# SDR-Enabled Multichannel Real-Time Measurement System for In Situ EMF Exposure Evaluation
## 2 Materials and Methods
### The Proposed Sensor
Note that the incident field on the two antennas (antenna 1 and antenna 2) have same amplitude, but different phases, depending on the direction of the plane wave. The field on the  
Figure 1: Schematic description of the proposed E-field sensor (**a**) and E- and H-field sensor (**b**). Figure 2: Circuit with antenna in receive mode; (**a**) schematic diagram; (**b**) circuit diagram Even if the re-irradiated energy is quite small, it can be received by antennas that are very close to the re-irradiating antenna, causing mutual coupling. The analysis of mutual coupling for receiving antennas is quite complex. In the following, we will introduce a simplified model that allows us to have a rough idea of the impact of the mutual coupling. two antennas will be called \(E_{i1}\) and \(E_{i2}\), wherein \(E_{i1}=E_{i2}\ exp(j\varphi)\), \(\varphi\) being the phase difference. Two different coordinate systems (\(z_{1}\) and \(z_{2}\)) are used for the two antennas to have voltage definitions coherent with the standard definitions using in-microwave scattering parameters \(S\), which will be used in the following discussion. Let us consider antenna 1. The incident field \(E_{i1}\) excites a voltage wave \({V_{1}}^{-}=k_{1}\ E_{i1}\) (\(k_{1}\) being a quantity representing the response in amplitude and phase of the antenna 1, this quantity depends on the direction of propagation and polarization of the incident field) traveling from the antenna toward the load (i.e., the receiver R\({}_{1}\)). In field level measurement systems, antennas are matched to the receiver. Accordingly, in the following, we suppose that half the energy is absorbed by the load, while the other half is re-radiated by the antenna. Accordingly, we suppose a voltage wave \({V_{1}}^{+}\), having amplitude equal to \({\mid V_{1}}^{-}{\mid}/\sqrt{2}\) amplitude, which feeds antenna 1, causing a field \(E_{521}\) incident on antenna 2. In practice, antenna 2 received two contributions, \(E_{i2}\) and \(E_{521}\). The voltage on the receiver (i.e., R\({}_{2}\)) is the interference between the two voltage waves \({V_{2i}}^{-}\) and \({V_{2s}}^{-}\) associated to \(E_{22}\) and \(E_{521}^{-}\). The antenna 1 and antenna 2 system can be modeled as a two-port linear microwave device, whose input-output relationships can be modeled using \(S\) scattering coefficients [32]. The scattering coefficient S\({}_{21}\) is defined as \(S_{21}={V_{2}}^{-}/{V_{1}}^{+}\) when no voltage waves travel from the receiver R2 toward antenna 2. Accordingly, we have that the scattered field \(E_{s}\) gives a voltage wave at the gap of antenna 2 equal to \({V_{2s}}^{+}={V_{1}}^{+}\ {S_{21}}=k_{2}\ E_{i1}\ {S_{21}}/\sqrt{2}\), while the incident field \(E_{i}\) on antenna 2 gives a wave voltage equal to \({V_{2i}}^{+}=k_{2}\ E_{i2}\). Note that \(k_{2}\) is generally different from \(k_{1}\) since they generally depend on the orientation of the antennas in the space with respect to the incident field. The voltage at the gap of the antenna is \({V_{2}}^{+}=(k_{2}\ E_{i2}+k_{1}\ E_{i1}\ {S_{21}}/\sqrt{2})\). To have a rough estimation, let us suppose that \(k_{1}\approx k_{2}=k\). In this case, we have \({V_{2}}^{+}\approx k\ E_{i2}\ (1+{S_{21}}\ exp(j\varphi)/\sqrt{2})\) instead of \(k\ E_{i2}\), that is, the value in absence of mutual coupling. Since the phase of the second term is not known, the relative measured quantity ranges in  
\[\Delta\ E\ \approx\ \Big{(}\mathbf{k}\Big{|}E_{i2}\Big{|}\Big{(}1\ \pm\ S_{21}/\sqrt{2}\ \Big{)}\Big{)}/(\mathbf{k}\big{|}E_{i2}\big{|})=1\ \pm\ S_{21}/\sqrt{2} \tag{1}\]  
It must be noted that this is a highly simplified model. For example, the presence of antenna 2 also modifies the currents on antenna 1, mismatches between the antenna and the load change the re-irradiated energy, short antennas used in field measurements usually have losses (that help to enlarge the bandwidth avoiding too much deep resonances), and so on.