Special Section: Data Acquisition & I/0
Digital data boosts accuracy
Analyzing accelerometer data through digital integration means less noise, better signal
By Matt Rose and Brandon Jones
When testing turbine engines at the Arnold Engineering Development Center, an analog system is instrumental in integrating large amounts of accelerometer data into velocity and displacement data. Because of the current system’s age, though, it needs frequent calibration to ensure data quality. To reduce needed calibrations, minimize system noise, and improve data signal integrity, the ideal would be a new system that digitizes and integrates accelerometer data as close to the source of the data as possible. Such a system would produce a digital signal rather than analog, reducing noise problems and eliminating analog component tolerances.
A vibration measurement system uses accelerometers, charge amplifiers, and data acquisition hardware to process and display vibration data from turbine engine tests. We use analog methods to accomplish all charge-to-voltage conversion and integration by the charge amplifiers. Digital integration techniques would eliminate the need for the aging charge amplifiers, reducing the requirement for frequent calibrations while improving data quality and overall system reliability. But implementing a digital integrator would have its own design challenges. Take a look at the results we obtained using this new digital integrator compared to data acquired using the current analog integrator.
Design requirements, challenges
We needed to design a digital integration filter with the same magnitude response as the analog integration filter currently being used inside the charge amplifiers. With such a capability in place, it would only be necessary to store the raw acceleration data directly in digital form, and thus derive any data desired.
Presently, the vibration system consists of a charge-producing vibration sensor, a charge amplifier, and a recording and analysis system. The sensor itself goes on the test article inside the test cell and connects to the input of a charge amplifier outside the test cell approximately 200 feet away. The output of the charge amplifier then connects to the input of a recording and analysis system. The charge amplifier simply receives a charge input from the accelerometer and converts the charge signal to a voltage signal.
While the acceleration signal is important, what we really need is processed variation of the acceleration signal, mainly velocity and displacement. The charge amplifier performs an analog integration to get velocity and integrates again to produce a displacement signal. It produces velocity and displacement proportional voltages simultaneously along with acceleration output. The input travels 200 feet to the analog charge amplifier because the charge amplifier cannot operate in the environmental conditions of an altitude test cell. The basis of the new design converts the analog signal to a digital signal at the accelerometer within the test cell, thus eliminating noise potential.
An ideal system design would consist of one that acquires acceleration data, digitizes it, integrates it, and sends a digital data stream to a recording and analysis system. This type of system minimizes noise pickup and eliminates hardware as compared to the existing analog system. The hardware needed for this system is currently available, but we need a filter design that integrates the digitized acceleration signal and provides an accurate output comparable to the velocity and displacement outputs from a charge amplifier.
To design a digital integrator, we researched the analog integration circuit contained with the charge amplifier and verified it to be correct when compared to an ideal integrator. We calculated the s-domain transfer function of the integrating circuit as:
The plot of the magnitude of equation 1 and an ideal integration curve verifies the output of the charge amplifier is the actual mathematical solution to the integral and is suitable to serve as the baseline for comparison when designing a digital integrator.
We also explored several classical techniques to design a digital integrator. Typical transfer functions of IIR integrators include:
We commonly call these filters Newton-Cotes integrators. While these filters are capable for most applications throughout the frequency band of interest (10Hz-10 KHz), because of the presence of a pole on the unit circle, they do not give the same accuracy the analog integrator shows. While these filters show the same accuracy as a charge amplifier at low frequencies, they are unacceptable for data measurement purposes due to inaccuracy in the frequency band up to 40% of the Nyquist.
We can achieve improved accuracy when sampling at a higher bandwidth using these filters; however, the amount of data we would obtain is immense, and likely too excessive to manage and analyze for engineering information. To limit the amount of data to process while preserving the accuracy of the integration, we designed a different integrating filter and operated it at a 40-KHz sampling rate while also decimating it by four after each integration stage to reduce the amount of data to display and store.
