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www.etasr.com Ghaly et al.: Design and Modeling of a Radiofrequency Coil Derived from a Helmholtz Structure 

 

Design and Modeling of a Radiofrequency Coil 

Derived from a Helmholtz Structure 
 

Sidi M. Ahmed Ghaly 

Electrical Engineering Department, 

College of Engineering, Al Imam 

Mohammad Ibn Saud Islamic 

University, Riyadh, Saudi Arabia 

smghaly@imamu.edu.sa 

Khalid. A. Al-Snaie 

Electrical Engineering Department, 

College of Engineering, Al Imam 

Mohammad Ibn Saud Islamic 

University, Riyadh, Saudi Arabia 

kalsnaie@imamu.edu.sa 

Asad. M. Ali 

Electrical Engineering Department, 

College of Engineering, Al Imam 

Mohammad Ibn Saud Islamic 

University, Riyadh, Saudi Arabia 
asadali@ imamu.edu.sa

 

 

Abstract—This paper focuses on radiofrequency (RF) coils that 

can produce a high electromagnetic field homogeneity to be used 

for magnetic resonance imaging (MRI) applications. The 

proposed structure is composed of four wire loops symmetrically 

located on an ellipsoidal surface. The main objective of this work 

is to improve field homogeneity compared to a standard 

Helmholtz coil. Numerical simulation was carried out to assess 

the RF electromagnetic field behavior of the proposed coil. 

Different electrical modeling and simulations were investigated, 

particularly the study of the whole modeling of the proposed 

structure taking into account all the couplings between the loops. 

The proposed coil was evaluated and compared with the standard 

Helmholtz coil. Simulation and experimental results confirmed 

the good performance of the developed coil in terms of 

electromagnetic field homogeneity, efficiency, sensitivity, and 

quality factor. 

Keywords-radiofrequency coils; modeling; simulation; design; 

electromagnetic field homogeneity; impedance measurements; MRI 

I. INTRODUCTION  

Since the demonstration of the nuclear magnetic resonance 
(NMR) phenomenon in 1946, the number of its applications 
grows steadily [1-3]. NMR imaging is based on the study of the 
magnetization modifications of the nuclei of a subject 
substance to a precise magnetic environment [4], in particular, 
a very strong magnetic field. The field homogeneity is needed 
over the observed area and it has been shown that the quality of 
the image obtained is, in particular, a function of the intensity 
of the applied static magnetic field and the homogeneity of the 
radiofrequency electromagnetic field produced by the 
transmitting coil and associated with the receiving coil [5-7]. 
The electromagnetic field homogeneity of radiofrequency coils 
is usually better with multiple radiating elements, of identical 
or different attributes [8, 9], and their design can be 
accordingly optimized. For example, birdcage coils [8] are the 
most popular in MRI, they show outstanding performance in 
terms of magnetic field and signal-to-noise ratio (SNR), and 
have a sufficient number (usually greater than 8) of parallel 
radiating strands, often periodically placed on the surface of a 
cylinder. The optimization of RF coils is the subject of 
continuous research. Currently, inductive coupling problems 
between radiating elements (conductors) are largely taken into 

account, and this task is partly facilitated for periodic structures 
[10-12]. For RF coils that do not have a specific periodicity, 
optimization and implementation are more challenging. This is 
the case of Helmholtz coils [13] which consist of two coaxial 
circular coils separated by a distance equal to their radius. An 
improvement in the homogeneity of the field passes necessarily 
by the addition of coaxial turns. Thus, original structures, 
generally contained in a spherical and ellipsoidal envelope and 
having four or more free element loops have already been 
proposed [10-12] and very interesting properties in terms of 
homogeneity and sensitivity are obtained. A certain number of 
possibilities nevertheless remain to be analyzed. This research 
work considers aspects related to electronic instrumentation 
and in particular, the design, modeling and optimization of RF 
coils for magnetic resonance imaging, presenting a certain 
simplicity of implementation and having good homogeneity of 
RF electromagnetic field. More precisely, the optimization and 
modeling of a symmetrical coil with four coaxial circular loops 
producing effective sixth order homogeneity of 
electromagnetic field are presented. In fact, the proposed 
structure includes only two coaxial circuits. In NMR imagery, 
we have already demonstrated the advantages of the free 
element four-coil system [10, 11] that provides fourth order 
homogeneity. In this work, the main objective is to increase the 
sensitivity of the coil and reduce the number of tuned loops 
from four to two, which can be done by connecting the two 
loops positioned in the same side in series with a shared tuning 
capacitance. This configuration looks like a distributed 
Helmholtz coil.  

