Proceedings of the 7th International Caribbean Conference on Devices, Circuits and Systems, Mexico, Apr. 28-30, 2008

Advanced Methods for Algorithmic Corrections of Errors in Immitance Measurement Oleg Starostenko

Jorge Rodríguez-Asomoza, Vicente Alarcon-Aquino

Research Center CENTIA, Universidad de las Américas, Cholula, Puebla, México [email protected]

Department of Computing, Electronics and Mechatronics Universidad de las Américas, Puebla, México [email protected], [email protected]

Abstract— This paper presents an analysis of some design concepts and development of advanced techniques for incrementing speed and accuracy of CLR meters of low cost. Some measuring systems have been designed in order to estimate performance of novel approaches, such as a structural method of error correction and iterating correction method of errors in immitance measurement. The expressions of functional conversion (transfer functions) and block diagrams of designed equipment are presented and discussed for these algorithmic methods. Using the best iterating correction method the E7-13a LCR meter has been designed and tested for estimation of its accuracy and speed during measurement of capacitance, inductance, resistance, and conductance. The obtained results are evaluated to define efficiency of proposed methods, their utility and performance. These results encourage more researches in this open problem of immitance measurement.

I. IMMITAMCE MEASURING METHODS AND DEVICES Actually, measuring devices and systems for analysis of complex parameters (impedance and admittance of electrical circuits on alternate current) are used in different applications due to their high speed (0.01-0.001s) and relative measurement error (0.01-0.1%) [1], [2]. However, an increment of accuracy and speed is achieved by incorporating new blocks or control units into measuring process [3], [4]. Therefore, a cost of novel technique will be raised up. To improve metrological characteristics of measurement facilities without increment of cost, the error correction methods are used. Usually implemented bridge measurement approach with very precise results is not useful in these applications due to a great number of equilibrated iterations, which significantly reduce a speed of measurement [5]. A novel approaches are based on linearization or correction of transfer functions of measuring converters in direct measuring and feedback control channels without reduction of operational speed [6]. Another advantage of correction methods is extension of operational frequencies of measured signals with low effective power of test or reference voltage.

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Any measured component could be represented by its equivalent circuit, which consists of serial and parallel connections of passive parameters, such as capacitance, inductance, and resistance (CLR). The equivalent or substitution circuit is varied according to a magnitude and frequency interval of test exciting signal. The joint term for full complex resistance from a serial equivalent circuit (impedance) and full complex conductivity from a parallel equivalent circuit (admittance) is denominated in this paper as immitance. There are a lot of reports about measuring systems for analysis and estimation of immitance features. To achieve the highest speed and accuracy the direct conversion methods of measured passive parameters are used in modern meters [4], [7]. This approach is simple in implementation, fully automatic, and may be combined with error correction blocks to achieve the accuracy of compensation methods. The generalized block diagram of direct conversion approach is shown in Fig. 1. The measurement procedure requires that the passive value X is transferred by some sequential converters, e.g., by the active/passive measuring converter, vector-scalar converter, and analog-to-digit converter. Initially, a complex passive value X {ReX, ImX} is interpreted as complex active voltage VX. Then a vector-scalar converter based on phase detector and low frequency filter selects and separates the measured complex voltage VX into two active ReVX and reactive ImVX components. Finally, they are converted into constant voltages or currents, which are transformed by analog-to-digit converter into a pair of digital components NReVx and NImVx proportional to the measured passive value [8]. There are some types of active measuring converters: conductance or resistance and normal or inverted converters, which are distinguished by presentation of measured components and their position in electrical circuit. The block diagrams of some active converters are shown in Fig. 2.

X {ReX,ImX}

VX

Active/Passive measuring converter

feedback. Structural methods for incrementing measurement accuracy on each conversion stage achieve total error of order no less than 0.01% [8], [9]. The accuracy of measurement is defined by magnitude of steady-state errors and precision of transfer converters in feedback channel of system. The simplest case of error reduction is provided by insertion of compensation resistor in feedback channel.