In order to design a digital integrator with accuracy comparable to the charge amplifiers currently in use, we followed the following strategy. First, we designed an IIR filter that provides the ideal amplitude and phase responses. You can calculate the approximation several ways, such as minimax, weighted least squares, and least squares. You need to stabilize the resulting filters by reflecting the poles outside the unit circle inside the circle. This operation does not alter the magnitude response, but it does change the phase. Finally, you need an all-pass filter to make the phase linear. After combining the filters, transfer function of high order ensues with significant group delay. It is possible to reduce the order by approximating a transfer function of the form: zt H(z), where t is the positive sample delay of the filter.
For an integrator sufficiently accurate to half the Nyquist frequency, it is necessary to have a 5th order equation. The order of the integrating equation must be substantially higher to increase this accuracy for larger frequency bands as it requires a 40th order equation for accuracy to 90% of the Nyquist frequency. For the integrator used for testing purposes here, we used the weighted least-squares algorithm.
For the fifth-order integrator imposed over 100 frequencies with delay of 4.47796 samples combined with a phase-correcting all-pass filter, the transfer function is:
DC offset removal
It is necessary to highpass filter any input signal before the integration filter because when a signal is passed through the integrator, any DC offset will cause the system to become unstable.
A DC blocking filter must have a pole at or near the unit circle; therefore, it is expressed by the equation:
In equation 6, the numerator is the operator depiction of a first-order backward difference operator and is an estimation of a digital differentiator, while the denominator is a digital integrator and exhibits a pole at a that determines the cut-on frequency. To be stable the coefficient, a must be inside the unit circle but very close to 1. The following equation can determine this coefficient, dependent on the cut-on frequency, Fc:
The impulse response of this filter decays to zero according to (a-1)*ak-1: the closer a is to 1, the longer the filter takes to stabilize. You must balance this with the fact that the pass-band frequency magnitude will ideally be flat as low as 10 Hz. With the cut-on frequency set to 2 Hz, the magnitude of the filter at 10 Hz was -2 dB before finally settling at a magnitude of 0 dB at 100 Hz. Other features of the filter are:
- The normalized cut-on frequency is identified at the -6 dB point
- The pass band of the filter is completely flat leading up to the Nyquist frequency
- The slope of the filter is 20 dB/decade
- The phase response is negligible
- The coefficient a must vary between 0 and 1
Comparing the magnitude responses of the ideal integrator to the new 5th-order integrator, the overall percent error is within 1% in the frequency band of interest.
When comparing sampled data, we used two different data sets. The first was a discrete frequency signal input from a calibrated source, and the second was non-linear data obtained from attaching an accelerometer to a speaker and sweeping the frequency from 1Hz to 20 KHz. We used the acceleration output of a TRIG-TEK 203TN charge amplifier as the input for the cascaded highpass and digital integration filters and compared them to the velocity output of the charge amplifier.
We input the discrete frequencies of 20 Hz, 500 Hz, 1000 Hz, 2000 Hz, and 5000 Hz and compared them. We calculated the overall percent error of the cascaded digital integrator and highpass filters as 8.7% when compared to the analog integrator with this data set. The digital integrating filter compares satisfactorily to the analog integrator in both data sets.
Positive results from testing digital integration show the total magnitude response is well within 10% of an ideal integrator, yet the phase response is lacking. We phase corrected all data by computing the phase differences and shifting the digital integrator data by this amount; therefore, we did not calculate phase errors in these percentages. Yet you should not be indifferent to phase error but calculate it by comparing the output phase to the input phase shifted 90°.
A true ideal integrator has a 90° phase shift while the digital integration filter does not. The TRIG-TEK charge amplifier does not have a 90° phase shift in its output data even though the integrator circuit shown in the first figure does. Additional circuitry inside the charge amplifier alters the phase of the output signal. We need to conduct further study to explore what the error is in the phase shift from a charge amplifier compared to the ideal and how to correct for the incorrect phase shift in the digital integrators.
ABOUT THE AUTHORS
C. Matt Rose is an engineering scientist at Aerospace Testing Alliance at Arnold Air Force Base in Tenn. E-mail him at firstname.lastname@example.org. J. Brandon Jones is also an engineering scientist at Aerospace Testing Alliance. E-mail him at email@example.com.