Theoretical background

In order to calculate the received power, the well-known radar equation needs to be adapted to the specific scenario of measurement of many small particles. The first important consideration are possible multiple scatterings inside the measurement volume which would have a significant influence on the calculation. However, multiple reflections can be neglected when $N_0\sigma_ts_v\ll 1$ is valid [Reference Skolnik8]. Here, N ₀ stands for the particle concentration, σ_t is the total scattering cross section of a single particle, and s_v is the side length of a cube representing the scattering volume. For the measurements investigated in this work, the particles are small in comparison to the wavelength and therefore the total scattering cross section of a single particle is comparably low. As a result, this condition can be assumed to be fulfilled in all measurements [Reference Goldhirsch9]. Consequently, the total received power of a volume with several small reflectors results from the sum of all individual reflectors or particles in this case. Applying this to the radar equation leads to the following variation for the total received power of a particle volume for this specific measurement setup:

(1)\begin{equation} P_R=P_T\frac{G_TG_R\lambda^2}{(4\pi)^3}\sigma_0V_{ef}. \end{equation}

In this context, σ ₀ represents the specific radar cross section (RCS) of the volume, P stands for power, G for gain, λ for the wavelength of the signal, and V_ef is the effective measurement volume. Indicators T and R mark the transmitted and received component, respectively. The RCS can be calculated using the particle concentration N ₀ in the volume, a size distribution function of the various particle diameters $f_x(r_p)$ inside the measurement volume and the RCS of all individual particles as follows:

(2)\begin{equation} \sigma_0=N_0\int_0^\infty f_x(r_p)\sigma_p(r_p)dr_p. \end{equation}

The effective measurement volume is dependent on the radiation pattern of the antennas that are implemented for the measurement. It is the superposition of the receiving and transmitting antenna patterns $F_{T,R}$ and can be determined computing the following integral:

(3)\begin{equation} V_{ef}=\int_V\frac{F_T^2F_R^2}{R_T^2R_R^2}. \end{equation}

The only parameter in Eq. (1) that depends on the diameter of the particles is the RCS. Rearranging Eq. (1) with respect to RCS yields:

(4)\begin{equation} \sigma_0=P_R \frac{(4\pi)^3} {G_TG_RP_T\lambda^2V_{ef}}. \end{equation}

It is evident that, except for the received power P_R, all parameters are system parameters that can be determined before the measurement itself. Consequently, with the measurement result and the knowledge of all system parameters, the RCS can be computed with the Eq. (4). However, Eq. (2) still has too many unknowns to determine the diameters from the measured results.

In order to improve this, a further measurement with a sensor at a different frequency can be performed. With two frequencies, two RCS can be calculated with the measured received power P_R and the system parameters enabling a comparison between these two to obtain additional information regarding the particle size. By dividing the two RCS, the differential RCS (DRCS) is calculated as follows:

(5)\begin{equation} \sigma_d=\frac{\sigma_{0,1}}{\sigma_{0,2}}=\frac{\int_0^\infty f_x(r_p)\sigma_{p,1}(r_p)dr_p}{\int_0^\infty f_x(r_p)\sigma_{p,2}(r_p)dr_p}, \end{equation}

where the unknown particle concentration N ₀ is already eliminated from the equation. Note that this is only valid when the dual-frequency measurement is simultaneous and therefore measures the same particle concentration inside the volume. However, analytical calculation with this equation is still not feasible due to the infinite integral. This can be avoided by making a priori calculations which can be compared to the measured results and allow conclusions to be drawn regarding the size of the particles. For these calculations, a further insight into the different scatterings of small particles is required.

Given that the measured particles in this work have a diameter below 1 mm and frequencies are between 90 and 180 GHz, the theories of Rayleigh and Mie scattering are primarily relevant for the examination. Both scattering theories are mainly applicable to spheres. The Rayleigh scattering is valid for spheres with a radius smaller than approximately $r=\frac{\lambda}{10}$ while the Mie scattering can be applied at higher radii [Reference Balanis10]. Moreover, the Mie theory provides an equation for the calculation of the normalized RCS. In Fig. 1, the calculated values are presented where the three relevant regimes are highlighted by different colors. In the Rayleigh regime (blue), the normalized RCS increases linearly with increasing radius. The second area is the Mie regime (green). Here, the normalized RCS is very dependent on the specific radius. For spheres with higher radius, the optical region begins where the normalized RCS approaches a constant value.

For the calculation of the particle radius inside a stream, the transition between the first two regions is of importance. As mentioned, the transition is not dependent on the absolute sphere radius but its ratio to the wavelength. Therefore, when measurements are performed with different frequencies, this transition occurs at different absolute sphere diameters. Additionally, when the particles are dielectric spheres, the refractive index or the permittivity has an influence on both the overall reflected power and on the position of the transition [Reference Ruck, Barrick, Stuart and Krichbaum11]. With the knowledge of the particles permittivity and the transmitting frequency, the RCS can be calculated with the Mie theory.