Vectorscalar converter

V0

ReVX ImVX Test signal generator

NReVx NImVx

Analogto-digit converter

Figure 1. Block diagram of system with direct measuring method

V0 a)

1

∼

CK R0

V0 b)

YX 2

CK

CK ZX

R0 V X OpAmp

NA

VX OpAmp

Figure 2. Electrical circuit of normal active converters a) for complex conductance and b) for complex resistance measurements

Both the stray capacitance and the input leakage conductance are always presented on an input of the measuring converter. For resistance-voltage converter the simplified circuit is presented in figure 2b). We will use only analysis of the normal conductance active converters, since the procedure for others is similar. For ideal complex conductance conversion ( Y X → V X ) using an OpAmp, the output voltage proportional to measured value is found as:

VX

1 -YX R0 1 1 V0 βA

-YX R0 V0

Using presented in Fig. 2 active converters, the digital representations of active and reactive components of measured complex value (real and imaginary parts of unknown measured voltage) NA and NR are obtained as function of bit numbers n of analog to digit converter, resistance of compensation resistor RC, active and reactive components of measured complex conductivity YX*=YA*+YR*, and active and reactive components of relative errorδ. Therefore, the (2) represents the digital equivalents of measured parameters, where adjustment of RC magnitude can be used to obtain the exact results.

(1)

The feedback gain βA depends on parameters of the OpAmp and due to its high value can be used for error reduction. The errors of test signal deviation and test resistor (R0) instability may be excluded due to their low amount (their errors are less than 0.001%) [8]. The errors of stray capacitance and input leakage conductance of a measuring converter, as well as the impedance of an input cable have additive character and can be removed by the initial “zero calibration” of the measuring system. Some novel approaches are proposed in this paper to detect and estimate best method for different CLR measuring architectures in 100 Hz – 1 MHz frequency intervals of test signal. II. STRUCTURAL METHOD OF ERROR CORRECTION A structural method of error correction is based on insertion into modified device additional high speed analog/digital units, which linearize total non-linear transfer function of a measurement system using deep negative

2

1

, NR

2

1

(2)

Initially, a system must be trained with different immitance basic etalons or standard reference components for calibration of forward channel. The proposed method in case of complex parameter measurements can be implemented as a system presented in Fig. 3. The block diagram of the transfer process is shown in Fig. 4. The transfer function of the first block (active measuring converter) is found as (3) where principal error is defined by instability of reference voltage V0 and by error of the digital generators composed by modules such as Clc, Cnt, Mem, RC of active/reactive components, and DAC. The second block consists of circuits for conversion of measured current to scalar voltages in phase and with π/2 displacement, LFFs, DCAs, and compensating feedback of digital generator. According feedback actions the transfer function in simplified form is obtained as ⁄

⁄1

1⁄

(4)

The last block includes ADCs of real and imaginary measured components describing by 2 ⁄

1

(5)

where n, VO, and δADC are number of bits, reference voltage, and relative error of ADC. Analysis of general transfer function with introduced units of compensation shows that principal error is defined by reference voltage and errors of compensation register, multiplicative errors of direct and alternate amplifiers gain factors, and ADC/DAC errors. All these errors can be reduced to minimum value by selecting of accurate components and circuit.

RC

VR DAC

RC

VA

DAC–digit-to-analog converter

sin

Mem

Cnt

Clc

ACA-alternate current amplifier Cnt–address counter

cos Mem

DAC

ADC

R DAC

DCA

LFF

NA

ACA

YX Measuring amplifier

VTEST

ADC DAC

LFF

DAC

NR

DCA RVG

VREFERENCE

LFF–low frequency filter Clc–digital clock generator DCA–direct current amplifier ADC–analog-to-digit converter Mem–Sine signal form table RVG–reference voltage generator

Figure 3. Immitance measuring system with structural error correction

VO

VA YX

bt VT kt

IX

K1K2ar RK

NA W3(jω)

VR

IK

VO

k–clock pulses number, t-clock pulses interval bk–feedback circuit transfer function K1K2–direct and alternate current amp. gains VO–reference voltage

bk W1(jω)

NR

bt–gain factor of voltage generator

W2(jω)

kt

ar–gain of current to voltage converter

Figure 4. Block diagram with transfer functions

The advantage of the structural algorithmic correction method is that the measurements are performed without speed reduction (during the same measurement cycle the propagation time for analog correction circuits (in the worst case: hundreds of microseconds) does not have a significant effect over measurements in millisecond cycles. The disadvantage of the approach is a significant number of ADCs and DACs, which must be low cost. That means, they have reduced dynamic range and accuracy. III.

