Evaluation of Maximum Voltage or Maximum Link Distance on Implantable Devices

1. Introduction

The current technology allows the implementation of electronic devices under the skin for therapeutic purposes. As a result the wireless technology can be used to monitor some illness and mitigate the patient`s discomfort. As an example, consider the case of diabetes patients that must undergo to daily painful blood collections to monitor the blood sugar levels. Therefore an electronic device capable of measuring the blood sugar level and transmit the result to an external device would greatly improve the quality of life of those patients. Nevertheless, the placement of an implantable device requires a minor surgery.

Once the implantable device is placed in a subject, the battery lifetime and its charge will dictate the system lifespan [1].

Consequently, it could be used the passive target technology, in which the device contains its own energy source. Thus the limitation imposed by batteries would not exist in addition to size reduction to the device. The radio frequency identification – RFID technology fits perfectly into that application.

An important parameter to be analyzed is the operation frequency. The communication frequency between the external reader and the implantable device should suffer the smallest interference due to the human tissue. The frequency of 13.56 MHz offers low attenuation and can be chosen for implantable devices. This chapter discusses the energy transfer from reader to tag by inductive coupling and its limitations. It presents a way to evaluate the target voltage provided by the inductive coupling by using basic circuit theory.

2. Basic inductive coupling circuit

Figure 1 shows a typical inductive coupling system. Observe that the system is composed by the reader and the tag. The reader is used to receive data from the tag and also to transfer energy to it.

It is chosen series resonance configuration, as shown in Figure 2, in order to raise the voltage at inductor L₁, since voltage in a series resonance inductor is larger than the voltage provided by the source. From basic circuit theory and by observing Figure 3, the impedance seen by the reader is given as:

$$Z=R_1+(j\omega L_1-j\frac{1}{\omega C_1})$$E1

Figure 1.

Typical inductive coupling system.

Figure 2.

Simplified inductive coupling circuit at the reader.

The resonance, for low frequency operation, is given as:

$$\omega L_1=\frac{1}{\omega C_1}.$$E2

However impedance Z becomes resistive and equal to R₁ that represents the inductor series resistance. Therefore, the reader impedance depends on resistance R₁ and voltage the inductor becomes:

$$E_1=jI\omega L_1=j\frac{V_1\varpi L_1}{R_1}=jQ{V}_1.$$E3

At this point it is important to define the quality factor of the resonant circuit, Q=wL₁/R₁. It defines the ratio of stored and dissipated energy in an inductor. It demonstrates that the reader efficiency increases by using series resonance, as summarized by expression (4).

$$\left|E_1{}_{}\right|=|V_1|Q$$E4

The reader receives data and power by the inductive coupling through L₁ and L₂. The reader takes advantage of the parallel resonance and high impedance that occurs at frequencies near the resonance. It indicates that, at resonance, the voltage over the LCcircuit is high and can be used to power up the target. Therefore the inductive coupling allows the use of passive target concept, or self-powering. Nevertheless tuned circuits have multiple resonances that cause a reduction in the energy transfer from reader to target. Thus a better understanding of the influence of resonance circuits may help to determine precisely the supply voltage at the target.

3. Concept of inductance

A current flowing in a straight infinite conductor generates a magnetic flow density (or magnetic induction) proportional to the current. That relationship was observed by Biot Savart, as given by (5).

$$\overrightarrow{B}=\frac{{\mu }_0}{4\pi }{\displaystyle \int i\frac{d\stackrel{}{\overrightarrow{l}X{\overrightarrow{a}}_r}}{{\left|\overrightarrow{r}\right|}^2}}$$E5

where:

$$d\overrightarrow{l}$$represents an elemental piece of conductor,$${\overrightarrow{a}}_r$$represents the unit directional vector,$${\left|\overrightarrow{r}\right|}^2$$represents the module of distance where the magnetic flow intensity is calculated,$${\mu }_0$$is the vacuum permeability constant.

