Low Power Portable Device For High Precision Electrical Impedance Detection

Low power portable device for high precision electrical impedance detection C. L. Petersen∗ SensaWave Technology Inc., BC, Canada A low power digital device for electrical impedance measurements has been developed. It is intended for embedded high precision sensor interface applications, but will also operate as a general purpose phase sensitive detector. It integrates an automatic least mean square (LMS) AC bridge with a digital dual phase lock-in amplifier, and is capable of measuring electrical impedance with a precision of 0.001% (10ppm). Capacitance measurements with attofarad resolution have been realized using the device.

I.

other arm of the bridge is driven by the AC voltage

INTRODUCTION

Many electrical sensors are interfaced with either resistive or capacitive bridge circuits. These circuits are balanced by adjusting the values of discrete passive components. This approach is specific to particular sensors, and full balance is often unobtainable due to the complex nature of the sensor impedance. In addition, long term sensor drift and component aging can cause significant changes in the circuit output offset over time. This paper describes an alternative approach to sensor interfacing that draws inspiration from several efforts to automate AC bridges1,2 . While these efforts led to precise auto-balancing solutions, they were bulky and had slow feedback that made real time measurements difficult. Faster feedback is here achieved by introducing a least mean square stochastic gradient technique3,4 , and a smaller form factor by integration on a single digital micro-processor, thereby providing a device suitable for embedded sensing applications. The device works with any type of impedance, and operates in two stages; The sensor bridge is first balanced using a fast error cancellation algorithm that quickly provides an approximation to the ratio (both in-phase and quadrature) of the sensor and reference impedance, and then the bridge error signal is tracked with a high-gain dual phase lock-in amplifier to provide a precise determination of the sensor impedance in real time.

VB = B cos(ωt + φ),

where the amplitude B and phase φ are adjusted automatically by the error cancellation algorithm until the bridge error signal is as small as possible. VB can also be expressed in terms of the in-phase and quadrature VA signals: VB = z1 A cos(ωt) + z2 A sin(ωt) = Re(zAe−iωt ),

CONCEPT AND OPERATION

Fig. 1 shows a block diagram of the electrical impedance detection system. It is connected to a two armed bridge consisting of an unknown (sensor) impedance and a reference impedance. Balance is achieved by adjusting the voltages on the arms of the bridge, instead of adjusting the values of bridge circuit elements as in conventional sensor interfaces. Fig. 2 shows the equivalent circuit elements of the bridge interface. The bridge has two arms with impedances ZA and ZB . An AC drive voltage VA = A cos(ωt)

(1)

is applied to one arm of the bridge. This drive voltage is kept constant during the balancing of the bridge. The

(3)

where z = z1 + iz2 is a complex number representing the bridge balance ratio. The error cancellation algorithm determines the bridge balance using an iterative stochastic gradient search algorithm4 , continuously adjusting z through znew = zold + χVerr Aeiωt ,

(4)

where Verr is the bridge error signal and χ is a constant factor chosen small enough to allow convergence of the iterative procedure. Referring again to Fig. 2, the bridge error signal, Verr , is given by Verr = Zk (

VA VB + ), ZA ZB

(5)

where Zk is the parallel combination of all impedances at the center of the bridge, Zk = (

II.

(2)

1 1 −1 1 + + ) . ZG ZA ZB

(6)

The impedance ZG represents stray capacitances and the bias current path of the input amplifier. As shown in Fig. 1, the bridge error signal is fed to two multiplier stages, which multiply the error signal with the output signal VA and its 90 degree phase shifted quadrature, respectively. The two multipliers form the the heart of the dual-phase lock-in amplifier. The output from the multipliers can be expressed: Vmul = Verr Aeiωt = Zk (

VA VB + )Aeiωt . ZA ZB

(7)

Vmul is next fed into low pass filters whose time constants are chosen to strongly attenuate the waveform frequency, resulting in a DC signal given by vlowpass =

1 z 1 + )A2 , Zk ( 2 ZA ZB

(8)

2

DC input

Sensor impedance

90 deg. Lock-In channel 1

Reference impedance

Lock-In channel 2 In-phase balance Quadrature balance

Error cancellation algorithm

FIG. 1: Block diagram of the electrical impedance detection device.

and the final lock-in amplifier output is verr =

2vlowpass z 1 + )A. = Zk ( A ZA ZB

(9)

In a real measurement situation Verr will contain broadband noise, DC offset etc., but such spurious components are eliminated in the low pass filter stage leading to Eq. 9. The error signal measured with the lock-in can be thus be expressed verr = Cz + D,

(10)

where C = Zk A/ZB and D = Zk A/ZA . The zero point balance ratio, z0 , fulfills Cz0 + D = 0.

This provides a very accurate method to track changes in electrical impedance. If the sensor bridge is primarily capacitive, Eq. 14 can conveniently be rewritten in terms of capacitance as CA = (verr

where CA = 1/iωZA and CB = 1/iωZB . The equations given above rely on interpolation between two measurement points. Linear regression can instead be used to determine C and D from a greater number of measurements, thereby increasing the precision of the method. The equations are easily modified to allow for this. III.

