High-precision inductance measurement system based on double-excitation auto-balancing bridges

AI Yu， FU Xiao， LI Yao， WANG Xianquan， DUAN Fajie， JIANG Jiajia

(1. State Key Lab of Precision Measuring Technology and Instruments， Tianjin University， Tianjin 300072, China； 2. Systems Engineering Research Institute, China State Shipbuilding Corporation, Beijing 100036, China)

Abstract： In order to achieve high precision measurement of inductance in a wide frequency range, a method of inductance measurement based on double-excitation auto-balancing bridge is proposed. In this method, the direct digital synthesizer (DDS) as signal generator is used as the bridge excitation source, and the bridge is automatically balanced by adjusting and measuring the voltage ratio. Using standard resistors, the system can achieve high precision measurement of four-terminal pair inductors in the frequency range of 100 Hz-100 kHz. Aiming at the low efficiency of bridge balancing, an iterative balancing algorithm based on the steepest descent method is proposed. In order to suppress the interference caused by the initial phase change and non-integer periodic sampling, the high-precision measurement of the complex impedance of inductance is realized based on the all-phase fast Fourier transform (apFFT). Finally, the corresponding measurement system is built and the inductance measurement experiment is carried out. The experimental results show that the relative error of the system for inductance measurement can be as low as 0.009%, and the optimal relative measurement uncertainty of the system can reach 9.89×10-5 compared with 5×10-4 of commercial impedance analyzer.

Key words： double-excitation auto-balancing bridge； inductance measurement; all-phase fast Fourier transform (apFFT)； relative measurement uncertainty

0 Introduction

As a basic electronic component, inductor is widely used in electric power, transportation, communication and other industries. The accuracy of its value has a great influence on the calculation and practical application of the circuit, so the high-precision measurement and calibration of inductor is becoming more and more important. Traditional commercial impedance analyzers are capable of automatic measurement with wide bandwidth and high resolution, but their relative uncertainty is rarely better than 5×10-4, which is not adequate to calibrate impedance standards. At present, the highest measurement accuracy of commercial digital AC bridge on the market only reaches 2×10-4magnitude[1]. In addition, although high-precision metering bridges can achieve small measurement uncertainty, they are time-consuming to set up, cumbersome to operate, expensive and not suitable for daily calibration[2-3]. Therefore, how to realize high-speed and high-precision inductance measurement at low cost is an urgent problem to be solved.

Aiming at the above problems, researchers proposed a variety of measurement methods based on digitally assisted bridges. In some of them, the impedance ratio relies on the high-precision and stable output of voltage. For example, Corney proposed a digital generator assisted impedance bridge, whose relative standard uncertainty can reach 10-5, but its measurement relies on manual operation, resulting in low measurement efficiency[4]. In another class of setups, the impedance ratio does not directly rely on the accurate generation of voltage but is rather dependent on the accurate measurement of voltage[5-7]. For example, the Chinese National Institute of Metrology (NIM) has proposed a dual-channel A/D sampling AC bridge that automatically achieves zero equilibrium, which has a measurement uncertainty of 10-5, but is not suitable for comparison of low impedance values[8]. Overney et al. proposed an RLC bridge based on an automatic synchronous sampling system, which can achieve a relative standard uncertainty of 12 μH/H in the frequency range of 400 Hz-5 kHz, and can achieve automatic bridge balancing and automatic value measurement. However, the bridge balancing efficiency is too low, and the measurement system must be sampled within the integer period of the measurement signal, which will increase the complexity of system design[9].

To sum up, although these two methods can achieve high-precision measurement of inductance, they still have shortcomings in bridge balancing efficiency, measurement range and system complexity. In order to solve the above problems, we propose a method of inductance measurement based on double-excitation auto-balancing bridge, and build a prototype system. Aiming at the low efficiency of bridge balancing, an iterative balancing algorithm based on the steepest descent method is proposed to achieve rapid bridge balancing. In order to enlarge the measuring range of the system, the high-precision DDS signal generator is used as the excitation source of the double-excitation balance bridge to realize the output of wide frequency range signal, and the multi-value measurement reference source is used to realize the inductance measurement of wide band and large range. To suppress the measurement error caused by the initial signal phase change and non-integer periodic sampling, the all-phase fast Fourier transform algorithm (apFFT) is used for data processing, which can accurately obtain the signal initial phase and solve the problem of non-integer periodic sampling, then the amplitude ratio and phase difference of the vector voltage of the device can be calculated with high precision, and the complexity of system design can be reduced. Through simulation analysis and experimental verification, it is proved that the system can meet the requirements of high-precision measurement of four-terminal inductance under different frequency and inductance.