From (5), there will be a magnetic flow area around the conductor. The magnetic flow is given by (6)

$$\varphi ={\displaystyle \int \overrightarrow{B}d\overrightarrow{S}}$$E6

where:

$$\overrightarrow{B}$$- represents the magnetic induction vector,$$d\overrightarrow{S}$$- represents the area vector inside the magnetic induction vector.

In that line, the magnetic flow is proportional to the current flowing in the conductor and there is a linear relationship between them, known as inductance, as shown by (7).

$$L\frac{di}{dt}=\frac{d\varphi }{dt}\Rightarrow L=\frac{d\varphi }{di}$$E7

The premise of infinite and straight conductor of the Biot Savart law is not obeyed in the construction of inductors, thus the inductance parameter of a component will depend on many factors such as geometric shape, number of coils and resistance of the conductor.

4. Concept of mutual inductance

Whenever two inductors are close, the magnetic flow generated by one influences the flow in the other, as indicated in Figure 3.

Figure 3.

Simplified inductive coupling.

Therefore, a time varying current i1 flowing on inductor 1 generates a magnetic flow 21 into inductor 2. The density of magnetic flow that reaches inductor 2 depends on the position and distance between the inductors. Thus mutual inductance is defined as the number of coils of inductor 2 that receives the magnetic flow generated by inductor 1, divided by current i1, as shown by (8).

$$M_{21}=\frac{N_2{\varphi }_{21}}{i_1}$$E8

where :

1. N₂ is the number of coils of inductor 2,

2. $$i_1$$

3. ₁₂ is the magnetic flow on inductor 2 due to inductor 1,

4. M is mutual inductance between inductors 1 and 2.

If there is any current on inductor 2, the same phenomenon occurs, as given by (9).

$$M_{12}=\frac{N_1{\varphi }_{12}}{i_2}$$E9

Consider two inductors of length D and sectional areas A₁ and A₂, much smaller than D, number of coils N₁ and N₂ for inductors 1 and 2, respectively, and placed on the same vertical axis, as shown in Figure 4. As can be observed, inductor 1 is placed inside inductor 2 and they have the same orientation. If it is applied a current into inductor 2, the mutual inductance is given by (9).

Figure 4.

Long inductors coupled on the same axis.

If the magnetic field is uniform inside inductor 2, then flow ₁₂ can be taken as (10).

$${\varphi }_{12}=B_2{A}_1$$E10

Expression (10) can be placed into (9), as:

$$M_{12}=\frac{N_1{A}_1{B}_{2_{}}}{i_2}$$E11

By considering that the inductors core is air and the coils are composed of close and simple layers, the Biot e Savart expression can be used, where:

$$B_2=\frac{{\mu }_0{N}_2{i}_2}{d}$$E12

Thus, by combining (11) and (12):

$$M_{12}=\frac{N_1{N}_2{\mu }_0{A}_1}{d_{}}$$E13

From the analysis of (13) it can be observed that for the auto inductance, the number of coils is the same, thus the expression becomes:

$$L=\frac{{\mu }_0{N}^{2}A}{d}$$E14

For the inductor, the coupling will be taken in its own area, meaning that they are equal, thus:

$$L_1=\frac{{\mu }_0{N}_1{}^{2}A}{d}$$E15

$$L_2=\frac{{\mu }_0{N}_2{}^{2}A}{d}$$E16

By combining (15) and (16), one obtains:

$$\begin{array}{l}M=\frac{{\mu }_{0}A}{d}\sqrt{\frac{L_{1}d}{A{\mu }_0}}\sqrt{\frac{L_{2}d}{A{\mu }_0}=}\sqrt{L_1{L}_2}\\ \end{array}$$E17

In real case, areas A₁ and A₂ are not equal, indicating that the magnetic flow is not covered completely by both inductors. Therefore, the coupling factor is reduced, as given by:

$$M=k\sqrt{L_1{L}_2}$$E18

where k is the magnetic coupling factor.

