#### 4.4.1. Influence of the Carbon Fiber Reinforcement Embedded in Concrete, 0.4 to 5 GHz

Figure 6a shows the measurement results for all concrete samples from 0.4 to 5 GHz. All samples show an oscillating behavior depending on the sample thickness, which can be explained by internal interference (i.e., the superposition of multiple reflections at the medium boundaries). The samples with carbon fiber reinforcements strongly attenuate the transmitted power up to 2 GHz. The non-reinforced samples do not show this behavior, independent of their thickness. Also, the thicker samples cause a higher attenuation than the thinner ones. The attenuation declines much faster compared to the reinforcement material (see

Figure 4), being effectively indiscernible already at about 2 GHz, whereas in

Figure 4 the influence is notable up to about 5 GHz. As in

Figure 4, the attenuation is dependent on polarization.

In order to explain this behaviour, we turn to the wire screen model [

33] again. The wire screen is now embedded in a dielectric. Recalling that the speed of light inside a medium depends on relative permittivity

${\epsilon}_{\mathrm{r}}$ and relative permeability

${\mu}_{\mathrm{r}}$:

with

${\epsilon}_{0}$ and

${\mu}_{0}$ the permittivity and permeability of free space. Inserting into Equation (

1), replacing

${c}_{0}$ with

c, and setting

${\mu}_{\mathrm{r}}=1$ (due to the dielectric nature of the medium), yields:

which is the shielding efficiency for a small aperture metal shield, embedded in a dielectric characterized by the frequency-dependent relative permittivity

${\epsilon}_{\mathrm{r}}\left(f\right)$.

In

Figure 6b the attenuation of the reinforced concrete slabs is shown, together with the predictions from Equation (

3) for wire screens with

$10.7$ mm and 13 mm gap widths (see

Table 2), with

${\epsilon}_{\mathrm{r}}$ from the fit curve in

Figure 5. As can be seen, the curves follow the predicted attenuations, although the measured attenuations are much higher. This can be attributed to the low thickness of the material and the increasingly low frequencies, so that a frequency-dependent effective permittivity can be assumed:

where

$0\le z\le 1$ is the fraction of the wavelength that is inside the dielectric. As can be seen, using an effective permittivity based model yields a much better fit, which, most importantly, does not underestimate the losses.

#### 4.4.2. 5 to 67 GHz

Figure 7 shows the loss experienced by the electromagnetic wave traveling through the concrete slabs from 5 to 67 GHz for the 10 mm samples, and 5 to 47.5 GHz for the 20 mm samples. As expected, the attenuation increases with frequency. For the 20 mm samples it increases twice as fast as for the 10 mm samples, due to the double thickness (double

y axis range in

Figure 7a vs.

Figure 7b).

Looking at theory, two mechanisms can explain this behaviour: loss attributed to the $tan\delta $ of the material, and loss due to reflections at the material boundaries.

In order to quantify the dielectric loss, we assume a non-magnetic medium

${\mu}_{r}=1$ with zero conductivity

$\sigma =0$. The

attenuation constant ${\alpha}_{\mathrm{d}}$ due to dielectric losses can be expressed as [

37]:

Using the relationships

we can restate using frequency-dependent terms:

The dielectric loss

${L}_{\mathrm{d}}$, in

$\mathrm{dB}$, can then be written as follows:

with

x the material thickness in

$\mathrm{m}$. Assuming homogeneous material, it is useful to normalize to a certain thickness for easier comparison. For 1 cm, Equation (

7) becomes:

To calculate the losses that occur at the medium’s boundary transitions, Fresnel’s equations (e.g., Reference [

38]) can be used. Two transmissions are necessary: from free space into concrete, and from concrete back into free space. Assuming perpendicular incidence, the (double) transmission loss

${L}_{\mathrm{T}}$ is:

Both

Figure 7a,b show

${L}_{\mathrm{d}}$ as well as

${L}_{\mathrm{T}}+{L}_{\mathrm{d}}$, with the values for

$tan\delta \left(f\right)$ and

${\epsilon}_{\mathrm{r}}\left(f\right)$ taken from

Figure 5. The measured curves of the 10 mm samples presented in

Figure 7a follow the curves as predicted by theory closely. The discontinuity at 50 GHz is caused by the switch-over to the different measurement setup (see

Section 3.3). The measured attenuation is a little lower than predicted up to 50 GHz; the other measurement setup yields values a little higher than predicted, always within a 4 dB range. As shown in

Figure 7b, the measured data of the unreinforced 20

$\mathrm{m}\mathrm{m}$ samples also closely follows predictions. Looking at both

Figure 7a,b, the attenuation of the un-reinforced samples is stronger than predicted by theory towards higher frequencies. This may be caused by surface roughness, as described above and shown in

Figure 2.

For the 20 mm samples shown in

Figure 7b, measurement data is only useable up to

$47.5$ GHz, due to the combined losses of free space path and material. It can be seen that the different samples group: both curves of each BZT 1, BZT 2, and “no reinforcement” are close to one another. The samples without reinforcement show the lowest attenuation, followed by BZT 2 and then BZT 1, each separated by about 4 dB. This is contrary to the expectation that the reinforced and unreinforced samples would group, and reinforcement would have no significant influence. This may be caused by additional internal scattering by the carbon fiber reinforcement. This effect can be captured by a 20% “penalty” to the normalized loss factor

${L}_{\mathrm{n}}$, also shown in

Figure 7b.

Looking again at

Figure 7a with Equation (

8) in mind, we can see that

${L}_{\mathrm{d}}={L}_{\mathrm{n}}$, due to the material thickness being 1 cm.

${L}_{\mathrm{n}}$ is very high, with 5 dB/cm at 10 GHz and 26 dB/cm at 60 GHz. In Reference [

17], a measurement of a concrete sample with

${L}_{\mathrm{n}}\approx \phantom{\rule{0.166667em}{0ex}}$ $3.6$ dB/cm at 10 GHz is presented. Unfortunately, details about the manufacturing process are not included. The value is in agreement with our measurements, also showing a very high loss of concrete.