 # Self-calibration

Last Modification: January 25, 2014

Electronic circuitry used in measurement equipment are subject to a certain deviation who influences the stability and accuracy. Offset, drift en variations in gain due to temperature changes, aging and power supply changes cause measurement uncertainties of sometimes unacceptable proportions. By making use a self-calibration circuit and protocol the accuracy of a measurement is vastly increased.

## Offset and gain

The graph in figure 1 shows three amplifier characteristics. The thick black trace represents an ideal gain of 100 and no input offset. The red trace is of an amplifier who has a deviating input offset of -25 mV, the gain is also exact 100. The blue trace represents an amplifier with a input ofset voltage of +50 mV and a deviating gain of 70.

Precisely this kind of of deviations in offset and gain or attenuation can be corrected with a self-calibration. Condition is that the signal processing electronics functions linearly.

## Principle

The principle of an automatic calibration is based on the continuously or periodically calibrating the signal processing hardware and/or sensors against two stable references. Before any measurement is done the input of the instrument is connected first to a low reference voltage and after that to a high reference voltage. With these two measurements the offset and the gain can be determined. The actual measurement that follows after this is influenced by the same offset and gain. Now the deviations of the hardware are known, the reading of the actual measurement can be corrected.

Most often the ground or 0 V will be used as low reference voltage. The high reference voltage could have a value that is close to the highest occurring input voltage.

## Measure cyclus

The three steps whereof a self-calibration cycle consists are: Fig. 2: Self-calibrating step 1: The input is connected to the lower reference.

#### Step 1:

The instruments input is connected to the low reference voltage (gnd = 0 V). Now only the input offset voltage of the amplifier is amplified and recorded by the meter M. This value gets the label Vnul. Fig. 3: Self-calibrating step 2: The input is connected to the higher reference.

#### Step 2:

The input of the instrument now is connected to the high reference voltage Vref. The reference voltage with the input offset voltage add to it is amplified and registered by the meter M and gets the label Vmax. Fig. 4: Self-calibrating step 3: The input is connected to the unknown voltage.

#### Step 3:

Finally the instruments input is connected to the unknown voltage Vx. This measured voltage has also the input offset voltage included. After amplification the output voltage is recorded by the meter M, labeled Vin.

With these three measured values Vnul, Vmax and Vin acquired in this cycle, the accurate value for the unknown voltage VX is obtained.
The exact gain: [equ. 1]
The unknown voltage: [equ. 2]
The recording meter M will normally be a microcontroller with AD-converter and a small piece of code can do the calculations.

## Periodic calibration

Every self-calibration cycle consists of three measurements. Normally the electronics in a measurement instrument is reasonable stable within a certain time period. There is no need to calibrate the instrument so often. Only if the instrument is subject to temperature changes the parameters may change. This is for example the case during warming up after a power up condition. It is therefore often sufficient enough to do an calibration only once in a while or if the temperature changes a certain amount

## Sensors

To measure non electrical parameters, sensors are used who converts the quantity to the electrical domain. Sensors are not only sensitive to the contemplate parameter, but also to other environmental conditions. If the sensors themselves are also be included to the calibration, the unwanted influences will be eliminated.

### Example: Capacitive fluid level sensor

Te clarify this method a capacitive fluid level sensor is used as example. The principle of the capacitive level sensor is based on the capacity change of a capacitor by the area ratio of two types of dielectrics. The two plates of the capacitor extend over the entire height of the vessel. The bottom part has a liquid as a dielectric, and the top part has air or another gas as dielectric. So the capacitance is dependent on the hight of the fluid level.

### Capacitive sensor

If the sensor exists of only this capacitor the measurement uncertainty is high because many parameters influences the dielectric properties of the liquid and gas and the physical dimensions. The temperature and composition of the liquid and the gas plays a major role. Beside this, the voltage of the AC voltage needed to measure the capacitance may vary what makes the measurement uncertainty larger.

### Circuit and self-calibration

So, there are many causes that decreases the accuracy of the sensors capacity. By including the sensor into the self-calibration the accuracy can be improved in a great manner.

Figure 5 shows the self-calibration measuring arrangement with capacitive sensors. Beside the actual level sensor Cm there are two reference sensors placed in the tank. The reference sensor Cref-high is placed on the bottom of the tank and is all times placed fully into the fluid. The reference sensor Cref-low is placed in the top part of the tank and will never touch the fluid. The measurement is only valid if the fluid column will remain between hmin and hmax.

The common plate of the capacitors is connected to an alternating sinusoidal voltage. The amplifier Ais of a special type: the input impedance is 0 Ω, and is converting a current into a voltage. In this way the current through the capacitor is measured.

The first two measurements of the self-calibration measure the capacitance of the two reference capacitors Cref-high and Cref-low. These two measurements determine the scale factor and offset. The following third measurement in the cycle measures the capacity Cm. By these three measurements the hight of the fluid level relative to the bottom of the actual measure sensor Cm can be determined by he measured currents and the hight hs of the measure sensor: [equ. 3]

In figure 5 the two reference capacitors are drawn smaller than the measure capacity, but they must have the same capacitor area. The shape can be varied so the two reverence capacitors can be made wider and less high. The equation 3 is valid if all three capacitors have the same size. If the capacitors don't have the same size, than this must be corrected in the calculation.

This type of capacitive sensors are often very long. To avoid one large measuring sensor and improve the accuracy, the measure capacities are made of multiple smaller capacitors. It is easy to determine which sensor sees the liquid-gas boundary by comparing the measurements and search for the capacitance where change occurs. The capacitor above the identified capacitor is than surrounded by gas and the one below is immersed is the liquid. These two sensors can now act as the low and high reference sensors.

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