5. Inductive coupling with simple resonance

Although we have discussed inductive coupling between reader and target, at resonance, that characterize a double resonance, we can use simple resonance. In this configuration the reader presents just an inductor magnetically coupled to the inductor at the tag. The simplified circuit is shown in Figure 5. Observe the resonant circuit is positioned at the tag. In order to simplify the analysis, consider the circuit in free space. By considering also inductors without loss, the circuit of Figure 5 can be equated by using (2), resulting into (19) and (20).

Figure 5.

Inductive coupling with simple resonance at the tag.

$$V_1=I_{1}j\varpi L_1-I_{2}j\varpi M$$E19

$$0=I_2(j\varpi L_2+\frac{1}{jw{C}_2})-I_{1}j\varpi M$$E20

where M is the mutual inductance between inductors L₁ and L₂.

From (20):

$$I_2=\frac{I_{1}j\varpi M}{j\varpi L_2+\frac{1}{j\varpi C_2}}$$E21

By replacing (18) into (21):

$$I{}_2=\frac{I_{1}j\varpi k\sqrt{L_1{L}_2}}{j\varpi L_2+\frac{1}{j\varpi C_2}}$$E22

Now, replacing (22) into (19):

$$V_1=I_{1}j\varpi L_1-\frac{I_{1}j\varpi k\sqrt{L_1{L}_2}j\varpi k\sqrt{L_1{L}_2}}{j\varpi L_2+\frac{1}{j\varpi C_2}}$$E23

Thus, the impedance seen by the generator is:

$$\begin{array}{l}{Z}_1=\frac{V_1}{I_1}=j\varpi L_1+\frac{{\varpi }^2{k}^2{L}_1{L}_2}{j\varpi L_2+\frac{1}{j\varpi C_2}}\\ \end{array}$$E24

From the analysis of (24), it can be observed that the inductance seen by the reader is inductance L₁ added to impedance Z_a reflected from the target, and represented by:

$$Z_a=\frac{{\varpi }^2{k}^2{L}_1{L}_2}{j\varpi L_2+\frac{1}{j\varpi C_2{}^{}}}$$E25

Therefore, by using the same analysis, the impedance Z_b reflected on inductor L₂ is:

$$Z_b=\frac{{\varpi }^2{k}^2{L}_1{L}_2}{j\varpi L_1}$$E26

The equivalent circuit reflected at the target is shown in Figure 6.

Figure 6.

Equivalent circuit reflected at the inductor L_2.

By considering the equivalent circuit at resonance, then:

$$0=(j\varpi L_2+\frac{{\varpi }^2{k}^2{L}_1{L}_2}{j\varpi L_1}+\frac{1}{j\varpi C_2})I_2$$E27

From the development of (27):

$$0=j\varpi L_2(1-k^2)+\frac{1}{j\varpi C_2}$$E28

From (28), the resonance frequency is given as:

$$F{o}_1=\frac{1}{2\pi \sqrt{L_2(1-k^2)C_2}}$$E29

Thus the resonance frequency at the target was modified by the term r given by:

$$r=\frac{1}{\sqrt{1-k^2}}$$E30

Consequently, the resonant frequency varies according to the coupling factor k. Thus, the distance between the inductors and their relative orientation can modify the amount of energy transferred by the inductive coupling. In order to validate equation (29), it was run a Spice simulation with a coupling factor ranging from 0.1 to 0.3, as shown in Figure 7. It can be observed that resonance frequency increases along with the coupling factor. As an initial condition, L₁ and L₂ were taken as 3,5 H and capacitor C₁ equals to 38 pF. Observe, from Figure 7, that voltage on inductor L increases along with the coupling factor.

Figure 7.

Spice simulation of coupling factor.

6. Inductive coupling with double resonance

There are two tuned circuits in the double resonance, where the primary is the reader and the secondary is the tag. Figure 8 presents the basic configuration for the double resonance inductive coupling.

