Here two different non -invasive current sensors are proposed, modeled and analyzed. The current sensors are based on the Ampere Force Law (AFL), defining the magnetic force between two parallel wire carrying currents. These current sensors can be used for detecting/sensing DC and AC currents as well as their combination in a single wire or mu ltiple wires, and they do not rely on any permanent magnets for operation. In the first configuration, there are two microbeams, in which one of them is at the vicinity of the wire and undergoes the mechanical vibrations due to the magnetic force between t he wire and the microbeam. The movement of the microbeam while it is generating a magnetic field induces a current inside another microbeam, which is stationary, as the output signal of the current sensor. In the second configuration, a single composite pi ezoelectric microbeam is used. The magnetic force between the wire and the piezoelectric microbeam leads the piezoelectric microbeam to move, thus it produces a voltage. Both configurations present extremely low power consumption, which is not dependent on the sensitivity of the current sensors. The dynamic response, sensitivity and power consumption of the current sensors are investigated, compared and discussed. Keywords: non-invasive current sensors; extremely low power consumption; high sensitivity ; dynamic response. 1 Introduction Nowadays the uncountable benefits of quantum effects and size -dependency phenomena have become a great motivation for scientists to miniaturize the electronic and photonic devices. This trend made the nanotechnologies to become the most dominant research field in all over the world. However, these miniaturized electronic and photonic nanodevices req uire highly accurate nano/microelectromechanical devices (NEMS/MEMS), such as current sensors, to precisely control their power in order to reach their maximum quantum efficiency (Cao, Jian Tian, & Wang, 2017; Kouhpanji, Behzadirad, Feezell, & Busani, 2018 ; Reza & Kouhpanji, 2017; W Xian et al., 2017; Zamani Kouhpanji, Behzadirad, & Busani, 2017; Zamani Kouhpanji & Jafaraghaei, 2017). Moreover, it should be emphasized that the applications of the current sensors are not limited only to the nanotechnologies. They are also a fundamental part of power networks, where any over/under flowing current damages the whole network (Sensitivity, 2014). A vast number of studies on the current sensors was mainly manifested by early studies of Edwin Hall in 1879 (Sherman, Wright, & White, 2016), followed by several intellectual non -invasive current sensors, such as magnetoresistive sensor (Leland, White, & Wright, 2008), Rogowski coil sensors (Chattock, 1887; Ward & Exon, 1993), and Hall effect sensors (Ramsden, 2011; Ziegler, Woodward, Iu, & Borle, 2009). The state -of-art of the non - * Corresponding author, Email: zaman022@umn.edu, Tel.: +1(618) -305-9636. 2 invasive current sensors is that they can measure the current flowing insi de a wire without the need of being inserted in the circuit as it used to be the case for the traditional current sensors. This advantageous of the non -invasive current sensors makes them an excellent candidate for health monitoring and diagnostics of ele ctrical circuits, from industrial and household scales to portable electronic devices such as cellphones. For diagnostic purposes of industrial equipment, the current sensor can be easily placed next to wires to measure the flowing current without the need of cutting the wire. This feature reduces diagnostic time and facilitates the process. For household applications, the current sensor can be used to monitor the electrical consumption for controlling power consumption at peak times. Or, if there is an ele ctrical discharge, the current flow suddenly changes, thus the current sensor can shut off the equipment for safety purposes. For portable electronic devices, these types of current sensors can monitor power consumption with different applications. In the case when the device is running out of battery, the current sensor can show notifications to the user to close the application for saving battery. In addition to the high -power consumption of the traditional current sensors, they need to encircle wires to sense the current. This practical limitation causes these current sensors not to be used when there is more than one current in the nanodevice or network. Attempts to improve the performance of the current sensor resulted in developments of a new designs o f the current sensors such as magnetic tunnel junction (Breth et al., 2011; Julliere, 1975), piezoresistive sensors (Lin & Du, 2011), and delta -f resonant sensors (Hui, Nan, Sun, & Rinaldi, 2013; Wang & Maeda, 2015). In