ITERATING COMPENSATION METHOD

The iterating compensation may be introduced as algorithmic correction method. It is used in order to improve metrological characteristics of immitance parameter measurement by error correction unit incorporated into feedback channel that generates correction signals according to results of previous step of iterating error reduction [8]. The simulation of iterating processes and comparison with well-known designed measuring systems has been carried out for different frequency intervals of test signal. The block diagram of system implementing proposed iterating approach is shown in Fig. 5. Using the active converters as

basic element of a measuring circuit, the results for active and reactive components in digital form on any iteration M may be derived as: 1

1

1

(6) 1

1

1 (7)

The accuracy limit of immitance measurement depends on the precision of a compensation resistor and the variation of transfer coefficients of the direct channel during changing the configuration of a measuring circuit (measuring or iterating mode). This variation is manifested due to differences of a resistance of switching contacts and an output resistance of the DAC, which works as test/compensation voltages generator.

RM Σ

PD+ADC

IX

IV

YX

Iteration block 1 for measurement of passive values

IAR

Σ

Cm-switch RK

Iteration block2 (active component)

cos DG

VT(VK)

PD-phase detector

Multiplier with Multiplexor

sin

Cm

DG-digital generator of sin and cos NA

IV-initial value register

NR

IAR-iteration approxim. register

Multiplier with Multiplexor

Σ+DAC

Figure 5. Block diagram of immitance meter with iterating correction of errors. TABLE I. Iterations (M) δA(M) δR(M)

0 24 -34

1

3

-1.9 5.9

4

2.5 -0.38

5

0.07 -0.89

6

-0.38 0.02

7

0.02 0.13

8

0.05 -0.01

0.01 0.00

NUMBER OF ITERATING STEPS FOR IMMITANCE MEASUREMENTS WITH RELATIVE ERROR 0.05 %.

1 2

2

-16 -5.7

TABLE II. f, kHz steps

RELATIVE ERRORS OF MEASUREMENT ON F=500 KHZ WITH I TERATING CORRECTION

2 2

5 2

10 2

Table I shows the values of relative errors during iterating correction process for measurement on 500 kHz. Table II presents results of the experiments for determination of number of necessary steps in order to achieve measurement errors less than ±0.05% at a wide range of frequencies. The advantage of proposed iterating method for immitance measurement is that, all the operations for generation of correction action in feedback channels are made in digital form. That is why, the feedback errors are very small and equal to half of the least significant bit of operating word of used DAC (±0.5/2n, where n is the number of input bits of DAC). The iterating correction may be used for accurate immitance measurement in a frequency range no more than 500 kHz in order to avoid a great number of iterating steps (Table II). This increment of required iterations is the result of tight converter channel interaction on high frequencies. The principal disadvantage of this method is its low speed for accurate measurement on high frequencies, because a great number of iterations is needed. For example, to reduce relative error to 0.01% during a measurement on a frequency of the test signal of 100 kHz, it is necessary to have more than 10 iterations.

20 2

50 2

IV.