Figure 1. Calculated monostatic RCS of a sphere depending on its radius in ratio to the signal wavelength.

Figure 2 shows the exemplary calculated DRCS of a dual-frequency approach with frequencies at $f_1=160$ GHz and $f_2=90$ GHz with spherical glass particles with a permittivity of $\varepsilon_r = 3.8+0.005j$ [Reference Chen, Nguyen and Afsar12]. To verify these calculated results, the monostatic RCS of a single sphere is computed for both frequencies with a full-wave simulation software and compared. The simulated results agree with the calculations, thus confirming this approach. Initially, a constant value can be observed in Fig. 2. This occurs at particle sizes within the Rayleigh scattering regime, thereby both measured RCSs are increasing linearly with particle size. Inside the Rayleigh regime, the RCS of a single particle can be calculated in a simplified way as follows [Reference Ruck, Barrick, Stuart and Krichbaum11]:

(6)\begin{equation} \sigma_{rayleigh}=\frac{128}{3} \frac{\pi^5r_p^6}{\lambda^4} \frac{(\varepsilon'-1)^2+\varepsilon^{\prime\prime 2}}{(\varepsilon'+2)^2+\varepsilon^{\prime\prime 2}} \end{equation}

where $\varepsilon'$ corresponds to the real part and $\varepsilon^{\prime\prime}$ to the imaginary part of the permittivity. Because both devices measure the same particle radius and permittivity, everything besides the wavelength can be reduced. Note, that is valid for the chosen particles of glass due to the almost constant permittivity at the investigated frequencies [Reference Chen, Nguyen and Afsar12]. In the given exemplary scenario, the constant DRCS value can then be calculated with Eq. (6) to:

(7)\begin{equation} \frac{\sigma_{d,1}}{\sigma_{d,2}}=\left(\frac{f_1}{f_2}\right)^4=\left(\frac{160\,\text{GHz}}{90\,\text{GHz}}\right)^4=9.99\,\text{dB}. \end{equation}

Figure 2. Calculated monostatic DRCS for a constant frequency $f_1=90$ GHz and $f_2=160$ GHz. The blue area marks the area of unambiguity.

Figure 3. Calculated monostatic DRCS for a constant frequency $f_1=90$ GHz and various frequencies between $f_2=110$ GHz and $f_2=180$ GHz.

The blue-marked area in Fig. 2 indicates the area of unambiguity. In this segment, the scattering of the higher frequency signal is already in the Mie regime, while the lower frequency signal still is within the Rayleigh regime. Here, a measured DRCS corresponds to one specific pre-calculated DRCS and can therefore be connected to the corresponding particle radius r_p. With further increasing particle radius, both signals are scattered in the Mie regime and the measured DRCS cannot be assigned to one distinct radius. The two frequencies of the measurement setup, therefore, have a significant influence on which particle diameters can be detected and should be chosen carefully depending on the measured particles. A dynamic choice of frequencies could also be advantageous to shift the area of unambiguity during the measurement to obtain further information. Furthermore, a bistatic approach also influences the steepness inside the area of unambiguity as the energy is not reflected equally to all angles in the Mie range. However, the position of the minimum does not change and the determinable maximum radius does not change.

These calculations are assuming a monodisperse particle stream, which is not always the case in practical applications. Several studies have shown that there are two applicable size distribution models for the naturally occurring particle streams [Reference Heintzenberg13, Reference Fang, Patterson and Turner14]. One is the log-normal or Galton distribution while the second one is the Waloddi Weibull distribution. These distributions can have a significant influence on the actual DRCS as they are included in Eq. (5). However, in real measurements, the distribution typically is not known beforehand making it significantly more complex to determine the radius.

Radar sensors

In this work, two fully integrated radar sensors are used for the simultaneous dual-frequency measurement. One sensor has a fixed frequency in the W-band at 90 GHz while the frequency of the second sensor can be varied over the entire D-Band. To show the influence on the DRCS, in Fig. 3, the calculation results over different particle radii r_p are shown. The detectable particle radius decreases with increasing frequency f ₂ which is expected due to the transition between Rayleigh and Mie scattering being at lower particle radius. Furthermore, the constant value at lower radius is changing as expected by Eq. (7).

