Theoretical analysis of enhanced radiation intensity of piezoelectric antenna

The piezoelectric antenna structure includes electrodes and substrates. Not only the dielectric loss and mechanical vibration loss caused by the piezoelectric material itself cannot be ignored, but also the electrode loss and connection loss with the RF interface cannot be ignored. Therefore, the MBVD equivalent circuit model [Reference Jin, Dong, Luo and Milne23, Reference Vorobiev and Gevorgian24] considering the losses is used for theoretical analysis of piezoelectric antenna, as shown in Fig. 1(a). Where C_m, L_m, R_m are motional capacitance, motional inductance and motional resistance respectively; R ₀ represents dielectric loss in the piezoelectric materials; R_s represents electrode loss and lead wire loss; C ₀ is a static capacitance, which represents the dielectric characteristics of the piezoelectric antenna [Reference Muralt, Conde, Artieda, Martin and Cantoni25].

Fig. 1.

(a) MBVD equivalent circuit model. (b) LLC matching circuit.

The equivalent electrical impedance of the piezoelectric antenna is calculated according to Fig. 1(a):

(1)$$\eqalign{Z_{in} = & R_s + \left({\displaystyle{1 \over {\,j\omega C_m}} + j\omega L_m + R_m} \right)/{/}\left({R_0 + \displaystyle{1 \over {\,j\omega C_0}}} \right)\cr = & R_s + \left[{\displaystyle{1 \over {R_0 + {1 \over {\,j\omega C_0}}}} + \displaystyle{1 \over {R_m + j\left({\omega L_m-{1 \over {\omega C_m}}} \right)}}} \right]^{{-}1} \cr = & R_s + \left[{( {R_0 + j\omega C_0} ) + \displaystyle{{\,j\omega C_m} \over {1-\omega^2C_mL_m + j\omega C_mR_m}}} \right]^{{-}1}} $$

When the resonance occurs on the C_m, L_m, R_m branches in Fig. 1(a), the whole circuit becomes that the resistance R_m is connected in parallel with the capacitor C ₀. At this time, the inductance L_{add- 1} is connected in series, and the whole circuit is changed from parallel circuit to series circuit. The inductance L_{add- 1} can effectively offset the capacitive part generated by C ₀ in the circuit, and can also generate a new resonant frequency with C ₀. In addition, when the L and C series circuit is connected in parallel with piezoelectric antenna, the inductive reactance is very small and the capacitive reactance is very large. The transmission loss of piezoelectric antenna is very small. Therefore, based on Fig. 1(a), the parallel LC circuit and the series variable inductance are used to regulate the radiation intensity and resonant frequency of the piezoelectric antenna. The equivalent circuit is shown in Fig. 1(b), which is called LLC circuit.

The equivalent electrical impedance of the piezoelectric antenna is calculated according to Fig. 1(b):

(2)$$\eqalign{Z_{in} = & R_s + \left({\displaystyle{1 \over {\,j\omega C_m}} + j\omega L_m + R_m} \right)/{/}\left({R_0 + \displaystyle{1 \over {\,j\omega C_0}}} \right)/{/}\left({\displaystyle{1 \over {\,j\omega C_{add}}} + j\omega L_{add-2}} \right) + j\omega L_{add-1} \cr = & R_s + \left[{\displaystyle{1 \over {\displaystyle{1 \over {\,j\omega C_m}} + j\omega L_m + R_m}} + \displaystyle{1 \over {R_0 + \displaystyle{1 \over {\,j\omega C_0}}}} + \displaystyle{1 \over {\displaystyle{1 \over {\,j\omega C_{add}}} + j\omega L_{add-2}}}} \right]^{{-}1} + j\omega L_{add-1}}$$

By comparing equations (1) and (2) can be seen that, it is necessary to realize the impedance transformation of the piezoelectric antenna by the matching network. The impedance of the piezoelectric antenna is changed to make it close to the output impedance, so as to achieve the best power transmission matching. On the other hand, the impedance of the piezoelectric antenna is adjusted by inductive components, and the phase angle of impedance tends to zero at the resonant frequency. The reactive component of transmission power is reduced and the transmission efficiency of piezoelectric antenna is enhanced.