a more realistic evaluation of these current sensors; however, they represented very small power consumption in some cases, there is always a tradeoff between the power consumption, the sensor’s sensitivity, and the magnetic field saturation of these current sensors (Sherman et al., 2016). To avoid the tradeoff, a new family of the current sensors emerged that they do not require any power supply (Leland, White, & Wright, 2006; Leland et al., 2008; Wang et al., 2017). This family of the current sensors consists of a permanent magnet and a pie zoelectric microbeam, where the permanent magnet is fixed on the tip of the piezoelectric microbeam. In these current sensors, the interaction between the permanent magnet and the magnetic field of the wire results in a mechanical force applying on the pie zoelectric microbeam. The main drawback of these current sensors is the durability and reliability of the sensors. That is why the characterization and performance of the sensors highly depend on the magnetic properties of the permanent magnet (Shang et al ., 2017) and the piezoelectric properties of the microbeam that degenerate over time (Jackson, Olszewski, O’Murchu, & Mathewson, 2017; Olszewski, Houlihan, Blake, Mathewson, & Jackson, 2017; Zamani Kouhpanji, 2018a). Furthermore, this type of current senso rs is mainly usable for paired -wires configurations, where there is a uniform magnetic field gradient. For the case of a single wire, not only the magnetic field gradient is nonuniform but it also is nonlinear in the space surrounding the wires, where the position and orientation of the current sensor must be precisely determined (Leland et al., 2008; Weikang Xian & Wang, 2017) in order to have a decent accuracy and sensitivity. Consequently, the first motivation of this work is to design and analyze two al ternative non -invasive current sensors without any limitations on the number of wires carrying unknown currents. The second motivation is to enhance the performance and reliability of the current sensors by avoiding the uncertainties due to the properties of the materials used in the current sensor. To do so, two different current sensor configurations are proposed, designed and analyzed. In both configurations, instead of using a permanent magnet, a microbeam carrying a current was used 3 to interact with th e magnetic field of the wire, which causes the microbeam to vibrate. Then, the mechanical vibration is converted to a voltage as the sensors’ output. 2 Modeling In this section, the two alternative configurations of the non -contact current sensors are presented. Fig. 1(a) shows the current sensor based on a parallel beam carrying current , and Fig. 1(b) shows the current sensor based on a single piezoelectric beam . For Fig. 1(a), the first configuration, the current sensor consists of two microbeams which can be made any conductive materials while Fig. 1(b), the second configuration , the current sensor has a layer of a piezoelectric material on top and bottom separated by a shim layer . Noted, the shim usually is chosen to be a metallic layer to help extracting the current produced by piezoelectric layers and reducing the equivalent sti ffness of the cantilever. For simplicity of the formula, it is assumed that the wire is very long, and its cross -section is much smaller than the cross -sections of the microbeams. However, the effects of the wire’s dimensions can be readily considered (Kouhpanji, 2017a, 2017b; Zamani Kouhpanji, 2018b) . Thus, the magnetic field around a wire carrying current can be obtained by   y zwe eriH ˆ sinˆ cos2   (1) where, iw is the current inside the wire, which is unknown, and it can be either DC or AC; ez is the unity vector along z-axis; ey is the unity vector along y -axis; r is the distance of each point in the space from the wire’s center, see Fig. 2. For convenience , each configuration is analyzed separately in designated sections as follows. Furthermore, all microbeams are along x -axis, and y-z plan is perpendicular to the microbeam s axis . (a): a schematic 2D image of the current sensors, including the dimensions and side views. 4 (b): a schematic 3D image of the current sensors. The left image shows the 1st configuration and the right image shows the 2nd configuration. Fig. 1. The front view and side view of the proposed configurations for the current sensors. In the first configuration, the microbeams have the same cross -section area. For the second configuration, the two piezoelectric layers have the same cross -section dimensions , and the total cross - section area is equal to the area of the first configuratio n’s microbeams. 