1 Measurement principle and system structure

1.1 Measurement principle

This measuring system is based on the double-excitation auto-balancing bridge, which is evolved from the Kelvin bridge. Kelvin bridge is shown in Fig.1(a). It is a kind of DC balanced double-arm bridge that can be used to measure low resistance (10-5Ω-1 Ω)[10]. It can reduce the influence of wiring resistance and contact resistance on the measurement results, and its measurement accuracy is high. When the Kelvin bridge is balanced, the potentials atbanddequal.

(a) Kelvin bridge

(b) Improved Kelvin bridge

(c) Double excitation balanced bridgeFig.1 Measurement principle of the system

By setting pointbin Fig.1(a) as the ground reference point, the Kelvin bridge is split and the circuit shown in Fig.(b) is obtained. Furthermore,Z3andI3,Z4andI4in Fig.1(b) can be equivalent to voltage sourcesUbandUt, respectively, and the circuit shown in Fig.1(c) can be obtained, that is, a double-excitation balancing bridge. At this point, the balance state of the bridge can be determined by adjusting the voltage sourceUbandUt, and judging whether the value ofVwis zero. When the double-excitation bridge is balanced, the following conditions can be satisfied as

(1)

whereZ1,Z2,Z3andZ4are the resistances of the Kelvin bridge arm,Zbis the inductance to be measured, andZtis the standard resistance. The balancing condition is only true whenVw=0. But in the actual situation, the value ofVwcan only be close to zero but cannot be equal to zero. Therefore, the double-excitation balancing bridge should meet

(2)

After calculation, we can get

(3)

Then, whenZ1/Z2=-(Vb-Vw)/(Vt-Vw), we have

(4)

Based on the above analysis, the double-excitation balancing bridge does not need to adjust the values ofZ3andZ4, but realizes the bridge balance by adjusting the controlled excitation sources at both ends. Therefore, compared with the Kelvin bridge, the circuit based on the double-excitation balancing bridge is more suitable to be realized by programmable logic devices, which is conducive to the rapid and high-precision balancing of the bridge. At the same time of adjusting the excitation source, the balance state is determined and the impedance value is calculated by measuring the voltage values atVb,VtandVw.

To sum up, the measurement system based on double-excitation auto-balancing bridge can use two high-precision DDS signal generators as excitation sources, a high-resolution analog-to-digital converter (ADC) to sample voltage values and a switchable analog switch to change the sampling point. The system uses 16 bits of amplitude quantization and 24 bits of phase quantization to meet the high-precision requirements of DDS. However, the traditional table method is difficult to achieve the above performance, and the DDS chip on the market cannot achieve this quantization accuracy. Therefore, The high-precision DDS signal generator is realized by using high-resolution digital-to-analog converter (DAC) based on coordinate rotation digital computer method (CORDIC) and field-programmable gate array (FPGA).

1.2 System structure

Based on the above analysis, the system structure can be designed as shown in Fig.2. FPGA controls the corresponding DAC chip to achieve the generation of double-excitation signals. At the same time, FPGA controls the ADC chip through analog switch to sample the state information of each point of the double-excitation balancing bridge, and then sends the data to the upper computer to calculate each value to realize the measurement of inductance.

Fig.2 System structure

During the measurement, the inductanceZband the standard resistanceZtare connected to the system respectively in a four-terminal pair mode. We can adjust the amplitude ratio and phase shift of the two signals to minimize the voltageVwat the intermediate point, and thus achieve Wagner balance. When we adjust the variable impedanceZ1andZ2of the Kelvin resistance network until the on or off effect of switch K onVwvalue is small enough to be negligible, Kelvin balance is realized[11]. At this point, the bridge gets balanced, and the voltage signals ofVb,VtandVware sampled, respectively. After performing FFT on the sampled numeric sequence, the ratio of impedance can be obtained by