100 3

200 4

500 7

1000 22

PRACTICAL IMPLEMENTATION OF ERROR CORRECTION METHODS The real implementation of error correction methods is shown in this paper on base of the E7-13a CLR meter, where the iterating correction method is used due to its lower cost and higher accuracy than structural method of error correction provides. The E7-13a meter is used for measurement of capacitance, inductance, resistance, conductance, and active/reactive parameters of the passive values including a Q-factor and tangent of loss. In Fig. 6 the block diagram of system is shown, where direct measuring method is applied for two voltage components measured in phase and with π/2 displacement with respect to alternate reference signal. The technical characteristics of the meter are presented in Table III, where CX, LX, RX are results presented on the display of the device, CLN, LLN, RLN are nominal limits of the CLR measurement. Table III shows the relative errors of the device. Additionally, the device has the following technical specifications: the test signal frequency is 1kHz ±10Hz and the magnitude is 200 mV; the measuring time including error correction process and visualization of result on 5 digit display is not more than 1s; the length of cable is 2.0 m; the power supply from adapter is 5 VA and from a battery is 1.5 W; weight is 2kG with dimensions 200x70x227 mm. The E7-13a CLR meter with implementation of the iteration correction method is presented in Fig. 7.

YX Test signal generator

Active converter

V*X

Vector-Scalar converter

Vtest

Error correction unit

Power supply

VIm

VRe

Vcorrection

ADC NIm

NRe Device display Figure 6. Block diagram of E7-13a CLR meter TABLE III. Measured parameter Capacitance

Inductance

Resistance

RELATIVE ERRORS OF MEASUREMENT OF C, L, R Range 1100 pF 1000 pF 10 nF 100 nF 1 mkF 10 mkF 1 mH 10 mH 100 mH 1H 10 H 100 H 10Ω 100 Ω 1000 Ω 10 kΩ 100 kΩ 1000kΩ

Resolution 0.1 pF 1 pF 0.01 nF 0.1 nF 0.001 mkF 0.01 mkF 0.001 mH 0.01 mH 0.1 mH 0.001 H 0.01 H 0.1 H 0.01 Ω 0.1 Ω 1Ω 0.01 kΩ 0.1 kΩ 1 kΩ

Relative error (%) 0.02 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.05 0.05 0.02 0.01 0.01 0.01 0.01 0.02

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

V. CONCLUSIONS Two error correction methods have been proposed and tested for measurement of immitance parameters of complex values. These methods require simple and inexpensive electronic components, without any reduction of conversion and data processing speed. In order to obtain accurate results the algorithmic error correction unit has been implemented, which for the same experiments provides error reduction to 0.01%. Based on the direct measurement method the E7-13a CLR meter has been designed and constructed. The principal disadvantage of proposed approaches is that a correction unit can be a complex circuit due to great number of arithmetic operations and complicated expressions for computing the correction actions. This problem may be solved by standardization and unification of analog circuits in specialized high-scale integrated circuits, which are used for execution of simple arithmetic operations in analog form. REFERENCES [1] [2] [3] [4] [5] [6]

[7]

[8]

[9]

Figure 7. E7-13a CLR meter

National Instrument Corp., Test and Measurement Industrial Automation, Instrument Reference Catalogue. USA, 2008 Instek LCR-821, High Precision LCR Meter, Products catalog, 2008. http://www.testequipmentdepot.com/instek/meters/lcr821.htm BK Precision Electronic instruments, Model 885 Synthesized Circuit LCR Meter, http://www.bkprecision.com/www/np_specs.asp?m=885 Wayne Kerr Electronics, Fast automatic LCR measurement to 3MHz 6440B, http://www.waynekerrtest.com/global/html/products/lcr.htm Bell Electronics, BK Precision 875B LCR Meter, Handheld, LowOhm, 2008, http://www.bellnw.com/products/0161/ V. Khoma, E. Pokhodilo, O. Starostenko. “Device for measurements of complex resistance components”, Patent of USSR Nº1778712, August,1, 1992. L. Schmalzel, “Electrical Measurement, Signal Processing, and Displays”, IEE Instrumentation & Measurement J., vol. 7, Issue 3, 86–87, 2004. V. Khoma, O. Starostenko, “Noise reduction in operational networks of immitance-voltage converters”, J. of National Univ. "Lvivska Polytechnica", section: Automatics, Measurements, and Control, Ukraine, vol. 445, 62-65, 2002. A. Daire, “Low-voltage measurement techniques: a three-point-delta method provides more accurate resistance measurements in lowvoltage applications”, EE-Evaluation Engineering J., vol. 44, Issue 6, 40, 2005