Multi-tunable microelectromechanical system (MEMS) resonators

1. A method for tuning a vibratory device including a cantilevered resonator comprising the steps of:

increasing a voltage V 0 supplied to said vibratory device to thereby increase the bandwidth of said vibratory device; and

keeping the resonant frequency of said vibratory device at substantially the natural frequency of said cantilevered resonator, wherein said vibratory device comprises:

a capacitor including a movable plate and a fixed plate spaced from each other, said movable plate being part of said cantilevered resonator;

a voltage source connected to said capacitor for providing said voltage V 0 across said capacitor to produce an attractive force between said movable plate and said fixed plate;

a circuit connecting said voltage source to said capacitor; and

a load resistor in said circuit having a resistance R L satisfying the following equation:

μ

=

ω

0

⁢

c

0

⁢

R

L

(

1

-

λ

)

where

μ is at least 10;

ω 0 is the beam constant for said cantilevered resonator;

c 0 is the capacitance for said capacitor; and

λ is the voltage dependent coupling parameter for said voltage V 0 .

2. The method of claim 1 , wherein the tuning of said vibratory device is actively tuned.

3. The method of claim 1 , wherein said vibratory device is part of an array.

4. The method of claim 1 , wherein said vibratory device is a MEMS resonator.

5. The method of claim 1 , wherein said bandwidth and said resonant frequency are independently tuned.

6. The method of claim 5 , wherein said resonant frequency is independently tuned up to match a chosen frequency.

7. The method of claim 5 , wherein said resonant frequency is independently tuned down to match a chosen frequency.

8. The method of claim 1 , further comprising the step of decreasing said voltage V 0 to thereby increase the bandwidth of said vibratory device.

9. A method for tuning a vibratory device including a cantilevered resonator comprising the steps of:

increasing a voltage V 0 supplied to said vibratory device to thereby increase the bandwidth of said vibratory device; and

simultaneously decreasing the resonant frequency of said vibratory device below the natural frequency of said cantilevered resonator, wherein said vibratory device comprises:

a capacitor including a movable plate and a fixed plate spaced from each other, said movable plate being part of said cantilevered resonator;

a voltage source connected to said capacitor for providing said voltage V 0 across said capacitor to produce an attractive force between said movable plate and said fixed plate;

a circuit connecting said voltage source to said capacitor; and

a load resistor in said circuit having a resistance R L satisfying the following equation:

μ

=

ω

0

⁢

c

0

⁢

R

L

(

1

-

λ

)

where

μ is no more than 0.1;

ω 0 is the beam constant for said cantilevered resonator;

c 0 is the capacitance for said capacitor; and

λ is the voltage dependent coupling parameter for said voltage V 0 .

10. The method of claim 9 , wherein the tuning of said vibratory device is actively tuned.

11. The method of claim 9 , wherein said vibratory device is part of an array.

12. The method of claim 9 , wherein said vibratory device is a MEMS resonator.

13. The method of claim 9 , wherein said bandwidth and said resonant frequency are independently tuned.

14. The method of claim 13 , wherein said resonant frequency is independently tuned up to match a chosen frequency.

15. The method of claim 13 , wherein said resonant frequency is independently tuned down to match a chosen frequency.

16. The method of claim 9 , further comprising the step of decreasing said voltage V 0 to thereby increase the bandwidth of said vibratory device.

17. A method for tuning a vibratory device including a cantilevered resonator comprising the steps of:

increasing a voltage V 0 supplied to said vibratory device to thereby increase the bandwidth of the vibratory device; and

varying the resonant frequency of said vibratory device by adjusting the self-inductance L of an inducer of said vibratory device, wherein said vibratory device comprises:

a capacitor including a movable plate and a fixed plate spaced from each other, said movable plate being part of said cantilevered resonator;

a voltage source connected to said capacitor for providing said voltage V 0 across said capacitor to produce an attractive force between said movable plate and said fixed plate;

a circuit connecting said voltage source to said capacitor, said circuit including said inducer; and

a load resistor in said circuit having a resistance R L satisfying the following equation:

μ

=

ω

0

⁢

c

0

⁢

R

L

(

1

-

λ

)

where

μ is no more than 0.1;

ω 0 is the beam constant for said cantilevered resonator;

c 0 is the capacitance for said capacitor; and

λ is the voltage dependent coupling parameter for said voltage V 0 .

18. The method of claim 17 , wherein said self-inductance L of said inducer satisfies the equation:

δ

=

ω

0

⁢

c

0

⁡

(

L

⁢

⁢

ω

0

)

(

1

-

λ

)

-

1

wherein |δ| is less than μ.

19. The method of claim 17 , wherein the tuning of said vibratory device is actively tuned.

20. The method of claim 17 , wherein said vibratory device is part of an array.

21. The method of claim 17 , wherein said vibratory device is a MEMS resonator.

22. The method of claim 17 , wherein said bandwidth and said resonant frequency are independently tuned.

23. The method of claim 22 , wherein said resonant frequency is independently tuned up to match a chosen frequency.

24. The method of claim 22 , wherein said resonant frequency is independently tuned down to match a chosen frequency.

25. The method of claim 17 , further comprising the step of decreasing said voltage V 0 to thereby increase the bandwidth of said vibratory device.