Figure 8.

Double resonance inductive coupling.

From the analysis of Figure 8 and by considering free space and inductors without loss, then:

$$V_1=I_1(j\varpi L_1+\frac{1}{j\varpi C_1})-I_{2}j\varpi M$$E31

$$0=I_2(j\varpi L_2+\frac{1}{j\varpi C2})-I_{1}j\varpi M$$E32

From (32) it can be obtained that:

$$I_2=\frac{j\varpi M{I}_1}{j\varpi L_2+\frac{1}{j\varpi C_2}}$$E33

By replacing (33) into (31) one obtains:

$$V_1=I_1(j\varpi L_1+\frac{1}{j\varpi C_1})+\frac{{\varpi }^2{M}^2}{j\varpi L_2+\frac{1}{j\varpi C{}_2}})$$E34

Therefore, the impedance seen by the generator is:

$$Z=\frac{V_1}{I_1}=(j\varpi L_1+\frac{1}{j\varpi C_1})+\frac{{\varpi }^2{M}^2}{(j\varpi L_2+\frac{1}{j\varpi C_2})}$$E35

It can be rewritten as:

$$Z=j\{\frac{(1-w^2{L}_1{C}_1)(1-w^2{L}_2{C}_2)+w^4{M}^2{C}_1{C}_2}{w{C}_1(1-w^2{L}_2{C}_2)}\}$$E36

Expression (36) allows the behavior analysis of the impedance at the generator, as shown in Figure 8. As previously shown, the mutual inductance depends on the coupling factor. Therefore, it can be verified the influence of the coupling factor in the response of a double tuned circuit. It was run a Spice simulation, as shown in Figure 9, where the coupling factor ranged from 0.1 to 0.2, and the frequency varied from 12 to 16 MHz. As an initial condition, L₁ and L₂ were taken as 3,5 H and capacitors C₁ and C₂ are equal to 38 pF. In that case, the resonance frequencies at primary and secondary are equal and it will be taken as an individual resonance.

Figure 9.

Spice simulation of double resonance.

From Figure 9 it can be observed that larger coupling factor causes larger deviation of the resonance frequency from the individual frequency. The energy transfer by RF coupling is directly dependent on the variation of the coupling and consequently dependent on the approaching between reader and tag, in other words orientation of the reader and tag inductances strongly changes the double resonance between reader and tag.

5. Mutual inductance measurement method

According to [3] and [4], the mutual inductance between two inductors can be measured. Considering two magnetically coupled inductors, as shown in Figure 10, the inductance seen by the generator is given as:

$$L_s\text{'}=L_1+M_{12}+M_{21}$$E37

Figure 10.

Magnetically coupled circuits.

Thus, the inductance seen at the generator terminals is the inductance plus the mutual inductance of inductor 1 over inductor 2, plus the mutual inductance of inductor 2 over inductor 1. Now, considering the series association, as shown in Figure 11, the total inductance is given as:

$$L_s`=L_1+L_2+M_{21}+M_{12}$$E38

Figure 11.

Series association of inductors.

If inductances L₁ and L₂ are associated in series, as shown in Figure 11, the resulting inductance is the sum of the individual inductance and the mutual inductances. If the measurement method is altered, as shown in Figure 12, where one inductor is inverted, the new series inductance is given as:

$$L_s``=L_1+L_2-M_{12}-M_{21}$$E39

Figure 12.

Series association of inductors, in which one inductor is inverted.

Now, subtracting L’ from L’’ results:

$$L_1+L_2+M_{12}+M_{21}-L_1-L_2+M_{12}+M_{21}=L_s`-L_s``$$E40

By considering M₁₂ = M₂₁ =M, then:

$$M=\frac{L_s`-L_s``}{4}$$E41

Once L₁ and L₂ are known, the coupling factor can be found as:

$$k=\frac{M}{\sqrt{L_1{L}_2}}$$E42

6. Model for the energy transfer based on the classic circuit theory

The energy transfer between reader and tag in free space can be modeled using the classic circuit theory. Consider Figure 13 that represents the magnetic coupling between reader and tag.

