The structural method of error correction is based on insertion into the direct or feedback channel of measuring system of additional high-speed analog or digital units, which linearize the total non-linear transfer functions of the designed circuit [13,14]. We introduce an error correction method which combines the simplicity and speed of direct channel measurements and the high accuracy of feedback systems. For this, the output voltage and relative error vector of the active converter (see Figure 1) with a presence of input cable capacitance C_K are calculated as: (2) V X = − Y X R 0 1 1 + i f / f T [ 1 + R 0 ( Y X + i ω C K ) ] V 0 ; δ = i f / f T 1 + R 0 ( Y X + i ω C K ) 1 + i f / f T [ 1 + R 0 ( Y X + i ω C K ) ]where A₀ is the direct current gain of the OpAmp without feedback, f_T is the frequency of unity amplification, f is the frequency of the test signal. For example, if C_K = 100 nF, ω = 2πf = 6,280 s⁻¹, Y_X = (1 + i0) μS, f_T = 10⁷ Hz then |δ| ≈ 0.02%. The transfer function of the whole active conductance converter without frequency-range dependence may be presented as: (3) G Y = X 1 + i X 2 1 + ( K 0 + iMF ) ( 1 + X 1 + i X 2 + iTF )where Y₁, Y₂ are the active and reactive components of the converted conductance Y_X, X₁ = Y₁R₀, X₂ = Y₂R₀, TF = 2πfC_KR₀, K₀ = 1/A₀, MF = f/f_T. The ideal transfer function is found now as G_ideal = X₁ + iX₂ and with G₁ = Re(G_ideal), G₂ = Im(G_ideal), the relative error vector of conversion for active and reactive components may be found as: (4) δ 1 = G 1 / X 1 − 1 − = φ 1 ( X 1 , X 2 , T F , M K , K 0 ) δ 2 = G 2 / X 2 − 1 − = φ 2 ( X 2 , X 1 , T F , M K , K 0 )

Thus, the analysis of errors is reduced to computing the functions ϕ₁ and ϕ₂. On the basis of previous equations, active and reactive components of conversion errors are calculated (a detailed description of error analysis may be found in [8]): (5) δ 1 = − ( 1 + X 1 − X 2 2 / X 1 ) K 0 + ( 2 + 1 / X 1 ) X 2 M F + Q 1 + 2 ( 1 + X 1 ) K 0 − 2 X 2 M F + Q δ 2 = − ( 1 + 2 X 1 ) K 0 + ( X 2 + X 1 / X 2 − X 1 2 / X 2 2 ) M F + Q 1 + 2 ( 1 + X 1 ) K 0 − 2 X 2 M F + Q ; Q = [ ( 1 + X 1 ) 2 + X 2 2 ] ( K 0 2 + M F 2 )

With the obtained Equations (5), the components of error vectors have been calculated for different combinations between X₁ and X₂. For example, if in the measured structure the active component predominates, then X₁ = 1.0 and X₂ = 0.1; otherwise, if the reactive component predominates, then X₁ = 0.1 and X₂ = 1.0. Taking into account two non-linear equations for active and reactive components of conductance converters transfer function (Equation 3) and applying their linearization according to: (6) X 1 = G 1 ( 1 − δ 1 / 1 + δ 1 ) ≈ G 1 ; X 2 = G 2 ( 1 − δ 2 / 1 + δ 2 ) ≈ G 2the approximate solution for the correction algorithm is defined as: (7) N 1 = G 1 + [ G 1 ( 1 + G 1 ) A 0 − G 2 ( T F + G 2 ) A 0 − G 1 ( T F + G 2 ) f T / f − G 2 ( 1 + G 1 ) f T / f ] N 2 = G 2 + [ G 2 ( 1 + G 2 ) A 0 − G 1 ( T F + G 2 ) A 0 − G 2 ( T F + G 2 ) f T / f − G 1 ( 1 + G 1 ) f T / f ]