2.1 The first configuration Using Eq. (1) and the parameters presented in Fig. 1, the total force applied on the moving microbeam can be determined using Ampere Force Law (AFL) as follows 22 0 2 2 2 2 22ˆˆ 2wm wm wm wmhhd t d t wm m y z hh ddii yzF L i H dydz L e e dydzz y z y        rr r (2) where, is the permeability of the vacuum ; im is the current inside the microbeam; t is the thickness of the moving microbeam; h is the width of the moving microbeam ; L is the length of the microbeams; and dwm is the distance between the wire and the moving microbeam . Due to the symmetry condition, the force in z -direction is equal zero (Zamani Kouhpanji, 2018b) . Therefore, the total force applied on the microbeam is in y -direction, and it equal to 2 2 11 0 224ln 2 tan 2 tan2 2 4 2 2wm wm wm wm wm wm wmh d t ii h h hF d d t Lh d d d t                (3) Noted, the total force applied on the microbeams is in y -direction, and its direction and frequency are dependent on the product of the currents flowing in the wire and the microbeams , moving microbeam for the 1st configuration and the piezoelectric microbeam for the 2nd configuration . 5 Fig. 2. Magnetic field generated around the wire due to its current. Next step is to model the dynamic response of the moving microbeam. The moving microbeam can be modeled as a simple mass connected to a spring and damper . Therefore, the equation of motion of the moving microbeam can be written as follows F kww wM  (4) where, M is the moving microbeam ’s mass;  is the damping coefficient; k is the spring constant of the moving microbeam equal to 32Eht3/L3, E is the Young’s modulus of the moving microbeam . Furthermore, w is the displacement of the center of the moving microbeam, and the dots represent the time derivative of the displacement. Thus, the transfer function of the moving microbeam is 2 221)( n ns sMFsW  (5) where,  is the damping ratio equal to kM2 ; and n is the natural frequency of the moving microbeam equal to Mk . It should be mentioned, according to (Kahrobaiyan, Asghari, Hoore, & Ahmadian, 2012) , the displacement of each point along the moving microbeam can be determined by multiplying its time response (determined by solving Eq. (4)) and the eigenfunction of the movi ng microbeam . The eigenfunction of a continuous microbeam under fixed - fixed boundary condition, (x), was given (Kahrobaiyan et al., 2012)  cosh cos( ) cosh cos sinh sinsinh sinnn n n n n nnx x x xxL L L L                                 (6) Here, n corresponds to the mode of resonant of the moving microbeam , which are the solutions of cosh(n)cos(n)=1. Once the moving microbeam starts vibrating, it induces a current inside the stationary microbeam because of their relative velocity and the magnetic field generated by its current (Nay feh & Brussel, 2015) . The voltage induced between the two ends of the stationary microbeam can be determined as follows (Zamani Kouhpanji, 2018b)              2 202 0 0 0 0 0, 21h ht d t dzL sL m dudvuvHw dxx ALdxHw Vms ms    (7) 6 where, w is the velocity at the center of the moving microbeam; As is the cross -section area of the stationary microbeam; dms is the distance between the moving microbeam and stationary microbeam; and Hz is the magnetic field generated by the moving microbeam in z-direction determined by (Feynman, Leighton, & Sands, 2013) 2 0 00 22 0 002( , )2ht m x h mi v vH v u dvduA u u v v     (8) where, Am is the cross -section area of the moving microbeam; and u-v are the ax es of the coordinate system located on the center of the upper face of the moving microbeam while the x -axis is along the moving microbeam ’s axial axis . Moreover, the (u, v) and (u0, v0) represent points on the moving and stationary microbeam, respectively , at this coordinate system . It is noteworthy to be mentioned that the frequency of the output signal in this configuration is a combination of the frequencies of the wire’s current and the moving microbeam’s current because the output signal is a result of products of the currents which have sinusoidal terms according to their frequency . However, if the wire’s current is a DC current, the output signal frequency is only dependent on the frequency of the microbeam’s current, and it has the following time-dependent term t t tdtd mm m m  2sin2sin sin  (9) If the wire has an AC current, then the output signal has the following time-dependent term (left side of Eq. (10) which can simplified to summation as the right side of following equation       t t t t t tdtd m wm w m wm w ww m m w  2 cos42 cos4cos2sin sin sin   (10) where, t is time; w is the frequency of the wire’s current; and m is the frequency of the moving microbeam’s current. 