Figure 13.

Magnetic coupling between reader and tag.

Observe that resistor R₂ represents the losses on L₂, R_tag represents the load at the tag and R₁ represents the losses at inductor L₁. The analysis can be greatly simplified if the reader circuit can be converted into the series equivalent circuit, since the reader load could be represented as the resistance losses of capacitor C₂. Therefore, the quality factor of capacitor C₂ can be expressed in terms of its losses resistance, as indicated in Figure 14. The equivalence between series and parallel circuit is granted if the real and imaginary parts are equal, thus:

$$y_p=G_p+j\omega C_p$$E43

$$R_s=\frac{G_p}{G_p{}^2+{(\omega C_p)}^2}$$E44

$$C_s=\frac{G_p{}^2+{(\omega C_p)}^2}{{\omega }^2{C}_p}$$E45

$$G_p=\frac{1}{R_p}$$E46

Figure 14.

Capacitive equivalent circuits.

That approximation allows the conversion of tag load R_p into R_s, as given by (44). The approximation is possible since the capacitor presents low losses at the frequency range. Thus, tag load provides the larger contribution in the energy dissipation in the magnetic coupling between reader and tag.

Observe also that the equivalent capacitor given by (45) is frequency dependent that implies in another source of variation in the resonance frequency at the tag. Based on that premise, the equivalent circuit is shown in Figure 15.

Figure 15.

Modified equivalent circuit.

The circuit of Figure 15 can be equated as:

$$R_t=R_2+R_s$$E47

Thus:

$$V_1=I_1(R_s+j\omega L_1+\frac{1}{j\omega C_1})-I_{2}j\omega M$$E48

where M is the mutual inductance between L₁ and L₂.

$$0=-I_{1}j\omega M+I_2(R_t+j\omega L_2+\frac{1}{j\omega C_2})$$E49

Combining (48) and (49) results into:

$$V_2=\frac{V_{1}M}{\begin{array}{l}[{(}^{\omega }+(Rt+j\omega L_2+\frac{1}{j\omega C_2})(R_s+j\omega L_1+\frac{1}{j\omega C_1})]C_2\\ \end{array}}$$E50

It can be observed from (5) the possibility of predicting the voltage at the tag, under free space conditions. Nevertheless, in real application, the tag shall be placed inside a human being, usually in the belly area, in order to measure any biological data. Since pork tissue presents electromagnetic properties similar to human tissue, it will be used in the measurements.

7. Comparison between simulated and measured data for free space

It was used the test set up shown in Figure 16, where L₁ =1.77 uH and L₂ =5.4uH. Observe that the inductors are axially aligned. That orientation was chosen to achieve the best magnetic coupling, considering the ratio between length and diameter of the inductor. The inductors series resistances R₁ =1.14Ω and R₂ =2.2 Ω (at 13.56 MHz) were obtained using a network analyzer. The experimental measurements and the simulations were conducted considering a generator having a 1V and a 50 Ω internal impedance. The tag load was taken as a 1 kΩ resistor. Table 1 presents the measurement results.

Figure 16.

Free space measurement test set up.

Measurement	Distance L1- L2	M
L’=10.00uH; L’’=6.33uH	5mm	0.917 uH
L’=9.48uH; L’’=6.71uH	10mm	0.692uH
L’=9.13uH; L’’=7.00uH	15mm	0.532uH
L’=8.74uH; L’’=7.26uH	20mm	0.370uH
L’=8.55uH; L’’=7.40uH	25mm	0.287uH

Table 1.

Mutual inductance measurements.

