# First SIMO Harmonic Radar Based on the SFCW Concept and the HR Transfer Function
## 2 The HR Equation and HR Transfer Function
### Realization of the HR Tag Matrix  
The development of the harmonic radar tag matrix was started with its basic building block--the HR tag cell. Due to economic reasons, a two-layer Isola Itera RF substrate with a thickness of 0.508 mm was chosen to realize the matrix. In a longer development process in Ansys HFSS to balance bandwidth, gain and return loss of the planar antennas on the thin substrate two RX/TX patch antennas for the tags were developed. The finally manufactured antenna visualized in Figure 6 has the following dimensions:  
RX antenna: length and width 32.1 mm, peeled corner 2 mm, slot length 5.6 mm and slot width 0.6 mm, TX antenna: length and width 16 mm, peeled corner 1 mm, slot length 2.3 mm and slot width 0.4 mm. Figure 5: **Left**: Photo of the RHCP-polarised 2.45 GHz and LHCP-polarised 4.9 GHz planar HR interrogator patch antenna array. **Right**: Back-side of the antenna containing the power splitters/combiners, phase and impedance matching networks, feedthroughs, and land patterns for P-SMP connectors. The corresponding matching values are \(-12\) dB for both bands in the band center. The feed point impedances were chosen so that the antennas could be matched over the widest possible bandwidth. The switchable frequency doubler, which also supports a tuning capability via its bias voltage, requires even more development effort than is reflected in its simple circuit topology shown in Figure 7. This nonlinear microwave circuit was simulated and optimized with the help of the harmonic balance simulation function of the RF circuit simulator Keysight _ADS_ - _Advanced Design System_ using the spice and scattering parameters provided of the components supplied by the manufacturer. Additional background information about the design and characterization of cooperative HR tags can be found in [18; 19; 29; 30; 31]. In this design, a BAT15 Schottky diode from Infineon was used as the nonlinear element of the tag, while the BAR64 PIN diode from Infineon was used as an RF switch for the illumination signal at the nonlinear element's input. A custom equivalent circuit model was fitted for the PIN diode from the available manufacturer data and the series inductances and pad capacities of the discrete components used were also taken into account. The antenna feed point impedance obtained by the simulation in HFSS were used as port impedance in the harmonic balance optimization process. In order to optimize the tag in the harmonic balance simulation for operation in the square-law transfer function region, it is important to obtain an approximate figure of the available illumination signal input power at the doubler's input, which can be calculated using classical link budget models. The path loss according to Friis transfer equation at 2 m distance from the tag reflection matrix is:  
\[F^{I} =20\cdot\log\left(\frac{4\pi\cdot 2\,\text{m}\cdot 2.45\,\text{GHz}}{c }\right)\,\text{dB}\quad,\text{ resulting in} \tag{12}\] \[F^{I}|_{2\,\text{m}} =46.3\,\text{dB}\enspace.\]  
When a transmission of an illumination signal with a transmit power of +27 dBm = 500 mW EIRP, equivalent to +17 dBm at the TX antenna's feedpoint at a frequency of  
Figure 6: Sketch of the produced patch tag antenna including frequency doubler circuit with the stubs and diode footprints. All elements together form one matrix cell element. 2.45 GHz is assumed, the input power to the doubler can be estimated with the transponder's illumination signal receiving antenna gain of 6.9 dBi to be  
\[P_{RXT}^{I}|_{2\,\mathrm{m}} =27\;\mathrm{dBm}-46.3\;\mathrm{dB}+6.9\;\mathrm{dB}\quad,\;\mathrm{ therefore} \tag{13}\] \[P_{RXT}^{I}|_{2\,\mathrm{m}} =-12.4\;\mathrm{dBm}\;,\]  
which is certainly an absolute best case figure and can be used as the worst case scenario for avoiding harmonic compression effects in the doubler.