

*Nuclear Magnetic Resonance*

[24] Paul CR. Inductance-Loop and Partial. John Wiley & Sons, Inc.; 2010.

[25] Brown RW, Cheng YN, Haacke EM, et al. Magnetic Resonance Imaging: Physical Principles and Sequence Design. John-Wiley & Sons, Inc.; 2014

[26] Dwight HB. Tables of Integrals and Other Mathematical Data. 4th ed. New

ISBN: 978-0-470-46188-4

York: Macmillan; 1961

[17] Xu B, Wei Q, Liu F. An inverse methodology for high-frequency RF coil design for MRI with de-emphasized B1 fields. IEEE Transactions on Biomedical Engineering. 2005;**52**(9). DOI: 10.1109/

[18] Abraham R, Ibrahim TS. Humanbody coil design for magnetic resonance imaging at 7 Tesla. In: IEEE Antennas and Propagation Society International Symposium. 2005. DOI: 10.1109/

[19] Li BK, Xu B, Hui HT, et al. A new approach for magnetic resonance RF head coil design. In: IEEE Engineering in Medicine and Biology 27th Annual Conference. 2005. DOI: 10.1109/

[20] Morey AM, Bhujade S, Bhuiya T. Design and development of surface coil for 1.5T MRI scanner. In: International Conference on Smart Technologies and

Communication, Controls, Energy and Materials (ICSTM). 2015. DOI: 10.1109/

[21] Shou G, Xia L, Liu F, et al. MRI coil design using boundary-element method

Zulkifli FY. Eight-channel phased array RF coils design for 3T parallel MRI system. In: IEEE 4th Asia-Pacific Conference on Antennas and Propagation (APCAP). 2015. DOI: 10.1109/APCAP.2015.7374269

[23] Wei S, Yang W. A high-temperature superconducting RF coil design for low field MRI. In: Asia-Pacific International

Symposium on Electromagnetic Compatibility (APEMC). 2016. DOI: 10.1109/APEMC.2016.7522954

**32**

with regularization technique: A numerical calculation study. IEEE Transactions on Magnetics. 2010;**46**(4)

[22] Basari ASH, Rahardjo ET,

Management for Computing,

TBME.2005.851514

APS.2005.1551696

IEMBS.2005.1615624

ICSTM.2015.7225428