(5)

whereRbandLbrepresent the series inductance and series resistance of the device under test,Rtis the resistance value of the standard resistance,ωis the angular frequency of the measured signal, andτis the time constant. The obtained parametersAandBcontain information about the amplitude ratio and phase relationship between the measured voltages, and the measured series inductance and resistance values can be obtained by

(6)

2 Algorithm for double-excitation balancing bridge

2.1 Auto-balancing algorithm

If the above methods only rely on manual balancing, the process is complicated and inefficient： During bridge balancing, firstly, keep the switch closed, keep the voltageVtconstant, and minimizeVwby adjustingVb； Then, turn off the switch and adjustZ1andZ2to minimizeVw； Finally, repeat the previous adjustment process until the on/off of the switch has negligible effect onVw. When the rough estimate ofZbis known,Z1andZ2can quickly reach the balanced position by manual adjustment. But forVb, the search method by manual adjustment, traversal or bitwise adjustment is too inefficient to achieve fast measurement of inductance. Therefore, the algorithm that can achieve fast bridge balance is the key. Based on this, we propose an iterative balancing method based on the steepest descent method to determine the value ofVbonly after a few calculations and realize the rapid measurement of inductance. The balancing process is shown in Fig.3.

Fig.3 Iterative balancing process

During bridge adjustment, we first keep the voltageVtof the standard resistor constant. An initial voltageVb0with random amplitude and phase is input toZb, the inductor to be measured, then we have

(7)

The initial complex impedance valueZb0can be obtained from the above equation. To achieve bridge balance, the root-mean-square (RMS) value ofVwshould be zero under ideal conditions. Therefore, by settingVwto zero, the voltageVb1in next state can be calculated reversely through Eq.(7). By adjusting the voltage ofZbtoVb1, and sampling data again for calculation, we get the complex impedance valueZb1under the voltage ofVb1. Afterwards, the threshold is set toε, and the iterative process of calculation, adjustment and re-calculation is repeated untilVw,nandVw,n+1after two consecutive adjustments meet

|RMS(Vw,n+1)-RMS(Vw,n)|<ε.

(8)

Then it is considered that the algorithm has converged and the bridge is balanced. At this time, the optimal voltage valueVb,bestcan be obtained, and the inductance value can be calculated by sampling data.

However, it is impossible for the balancing algorithm to adjust the bridge to the ideal balance state under the practical application conditions. The bridge usually works under the condition of imperfect balance, so the threshold value needs to be set reasonably to ensure that the bridge can achieve high precision and avoid the problem that the algorithm cannot converge.

2.2 Processing of sampled data

As seen from the above analysis, the estimation accuracy of FFT on the amplitude and phase of each signal will directly affect the accuracy of the final measurement results. The traditional FFT transform is to truncate the measured signal and then perform FFT after the period extension to obtain the spectral analysis result. Therefore, if the truncated signal is not integer period, it will lead to spectrum leakage effect of the signal, and then reduce the measurement accuracy. However, it is difficult to realize accurate integer periodic sampling of the signal under actual working conditions, and the initial phase of the signal will also affect the calculation results of FFT.

Therefore, the system uses the apFFT to suppress the interference caused by the initial phase and non-integer period[12-14]. There exists the square relation between the spectral amplitude of apFFT and that of traditional FFT, which means that the ratio of the side spectral line to the main spectral line also decays in accordance with the square relation. It makes the main spectral line more prominent, and thus apFFT has a good function of suppressing spectrum leakage. The apFFT also has phase invariance, that is, in non-integral period sampling, the phase information of the signal can be directly extracted from the spectral analysis results without any additional correction measures. The apFFT is realized by all-phase preprocessing the sampled data and then performing FFT. The data processing steps are as follows：

1) Periodic extension is carried out for each input data vector at the original position respectively；

2) A new periodic sequence is formed by summing the extended sequence vertically；

3) The new periodic sequence is truncated by a rectangular window to generate an all-phase input sequence.

The amplitude and phase of sinusoidal signals with an amplitude of 1 V and a frequency of 1 024 Hz are analyzed by FFT and apFFT, respectively. Setting the sampling frequency as 1.6 kHz and FFT pointNas 512 points, the results are shown in Fig.4.

