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, responsive to an applied pressure, an electrical measurement signal from a sensor having a protective screen, wherein the measurement signal includes a distortion due to the protective screen;
applying, to the measurement signal, a complementary compensation filter circuit adapted to compensate for the distortion in the measurement signal, wherein applying the complementary compensation filter circuit reduces the distortion to less than 1 dB for frequencies ranging from zero up to 60% of a resonant frequency and less than 2 dB for frequencies ranging from over 60% to 90% 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 second order transfer function to the measurement signal to produce a compensated frequency response signal 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; and
determining a quality factor complementary to the frequency response of the sensor.
4. The method of claim 3 , further comprising:
determining a compensated frequency response of the sensor based on one or more of the determined resonant frequency and the determined quality factor.
5. The method of claim 1 , wherein the complementary compensation filter circuit is an analog circuit.
6. 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 a frequency variable, f n is the resonant frequency of the sensor, and Q is a quality factor associated with the sensor frequency response.
7. 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 sensor, and Q is a quality factor associated with the sensor frequency response.
8. The method of claim 1 , wherein the sensor comprises a diaphragm having a resonant frequency modeled by
f
=
3
6
2
π
Et
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, ν is Poisson's ratio of the diaphragm material, and ρ is the applied pressure measured in Pascals.
9. The method of claim 1 , wherein the distortion is further due to a resonance of the sensor.
10. A filter comprising:
a measurement input terminal adapted to receive an electrical measurement signal from a sensor having a protective screen responsive to a pressure applied to a diaphragm of the sensor, wherein the measurement signal includes a distortion due to the protective screen associated with the sensor;
a complementary filter compensation circuit adapted to compensate for the distortion in the measurement signal, the complementary filter compensation 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 up to 60% of a resonant frequency and less than 2 dB for frequencies ranging from over 60% to 90% of the resonant frequency; and
an output terminal for outputting the compensated signal, wherein the compensated signal reduces a resonance associated with the sensor.
11. The filter of claim 10 , wherein the compensation circuit comprises second order transfer function implemented by a double integrator with feedback.
12. The filter of claim 10 , wherein a response of the compensation circuit is configured based on one or more of the resonant frequency and a quality factor associated with the sensor.
13. The filter of claim 10 , 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.
14. The filter of claim 10 , 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.
15. The filter of claim 10 , further comprising a sensor, wherein the sensor comprises a diaphragm having a resonant frequency modeled by
f
=
3
6
2
π
Et
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, ν is Poisson's ratio of the diaphragm, and ρ is an applied pressure measured in Pascals.
16. The filter of claim 10 , wherein the compensation circuit comprises a low-pass filter configured to attenuate measurement signal frequencies or compensated signal frequencies above the resonant frequency.
17. The filter of claim 16 , wherein the low-pass filter comprises a 6-pole filter having a 48 dB/octave roll-off.
18. The filter of claim 10 , further comprising a sensor, wherein the sensor comprises an accelerometer.
19. The filter of claim 18 , wherein the sensor includes one or more piezoresistive elements in communication with a diaphragm.
20. The filter of claim 10 , wherein the compensation circuit is an analog circuit comprising one or more of an amplifier, a summing block, an integrator, and a band-pass filter.