In Equations (7) the components in brackets are correction actions which must be added to corresponding components of the transfer functions G₁ and G₂ in order to obtain accurate results. These equations may be simplified without significant reduction of precision due to 1/A₀≈0: (8) N 1 = G 1 − [ G 1 ( T F + G 2 ) + G 2 ( 1 + G 1 ) ] M F ; N 1 = G 2 − [ G 2 ( T F + G 2 ) − G 1 ( 1 + G 1 ) ] M F

Therefore, the improvement of metrological characteristics is achieved by computing components of the systematic error and its compensation by additional operations according to the described algorithm. We propose to name this technique the “structural algorithmic correction method”. The additional arithmetic operations in the Equation 8 are implemented using analog adders and multipliers based on OpAmp. Input variables such as R₀ or TF are simulated by reference direct-voltage dividers and frequency-to-voltage converters, respectively [8].

The plots for error behavior without the correction algorithm δ₁ and δ₂ according Equation (5) are shown in Figure 3 as the functions δ₁=ϕ₁(f/f_T) and δ₂=ϕ₂(f/f_T). In the same plot the results with correction e₁ and e₂ computed according to expression (9) are shown for conductance measuring converters, where: (9) e 1 = N 1 X 1 − 1 , e 2 = N 2 X 2 − 1

In a wide range of frequencies, the errors of the measuring converter sometimes change their sign, so the absolute values of errors Mod[e₁], Mod[e₂] are presented in Figure 3.

Analysis of errors has been made for cheap OpAmp with a small gain factor A₀ = 20,000, the frequency of unitary amplification f_T = 10 MHz, and test signal frequency range from f = 1 Hz to 1 MHz corresponding to MF = 10⁻⁷−10⁻¹. The results of the experiments presented in Figure 3 show specific intervals of frequencies, where the error can be less than 0.001%. For example, in order to achieve an error of about 0.01%, the measurements must be done in the 40–100 kHz frequency range. This interval depends on the parameters of OpAmp, the magnitude of the reference resistor, and the frequency of the test signal. Therefore, the low error band may be computed and selected for a particular application! As result, it is possible to design cheap and high performance impedance meters with very low errors.

Another unexpected result of implementation of the structural algorithmic correction method is the measurement frequency range extension with the same level of errors. For example, a measurement error of 0.5% without correction (curves δ₁ and δ₂) can be achieved only in the frequency interval up to f/f_T ≈ 0.0003/0.0004 (f ≈ 3/4 kHz). For conductance conversion with correction, the errors e₁ and e₂ have the same level up to the frequency interval f/f_T ≈ 0.02 (f ≈ 200 kHz). This means that the error correction method reduces the error level and maintains it in a wide test signal frequency range. In the previous example with a conductance converter the frequency extension covers two orders of magnitude; this is significant improvement! Another advantage of the structural algorithmic correction method is that the measurements are performed without speed reduction (during the same measurement cycle). The propagation delay for analog correction circuits (in the worst case: hundreds of microseconds) does not have a significant effect over measurements in millisecond cycles. Moreover, the impedance meter is quite cheap due to the basic quality of electronic components used in measuring converters and correction unit. For example, as shown in Figure 3, errors as low as 0.001% have been achieved with OpAmps with a gain factor only about 20,000 (modern OpAmps have A₀ gain factors of more than 10⁶).

The principal disadvantage of this method is that a correction unit can be a complex circuit because it must support many arithmetic operations for generation of corrective actions. This problem may be solved by standardization and unification of analog circuits for simple arithmetic operations in high-scale integrated circuits. There is no correlation between measured active (real) and reactive (imaginary) components of measured value as well as between the components of systematic errors that appear during impedance measuring processes. This is because the result basically depends on the physical characteristics of the measured value and not on the proportion between real and imaginary parts of measured value. If a measured value is a characteristic of semiconductor structure, the impedance measured components are better used for describing its features, such as substrate doping, flat band voltage, mobile ion concentration, series of resistance effects, charge densities, loss in the MOS structure due to charging and discharging of interface traps, etc.

The structural algorithmic correction method has been introduced for linearization of the transfer function of a complex conductance measuring converter. However it is only the first stage of the measurements. The consecutive vector-scalar and analog-to-digital converters introduce non-linear errors which must be removed by their own correction structures that may occupy more than 30% of all the measuring equipment. As usual, these structures are simple to implement and may be used for incrementing the performance of cheap measuring systems.

