SBOA601 January 2025 LOG200

6 Error Sources and Uncalibrated Error Analysis

This section shows an example of the LOG200 circuit estimated DC error analysis. These calculations use the typical LOG200 data sheet specifications to calculate the device's uncalibrated DC accuracy. This error estimate only includes the errors of the LOG200 device stand-alone.

This example assumes a photosensor application requiring a current range of 20nA to 2.5mA, where the LOG200 connects to a 12-Bit pseudo-differential input ADC powered with a 3V supply. Figure 6-1 shows the LOG200 circuit connections to the ADC.

Figure 6-1 LOG200 Circuit connected to Pseudo-Differential Input ADC

Table 6-3 show the typical LOG200 specifications for error calculation:

Table 6-1 LOG200 Specifications for Error Calculation

Parameter	Description/Comment	Error Typical	UNIT
K_Error	Scaling factor error 100pA to 1mA	±0.15	%
LCE_Error	Logarithmic conformity error 10nA to 1mA	±0.050	%
I_{REF_Error}	Reference current error ±0.3% of I_REF = 1µA	±0.003	µA
V_OSO	Output offset error of logarithmic amplifier	±1.300	mV

Equation 9 shows the LOG200 designed for transfer function:

Equation 9.

V_{L O G_I d e a l} = K \times {l o g}_{10} (\frac{I_{P D}}{I_{R E F}}) + V_{R E F}

Equation 10 uses the designed for LOG200 transfer function to calculate the logarithmic amplifier output while adding the typical data sheet error parameters:

Equation 10.

V_{L O G O U T_A c t u a l} = (K + ∆ K) \times {l o g}_{10} (\frac{I_{P D}}{(I_{R E F} - {∆ I}_{R E F})}) + (V_{R E F} \pm {∆ V}_{R E F}) \pm {L C E}_{E r} \pm V_{O S O}

On a typical photodiode measurement application, where the photosensor connects to I1, V_OS does not contribute a significant error on the log amplifier transfer function. When the photodiode is biased with a proper reverse voltage, the sensor produces an input current virtually independent of the small input offset voltage V_OS. Therefore, in this photodiode measurement, VLOG_Actual is only dependent on the output offset error (V_OSO) of the logarithmic amplifier.

In this example, the REF165 1.65V reference connects to the negative input of a pseudo-differential input ADC, where the ADC converts only the difference between the ADC positive and negative inputs (AINP-AINN). Therefore, the accuracy of the voltage reference does not contribute to an error in the overall ADC conversionEquation 11 provides an estimate the total uncalibrated output error for a given photodiode current in this example:

Equation 11.

V_{L O G O U T_A c t u a l} = (K + ∆ K) \times {l o g}_{10} (\frac{I_{P D}}{(I_{R E F} - {∆ I}_{R E F})}) + V_{R E F} \pm {L C E}_{E r} \pm V_{O S O}

Table 6-2 summarizes the primary error sources in the LOG200 application. The calculations assume a photosensor application requiring a current range from I_min = 20nA to I_max = 2mA.

Table 6-2 Estimated V_LOG Uncalibrated Error Calculations From Typical Spec

Parameter	Equation	Result
Uncalibrated DC accuracy at I_min = 20nA, at 25 °C
Nominal log amp output at I_min V_{LOG_Imin}	$V_{L O G_I m i n} = K \times {l o g}_{10} (\frac{I_{m i n}}{I_{R E F}}) + V_{R E F}$	1.2253V
Log conformity error at I_min LCE_{Error_Imin}	${Δ L C E}_{E r_I m i n} = {L C E}_{E r r o r} \times (V_{L O G_I m i n} - V_{R E F})$	-0.212mV
Actual log amp output at I_min V_{LOG_Actual_Imin}	$V_{L O G_A c t u a l_I m i n} = (K + ∆ K) \times {l o g}_{10} (\frac{I_{P D}}{(I_{R E F} - {∆ I}_{R E F})}) + V_{R E F} - {Δ L C E}_{E r_I m i n} - V_{O S O}$	1.223V
Log amp error at at I_min V_{LOG_Error_Imin}	$V_{L O G_E r_I m i n} = V_{L O G_I m i n} - V_{L O G_A c t u a l_I m i n}$	-2.598mV
Uncalibrated DC accuracy at I_max = 2.5mA, at 25 °C
Nominal log amp output at I_max V_{LOG_Imax}	$V_{L O G_I m a x} = K \times {l o g}_{10} (\frac{I_{m a x}}{I_{R E F}}) + V_{R E F}$	2.4995V
Log conformity error at I_max LCE_{Error_Imax}	${Δ L C E}_{E r_I m a x} = {L C E}_{E r r o r} \times (V_{L O G_I m a x} - V_{R E F})$	0.425mV
Actual log amp output at I_max V_{LOG_Actual_Imax}	$V_{L O G_A c t u a l_I m a x} = (K + ∆ K) \times {l o g}_{10} (\frac{I_{m a x}}{(I_{R E F} - {∆ I}_{R E F})}) + V_{R E F} + {Δ L C E}_{E r_I m a x} + V_{O S O}$	2.504V
Log amp error at at I_max V_{LOG_Error_Imax}	$V_{L O G_E r_I m a x} = V_{L O G_I m a x} - V_{L O G_A c t u a l_I m a x}$	4.526mV
Total uncalibrated error as % of full-scale
% Full-scale error at I_min V_{LOG_Error_Imin_FS}	$V_{L O G_E r_I m i n_F S} = \frac{V_{L O G_E r_I m i n}}{V_{L O G_I m a x} - V_{L O G_I m i n}} \times 100 %$	-0.204%
% Full-scale error at I_max V_{LOG_Error_Imax_FS}	$V_{L O G_E r_I m a x_F S} = \frac{V_{L O G_E r_I m a x}}{V_{L O G_I m a x} - V_{L O G_I m i n}} \times 100 %$	0.355%

The typical error specifications are added into the LOG200 transfer function to calculate a total compounded error. Since the absolute values are added together to calculate the total error, calculations using these typical values can yield a conservative error analysis, as the summation of uncorrelated errors tends to result in a larger compounded total predicted error than the actual total error observed on a real system. In many applications, the circuit designer can perform a two point linear calibration to reduce the errors due to offset and scale factor, resulting in a smaller DC error.