(a) Estimated amplitude and error of FFT variying with initial signal phase

(b) Estimated phase of FFT varying with initial signal phase

(c) Estimated phase of FFT varying with signal frequency

(d) Estimated amplitude and error of apFFT varying with initial signal phase

(e) Estimated phase and error of apFFT varying with initial signal phase

(f) Estimated phase and error of apFFT varying with signal frequencyFig.4 Amplitude and phase of signal estimated by FFT and apFFT

It can be concluded from Fig.4(a) and (d) that the maximum estimation error of FFT for signal amplitude can reach 3.294×10-3, while that of apFFT for signal amplitude is 1.097×10-5. Therefore, the result of apFFT is more accurate to estimate the amplitude ratio. Fig.4(b) and (c) show that when the initial phase or frequency of the signal changes, FFT estimation error of signal phase is so large that it cannot be ignored. According to Fig.4(e) and (f), for apFFT, the phase estimation is accurate with the change of the initial signal phase, and the maximum estimation error is 7.416×10-4. When the initial signal phase is unchanged but the signal frequency changes, the maximum error of apFFT phase estimation is only 6.361×10-13.

It can be seen from the above analysis that the results obtained by apFFT are more accurate than those obtained by FFT, and the influence of signal initial phase change and non-integer periodic sampling on the calculation results is minimal and can be ignored.

3 Experimental verification

3.1 Evaluation of uncertainty

The ambient temperature of this experiment is 25 ℃, the relative humidity is 55%, and there is no obvious strong influence of external magnetic field and mechanical vibration around the system. When the bridge is balanced,

(9)

whereZbis the impedance value of the inductance,Ztis the standard resistance value,Vbis the voltage of the component to be measured, andVtis the voltage of the standard resistance. The inductance valueLcan be calculated from impedanceZband the result is

(10)

wherefis the signal frequency. Table 1 lists the measurement uncertainty components of the mathematical model, and the synthetic relative uncertainty of each inductor at each frequency is listed together with the measurement results in Table 2.

Table 1 Measurement uncertainty components

3.2 Correction and measurement results

Tonghui TH2826 LCR digital bridge was used to calibrate the system, and the corrected curves of the system were obtained at frequencies of 100 Hz, 1 kHz, 10 kHz and 100 kHz as shown in Fig.5.

(a) Corrected currve at a frequency of 100 Hz

(b) Corrected curve at a frequency of 1 kHz

(c) Corrected curve at a frequency of 10 kHz

(d) Corrected curve at a frequency of 100 kHzFig.5 Corrected curves of the system at each frequency

The nominal inductors of 0.1 mH, 1 mH and 10 mH were measured at 100 Hz-100 kHz frequency, and the measurement results were corrected by the corrected curves. The nominal value error of the standard resistance used for measurement is 10-5and the temperature coefficient is 5×10-6/℃. The measured value of TH2826 is used as the true value of inductance to compare with the measured value of this system, and the measurement results are shown in Table 2. The measurement data show that the inductance measurement precision of the system is high, the minimum relative error can reach 0.009%, and the optimal relative measurement uncertainty of the system can reach 9.89×10-5. After a detailed analysis of the measurement results, the possible causes of errors are mainly from two aspects： First, the manufacturing process of common inductor used in measurement affects the accuracy of measurement； Second, there is interference in the layout of circuit components, leading to the deviation of measurement results. The measurement accuracy can be further improved by subsequent improvement.

4 Conclusions

In this study, an inductance measurement system based on double-excitation auto-balancing bridge is designed, which can achieve high-precision inductance measurement in the wide frequency range of 100 Hz-100 kHz. The whole system is controlled by computer to realize the automation of bridge balancing and inductance measurement. In order to improve the measurement efficiency and accuracy, a bridge iterative balancing algorithm based on the steepest descent method is proposed, and apFFT algorithm is used to achieve high-precision estimation of signal amplitude ratio and phase difference. Obtained by experimental verification, the minimum relative error of the system measurement inductance is 0.009%, and the relative measurement uncertainty of the system can reach 9.89×10-5. The measurement results show that the system has small relative error, good real-time performance, and relative uncertainty better than the traditional commercial impedance analyzer. It provides a low-cost, high-speed and high-precision measurement method for inductance measurement, and has a wide application prospect.