Theoretical analysis of resonant frequency variation of piezoelectric antenna

In order to meet the needs of modern wireless communication systems, the antennas must develop in the direction of miniaturization, multi-function and high performance. In many cases, a wireless communication device needs to work in multi-band, so as to simplify the design of the communication system and reduce the cost. In order to solve the problem, multi-band antenna becomes the preferred solution. The advantage of multi-band antennas is that a single antenna can achieve the performance of multiple antennas. In practical applications, the wireless communication system utilizes multi-band combination of antennas, which has good working performance. Especially in dealing with various complex environments, it has obvious advantages.

At present, one of the challenges of piezoelectric antennas is that their operating bandwidth is narrow, which only has the best performance at the resonant frequency. The design of multi-band performance can broaden the operating frequency of piezoelectric antennas to deal with various complex application environments.

The MBVD model of the original piezoelectric antenna in Fig. 1(a) includes series resonant branches and parallel resonant branches, which cause the series resonant frequency and parallel resonant frequency of the piezoelectric antenna respectively. It is worth noting that the working frequency of the piezoelectric antenna is at the series resonant frequency, so the proposed method is mainly to change the series resonant frequency of the piezoelectric antenna. The tunable circuit is designed based on the equivalent circuit of piezoelectric antenna, which is connected in series with inductance or capacitance respectively, as shown in Fig. 2.

Fig. 2.

(a) Equivalent circuit of piezoelectric antenna with series inductance. (b) Equivalent circuit of piezoelectric antenna with series capacitance.

The series resonance frequency f_s and parallel resonance frequency f_p are calculated in equivalent circuit of Fig. 1(a):

(3)$$\eqalign{\,f_s = & \displaystyle{1 \over {2\pi \sqrt {L_mC_m} }} \cr f_p = & \displaystyle{1 \over {2\pi \sqrt {\displaystyle{{L_mC_mC_0} \over {C_m + C_0}}} }}} $$

The series resonance frequency f_s and parallel resonance frequency f_p are calculated in equivalent circuit of Fig. 2(a):

(4)$$\eqalign{& f_s = \displaystyle{1 \over {2\pi \sqrt {( {L_m + L_{add}} ) C_m} }} \cr & f_p = \displaystyle{1 \over {2\pi \sqrt {\displaystyle{{L_mC_mC_0} \over {C_m + C_0}}} }}} $$

The series resonance frequency f_s and parallel resonance frequency f_p are calculated in equivalent circuit of Fig. 2(b):

(5)$$\eqalign{\,f_s = & \displaystyle{1 \over {2\pi \sqrt {L_m\left({\displaystyle{{C_mC_{add}} \over {C_m + C_{add}}}} \right)} }} \cr f_p = & \displaystyle{1 \over {2\pi \sqrt {\displaystyle{{L_mC_mC_0} \over {C_m + C_0}}} }}} $$

By comparing equations (3)–(5), it can be seen that when the series resonance is occurred in the series resonance branch, the series inductance increases the equivalent inductance of the whole circuit. The series inductance will cause the f_s to shift to the low frequency, and the f_p remains unchanged. In addition, the series inductance interacts with the inherent capacitance C ₀, which can generate new resonant frequency. When the series resonance is occurred in the series resonance branch, the series capacitance reduces the equivalent capacitance of the whole circuit. The series capacitance will cause the f_s to shift to the high frequency, and the f_p remains unchanged. Because the series capacitance and the inherent capacitance C ₀ are capacitive, no new resonant frequency is generated.

Design and fabrication of variable inductance

The distributed capacitance and internal resistance in the real inductance greatly weaken the performance of piezoelectric antenna. The inductances with different values need to be replaced to adjust the resonant frequency of the piezoelectric antenna, and the variation of resonant frequency is discontinuous. In order to solve the above problems, the analog inductance can be realized by using a synthetic circuit [Reference Wang, Chen and Wen28], whose sketch is shown in Fig. 6(a).

