3.3.1. A. Detection of the Angular Position, θ, of the Steel Strand
The parameters of the experiments were identical to the simulations. The capacitance was measured using an impedance analyzer (Agilent 4294A, Beijing, China), and a diagram of the experimental device is shown in
Figure 9.
In the first round of measurement for the angular position, the experimental capacitance values between E1 and E4 were negative, as shown in
Table 3, which were invalid measurements. It is the stray capacitance that influenced the experimental results. When the steel bar appeared in the sector that was covered by the sensing area of the capacitive sensor, four capacitors were formed. For example, when E1 was used for excitation, the original capacitance,
C, was formed by E1 and E4. The stray capacitance,
Cs1,
Cs2, and
Cs3, were formed by E1 and the steel bar, E1 and the shielding layer, and E1 and the cement, respectively. As a result, the measured capacitance between E1 and E4 dropped below zero. Therefore, the quality factor (Q-factor) was introduced for further experiments of the angular position’s identification.
In our previous research, the boundary detecting method for post-tensioned pre-stressed ducts based on Q-factor analysis was introduced, which could effectively identify the boundary position of the three-phase duct model. Generally,
Q is calculated for a capacitor as follows:
where
ω0 is the resonance frequency (radians per second),
C is the capacitance,
XC is the capacitive reactance, and
RC is the series resistance of the capacitor.
The frequency sweep was performed using an Agilent 4294A. The frequency range was from 40 Hz to 100 MHz. The frequency sweep was conducted for each
α (0°, 45°, 90°…). The maximum value of the Q-factor was defined as MAX, and the corresponding frequency was
fmax. Since MAX and
fmax were unstable within the frequency range, trace bandwidth analysis was performed. The cutoff point was obtained by dividing the MAX by 2. The cutoff points were searched for toward both sides of the measurement parameter axis, using the current position of the MAX as the center. The bandwidth (distance between the two cutoff points), center value (midpoint of the two cutoff points), and corresponding frequency,
fc, were obtained. The steps for calculating the center frequency,
fc, were introduced in detail in our previous work [
9].
The center frequency,
fc, was selected for angle position detection, and the measured data are listed in
Table 4. Normalization was conducted for better identification of the steel strand angle position. The sensor positions for
θ = 15°,
r = 20 mm and
θ = 90°,
r = 20 mm are also shown in
Figure 10.
For θ = 15° and r = 20 mm, three characteristic points were marked, named A1, A2, and A3. A1, A2, and A3 corresponded to the normalized capacitance value, Cn14, when the position of the sensor, α, was 0°, 180°, and 225°, respectively. (i) A1 was the lowest point among the normalized values. Therefore, the steel bar was in the area of the sensor electrode at 0°, which was in the −45° to 45° range. (ii) A2 and A3 were the lowest points other than A1, and their normalized values were around 0.3, which means that the steel strand was in the opposite area of α = 180° and α = 225°, corresponding to the −45° to 90° range, respectively. (iii) Note that A2 was approximately equal to A3, so the steel bar was in the middle region of −45° to 90°, that is 0° to 45°. Therefore, the steel bar could be located in the 0° to 45° segment.
For θ = 90° and r = 20 mm, four characteristic points were marked, named B1, B2, B3 and B4. B1, B2, B3, and B4 corresponded to the normalized capacitance value, Cn14, when the position of the sensor, α, was 90°, 225°, 270°, and 315°, respectively. (i) B1 was the lowest point among the normalized values. Thus, the steel bar was in the area of the sensor electrode at 90°, which was in the 45° to 135° range. (ii) B2, B3, and B4 were the lowest points other than B1, and their normalized values were around 0.3, which means the steel strand was in the opposite area of α = 225°, 270°, 315°, i.e., the range from 0° to 180°. (iii) It is remarkable that the steel bar was in the symmetrical position when the sensor electrode was at 45° and 135°, 0° and 180°, and 225° and 315°, since their values were approximately the same. Therefore, the steel bar was in the middle of the 45° to 135° range, and its position angle was 90°.
By the method above, we could locate the steel bar at least in one 45° segment, and for the special angles (90°), the proposed method could even determine the exact position of the steel bar. Furthermore, the accuracy of the positioning could be improved by reducing the scanning interval of the sensor. For instance, we could set α = 30° or 15°, and the steel bar could be located in one segment of 30° or 15°.
3.3.2. B. Detection of the Center Distance, r, of the Steel Strand
In the second-round of measurements for the center distance, only one configuration was studied:
θ = 90° and
α = 90°, where
r ranged from 0 to 25 mm with increments of 5 mm, due to its symmetrical structure. The capacitance between E2 and E3 (
C123), and the capacitance of
θ + 180° (
C223) were measured using an Agilent 4294A, as described above. The excitation frequency was 1 MHz. The measured capacitance and relative proportion of the capacitance between E2 and E3 (
Pc) are demonstrated in
Table 5 and
Figure 11. In addition, normalization was conducted for a better comparison between the simulation and measurement, as illustrated in
Table 6 and
Figure 12.
(i) As the center distance, r, of the steel bar increases, the capacitance between E2 and E3 (C123) decreased gradually, and the capacitance of θ + 180° (C223) increased slowly, which contributed to the slight decrease in C123/(C123 + C223).
(ii) In
Figure 11, the experimental and simulated results (
Pc) followed the same decreasing trend as
r increased, indicating the validity of the experiments. The measured
Pc decreased more slowly than the simulated one because the stray capacitance weakened the influence of the center distance,
r, on
Pc.
(iii) For
Figure 12, the normalized
Pc of the experiments and simulations shared the same decreasing tendency. There were gaps between the three measured values and the simulation, that is
r/30 = 0.333, 0.5, and 0.667. On the one hand, due to the small variation of
Pc in the measurements, the normalization amplified the variable quantity. On the other hand, the position error of the steel bar caused by the twisting of the steel strand and the setting of the cement contributed to the differences between the experiments and simulations.
Therefore, we set Pc = C123/(C123 + C223) = 0.486 as the threshold of r = 15 mm (r/30 = 0.5, R3 = 22.5 mm) to determine whether the steel bar was located in the inner circle (0 mm ≤ r ≤ 15 mm) or the outer ring (15 mm ≤ r ≤ 30 mm). With the two rounds of measurements, the steel bar could be located in 1 of 16 segments. In the first-round of measurements, its angle (θ) was inspected in the 45° range. In the second round of measurements, its center distance (r) could be identified in one of the two rings. Thus, the improved sensor structure could locate the steel bar positions.