Systems and methods for extending frequency response of resonant transducers
Certain implementations of the disclosed technology may include systems and methods for extending a frequency response of a transducer. A method is provided that can include receiving a measurement signal from a transducer, wherein the measurement signal includes distortion due to a resonant frequency of the transducer. The method includes applying a complementary filter to the measurement signal to produce a compensated signal, wherein applying the complementary filter reduces the distortion to less than about +/−1 dB for frequencies ranging from about zero to about 60% or greater of the resonant frequency. The method further includes outputting the compensated signal.
1. A method comprising:
receiving an electrical measurement signal from a sensor having a protective screen, wherein the measurement signal includes distortion due to a resonant frequency of the sensor;
applying, to the measurement signal, an analog complementary compensation filter circuit adapted to compensate for distortion in the measurement signal caused by the protective screen, the complementary compensation filter comprising a second order transfer function to the measurement signal to produce a compensated frequency response signal, wherein applying the complementary compensation filter circuit reduces the distortion to less than 1 dB for frequencies ranging from zero to 60% of the resonant frequency; and
outputting the compensated signal wherein the compensated signal reduces a resonance associated with the sensor.
2. The method of claim 1 , wherein applying the complementary compensation filter circuit to the measurement signal comprises applying a filter having a frequency response based on an inverse frequency response of the sensor, wherein the second order transfer function is implemented by a double integrator with feedback.
3. The method of claim 1 , further comprising;
determining the resonant frequency associated with a frequency response of the sensor.
4. The method of claim 3 , further comprising:
determining a frequency response complementary to the frequency response of the sensor.
5. The method of claim 3 , further comprising:
determining a quality factor associated with a frequency response of the sensor.
6. The method of claim 5 , further comprising:
determining parameters of the compensation filter circuit corresponding to a frequency response and quality factor complementary to the frequency response of the sensor.
7. The method of claim 1 , wherein the complementary compensation filter circuit includes a Helmholtz frequency response, T c (f), represented by the equation:
T
c
(
f
)
=
-
f
2
+
i
f
f
n
Q
+
f
n
2
-
f
2
+
i
f
f
n
+
f
n
2
wherein f is the frequency variable, f n is the resonant frequency of the transducer, and Q is a quality factor associated with the sensor frequency response.
8. The method of claim 1 , wherein the complementary compensation filter circuit includes a seismic mass frequency response, T c (f), represented by the equation:
T
c
(
f
)
=
-
f
2
+
i
f
f
n
Q
+
f
n
2
f
n
2
wherein f is the frequency variable, f n is the resonant frequency of the transducer, and Q is a quality factor associated with the sensor frequency response.
9. The method of claim 1 , wherein the sensor comprises a diaphragm having a resonant frequency modeled by
f
=
36
2
π
E
t
2
12
ρ
a
4
(
1
-
v
2
)
and wherein the diaphragm is flat and square having a length a and thickness t, and where E is Young's modulus, v is Poisson's ratio of the diaphragm material, and p is the applied pressure measured in Pascals.
10. The method of claim 1 , wherein the complementary compensation filter circuit further reduces the distortion to less than 2 dB for frequencies ranging from 60% to 90% of the resonant frequency.
11. The method of claim 1 , wherein the receiving the measurement signal from the sensor comprises receiving a pressure measurement signal.
12. A filter comprising:
a measurement input terminal adapted to receive an electrical measurement signal from a sensor having a protective screen, wherein the measurement signal includes distortion due to a resonant frequency associated with the sensor;
an analog complementary filter compensation circuit adapted to compensate for distortion in the measurement signal caused by the protective screen, the complementary compensation filter comprising a second order transfer function, the circuit adapted to receive and compensate a frequency response of the measurement signal and to reduce the distortion to less than 1 dB for frequencies ranging from zero to 60% of the resonant frequency; and
an output terminal for outputting the compensated measurement signal, wherein the compensated signal reduces a resonance associated with the sensor.
13. The filter of claim 12 , wherein the compensation circuit comprises a double integrator with feedback.
14. The filter of claim 12 , wherein the compensation circuit comprises an anti-resonant frequency corresponding to the resonant frequency of the sensor.
15. The filter of claim 12 , wherein the compensation circuit comprises a quality factor that is about equal to or complementary to a quality factor associated with the sensor.
16. The filter of claim 12 , wherein the compensation circuit comprises a Helmholtz frequency response, T c (f), represented by the equation:
T
c
(
f
)
=
-
f
2
+
i
f
f
n
Q
+
f
n
2
-
f
2
+
i
f
f
n
+
f
n
2
wherein f is the frequency variable, f n is the resonant frequency of the sensor, and Q is a quality factor associated with a frequency response of the sensor.
17. The filter of claim 12 , wherein the compensation circuit comprises a seismic mass frequency response, T c (f), represented by the equation:
T
c
(
f
)
=
-
f
2
+
i
f
f
n
Q
+
f
n
2
f
n
2
wherein f is the frequency variable, f n is the resonant frequency of the sensor, and Q is a quality factor associated with a frequency response of the sensor.
18. The filter of claim 12 , further comprising a sensor, wherein the sensor comprises a diaphragm having a resonant frequency modeled by
f
=
36
2
π
E
t
2
12
ρ
a
4
(
1
-
v
2
)
wherein the diaphragm is flat and square having a length a and thickness t, and where E is Young's modulus, v is Poisson's ratio of the diaphragm material, and p is the applied pressure measured in Pascals.
19. The filter of claim 12 , wherein the compensation circuit is further configured to reduce the distortion to less than 2 dB for frequencies ranging from 60% to 90% of the resonant frequency of the sensor.
20. The filter of claim 12 , wherein the compensation circuit comprises a low-pass filter configured to attenuate measurement signal frequencies or compensated signal frequencies above the resonant frequency.
21. The filter of claim 20 , wherein the low-pass filter comprises a 6 pole filter having a 48 dB/octave roll-off.
22. The filter of claim 12 , wherein the measurement signal comprises a pressure measurement signal.
23. The filter of claim 12 , further comprising a sensor, wherein the sensor comprises an accelerometer.
24. The filter of claim 12 , further comprising a sensor, wherein the sensor includes a diaphragm.
25. The filter of claim 24 wherein the sensor includes one or more piezoresistive elements in communication with the diaphragm.
26. The filter of claim 24 wherein the sensor includes a housing, comprising one or more of a cavity and a mounting substrate.
27. The filter of claim 12 , wherein the compensation circuit is an analog circuit.
28. The filter of claim 12 , wherein the compensation circuit comprises one or more of an amplifier, a summing block, an integrator, and a band-pass filter.