This work, which includes modeling, simulations and 
experimental results, intends to be an important input for RF 
coils resulting from Helmholtz structure which can be used for 
MRI applications. 

II. ELECTRICAL MODELING 

In this section the electrical modeling of the proposed coil 
that includes all inductive and capacitive couplings will be 
discussed. The electrical modeling of the structure is necessary 
to demonstrate its operation and feasibility. The proposed 
structure is shown schematically in Figure 1. The proposed 
structure has the advantage to reduce the number of tuned 

Corresponding author: Sidi M. Ahmed Ghaly 



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circuits from four to two by connecting the two loops 
positioned in the same side with a shared tuning capacitance for 
each single circuit (two series circuits). This choice facilitates 
the comparison with the preceding proposed structures [10, 11]. 
Thus, the developed coil looks like a distributed Helmholtz 
coil. To confirm the feasibility of such a coil configuration, a 
whole electrical model of the four-loop system compelling all 
capacitive and inductive couplings is investigated. The 
coupling mode of the rings is an inductive coupling [14] 
between all loops and a capacitive coupling [15] between 
adjacent left or right loops (connected in series). The equivalent 
electrical diagram of the system in the ideal case (without 
losses) is shown in Figure 2. 

 

 

Fig. 1.  Structure of the proposed coil 

 

Fig. 2.  Electrical diagram for the proposed coil (without losses) 

Because of the probe symmetry, there are some obvious 
simplifications: Inductances L4=L1, L3=L2, capacitances C1=C2, 
mutual inductances M34=M12 and M24=M13. As for the 
Helmholtz coils [13] and using Kirchoff`s voltage law, the two 
electrically isolated circuits crossed by the currents I1 and I2 are 
characterized by the following equation system: 

[
𝑍 𝑗(𝑀14 + 𝑀23 + 2𝑀13)𝜔

𝑗(𝑀14 + 𝑀23 + 2𝑀13)𝜔 𝑍
] [

𝐼1
𝐼2

] = [0
0

]  (1) 

Two modes of resonance can be distinguished: a co-current 
symmetrical mode with I1=I2, which is the mode of interest and 
a mode counter current which is not symmetric with I1=-I2 
(gradient mode). For the co-current mode (the mode of 
interest), (1) can be reduced to: 

𝑍𝐼1 + 𝑗(𝑀14 + 𝑀23 + 2𝑀13)𝜔𝐼2 = 0  (2) 

Introducing the coupling coefficients kij [16], the resonance 
equation becomes: 

1 2 12 1 2

2

13 1 2 14 1 23 2

2

2 1

(

)

C L L k L L

k L L k L k L 

+ + +

+ + =
  (3) 

Thus, it has a real positive solution: 

𝑓1 =
1

2𝜋√𝐶(𝐿1+𝐿2 +2𝑘12√𝐿1𝐿2+2𝑘13√𝐿1𝐿2+𝑘14𝐿1+𝑘23𝐿2)

 (4) 

For the counter-current mode (I1=-I2), there is a frequency 
of: 

𝑓2 =
1

2𝜋√𝐶(𝐿1+𝐿2 +2𝑘12√𝐿1𝐿2−2𝑘13√𝐿1𝐿2−𝑘14𝐿1−𝑘23𝐿2)

 (5) 

As for the Helmholtz coil, it is necessary to ensure 
sufficient decoupling of the modes by separating the loops 
sufficiently. 

III. OPTIMIZATION OF THE FIELD HOMOGENITY 

For the calculation of the electromagnetic field (B1) 
produced by the proposed structure which consists of two 
symmetric elements and has small dimensions compared to the 
wavelength, the superposition principle can be applied to 
obtain the total electromagnetic field [17]. The component 
along the y axis of the magnetic field shown in Figure 1 
produced by a thin ring of wire, of radius a, positioned in the xz 
plane, centered on the origin O and carrying a current I is easily 
determined by using Biot-Savart’s law [18]: 

0

1 3

2 2

2 1

y
B

y







=

  
 +  
   

    (6) 

The extension to the proposed structure permits to obtain 
the total field B1y along the coil axis as shown:  

𝐵1𝑦 = 𝜇ₒ𝐼 ∑
1

2𝛼𝑖(1+[
𝑦−𝑦𝑖

𝛼𝑖
]

2
)

3
2

4
𝑖     (7) 

where yi is the ordinate of the position of the loop i on the axis 
and αi is the radius of the loop (i). 

Due to the symmetry of the structure, B1y component can be 
expressed as a Taylor expansion around the origin [19]. The 
odd orders of the function B1y should be zero and the Taylor’s 
expansion gives the even. For a given ellipsoidal structure, the 
symmetry returns the computation parameters to two and thus, 
the spacing between the loops (inner or outside rings) or the 
diameter of the loops can be adjusted. The angles αi can be 
introduced with respect to the axis (Oy), i.e. for loops 1 and 2, 
angles α1 and α2 (α1<α2) and for loops 3 and 4, angles π-α2 and 
π-α1 respectively as shown in (8). It should be noted that the 
field calculation depends only on the geometry of the coils.  

𝐵1𝑦 = 2𝐼𝜇ₒ {(
𝑠𝑖𝑛𝛼1

3

𝑎1
) + (

𝑠𝑖𝑛𝛼2
3

𝑎2
) + ((

𝑠𝑖𝑛𝛼1
2

𝑎1
) 𝑃91 (𝑐𝑜𝑠𝛼1)) (

𝑦

𝑟1
)

8

+

((
𝑠𝑖𝑛𝛼2

2

𝑎2
) 𝑃91 (𝑐𝑜𝑠𝛼2)) (

𝑦

𝑟2
)

8

+. . }   (8) 

Indeed, on the electrical level, the transformation ratio 
between currents is forced to one and it is possible to modify 
the solution by simply scaling factor. After the numerical 
calculations, there is a certain range of possibilities for fourth 
order homogeneity field which advantageously presents a 
unique and optimal six-order solution for the proposed coil. 
There is a unique solution for which the six-order derivative of 



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the electromagnetic field (B1) is zero. For this optimal case, the 
secondary geometric data for the proposed coil are given in 
Table I. 

TABLE I.  SECONDARY GEOMETRIC DATA FOR THE PROPOSED COIL 

Main axis a (cm) Secondary axis b (cm) α1 (rad/deg) α2 (rad/deg) 

3.042 2.459 0.734/42.09 1.299/74.42 

 

The eccentricity e is given by: 

𝑒 = √1 − (
𝑏2

𝑎2
) = 0.587   (9) 

According to Table I, the Taylor expansion in (8) can be 
reduced to: 

𝐵𝑖𝑦 = (𝐼𝜇ₒ/2){106.3-1.247 10
15 𝑦8 + ⋯} (10) 

Figure 3 shows the obtained dimensions of the ellipsoidal 
coil and the corresponding normalized electromagnetic field 
profile: 

 

 

Fig. 3.  (a) Dimensions of the proposed structure, (b) Normalized field 
profile 

It is clear from Figure 3(b) that profile widths at 1% and 
10% can be calculated and compared with those obtained from 
Helmholtz coil shown in Table II. 

TABLE II.  PROFILE WIDTHS AT 1% AND 10% FOR BOTH COILS 
(HELMHOLTZ AND PROPOSED) 

Structure Profile width at 1%(cm) Profile width at 10%(cm) 

Helmholtz 1.50 2.89 

Proposed 2.92 4.44 

 

By comparison with Helmholtz coil, the proposed structure 
has a significant increase in the profile. Thus, in considering 
uniformity of 1%, profile width has increased approximately 
97% and at 10% uniformity the profile width increased 58% 
more than the Helmholtz coil. The performance described 
above is very general and does not depend on the size of the 

coils. Indeed, it suffices to apply a simple scale factor to the 
ellipsoidal structure to obtain coils with the same 
performances. 

IV. SIMULATION 

In this part, the electrical behavior of the system will be 
investigated for the proposed structure (ellipsoidal coil). The 
system comprises of four coaxial wires with inductive coupling 
between all loops and capacitive coupling between adjacent left 
or right loops. It consists of two identical tuned circuits 
resonant at the same frequency.  

A. One Tuned Circuit Having Two Rings Coupled in Series 

In this case, assume a circuit tuned at frequency f0 and 
consisting of two rings with inductances L1 and L2 mutually 
coupled as shown in Figure 4. 

 

 

Fig. 4.  Two mutually coupled inductances 

In the case without losses, the value of the self-resonance 
frequency is given by: 

𝑓0 =
1

2𝜋√(𝐿1+𝐿2 +2𝑘12√𝐿1𝐿2)𝐶

   (11) 

B. Case of the Proposed Structure (2 Tuned Circuits): 

Coaxial wire loops are characterized by their self-
inductance and their mutual inductances, which depend at high 
frequencies on the geometrical configuration: loop diameter, 
wire diameter, and distances between the loops [20]. The two 
tuning capacitors are supposed to be identical. If we consider 
that the proposed structure is sourced by an ideal voltage 
source, then the equivalent electrical diagram is shown in 
Figure 5. 

 

 

Fig. 5.  Electrical diagram for the proposed coil (with losses) 

The tuning capacitor is determined from the established 
formula of the resonance frequency f1 of the coil as shown in 
(2). For the co-current mod, the tuning capacitor is deduced as: 

𝐶 =
1

2𝜋2𝑓1
2(𝐿1+𝐿2 +2𝑘12√𝐿1𝐿2+2𝑘13√𝐿1𝐿2+𝑘14𝐿1+𝑘23𝐿2)

 (12) 

The resonance frequency f1=100.24MHz is the Larmor 
frequency at static magnetic field B0=3T [7]. Table III presents 

(a) 

 
 

 

 
 

 

 
 

 

 

 
(b) 



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the data necessary for the preliminary adjustment of the 
ellipsoidal structure obtained from the equivalent electrical 
model in Figure 5. Tuning capacitor C and self-resonance 
frequency f0 of tuned circuits are presented in Table IV. 

TABLE III.  ELECTRICAL PARAMETERS AND TUNING ELEMENTS 

 Parameters (inductances Li coupling kij) 

 L1=L4 L2=L3 k12 k13 k14 k23 

Proposed Structure 83.3 117.6 0.16 0.06 0.02 0.20 

TABLE IV.  TUNING ELEMENTS 

f0 C 

108.37MHz 9.269pF 

 

The simulation results give the quality factor of the 
proposed structure which equals to 305 with losses when 
R1=R2=0.82Ω. Figure 6 presents the simulation of the input 
impedance curve of the proposed structure. As seen, there are 
two resonance series: at 100MHz and 123MHz. As the quality 
factor is high, then the energy absorbed at f0 is low. On the 
other hand, since R is small, the energies absorbed at f1 and f2 
are relatively large [7]. For the resonance frequencies, Table V 
compares the simulation results with the theoretical (without 
losses) for the ellipsoidal coil.  

 

 

Fig. 6.  Simulation of the input impedance vs frequency curve for the 
ellipsoidal coil. 

TABLE V.  SIMULATION AND THEORETICAL RESULTS WITHOUT 
LOSSES 

 Ellipsoidal structure 

Modes of resonance 1 2 

Theoretical f1 (Mhz) 100.24 118.85 

Simulation f1 (Mhz) 100.23 118.98 

 
A simulation of the currents variations in the ellipsoidal coil 

which includes two circuits only (carrying currents Ic1 and Ic2) 
is given in the Figure 7. It should be noted that at the two 
resonance frequencies f1 and f2, the currents Ic1 and Ic2 have 
maximum magnitude (almost identical) and the difference 
between the two modes remains essentially at the phase level: 
in-phase currents for f1 and in opposition of phase for f2. 

C. Matching of the Coil at 50Ω: 

The proposed coil is matched using an inductive coupling 
[14]. This kind of matching uses an attack loop with the same 

dimensions as the first loop of the proposed coil and this attack 
loop is tuned to f1 as shown in Figure 8. 

 

 

Fig. 7.  Simulation of the currents variation vs frequency for the 

ellipsoidal coil. 

 

Fig. 8.  Matching of the ellipsoidal coil with inductive coupling 

The attack coil is inductively coupled with the four loops of 
the structure. The matching of the coil at 50Ω is performed by 
optimizing the coupling loop position. The associated coupling 
coefficients (k0i, i=1, .., 4) allowing the matching are presented 
in Table VI. The simulation of the matching at 50Ω (Figure 9) 
gives a corresponding distance between the attack loop and the 
ellipsoidal coil equal to 2.56cm. The inductive coupling results 
in an impedance inversion: the series resonance modes become 
parallel. 

TABLE VI.  ASSOCIATED COUPLING COEFFICIENTS 

Coupling kij k01 k02 k03 k04 

Ellipsoidal structure 0.0736 0.0216 0.0096 0.0018 

 

 

Fig. 9.  Μagnitude of the input impedance vs frequency.Simulation 

of the matching for the ellipsoidal coil at f1=100MHz, R=50Ω.  



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V. EXPERIMENTAL PART 

In this part, we present the experimental results obtained 
with the developed coils (Helmholtz and ellipsoidal). We 
conducted out-of-field tests using a network analyzer. Relative 
measurements of the RF electromagnetic field (B1) along the 
main axis of the coils were carried out and were compared with 
those found in theory. 

A. Prototype Dimensions 

To facilitate comparisons, both standard Helmholtz coil and 
the proposed (Figure 10) had the same radial diameter d equal 
to 4.80cm (Table VII).  

TABLE VII.  COIL DIMENSIONS 

 Diameter Distances 

Structure Inner loops Outer loops Inner loops Outer loops 

Helmholtz 4.80 -- 2.40 -- 

Ellipsoidal 4.80 3.60 1.33 4.05 

 

 
(a) 

 
(b) 

Fig. 10.  The developed (a) Helmholtz and (b) ellipsoidal coils 

B. Matching and Tuning of the Coils 

It is possible to proceed directly to the tuning of the coils, 
but it requires a certain control of the various influential 
parameters (stepwise adjustment of the tuning capacitors). The 
tuning of each loop at f0 is performed individually by shorting 
the other loops and bringing it closer to a test loop. By 
predetermination of fo (see (4)), a vacuum presetting of the coil 
is carried out. Figure 11 gives an explanatory diagram. 

 

Fig. 11.  Individual tuning of ellipsoidal coil 

Figure 12 shows the variations of the module and phase of 
the input impedance seen by the source as a function of the 
frequency. 

 

Fig. 12.  Photograph of the tuning of the ellipsoidal coil 

Two parallel resonance frequencies are observed for the 
ellipsoidal coil. Table VIII gives the comparison between the 
values of the experimental measured frequencies (with losses) 
and those in theory (without losses) for the ellipsoidal coil: 

TABLE VIII.  EXPERIMENTAL AND THEORETICAL VALUES 

Ellipsoidal coil 

Resonance mode i 1 2 

fi theoretical (MHz) 100.25 118.85 

fi experimental (MHz) 100.24 117.91 

 

It can be observed that we have almost the same value of 
the frequency f1, while there is a small error for the other 
frequencies. For the last mode of resonance, the error is almost 
3% for the ellipsoidal coil. Impedance matching allows optimal 
energy transfer between coils and the transmission and 
reception system [14]. For optimal transfer in the coaxial 
transmission cable during excitation, the impedance of the 
system should be 50Ω [14]. The matching of the coils can be 
realized, either using capacitors (capacitive coupling) or using 
a coupling loop (inductive coupling). In our case, we opted for 
the matching by inductive coupling because it allows having a 
electrically balanced circuit. The matching of the coil is 
achieved by optimizing the position of the attack loop with the 
coil. Thus, the value of the distance to have an impedance of 
50Ω is equal to 1.9cm (1.91cm in theory) for the ellipsoidal 
coil. The impedance curve obtained for the matched ellipsoidal 
coil with impedance measurements using Network Analyzer is 
given in Figure 13. 

 

 

Fig. 13.  Impedance curve obtained for the matched ellipsoidal coil 



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C. Results 

Comparative tests have been carried out between the 
ellipsoidal and Helmholtz coils. Thus, relative field 
measurements along the axis of the coils were made using a 
small test loop connected to the Network Analyzer. Figure 14 
shows the measurement bench. We were able to take the 
electromagnetic field along the axis of the coils (Oy). The 
curves of this normalized field produced by both coils are given 
in Figure 15. 

 

 

Fig. 14.  The measurement bench 

 

Fig. 15.  The curves of the normalized field produced by both coils 

To compare the axial profiles, we determined the relative 
widths of the lobe at 10%. Thus, compared to the Helmholtz 
coil, the relative width of the lobe at 10% is 1.54 times larger 
for the ellipsoidal coil (see Table IX), which is in accordance 
with the theoretical forecasts (Table II). Similarly, for the same 
absorbed power, it is possible to check the efficiency of the 
coils [22]. Compared to the Helmholtz coil, the center field is 
1.42 times larger for the ellipsoidal coil as shown in Table IX. 
By scanning with frequency and using two sufficiently distant 
coils (one transmitting and the other receiving), an 
improvement of the quality factor of 47% for the ellipsoidal 
coil is noted. It is increased from 215 for the Helmholtz coil to 
316 for the ellipsoidal coil (Table IX). 

TABLE IX.  COIL PERFORMANCE COMPARISON 

Coil Helmholtz Ellipsoidal 

Relative widths of lobe at 10% 1 1.54 

Efficency 1 1.42 

Quality factor 215 316 
 

VI. CONCLUSION 

We presented the design and testing of a prototype 
magnetic resonance imaging coil composed of two free 

distributed elements contained in an ellipsoidal envelope and 
connected directly in series via a tuning capacitor. Compared to 
classic Helmholtz coils, the goal was to improve the 
homogeneity of the RF electromagnetic field produced by the 
coil to get an effective sixth order homogeneous field. A whole 
complete electrical modeling of the prototype coil was 
investigated, taking into account all couplings (inductive and 
capacitive). For a given resonance frequency, predetermination 
of the tuning frequency of the coil’s elements is performed, as 
well as the calculation of the associated tuning capacitors. The 
matching of the prototype coil by inductive coupling allowed 
great flexibility for settings and a real simplicity of design.  

Comparative tests and measurements between a reference 
Helmholtz coil and the prototype coil with the same radial 
diameter, have been carried out and were in good agreement 
with the theoretical predictions. Thus, the observed width of 
the field homogeneity lobe at 10% is considerably increased for 
the proposed coil. A significant improvement in the quality 
factor was also observed for the ellipsoidal coil. Moreover, for 
the developed ellipsoidal coil, efficiency and signal-to-noise 
ratio are higher as well. These good performance values for the 
ellipsoidal prototype coil give motivation to test the coil with 
load (with real samples) for MRI applications. 

ACKNOWLEDGMENT 

Authors would like to thank the deanship of Academic 
research at Al Imam Mohammad Ibn Saud Islamic University 
for the financial support of the project: No 371411/1437H. 

REFERENCES  

[1] F. Remeo, D. I. Hoult, “Magnet field profiling: Analysis and correcting 
coil design”, Magnetic Resonance in Medicine, Vol. 1, No. 1, pp. 44- 65, 
1984 

[2] C. E. Hayes, W. A. Edelstein, J. F. Schenck, O. M. Mueller, M. Eash, 
“An efficient, highly homogeneous radiofrequency coil for whole-body 
NMR imaging at 1.5 T”, Journal of Magnetic Resonance, Vol. 63, No. 3, 
pp. 622-628, 1985 

[3] J. T. Vaughan, H. P. Hetherington, J. O. Out, J. W. Pan, G. M. Pohost, 
“High frequency volume coils for clinical NMR imaging and 
spectroscopy”, Magnetic Resonance in Medicine, Vol. 32, No. 2, pp. 
206-218, 1994 

[4] K. Asher, N. K. Bangerter, R. D. Watkins, G. E. Gold, “Radiofrequency 
Coils for Musculoskeletal MRI”, Topics in Magnetic Resonance 
Imaging, Vol. 21, No. 5, pp. 315-323, 2010 

[5] Q. X. Yang, S. Li, M. B. Smith, “A Method for Evaluating the Magnetic 
Field Homogeneity of a Radiofrequency Coil by Its Field Histogram”, 
Journal of Magnetic Resonance, Series A, Vol. 108, No. 1, pp. 1-8, 1994 

[6] S. Li, Q. X. Yang, M. B. Smith, “RF coil optimization: Evaluation of B1 
field homogeneity using field histograms and finite element 
calculations”, Magnetic Resonance Imaging, Vol. 12, No. 7, pp. 1079-
1087, 1994 

[7] B. Gruber, M. Froeling, T. Leiner, D. W. J. Klomp, “RF coils: A 
practical guide for nonphysicists”, Journal of Magnetic Resonance 
Imaging, Vol. 48, No. 3, pp. 590-604, 2018 

[8] J. Mispelter, M. Lupu, A. N. M. R. Briguet, NMR Probeheads for 
Biophysical and Biomedical Experiments: Theoretical Principles & 
Practical Guidelines, Imperial College Press, 2006 

[9] B. J. Dardzin, S. S. Li, C. M. Collins, G. D.Williams, M. B. Smith, “A 
Birdcage Coil Tuned by RF Shielding for Application at 9.4 T”, Journal 
of Magnetic Resonance, Vol. 131, No. 1, pp. 32-38, 1998 

[10] S. M. A. Ghaly, S. S. Al-Sowayan, “A high B1 field homogeneity 
generation using free element elliptical four-coil system”, American 
Journal of Applied Sciences, Vol. 11, No. 4, pp. 534-540, 2014 



Engineering, Technology & Applied Science Research Vol. 9, No. 2, 2019, 4037-4043 4043  
  

www.etasr.com Ghaly et al.: Design and Modeling of a Radiofrequency Coil Derived from a Helmholtz Structure 

 

[11] S. M. A. Ghaly, K. A. Al-Snaie, S. S. Al-Sowayan, “Design and Testing 
of Radiofrequency Spherical Four Coils”, Modern Applied Science, Vol. 
10, No. 5, pp. 186-193, 2016 

[12] S. M. A. Ghaly, K. A. Al-Snaie, O. K. Mohammad, “Spherical and 
Improved Helmholtz Coil with High B1 Homogeneity for Magnetic 
Resonance Imaging”, American Journal of Applied Sciences, Vol. 13, 
No. 12, pp. 1413-1418, 2016 

[13] J. Wang, S. She, S. Zhang, “An improved Helmholtz coil and analysis of 
its magnetic field homogeneity”, Review of Scientific Instruments, Vol. 
73, No. 5, pp. 2175-2179, 2002 

[14] M. Decorps, P. Blondet, H. Reutenauer, J. P. Albrand, C. Remy, “An 
inductively coupled, series tuned NMR probe”, Journal of Magnetic 
Resonance, Vol. 65, No. 1, pp. 100-109, 1995 

[15] J. Murphy-Boesch, A. P. Koretsky, “An in vivo NMR probe circuit for 
improved sensitivity”, Journal of Magnetic Resonance, Vol. 54, No. 3, 
pp. 526-532, 2003 

[16] R. F. Lee, R. O. Giaquinto, C. J. Hardy, “Coupling and Decoupling 
Theory and Its Application to the MRI Phased Array”, Magnetic 
Resonance in Medicine, Vol. 48, pp. 203-213, 2002 

[17] C. M. Collins, Z. Wang, “Calculation of Radiofrequency 
Electromagnetic Fields and Their Effects in MRI of Human Subjects”, 
Magnetic Resonance in Medicine, Vol. 65, No. 5, pp. 1470-1482, 2011 

[18] Y. E. Esin, F. N. Alpaslan, “MRI image enhancement using Biot–Savart 
law at 3 tesla”, Turkish Journal of Electrical Engineering & Computer 
Sciences, Vol. 25, pp. 3381-3396, 2017 

[19] D. M. Ginsberg, M. J. Melchner, “Optimum Geometry of saddle shaped 
coils for generating a uniform magnetic field”, Review of Scientific 
Instruments, Vol. 41, No. 1, pp. 122-123, 2010 

[20] H. Fujita, T. Zheng, X. Yang, M. J. Finnerty, S. Handa, “RF Surface 
Receive Array Coils: The Art of an LC Circuit”, Journal of Magnetic 
Resonance Imaging, Vol. 38, No. 1, pp. 12-25, 2013