# Calibration Improves Accuracy and Stability of MEMS Gyroscopes

Gyroscopes are often used for inertial navigation or other applications where high accuracy or stability is required. To maximize performance, the gyros used in these applications are generally calibrated and externally temperature compensated. This article outlines a method for performing calibration and temperature compensation.

If a gyro is used in a temperature stable environment (or if it is kept at a constant temperature by means of a temperature control system), a single temperature point calibration is sufficient. To do this, simply bring the gyro up to the intended operating temperature (allowing sufficient soak time) and measure the null output voltage (V

To determine the scale factor (S), some method of applying angular rate must be used (in general a rate table). For instance, ADI’s ADXRS6xx series of gyros has very little nonlinearity, measure the output at one rotational rate (preferably one near full scale). The scale factor can then be calculated by

The calibration information required is similar to the single temperature point method described previously except that V

Once temperature and null voltages are measured and the scale factor is calculated at the three temperature points, it is convenient to reduce the calibration data down to calibration coefficients that can be applied to a general equation for temperature compensation. There are several ways to do this. The method outlined in this was article is arbitrarily chosen.

Define any gyro output parameter by the following equations:

where: V

The a and b coefficients are described by Equation 5 and Equation 6:

Once the “a” and “b” coefficients are calculated, all parameters may be expressed using the general equations previously presented.

A given gyro is measured at ?40°C, +25°C, and +85°C. The test results are shown in Table 1.

For this gyro the null output coefficients are show in Equation 7 and Equation 8:

In this example, the null output of the gyro at any temperature can be described by Equation 9:

Similarly, the scale factor coefficients are Equation 10 and Equation 11:

This gyro’s scale factor at any temperature can be described by the following equation:

Once the calibration coefficients are calculated, it is easy to convert the gyro output to temperature compensated angular rate data in degrees/second using the following four-step process.

1. Read the temperature output in volts. Determine the calculated null at the actual temperature using Equation 9.

2. Read the rate output in volts and calculate the null temperature corrected rate output using Equation 14.

3. Calculate the scale factor at the device temperature using Equation 12.

4. Using the previous information, calculate the actual angular rate using Equation 15

For the example gyro presented previously at 85°C, if the rate output is measured as 3.00 V, the actual angular rate is calculated as follows.

The calculated null at 85°C is:

The corrected rate voltage at 85°C is:

The calculated scale factor at 85°C is:

Thus, the angular rate in degrees/second is:

Temperature compensation of the

**CALIBRATION**If a gyro is used in a temperature stable environment (or if it is kept at a constant temperature by means of a temperature control system), a single temperature point calibration is sufficient. To do this, simply bring the gyro up to the intended operating temperature (allowing sufficient soak time) and measure the null output voltage (V

_{0}).To determine the scale factor (S), some method of applying angular rate must be used (in general a rate table). For instance, ADI’s ADXRS6xx series of gyros has very little nonlinearity, measure the output at one rotational rate (preferably one near full scale). The scale factor can then be calculated by

**TEMPERATURE COMPENSATION**

If the gyro will be operated over a range of temperatures, the best performance will be realized using temperature compensation. Gyros that incorporate an on-chip temperature sensor will facilitate this.The calibration information required is similar to the single temperature point method described previously except that V

_{0}and S must be measured at different temperature points. In addition, the temperature sensor output (VT) must also be recorded. For instance, because the temperature performance of gyros such as the ADXRS6xx is somewhat nonlinear, at least three temperature points should be used for compensation. Using more temperature points results in improved accuracy and null stability; however, the calculations become more complex.**THREE-POINT CALIBRATION**Once temperature and null voltages are measured and the scale factor is calculated at the three temperature points, it is convenient to reduce the calibration data down to calibration coefficients that can be applied to a general equation for temperature compensation. There are several ways to do this. The method outlined in this was article is arbitrarily chosen.

Define any gyro output parameter by the following equations:

where: V

_{P0}is a parameter value at the temperature sensor value ambient temperature (V_{T0}). V_{P1}is the same parameter value at the temperature sensor value (V_{T1}). V_{P2}is the same parameter value at the temperature sensor value (V_{T2}).The a and b coefficients are described by Equation 5 and Equation 6:

Once the “a” and “b” coefficients are calculated, all parameters may be expressed using the general equations previously presented.

**EXAMPLE**A given gyro is measured at ?40°C, +25°C, and +85°C. The test results are shown in Table 1.

For this gyro the null output coefficients are show in Equation 7 and Equation 8:

In this example, the null output of the gyro at any temperature can be described by Equation 9:

Similarly, the scale factor coefficients are Equation 10 and Equation 11:

This gyro’s scale factor at any temperature can be described by the following equation:

**CALCULATING TEMPERATURE COMPENSATED ANGULAR RATE**Once the calibration coefficients are calculated, it is easy to convert the gyro output to temperature compensated angular rate data in degrees/second using the following four-step process.

1. Read the temperature output in volts. Determine the calculated null at the actual temperature using Equation 9.

2. Read the rate output in volts and calculate the null temperature corrected rate output using Equation 14.

3. Calculate the scale factor at the device temperature using Equation 12.

4. Using the previous information, calculate the actual angular rate using Equation 15

For the example gyro presented previously at 85°C, if the rate output is measured as 3.00 V, the actual angular rate is calculated as follows.

The calculated null at 85°C is:

The corrected rate voltage at 85°C is:

The calculated scale factor at 85°C is:

Thus, the angular rate in degrees/second is:

**CONCLUSION**Temperature compensation of the

*i*MEMS series of gyros is straightforward. Using a simple curve fit to calculate calibration coefficients, only a minimum of calibration information must be stored in NVRAM and calculation of the actual rate requires only simple equations. A 3-point temperature calibration corrects for null drift sufficiently to achieve good null stability. Better null stability can be achieved using greater than three calibration temperatures and more complex calibration equations.**Harvey Weinberg**, BSEE, is the leader of the Applications Engineering group for inertial products at Analog Devices' MEMS and Sensor Technology Group, where he has worked for 12 years. Prior to ADI, he worked for 10 years as a circuit and systems designer specializing in process control instrumentation. He can be reached via email at harvey.weinberg@analog.com [1].