## 3 Results and Discussion

Table 2: The real and imaginary part of complex permeability and complex permittivity at 9 GHz, and the conductivity of spinel ferrites (NiZn)\({}_{1-\mathrm{x}}\)Mn\({}_{\mathrm{x}}\)Fe\({}_{2}\)O\({}_{4}\) and their nanocomposites with CNTs. EMI shielding effectiveness (\(SE\)) can be quantitatively expressed using the following equation [38]:  
\[SE=10\lg\left(\frac{P_{t}}{P_{t}}\right) \tag{2}\]  
where \(P_{t}\) and \(P_{t}\) are the power of the incident wave and the wave that travels through the material in decibels (dB), respectively. In the theory of EMI shielding, as an electromagnetic wave interacts with the shielding material, the incident power undergoes division into reflected, absorbed, and transmitted power. The corresponding power coefficients of absorbance (\(A\)), reflectance (\(R\)), and transmittance (\(T\)) satisfy the condition \(A+R+T=1\)[39]. In the microwave absorption model [40], the transmission waves are always negligible and reflection loss (\(RL\)) is the difference between the initial incident wave and the final reflected wave, including all the back-propagating EM waves reflected on different surfaces and interfaces. \(RL\) can be calculated using the measured complex permittivity and complex permeability data for the given frequency and absorber thickness using the following equations [38; 39]:  
\[RL(dB)=20 log\left|\frac{Z_{in}-Z_{0}}{Z_{in}+Z_{0}}\right| \tag{3}\]  
where \(Z_{in}\) is the input impedance of absorbent and \(Z_{0}\) is the intrinsic impedance in air; and  
\[Z_{in}=Z_{0}\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}}tanh\left(j\frac{2\pi fd}{c} \sqrt{\mu_{r}\varepsilon_{r}}\right) \tag{4}\]  
where \(\mu_{r}=\mu^{\prime}-i\mu^{\prime\prime}\) is the relative complex magnetic permeability (\(\mu^{\prime}\), \(\mu^{\prime\prime}\)--real and imaginary components of the complex permeability, respectively), \(\varepsilon_{r}=\varepsilon^{\prime}-\varepsilon^{\prime\prime}\) is the relative complex dielectric permittivity (\(\varepsilon^{\prime}\), \(\varepsilon^{\prime\prime}\)--real and imaginary components of the complex permittivity, respectively), \(f\) is the electromagnetic wave frequency, \(d\) is the thickness of sample, and \(c\) is the velocity of light in a vacuum. The results presented in Figure 5a demonstrate the change in absorption in the whole frequency range at different manganese ion contents in (NiZn)\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) ferrite at x = 0, 0.5, and 1. A significant increase in the bandwidth (\(RL\)) and its broadening at 9 GHz has been observed; namely, for samples at x = 0 and x = 1, the bandwidth corresponds to 4.4 GHz and 14 GHz, respectively. At the same time, a change in reflection loss from \(-10.5\) dB to \(-15.8\) dB was observed for the (NiZn)\({}_{1-x}\)Mn\({}_{x}\)Fe\({}_{2}\)O\({}_{4}\) substituted ferrite samples. The EM wave absorption processes in nanocomposites based on carbon nanotubes decorated with substituted ferrites (Figure 5b) are more efficient in the frequency range of 12 GHz to 20 GHz, most likely as a result of better impedance matching characteristics than for unmodified ferrite samples. Meanwhile, the minimum values of \(RL\) shifted to the low frequency region and the values also decreased due to an increase in the conductivity of the composites. Analyzing the mechanisms of electromagnetic wave absorption shows that the increase in losses, associated to both eddy currents and conduction losses, is a result of increased conductivity [41]. However, in the case of high conductivity, the impedance of the material is relatively small compared to the impedance of air and, as a result, almost all of the electromagnetic waves will be reflected. The loss of energy of the incident EM wave by absorbing materials is realized due to the joint action of the absorption and interference mechanism. In this case, the absorption mechanism is mainly determined by the impedance matching and the ability to attenuate the EM wave depending on the ratio of dielectric losses and magnetic losses.