The two integrated radar systems have been previously introduced in prior articles [Reference Kueppers, Jaeschke, Pohl and Barowski15, Reference Jaeschke, Bredendiek, Kueppers, Schulz, Baer and Pohl16]. Figure 4 shows a top view of the sensors. The W-band sensor consists of only the sensor itself with a transition from microstrip to rectangular waveguide, which is located in the upper half of Fig. 4(b), where the golden plates are shown [Reference Hansen, Kueppers and Pohl17]. On the other hand, the D-band sensor is integrated in a setup which provides further possibilities of connectivity and offers an integrated data acquisition unit which can be seen in Fig. 4(a). The waveguide transition is on the top of the picture. Both sensors are installed inside a dust proof metallic housing.

Figure 4. View of the fully integrated sensor of the (a) D-band and (b) W-band.

Figure 5. Simulated scattering pattern of one sphere when the radius is (a) inside the Rayleigh regime and (b) inside the Mie regime.

The W-band sensor exhibits an output power of approximately −2 dBm at a stable operating frequency of 90 GHz. For the processing of the measurement results, an external data acquisition unit is added to digitize the signal with a resolution of 24 bit and a sampling frequency of 96 kHz. The D-band sensor has a data acquisition unit with a resolution of 24 bit and a sampling frequency of 256 kHz, which is connected to RF front end as seen in Fig. 4. With its software, various operational parameters can be adjusted. For instance, the transmitting frequency can be swept over the entire band and the internal amplifier can also be set to a specific value. The output power is approximately −8 dBm. Both receivers have a noise figure of app. 17 dB.

In this measurement, rectangular horn antennas are used due to the well-defined radiation pattern. For the manufacturing of the antennas, a negative is milled out of conductive metal and the antenna is then made with electroplating. This process leads to reliably low tolerances in dimensions. The dimensions of the aperture are $a_D=7.7$ mm, $b_D=5.8$ mm with a length $l_D=29$ mm for the D-band antenna and $a_W=14.5$ mm, $b_W=10.2$ mm with a length of $l_W=58$ mm for the W-band antenna. In contrast to the measurements presented in [Reference Braasch, Bruhn, Teplyuk, Freiwald, Durdaut, Vogelsang, Pohl and Höft1], a quasi-monostatic approach is implemented for both sensors. Figure 5(a) and (b) shows the exemplary scattering pattern of one sphere when the diameter is inside the Rayleigh or Mie regime, respectively. Note that the source for this scattering is located at ϕ = 0 and θ = 0. Inside the Rayleigh regime, the power is scattered in the shape of a dipole as shown in Fig. 5(a). The measured power with a bistatic setup would therefore be almost constant. However, in the Mie regime, there are strong minima depending on the angle and the particle diameter as seen in Fig. 5(b) which leads to differences of the received power of up to 20 dB [Reference van de Hulst18]. This is avoided with the quasi-monostatic setup implemented here, which also still includes a separation of transmitting and receiving path.

Measurement setup

In comparison to the presented work in [Reference Braasch, Bruhn, Teplyuk, Freiwald, Durdaut, Vogelsang, Pohl and Höft1], the setup is adjusted to further improve the repeatability of the measurements and to have a more stable particle flow. Figure 6 shows a schematic of the overall setup. The presented setup is a proof-of-concept for this method to show its possibilities for future applications. At the top right, a fan accelerates the air as marked to the velocity v_air which is estimated to be around 9 $\frac{\text{m}}{\text{s}}$. The particles are induced at the top with a velocity v_particles and experience further acceleration due to the air flow as well as the gravity. This velocity is estimated to be around 3 $\frac{\text{m}}{\text{s}}$ inside the measurement volume. In order to improve the distribution over the cross-section of the tube, the narrow part as seen in the schematic figure is added which essentially realizes a de Laval nozzle [Reference Rezende19]. This leads to a higher velocity of the air and the particles inside the narrower tube, and with the continuous enlargement afterward, the particles and velocity are more evenly distributed. The pipes above the de Laval nozzle are made out of sheet steel. Below the nozzle where the actual measurements take place, the pipe is made out of transparent acrylic glass with a permittivity of 2.6 and a thickness of 4 mm. The acrylic glass pipe has a length of 2 m, resulting in an overall length of over 3 m for the complete setup. The particles are induced by a dispenser with a constant mass flow. At the bottom of the tube, the particles are captured so they are not induced into the environment.

Figure 6. Schematic representation of the measurement setup.

The realized setup with the attached radar sensors is shown in Fig. 7. Aluminum profiles are used for the supporting frame due to its flexibility for adaptation. Fastened to the aluminum profiles are attachments to secure both mounts for the measuring volume as well as for the W-band sensor. Furthermore, the sensor is attached to a special fixture that allows the adaption of the angle. The D-band sensor is mounted on a tripod which offers a similar possibility to freely adapt the angle and position. With a digital protractor, this angle can then be precisely set.

Figure 7. Measurement setup with both radar sensors.

Figure 8. Directivity of the D-band horn antennas.