(6)$$L = \displaystyle{{R_1R_3R_4C_1} \over {R_2}}$$

Fig. 6.

(a) The sketch of the analog inductor. (b) The prototype of the analog inductance.

The analog inductance consists of three constant resistors R ₁, R ₂ and R ₃, an adjustable resistor R ₄, a capacitor C ₁ and two OPA445AP operational amplifiers [Reference Ruby, Bradley and Larson29]. The equivalent inductance of this circuit is calculated by equation (6) [Reference Zhang, Xiao, Wen, Yu and Wen30], and the inductance value L can be adjusted by selecting appropriate R ₄. The adjustable range of equivalent inductance L is 0~20 mH in Fig. 6(b).

Experimental platform construction

The experimental test platforms are established to test the resonant frequency and radiation intensity of the piezoelectric antenna with external circuit. The experimental platforms are shown in Fig. 7. Figure 7(a) is the test schematic diagram for transmission distance and radiation magnetic field of piezoelectric antenna. Firstly, the sine wave signal is generated by the signal generator, which is amplified by the power amplifier and then loaded into the piezoelectric antenna to generate electromagnetic waves. The electromagnetic wave is received by the copper coil and transmitted to the oscilloscope to display in the form of voltage. The copper coil is made of enameled wire with a diameter of 0.35 mm, and the number of turns is 1000. According to the electromagnetic induction principle, the induced voltage of the copper coil is proportional to the magnitude of the alternating magnetic field. The unknown alternating magnetic field can be calculated by measuring the output voltage of the copper coil. Figure 7(b) is the test platform for transmission distance and radiation magnetic field of piezoelectric antenna and the experimental equipment is composed of signal generator (Tektronix AFG3252C), oscilloscope (Keysight InfiniiVision MSQX3024T), power amplifier, copper coil. Figure 7(c) is the test platform for resonant frequency and impedance characteristics of piezoelectric antenna and the experimental equipment is the Precision LCR Meter produced by QuadTech Company. Figure 7(d) is the test platform for S-parameter of piezoelectric antenna and the experimental equipment are the vector network analyzer (Rohde & Schwarz^®ZNB8-B34) and coaxial cable.

Fig. 7.

(a) Electromagnetic radiation test schematic diagram. (b) Electromagnetic radiation test platform. (c) Impedance test platform. (d) S-parameter test platform.

Experimental study on piezoelectric antenna tuning

Based on the analysis of tuning mechanism in section ‘Theoretical analyses’, the piezoelectric antenna with the series inductance and capacitance are tested in Fig. 7(c) and the test results are shown in Fig. 8.

Fig. 8.

(a) Influence of series capacitance on resonant frequency of piezoelectric antenna. (b) Influence of series inductance on resonant frequency of piezoelectric antenna.

It can be seen from Fig. 8(a) that the series capacitance will cause the series resonance frequency to shift to high frequency, while the parallel resonance frequency remains unchanged. The resonant frequency of the original piezoelectric antenna is 78.6 kHz. The piezoelectric antenna is connected in series with a 1.5 nF capacitance, and the resonant frequency is 89.3 kHz, which increases by 13.6%. The piezoelectric antenna is connected in series with a 2.2 nF capacitance, and the resonant frequency is 86.4 kHz, which increases by 10%. It can be seen from Fig. 8(b) that the series inductance will cause the series resonant frequency to shift to low frequency, while the parallel resonance frequency remains unchanged. The piezoelectric antenna is connected in series with a 1.7 mH inductance, and the resonant frequency is 59.7 kHz, which is reduced by 24% compared with the original piezoelectric antenna. With the increase of series inductance, the series resonant frequency decreases. When the series inductance of the piezoelectric antenna increases by 7.7 times, the series resonant frequency decreases by 22.4%. In addition, the series inductance interacts with the inherent capacitance C ₀, which can generate new resonant frequency. The piezoelectric antenna is connected in series with a 1.7 mH inductance, and the new resonant frequency is 113.1 kHz, which is increased by 44% compared with the original piezoelectric antenna. Compared with Figs 8(a) and 8(b), it can be found that the series inductance has more than significant effect on the series resonant frequency of piezoelectric antenna. The series inductance will generate a new resonant frequency, while the series capacitor will not generate a new resonant frequency.