(12)

These equations determine C and D through C=

v2 − v1 , D = v1 − Cz1 , z2 − z1

(15)

(11)

The error cancellation algorithm will settle on a value zauto close to z0 . We can then determine z0 with greater precision by measuring the lock-in error signal at two settings of the balance near zauto , z1 = zauto − ∆z and z2 = zauto + ∆z: v1 = Cz1 + D and v2 = Cz2 + D.

z2 − z1 − z0 )CB , v2 − v1

(13)

and with Eq. 11 we can therefore determine z0 = −D/C. This leads to a significantly better approximation to the zero balance point than zauto , due to the extreme selectivity and noise suppression of the lock-in amplifier, and the use of high signal amplification leading into the lock-in amplifier, made possible by the initial balancing performed by the error cancellation algorithm. Once the zero balance point z0 is determined, changes in the relative impedance can be continuously derived from the lock-in error signal through: z2 − z1 C C ZA = = (verr − z0 )−1 . (14) = ZB D verr − Cz0 v2 − v1

IMPLEMENTATION

The impedance detection scheme was implemented on an ARM based micro-controller using 32 bit integer arithmetics. Input and output signal were generated by 12 bit DAC and ADC devices, and super-sampling used to provide a theoretical 20bit balance ratio resolution. The frequency range for the device was 10 Hz - 10 kHz, and the output signal amplitude was 0 - 4 V peak-topeak. The analog output stage consisted of two DCcoupled micro-power buffer amplifiers, while the input stage was a DC-coupled actively guarded ESD-protected micro-power operational amplifier with an low input bias current of 200 fA. A 100 MΩ resistor to ground served as the input bias current path. This input design was chosen to allow direct interfacing to a wide range of sensor types. A secondary AC-coupled input amplifier with a fixed gain of 1,000 was used to amplify the error signal leading into the dual phase lock-in amplifier. The use of micro-power devices throughout enabled the entire device to operate from a single 5 V 150 mA supply, making it possible to power it directly from a standard USB interface. The device was controlled from a personal computer via a serial (RS232) connection, and was able to operate

3

ZG

VA

ZA

Verr

Acos(ωt) A Error cancellation algorithm

ZB VB

B, φ Bcos(ωt + φ)

FIG. 2: The bridge error cancellation concept. 2000 Lock-in error signal [µV]

1500

Re verr

Zero point balance

1000 500 0

V. -500 -1000

-2000 0.0225 0.025 0.0275 0.03 0.0325 0.035 In-phase balance ratio

FIG. 3: The bridge lock-in error signal near the zero balance point of a capacitive bridge.

in three distinct modes; as a dual phase lock-in amplifier, an auto-balancing bridge, and a simple DC voltmeter, as illustrated in Fig. 1. IV.

RESULTS

Fig. 3 shows how the measured lock-in error signal varies with the balance ratio in the vicinity of the balance set by the error cancellation algorithm. This measurement was made on a bridge consisting of a capacitive sensor and a 5 pF silvered mica reference capacitor at a frequency of 1 kHz and a signal amplitude of

1

2 3

CONCLUSIONS

Im verr

-1500

∗

500 mV. A linear dependence is observed, as expected from Eq. 10, and a precise determination of the zero point balance, and hence the capacitance ratio, is possible through Eq. 15. In this example the sensor quadrature balance ratio was negligible and the in-phase zero point balance was 0.0293, corresponding to a sensor capacitance of 146.5 fF. The resolution of the measurement was 10 aF, corresponding to ratio resolution of approximately 0.001% (10ppm). Convergence of the error cancellation algorithm was achieved after 100 ms of operation, after which the error signal amplitude was typically 100 µV. The balance determined by the error cancellation algorithm was typically only accurate to approximately 0.01%. The use of the lock-in amplifier thus improved the resolution by an order of magnitude. The device demonstrated similar performance on inductive and resistive sensor bridges.

Electronic address: [email protected] R. E. Cavicchi and R. H. Silsbee, Rev. Sci. Instrum. 59, 176 (1988). J. Vejdelek and S. Dado, Meas. Sci. Rev 3, 65 (2003). B. Widrow and M. E. Hoff, IRE WESCON Convention

A low power portable device capable of high precision impedance measurements has been developed. It works by first using an automatic error cancellation algorithm to seek the zero point balance in a two-arm AC bridge and then subsequently tracking lock-in error signal variations near the balance point. The device is capable of measuring electrical impedance with a precision of 10ppm. The bridge can always be balanced because the relative phase of the two drive signals VA and VB can be arbitrarily set by the error cancellation algorithm. For example, this means that a capacitor can be used as reference even if the sensor impedance is resistive, thereby potentially reducing Johnson noise and improving temperature stability of the measurements. In combination with the simple hardware implementation, the device thus provides a general purpose, precise and portable sensor interface suitable for a range of embedded applications. The sensitivity of the interface can be further increased by adding a 10-100× gain input amplifier. This will increase the total gain in the lock-in channel to 10,000 - 100,000, and provide an extremely sensitive detection system.

4

Record, New York 4, 96 (1960). M. Dutta, A. Rakshit, and B. S. N., IEI Journal 84, 26 (2003).