2.2 The second configuration The force applied on each layer of the piezoelectric micro beam can be calculated using Eq. (3), and the total force applied on the piezoelectric microbeam is the summation of all forces. Noted that the total current flowing in the piezoelectric microbeam is the summation of all current flowing in each layer. To determ ine each layer current, the piezoelectric microbeam can be modeled as three parallel resistances. Furthermore, it should be mentioned that the force applied on the piezoelectric microbeam is mainly due to the input current, not the current induced inside i t by mechanical vibration. That is why the induced current is usually several orders of magnitude smaller than the input current. Modeling the dynamical vibration of the piezoelectric microbeam is completely different from a regular microbeam, such as the 1st configuration. First, the piezoelectric microbeam is made of three layers where each layer can have different mechanical properties and thicknesses . Therefore, the equivalent stiffness , (EI) eq, of the microbeam must be considered to take into account the variations of the materials properties and sized on the mechanical stiffness. The equivalent mechanical stiffness can be determined using 7 3 22 2 1 13 13 1121 22 1212 htEttht ht E IE EId PZ jjj eq           (11) where, t1 is the thickness of the piezoelectric layers; h is the width of the piezoelectric layers which it is equal to the shim layer width ; t2 is the thickness of the shim layer; EPZ is the Young’s modulus of the piezoelectric layers; and Ed is the Young’s modulu s of the shim layer. Second, because of the existence of the charges on the piezoelectric layers impact the mechanical behavior of the piezoelectric, the governing equations of motion for piezoelectric microbeam are electromechanically coupled. The govern ing equations of m otion for piezoelectric microbea ms under fo rced vibration was derived by (Roundy & Wright, 2004) using analogy within th e electrical circuit and mechanical structures. (Roundy & Wright, 2004) derived the equation of motion of a piezoelectric microbeam as 1 31 2 2 3 31 1 22sp nn eq eq pk a d FS S S VM k M a c d tVSa      & & & & & (12) where , S is the axial strain inside the piezoelectric layers and the dots again show the time derivative of the axial strain ; V is the voltage between the two ends of the piezoelectric microbeam; Meq is the equivalent mass of the piezoelectric microbeam which is equal to the summation of all layers masses; d31 is the piezoelectric coefficient of the piezoelectric layers; cp is the capacitance induced between the piezoelectric layers due to accumulation of charges; the a1, a2 and a3 are the coefficients related to the configurations b etween the piezoelectric lay ers; ksp is the spring constant of the piezoelectric microbeam equal to 384(EI) eq/L3. Lastly, k2 is a geometric coefficient depending on the boundary condition , dimension and the length of the piezoelectric microbeam at boundaries, Le. The k2 for a fixed -fixed boundary condition is equal to 2 2 14 23 21 32e eL LL LtLk  (13) By solving the Eq. (1 2), the AC voltage induced between the two ends of the piezoelectric microbeam can be achieved as (Leland et al., 2008) 31 1 3 31 22 221 21p pz p eq nnd c t aVFdc k M ass     (14) where,  is the dielectric constant of the piezoelectric layer s. The force applied on the microbeam in the 2nd configuration is due to interaction between the magnetic field induced by the current of the wire and microbeam. According to Equ. (3), this force has the product of the both currents. If the wire carries a n AC current, its output signal will have the product of the si nusoidal functions as sin(wt)sin(pzt), which can be readily simplified as follows 8 t t t tpz w pz w pz w       cos21cos21sin sin (15) where, pz is the frequency of the piezoelectric microbeam’s current. And, if the wire has a DC current, then the frequency of the output signal will be equal to the frequency of the piezoelectric microbeam’s current. If the frequency of the wire is known, the frequ ency of the microbeam must be selected such as m=w-n or m=n-w, where n is the resonance frequency of the microbeam. In this case, the force applied on the microbeam will have at least one component at the resonance frequency of the microbeam leading in the largest mechanical vibration amplitude for the microbeam, consequently, highest signal -to-noise ratio at the outpu t. In practical application, if the frequency of the wire is unknown, the user must conduct a sweep mode frequency measurement to see the response of the current sensor, as it must be done for all current sensors. Once a sweep mode frequency measurement is conducted, there will be an output response associated with each of these frequencies. Then, by using the provided formulas, one can easily determine the frequency of the wire, even if it is unknown, as well as the required frequency for the microbeam for further detections. The similar argument is valid for the 1st current sensor configuration. 