In summary, the resonant frequency of the piezoelectric antenna can be effectively adjusted by the series inductance or capacitance without changing the structure of the piezoelectric antenna, which broadens the working bandwidth of the piezoelectric antenna. On the other hand, the new resonant frequency is generated by series inductance, which can make the piezoelectric antenna work at multi-band, and further broaden the application field of piezoelectric antenna.

Experimental study on improving radiation intensity of piezoelectric antenna

Based on the theoretical analysis of improving radiation intensity of piezoelectric antenna in section ‘Theoretical analyses’, the piezoelectric antenna with the external LLC matching circuit is tested in Figs 7(b) and 7(d), and the test results are shown in Fig. 9. The matching diagram of LLC circuit and piezoelectric antenna is shown in Fig. 9(a). The series inductance L _add-1 and parallel capacitance C _add are constant, which are 1 mH and 1.5 nF respectively. The parallel inductance L _add-2 is an adjustable analog inductance, as shown in Fig. 6(b). The resonant frequency and radiation intensity of piezoelectric antenna are adjusted by setting different L _add-2. In this section, the parallel inductance L _add-2 with 0.15 and 0.47 mH is taken as an example to contrastive analysis the influence of LLC circuit with different L _add-2 on the resonance frequency and radiation intensity of the piezoelectric antenna.

Fig. 9.

(a) The matching diagram of LLC circuit and piezoelectric antenna. (b) The S₁₁ parameters of the piezoelectric antenna. (c) The radiated magnetic fields at the three resonant frequencies.

The S₁₁ parameters of the piezoelectric antenna measured by vector network analyzer with or without an external LLC circuit are showed in Fig. 9(b). As can be seen from Fig. 9(b), when the piezoelectric antenna without external LLC circuit, the piezoelectric antenna has only one resonant frequency at 78.6 kHz, and S₁₁ is −11.2 dB. When the piezoelectric antenna with LLC matching circuit and L _add-2 is 0.15 mH, the piezoelectric antenna has three resonant frequency, which are 78.6, 134.8 and 304.5 kHz respectively, and the S₁₁ at the resonant frequencies are all lower than −10 dB. When the piezoelectric antenna with LLC matching circuit and L _add-2 is 0.47 mH, the piezoelectric antenna also has three resonant frequency, which are 95.8, 150.4 and 325.9 kHz respectively, and the S₁₁ at the resonant frequencies are all lower than −10 dB. Compared with the no external LLC circuit, the resonant frequency of the piezoelectric antenna with LLC circuit is increased to three, and the radiation intensity of the piezoelectric antenna is also greatly improved. The piezoelectric antenna has the dual functions of multi-band and the improvement of electromagnetic radiation.

In order to verify the effectiveness of multi-band of the piezoelectric antenna with LLC circuit, the radiated magnetic field of the three resonant frequencies of the piezoelectric antenna is analyzed when the piezoelectric antenna with the LLC circuit and the L _add-2 is 0.15 mH. The test results are obtained from the test platform in Fig. 7(b). The distance between the piezoelectric antenna and the receiving cooper coil is constant, which is 10 cm. The radiated magnetic fields at the three resonant frequencies are obtained, as shown in Fig. 9(c). As can be seen from Fig. 9(c), compared with the original piezoelectric antenna, at the common resonant frequency of 78.6 kHz, the radiation magnetic field of the piezoelectric antenna with the LLC circuit is 42.3% higher. In addition, when the piezoelectric antenna with the LLC circuit, the two new resonance frequencies are 134.8 and 304.5 kHz, respectively, and the radiated magnetic fields are 25.9 and 75.4 nT, respectively. In summary, when the piezoelectric antenna is connected with LLC circuit, it can not only make the piezoelectric antenna work at multi-band, but also enhance the electromagnetic radiation of the piezoelectric antenna.