Before the measurements itself are conducted, the system parameters for both devices can be determined with Eq. (4). Both the gain as well as the effective scattering volume V_ef are dependent on the radiation pattern of the antennas [Reference Teplyuk, Knöchel and Khlopov3]. The measured radiation patterns of the manufactured antennas are shown in Figs. 8 and 9, respectively. Furthermore, the gain of the antennas are measured to be 20 dB for the W-Band antenna and 19 dB for the D-Band antenna.

For the calculation of the effective measurement volume V_ef, a second degree polynomial is used to approximate the main lobe of the antennas. These functions are implemented for F_T and F_R in Eq. (3). Due to the implementation of identical antennas, $G_R = G_T$ is valid in both cases. In order to incorporate the angle of the antennas in the calculation, a local coordinate system is used that transforms the global Cartesian coordinates depending on the angle of the antennas. Due to the quasi-monostatic setup, the distance between receiving and transmitting antenna is assumed to be equal. The resulting function for the two sensors in the W-band and the D-band:

(8)\begin{equation} V_{ef}=\int_V\frac{F_{T}^2 \cdot F_{R}^2}{R_{T,R}^2} \end{equation}

is then integrated. The limits of the integral are set at a radius of 75 mm around the center of the tube because most of the particles are concentrated in the center. Table 1 shows a comparison of the system parameters for both sensors where the angle of the antennas is set to 30^∘.

Figure 9. Directivity of the W-band horn antennas.

Table 1. Comparison of the two systems of the W-band and the D-band

With these calculations, the systemic parameter can be determined before the measurement itself is conducted. All parameters besides the received power in Eq. (4) are known and will be combined to a systemic value S. The measured results are as follows:

(9)\begin{equation} S_W = \frac{(4\pi)^3}{G^2P_T\lambda^2V_{ef}} \approx 2.52\cdot10^5\frac{\text{m}^2}{\text{W}}, \end{equation}

(10)\begin{equation} S_D = \frac{(4\pi)^3}{G^2P_T\lambda^2V_{ef}} \approx 2.36\cdot10^6\frac{\text{m}^2}{\text{W}}, \end{equation}

where G_T and G_R are combined to G ² due to the identical transmitting and receiving antenna.

Measurement results

A measurement is carried out with glass particles with a mean diameter of 0.875 mm to prove the concept. Before the actual measurement itself, the particle flow of these particles with the implemented particle dispenser is measured with the help of a precise weighing scale. The particles are dispensed into the scale for one minute. The determined mass flow is 1.5 gs⁻¹. To ensure that the measured particle concentration is the same with both sensors, the measurement volume must be within the same range for each. Therefore, the elevation angle of the sensors should be identical for both setups and are set to an angle of 30^∘ as shown in Fig. 6.

The signal to noise ratio (SNR) of the measurement results is shown in Figs. 10 and 11, respectively. In order to minimize the impact of random noise on the measurement, 1000 spectra of constant particle streams are measured and averaged in the presented results. Both measurements show a very prominent peak with an SNR above 20 dB. The Doppler frequency varies between the two results as expected from $f_D=\frac{2v}{\lambda}$.

Figure 10. SNR of the measurement with the W-band sensor.

Figure 11. SNR of the measurement with the D-band sensor.

The SNR of this result can be integrated in order to obtain the received power from scattering of the particles inside the measurement volume. Note that the amplification inside the data acquisition unit has to be considered. With the determined received power P_R, the two individual RCS can then be calculated by:

(11)\begin{equation} \sigma_{0,W,D}=P_R\cdot S_{W,D}. \end{equation}

The next step is the calculation of the DRCS with these two values as shown in Eq. (5):

(12)\begin{equation} \sigma_d=\frac{\sigma_{0,D}}{\sigma_{0,W}}=0.5848=-2.33\,\text{dB}. \end{equation}

This value can now be compared to the predetermined plot that is shown in Fig. 2. For the given frequencies of 160 and 90 GHz, the calculated DRCS corresponds to a particle radius of either 0.396 mm or 0.44 mm. The latter value is very close to the expected mean radius of 0.4375 mm with a deviation of under 2 %. Since the measured particles lie outside the area of unambiguity, the distinction between the two radii in question can only be made using additional information. Typically, there is some pre-knowledge regarding an expected particle size for an application, which can be used to select proper frequencies for the measurement. However, additional information could also be obtained by adding more measurements like a third sensor at a different frequency. The difference between the calculated and the real values is possibly due to the unknown distribution of the particle radii. As discussed in the theoretical chapter, these calculations are mainly for monodisperse particle streams and different particle distributions can potentially effect the results of the measurement. These results not only prove that this approach offers a sufficient method to estimate the dimension of particles even when the exact distribution of the different sizes is not known, but it also proves that the utilized fully integrated SiGe-based sensors are sufficient for the measurements of particles with these dimensions.