3 Results and discussions The materials properties were taken from (Xiang & Shi, 2009) , for the piezoelectric layers, and (Rumble, 2017) for the silicon microbeams and a luminum shim . Table. 1 provides the physical parameters used for modeling and evaluating the designs. Table. 1: used parameters for modeling of the current sensors. Parameters 1st Configuration 2nd Configuration Thickness of the moving and stationary beam ( t) 1E-6 m - Thickness of top piezoelectric later ( t1) - 0.33E -6 m Thickness of the bottom piezoelectric layer ( t2) - 0.33E -6 m Width of the microbeams ( h) 1E-6 m Length of the microbeams ( L) 1E-3 m Distance between the moving microbeam and the wire ( dwm) 2E-6 m Distance between the stationary microbeam and the wire ( dms) 2E-6 m - Figs. 3 and 4 demonstrate the dynamic response of the current sensors by presenting the gain and phase shift of the current sensors. As it can be seen, the maximum gains decrease as the damping ratio increases. In the proposed current sensors, the damping is due to the structural damping and viscous damping induced by the interaction between the microbeams and the air surrounding the current sensors , where the structural damping has the dominant impact on the performance of the current s ensors . The structural damping can be significantly reduced by either increasing the length of the microbeams or decreasing the cro ss-section dimensions of the microbeams. Noted the phase shift of the two configurations are completely different from each other even though their gains are almost the same. For the 1st 9 configuration, the phase shift is equal to zero at the resonance freq uency, and it varies smoothly at the vicinity of the resonance frequency . However, for the 2nd configuration, the phase shift has a discontinuity at the resonance frequency, and it has 180 degrees phase shift. (a): Gain of the output voltage of the 1st current sensor. (b): Phase shift of the output voltage of the 1st current sensor. Fig. 3. Dynamical response of the 1st current sensor. (a) Gain, (b) phase shift. 10 (a): Gain of the output voltage of the 2nd current sensor. (b): Phase shift of the output voltage of the 2nd current sensor. Fig. 4. Dynamical response of the 2nd current sensor. (a) Gain, (b) phase shift. It should be mentioned that the position of the maximum gain of the 1st current sensor is only dependent on the moving microbeam nat ural frequency and the damping ratio, while it depends on the piezoelectric coefficient, dielectric constant, and piezoelectric capacitance in addition to the natural frequency and damping ratio of the 2nd current sensor , see Eq. (14) . This fact causes the position of the maximum gain of the 2nd current sensor to be more dependent on the damping ratio which can be seen as a small shift on the position of the maximum gain of the 2nd current sensor comparing to the 1st current sensor , see Fig. 5. Fig. 5. Comparing the gain of the current sensors. One of the critical factors in designing any sensor is the real-time sensitivity of the sensor. The real-time sensitivity is defined as derivative of the output signal to the input signal of the sensor. According to Eqs. (7) and (1 4), the output signal in both current sensors are linearly proportional to the input currents, which is one of the superior advantages of these configurations comparing to the prior current sensors. This linear behavior of the c urrent sensors results the real -time sensitivity of the current sensors to be the same as their time responses without any dependency 11 on the wire frequency, wire current amplitude or physical parameters of the wire. This superior advantage of these current sensors makes it possible to optimiz e an initial design of these current sensors, then it can be used for any input signal with the same performance. Therefore, the real-time sensitivity of the proposed current sensors is  dIVdy Sensitivit (16) where, V is the output voltage of the current sensors; and I is the product of the amplitudes of the unknown current inside the wire and the current of the microbeams. Fig. 6 depicts the time response of the current sensors for a sinusoidal input signal . In Fig. 6, all physical parameters are assumed to be the same for both current sensors, where the total thickness of the piezoelectric microbeam is the same as the moving and /or stationary microbeams’ thickness. The initial parts of the time response, equi valently , the real-time sensitivities , of the current sensors are affected by the transient response of the