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.
- magnetic resonance
- magnetic field
- transfer function
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 . 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 . 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  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 . 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, 7, 8, 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 . A good frequency resolution has been reached by using an optimization algorithm of strip-line chip . 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 . 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 . 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 . 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  by dropping the temperature up to 77 K.
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 . The field is the magnetic field directed towards the direction feet to head of the patient, determining the magnetic polarization of the human body. The field is produced by the uniform axial magnetic field and by the gradient coil. The transmitter coil produces a magnetic field called , directed along the transversal direction to the magnetic field , and is a function of time because of the impulse characteristic that has a function 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 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 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):
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:
The distance R is calculated from the wire element; the point can be represented as:
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:
By changing variables as , it has the integral (6):
from where the expression of magnetic field:
This vector generates a flow through the surface shown in Figure 1 as:
that returns the expression in (9):
from where the auto-inductance is:
By using the property, it is defined as:
Eq. (10) is defined as:
Eq. (10) can be simplified in the hypothesis that :
In case the wire has a rectangle cross section with as shown in Figure 3 ,
the expression of self-inductance is :
that, when simplified according to the assumption , returns :
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 .
that based on the condition returns the approximated expression:
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 the difference is significant. In the same figure, it is clear also that an increasing of d equal to and the two expressions tend to become coincident validating the assumption .
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
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:
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.
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 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:
where and .
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:
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):
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 that will produce the singularity in the magnitude diagram in a bode plot according to the transfer function (tf) is described in Eq. (27):
The FR is carried out as shown in Figure 18 for a loop having a square shape with and 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 differentiating the tf in Eq. (27), the condition of these two regions separated by the value 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 in order to perform some analysis that allows to understand better the dynamic of a single receiver coil. A desired value of fr called that is found to be at the right of the fr corresponding to the capacitive C 0 that we call satisfies the condition that . In this case we write the system with , , and carrying out the relation as:
Eq. (27) establishes that in order to reach the desired frequency , the capacitive value has to decrease since . 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 :
Eq. (30) establishes that for the magnitude of the FR decreases or rather .
The case where produces opposite sign in the differences and , but the relation between the and remains the same as in Eq. (31):
One has the infinitesimal differentiation for from where it has
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.
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.