In order to explore the power of other error correction techniques an iterating compensation method is also proposed. Iterating methods use error correction units incorporated into the feedback channel which generates correction signals according to the results of previous measurement steps [15]. The functional block diagram of the proposed system is shown in Figure 4. The system consists of a digital generator (DG) of orthogonal signals, a measuring unit of varying configuration with OpAmp for direct measurement and iterating correction modes, a direct channel for measuring the active and reactive components with its corresponding ADCs, two iterating blocks for active and reactive components, and feedback units for the generation of compensation voltages. The measuring circuit formed by an active converter of conductance Y_X generates a proportional complex current I_X and then transfers it into a complex voltage V_X. In other words, this circuit is a vector converter, which consists of an OpAmp with scale feedback resistor R_M, compensation resistor R_Comp, and switch Sw. DG forms orthogonal sine and cosine signals which are defined by two 10-bits codes for sine N_S and cosine N_C functions: (10) N S ( k T ) = sin ( 2 π kTf ) ; N C ( k T ) = cos ( 2 π kTf )where k is the number of clock pulses, and f is test signal frequency. These codes are used as reference signals for phase detectors (PD), for generation of analog test (V_T) and compensation (V_Comp) voltages. The measurement process is divided into two steps: impedance measurement without correction and iterative correction of the result. First, the sine V_T voltage is generated at the output of DAC by a look-ahead digital adder Σ. For this, the multiplier M_A repeats the sine codes N_S(kT) on its output because the set of ones “1” from measurement mode “Meas” of M_A is selected. These ones repeat the codes N_S (sine signal) in the output of M_A in measurement mode. The cosine signal codes are multiplied by zero from input “Meas” of M_R. Therefore, the V_T is computed as V_T(kT)=bV₀cos(2πkTf) where b is the transfer coefficient and V₀ is the reference voltage of DAC. V_T is applied to the measured impedance Y_X through switch Sw. The output voltage V_X and transfer function G_VC of vector converter are obtained as: (11) V X = I X G V C = ( Y A + i Y R ) G V C V T ; G V C = R M 1 + i f / f T [ 1 + R M ( Y A + i Y R + 1 / R K + i 2 π f C K ) ]where Y_A and Y_R are active and reactive components of measured value Y_X, f_T is the frequency of unitary amplification, and C_K is capacitance of the input cable.

The phase sensitive detectors PD_A, PD_R convert V_X into active and reactive components by comparing them with the signals of DG in phase and with a 90° phase shift (sine and cosine signals codes). This conversion is described by equations: (12) V A = c [ Re { V X } cos { ψ A } − Im { V X } sin { ψ A } ] ; ψ A = 2 π f ( t D + 2 Im { V X } / S ) ; V R = c [ Im { V X } cos { ψ R } + Re { V X } sin { ψ R } ] ; ψ R = 2 π f ( t D + 2 Re { V X } / S )where c is the transfer coefficient of the phase detector, ψ_A and ψ_R are parasitic phase shifts during the selection of active and reactive components (in Figure 1 they are the angles α and β), t_D is internal comparator delays, S is the speed of increasing the output voltage of OpAmp.

The digital codes of the measured components are obtained by ADC_A and ADC_R with their nominal transfer coefficient q_N, and absolute additive Δ_Ad and relative multiplicative δ_Mul errors: (13) N A ( 0 ) = q N ( 1 + δ Mul ) V A + Δ A d ; N A ( 0 ) = q N ( 1 + δ Mul ) V R + Δ A d

The magnitudes of components N_A and N_R are stored into the initial values registers IVR_A, IVR_R and into the iterating approximation registers IAR_A, IAR_R for iterating compensation cycles. These codes may be used as final results of the measurements. If higher accuracy is required of the results, the iterating process is applied. The total errors of first impedance measurement components are computed according to (13) as: (14) N A ( 0 ) = b α N ( 1 + δ A ( 0 ) ) Y A ; N R ( 0 ) = b α N ( 1 + δ R ( 0 ) ) Y Rwhere α_N is the nominal coefficient of direct channel transfer. In Table 1, the relative errors δ⁽⁰⁾_A, δ⁽⁰⁾_R of conductance Y_X = (100 + i100) S measurements are shown, R_K = 1 kΩ, f_T = 10 MHz, S = 50 V/mks, t_D = 10 ns, scale-resistor non-stability δ_M = 5%, Δ_A = 0.5 units of the least significant digit.