sensors, and they reach the steady state conditions after a few cycles. It should be emphasized that the 1st configuration provides a faster dynamic response comparing to the 2nd configuration for the same conditions . Fig. 6. Time response of the current sensors for a sinusoidal input current with = 0.5 , the frequency of the wire and microbeam is assumed to b e equal to the half of the resonance frequency of the microbeam . The power consumption of the current sensors is also analyzed which is another critical factor in designing high efficient current sensor s. In both configurations, the input power is related to the power consumed in the microbeams for generating the mechanical vibration. Therefore, 1st configuration: mm in Ri P2 (17) 2nd configuration: pz pz in Ri P2 where, ipz it the input current in the piezoelectric microbeam; Rm is the electrical resistance of the moving microbeam; and the Rpz is the electrical resistance of the piezoelectric microbeam. Another superior advantage of these configurations comparing to previous current sensors is that the consumed power is only dependent on the electrical resistance of the microbeams and the input current. On the other hand, according to Eqs. (7) and (1 4), since the output 12 signal is independent on the electrical resistance of the microbeams, the power consumption can be minimized to zero by increasing the doping level into the microbeams or using a metallic moving microbeam for the 1st configuration. To investigate the power efficiency , Pout/Pin, of the current sensors, here the output power of the current sensors is calculated. Since the output signal is a voltage, thus the output power is 1st configuration: sm outRVP2 (18) 2nd configuration: pzpz outRVP2 where, Rs is the electrical resistance of the stationary microbeam which is the same as Rm. Fig. 7 shows the normalized power efficiency of the current sensors. Fig. 7. Power efficiency of the current sensors normalized to the power efficiency at = 0.10 . For sim plicity, it is assumed that the wire carries a DC current , and the microbeams carry a current at their resonance frequency. As it can be seen from Fig. 7, the power efficiency of the current sensors is the same for the same physical parameters, and it only depends on the damping ratio of the sensors. Noted , even thoug h the power efficiency of the current sensors is the same, their output power could be different due to different electrical resistance of the microbeams, see Eq s. (18). Table 2 is provided to summarize the state -of-art of the proposed current in a nutshell. For the sake of comparison, it is assumed that the both configurations have the same dimensions. Table 2: Qualitative comparison between the two configurations. Practical concern 1st configuration 2nd configuration Sensitivity Good Medium Power consumption Good Good Signal to noise ratio Medium Good Linearity in response Good Medium Fabrication process Medium Complicated 13 4 Conclusion Here two different non -invasive current sensors are proposed, designed and analyzed based on the Ampere Force Law (AFL). The current sensors are able to detect/sense both DC, AC or their combinations in a single wire or multi -wire network. For the same physical parameters, the 1st configuration (current sensor using two parallel microbeams) presents a faster d ynamic response comparing to the 2nd configuration (current sensor using a piezoelectric microbeam). However, the 2nd configuration presents a higher gain comparing to the 1st configuration. Furthermore, experimentally wise, the 2nd configuration is more d ifficult to be fabricated due to the required extra steps for implementing the piezoelectric material on the microbeam. The dynamic response and the sensitivities of the current sensors are linearly dependent on the wire current and wire frequency. The pow er consumption of the current sensors is examined, and it was found that the power consumption of the current sensors is only dependent on the electrical resistance of the microbeams, which demonstrates extremely low power consumption for the current senso rs. Furthermore, studying the power efficiency of the current sensors showed that there is not any tradeoff or dependency between the power consumption and the sensitivity and the dynamic response of the current sensors. Therefore, it can be concluded that the sensitivity and dynamic response of the current sensors can be maximized without affecting the power efficiency of these current sensors. Furthermore, the dynamic response analysis, sensitivity analysis, and power consumption analysis of the proposed current sensors confirm the practical realization of the proposed current sensors in both industrial and household applications.