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The structure shown below is realized on a silicon chip. A mass M is suspended above the surface of the chip by means of 4 springs with spring constant K1 = K2 = K3 = K4 = K.
As a result, the mass can move in the x-direction. At one side, a comb structure is attached to the mass. