**3.2. Receiver coil L2**

For reception of the spectrum magnetic field, also use a flat coil as the receiving antenna, built on a phenol board as printed circuit with the following characteristics almost as similar to the transmitting antenna:


#### **Figure 8.** Coil and shield used

112 Advanced Aspects of Spectroscopy

**Figure 7.** Coil and Shielding employee

**3.2. Receiver coil L2** 

to the transmitting antenna:

this shield was "grounded".

For generating a spectral magnetic field a flat coil was used as a transmitting antenna built

 Spiral coil [12] internal diameter of 2 cm and an outer diameter of 9.5 cm with 20 turns, with cooper tracks width of 500 μm , and an equal distance between them, see Figure 7 24 μH Inductance measured experimentally in a frequency range of 100 KHz to 5 MHz Shielding is used to minimize capacitive coupling at the top and bottom of the exciting

The Network Analyzer employee served as a source of power in a frequency range of 100 KHz to 5 MHz applied. The combination coil and capacitance of the cables, that although were coaxial cables it presented a resonance frequency of 4.612 MHz This initial structure was proceeded with a series of measurements with actual physical capacitors with a dual role, first

For reception of the spectrum magnetic field, also use a flat coil as the receiving antenna, built on a phenol board as printed circuit with the following characteristics almost as similar

 Spiral coil internal diameter of 2 cm and an outer diameter of 9.5 cm with 20 turns, with cooper tracks width of 500 μm, and an equal distance between them, see Figure 8 24μH Inductance measured experimentally in a frequency range of 100 KHz to 5 MHz Shielding is used to minimize capacitive coupling in the bottom of the receiving coil,

see the system answer at different values, and the second as reference calibration [13].

on a phenol board as printed circuit with the following characteristics:

coil, forming a sandwich, this shielding was "grounded".

**3.1. Drive coil L1** 

 In addition to the above conditions and in order to minimize inductive coupling both coils are placed in perpendicular way see Figure 9, so that the only magnetic field received were the projected by the coil sensor.

**Figure 9.** Mechanical available of the coils to minimize Inductive coupling between transmitter coil and receiver coils

Network Analyzer employee close the system, as seen in Figure 9, the application is through the coil L1, placed horizontally, which serves as basis for the deposition of both the saline and the samples biological tissue. Receiving and monitoring of the signal through the coil L2, vertically positioned and perpendicular to L1

#### **3.3. Mirror-sensor coil**

As a passive coil mirror-sensor used a flat square coil of about 7.92 μH, see Figure 10, with a measured value of 9.716 μH, at a frequency applied of 1 MHz, Coil with a capacitor of 330 nF added to make a 1 MHz resonant circuit, finally resulting a resonant frequency of 1.14 MHz, like the previous coils was constructed on phenol board as a printed circuit having the following characteristics:

 The inductance L, was determined by experimental measurements, confirming the calculations used in the approximation developed by Ong et al[14].

**Figure 10.** Layout of the Mirror-Sensor coil with interdigital capacitor (not shown)


#### **4. Characterization of the system**

The technique of network development has in its whole with the element to explore interesting properties, which generates all possible parameters of the immittance associated with the two-port network, Figure 11.

**Figure 11.** Electronic models of the transmitter coil and receiver coil

$$V\_1 = Z\_{11} \ast I\_1 + Z\_{12} \ast I\_2 \tag{3}$$

$$V\_2 = Z\_{21} \* I\_1 + Z\_{22} \* I\_2 \tag{4}$$

$$V\_1 = \; j\omega L\_1 \ast I\_1 + \; j\omega \mathbf{M} \ast I\_2 \tag{5}$$

$$V\_2 = \text{ } \jmath \omega \text{M} \, \ast \, I\_1 + \text{ } \jmath \omega \text{L}\_2 \, \ast \, I\_2 \tag{6}$$

$$\mathcal{M}\_{12} = \frac{\Phi\_{12}}{l\_1} \tag{7}$$

$$M\_{21} = \frac{\Phi\_{21}}{l\_2} \tag{8}$$

$$
\delta H = \frac{l \delta l \sin \theta}{4 \pi r^2} \tag{9}
$$

$$H \cdot 2\pi r = i, \text{o } B = \frac{\mu l}{2\pi r} \tag{10}$$

$$
\Phi\_2 = B\_1 A\_2 = \left(\frac{\mu\_0 l\_1}{2R\_1}\right) \pi R\_2^2 = \frac{\mu\_0 \pi l\_1 R\_2^2}{2R\_1} \tag{11}
$$

$$M = \frac{\Phi\_2}{l\_1} = \frac{\mu\_0 \pi R\_2^2}{2R\_1} \tag{12}$$

This expression shows that M depends only on geometric factors, R1 and R2, and is independent of the current in the coil. As regards the expression ܯൌ݇כ ඥܮଵ כ ܮଶ to obtain an approximation of mutual inductance between coils, since low values of coupling coefficients "k" with air-core coils are obtained usually in the order of 0.001 to 0.15 [17] experimentally is considered as the value of k1 = 0.00167 (caused by a misalignment both angular and lateral between coils L1 and L2), and a value of k2 = k3 = 0.465 (caused by an angular misalignment between coils L1, L2 over L3 ), so we get the following values:

116 Advanced Aspects of Spectroscopy

from B1.

which form the coils.

The mutual inductance is then:

**Figure 12.** Approximate representation of variable in time magnetic induction in the coils system

(Sine θ merely states that if power is not in an optimal direction, then the field at that point decreases.) This is confirmed by Ampere's rule, � ��� = �; The line integral of the magnetic field in a closed loop is equal to the electric current flow through the closed loop" [16]. In particular, the magnetic field strength H in a circular path of radius *r* around an electric current *i* in the center is

� � ��� = �� ��� = ��

Because L1 and L2 have very similar characteristics, and considering the expression [9] may approximate the magnetic field concentric loops through both coils (see Figure 8a), B1 is the magnetic field of the first ring, the flow magnetic Φ2 through second ring we can determine

> ��� � ���

where μ0 is the magnetic permeability of free space, R1 and R2 are the radii of the rings

<sup>=</sup> ����� � ���

� = �� ��

� <sup>=</sup> �������

� ���

�� = ���� = �����

��� (10)

(12)

(11)

$$\begin{aligned} \mathbf{M}\_{31} &= \mathbf{M}\_{13} = 7.1 \,\upmu\text{H} \\\\ \mathbf{M}\_{23} &= \mathbf{M}\_{32} = 7.1 \,\upmu\text{H} \\\\ \mathbf{M}\_{21} &= \mathbf{M}\_{12} = 0.04 \,\upmu\text{H} \end{aligned}$$

By obtaining these values, we proceeded to simulate the circuit in PSpice [18], see Figure 13, to verify that they were not far from the actual physical model, which was obtained following graphs.

**Figure 13.** System coils with their respective coupling factors for achieving the simulation circuit had to be "grounded" L3 but with the highest permissible value, simulating a physical disconnection between the two coils.

In Figure 14 we can observe the system performance in terms of voltage induction refers to both L2 and L3.

$$
\begin{bmatrix} V\_1 \\ V\_2 \\ V\_3 \end{bmatrix} = j\omega \begin{bmatrix} L\_1 M\_{12} M\_{13} \\ M\_{21} L\_2 M\_{23} \\ M\_{31} M\_{32} L\_3 \end{bmatrix} \begin{bmatrix} l\_1 \\ l\_2 \\ l\_3 \end{bmatrix} \tag{13}
$$

$$
\begin{bmatrix} V\_1 \\ V\_2 \\ V\_3 \end{bmatrix} = j\omega \begin{bmatrix} Z\_{11}Z\_{12}Z\_{13} \\ Z\_{21}Z\_{22}Z\_{23} \\ Z\_{31}Z\_{32}Z\_{33} \end{bmatrix} \begin{bmatrix} l\_1 \\ l\_2 \\ l\_3 \end{bmatrix} \tag{14}
$$