It was conducted a set of simulations of equations (44), (45) and (50) for the range of 12 to 16 MHz and the result is shown in Figure 17. It can be observed that the tag voltage is approximately 1V and the resonance frequency varied with the mutual impedance. Thus a variation in the resonance frequency will cause a change in the inductor series resistance. Therefore, in case of frequency deviation at the generator, the resistance losses can change and may alter the measurement data. As a consequence, variable capacitors C₁ and C₂ were added to adjust the tag voltage level at the same operation frequency. That procedure maintain the inductor characteristics regardless any frequency variation. Table 2 shows a comparison between free space measurement and simulation data.

Figure 17.

MATLAB simulation.

Distance	Measured	Simulated	M
25mm	769mV	1001mV	0.287uH
20mm	943mV	1061mV	0.370uH
15mm	1038mV	1063mV	0.532uH
10mm	1026mV	1055mV	0.692uH
5mm	1007mV	1052mV	0.917uH

Table 2.

Comparison between free space measurement and simulation data.

As it can be observed, there is an error between the measurement and the simulation data, as it is shown in Table 3. Note that the error is smaller than 4.3% for distances smaller than 15mm. The error becomes larger for distance over 15mm, since there is magnetic field scattering due to coil imperfections. The connection cables between generator and reader may cause a magnetic field distribution, thus increasing the error.

Distance	Error
5mm	-4.3%
10mm	-2.75%
15mm	-2.34%
20mm	-11.12%
25mm	-23.17%

Table 3.

Error between measurement and simulation data for free space.

8. Comparison between free space and animal tissue

Once the tag voltage for free space is known, it can be compared to implanted conditions, where there is tissue between reader and tag. Figure 18 shows the set up used for the measurements, and Table 4 provides the tag voltage comparison between free space and pork tissue conditions. Since pork tissue presents electromagnetic properties similar to human tissue, it will be used in the measurements.

Distance	Free Space	Pork Tissue
10mm	1932mV	1869mV
15mm	1869mV	1787mV
20mm	1671mV	1549mV
25mm	1247mV	1202mV

Table 4.

Comparison between free space and pork tissue data.

The generator voltage was raised to approximately 1.94V in order to assure that low voltage measurements are not spoiled by noise. The pork tissue thickness is approximately 10mm. As it can be directly observed form Table 4, the free space data is different form the pork tissue data. The small difference suggests that at 13.56MHz the pork tissue causes s variation in the magnetic permeability as compared to free space. The error between free space and pork tissue data is presented in Table 5. The 5mm measurement was not conducted due to the pork tissue thickness The first measurement, 10mm, was conducted with the reader in direct contact with the pork tissue.

Figure 18.

Test set up using pork tissue.

Distance	Error
10mm	-3.26%
15mm	-4.39%
20mm	-7.30%
25mm	-3.61%

Table 5.

Error between free space and pork tissue data.

9. Conclusions

The classic circuit theory can be used to obtain the tag voltage from the mutual inductance between reader and tag in a RFID system. Therefore, based on the estimate of the mutual inductance between reader and tag, it is possible to estimate voltage at the tag. Nevertheless the main distress is the variation on the resonance frequency due to the mutual inductance, and ultimately, the coupling factor.

Two approximations must be conducted to estimate the tag voltage. The first one is due to the magnetic scattering between reader and tag inductors that becomes more relevant as the distance between them increases. The second approximation is due to the tissue that alters the magnetic permeability and consequently reduces the tag voltage. Thus, by considering the simulation errors and the errors due to the tissue, it is possible to estimate the voltage at the tag. Table 6 presents the compiled errors.

Distance	Error
10mm	-5.92%
15mm	-6.66%
20mm	-17.60%
25mm	-26.90%

Table 6.

Total errors between simulated and measured data.

Therefore, by taking into account both approximations it is possible to estimate the voltage drop at the tag and consequently, the power reduction. Once the power reduction is known, it can be taken into account during the tag circuit design. Consequently, the use of classic theory of magnetically coupled circuits is a simple way to estimate the voltage in implanted devices. Nevertheless additional studies regarding the resonance frequency variation is still required.