Iterating error correction begins with the generation of the first iterating compensation voltage V⁽¹⁾_Comp based on the codes N⁽⁰⁾_A and N⁽⁰⁾_R from IAR_A, IAR_R. After the modulation of these codes by DG-codes producing N_C(kT) and N_S(kT) the magnitude of the first compensation voltage is found as: (15) V Comp ( 1 ) ( k T ) = b V 0 [ N ∼ A ( 1 ) cos ( 2 π T f ) + N ∼ R ( 1 ) sin ( 2 π T f ) ]

The voltage from Equation (15) through the normally closed contacts of Sw is applied to the compensation resistor R_Comp. The output voltage V⁽¹⁾_X of measuring converter is appears as: (16) V X ( 1 ) = I X ( 1 ) G V C = V Comp R Comp G V C

The first iterating approximation result of V⁽¹⁾_X is found after analog-to-digital conversion as: (17) N A ( 1 ) = q N ( 1 + δ Mul ) V A ( 1 ) + Δ A d ; N R ( 1 ) = q N ( 1 + δ Mul ) V R ( 1 ) + Δ A d

The feedback iterating blocks, which consist of IVRs, IARs, subtractors S_A, S_R, and adders Σ_A, Σ_R generate the correction signals: Δ N A ( 1 ) = N A ( 0 ) − N A ( 1 ) ; Δ N R ( 1 ) = N R ( 0 ) − N R ( 1 ) which then are added to previous results of measurements from IAR registers producing: (18) N ∼ A ( 1 ) = N ∼ A ( 1 ) + Δ N ∼ A ( 1 ) ; N ∼ R ( 1 ) = N ∼ R ( 1 ) + Δ N ∼ R ( 1 )

The iterating-measuring process is continued according to the described algorithm (Equations 15–18). In general form, the iterating correction values of the (n + 1)-th approximation are found as: (19) N ∼ A ( n + 1 ) = N ∼ A ( n ) + ( N ∼ A ( 0 ) − N ∼ A ( n ) ) ; N ∼ R ( n + 1 ) = N ∼ R ( n ) + ( N ∼ R ( 0 ) − N ∼ R ( n ) )

According to the convergence condition of the iterating correction algorithm, the increment of approximation numbers leads to the fact that N⁽ⁱ⁾_A, N⁽ⁱ⁾_R approach to N⁽⁰⁾_A, N⁽⁰⁾_R. At the step I = m when the conditions N A ( m ) = N A ( 0 ) and N R ( m ) = N R ( 0 ) are satisfied with an accuracy about the least significant digit of the compared codes the correction is finished and the values: (20) N A ( m ) = β α N R Comp ( 1 + δ A ( m ) ) N ∼ A ( m ) ; N R ( m ) = β α N R Comp ( 1 + δ R ( m ) ) N ∼ R ( m )give the accurate results of the iterating impedance measurements. The δ^(m)_A and δ^(m)_R are the relative errors of the conversion at step m. The accuracy limit of measurement depends on the precision of the compensation resistor, time instability of direct and feedback channels, and non-linearity of the DAC.

Table 2 shows the relative errors during the iterating correction process for measurements on a test signal frequency equal to 500 kHz. Table 3 presents the results of experiments for the determination of necessary steps to achieve measurement errors of less than ±0.05% for a wide range of frequencies

The advantage of the proposed iterating method consists in the generation of correction actions in feedback channels in digital form. That results in a very small error equal to half of the least significant bit of the operating word of the DAC (±0.5/2ⁿ where n is the number of input bits of DAC). The iterating correction may be used for accurate measurements up to 500 kHz (Table 3) due to the increment of required iterations as a result of the tight interaction of converter channels at high frequencies.