$$Z\_{12} = \frac{V\_1}{l\_2} \left| l\_1 = 0 \right\rangle \\ Z\_{21} = \frac{V\_2}{l\_1} \left| l\_2 = 0 \right.$$

$$\frac{V\_3}{l\_3} = Z\_3 \tag{15}$$

$$V\_2 = j\omega \mathcal{M}\_{21} i\_1 + j\omega \mathcal{M}\_{23} \frac{V\_3}{Z\_3} \tag{16}$$

$$Z\_P = R\_S + \frac{R\_p}{1 + j\omega R\_P C} \tag{17}$$


$$Z\_3 := \frac{R\_3}{1 + (j \cdot R\_3 \cdot C\_3)} = 97.459 - 17.73i \tag{19}$$


$$V\_3 = \frac{j\omega\mathcal{M}\_{23}\mathcal{l}\_2}{1\text{--}\frac{j\omega\mathcal{L}\_S}{Z\_2}}\tag{20}$$

$$W\_1 = j\omega M\_{12} i\_2 + j\omega M\_{13} \left(\frac{j\omega M\_{23} i\_2}{Z\_3 - j\omega L\_S}\right) \tag{21}$$

$$Z\_{12} = j\omega \mathcal{M}\_{12} - \omega^2 \left(\frac{\mathcal{M}\_{23}\mathcal{M}\_{31}}{\sqrt{\left(\frac{1}{\hbar\_P}\right)^2 + \{\omega C\}^2} - j\omega \mathcal{L}\_S}\right) \tag{22}$$

The values obtained for both measurement and calculation as shown in Table 2 allows us to make the cell preparation of constant section for measuring conductivity. For this preparation had to consider the mechanical error sources such as physical dimensions of the saline cell, the effects of temperature on the cell, and error handling and positioning of it. It also ponder the discrepancy between the measurements of a real physical element such as a capacitor in parallel with a resistor, and the resistivity and permittivity expected of an electrolytic cell, it was therefore necessary to have a truer reference of known salt solution to evaluate deviation of our measurements and bring the measuring physical model to a virtual model and simulate their behavior in this way to "induce" their performance, come to perform these measurements with electrodes.

In Figure 19 shows a sample obtained from a series of measurements made under the scheme of Figure 18, changing the values of the capacitances in parallel with resistors first small values, reaching values used to 1 kΩ .

**Figure 19.** Comparisons between reference measurements with physical components and estimated values

Although performing multiple measurements and their subsequent acquisition of information due to the speed and flexibility of use of equipment to automate this process, include only the most representative of the behavior of a saline cell, as seen in Figure 20, with various degrees of salinity.

**Figure 20.** Measurement of Z12 of four saline cells

Values obtained with the saline cell, Figure 20, as well as the physical components are between 4% and 8% above the Z12 average obtained from the saline cell, provides that comparatively physical model is closer to cell model representing a saline cell of Figure 21, which is formed by a resistance of 100 Ω in parallel with a capacitor of 1 nF, with a response very "right" to the resonant frequency of the coil system which is approximately 3 MHz

**Figure 21.** Physical model equivalents to a 500 mL saline cell at a frequency of ~ 3 MHz

The conductivity of a saline solution with 2 g of salt dissolved in 1L of deionized water having a conductivity measurement 5.8μS/cm, presents a σ = 3.9 mS / cm at 25 ° C and σ = 3.4 mS / cm at 20 ° C, this represents a 0.2% concentration and 0.034 M.

If we use this solution as the first reference electrolytic cell and according to equation [1], with an equal volume to 500 mL and a length of 85 mm, we calculate a 37.05Ω impedance (Z) , with these values and data from our physical model equivalent gives a σ = 2.29 mS / cm which leads us to obtain a correction term 3.9/2.29 = 1.7.
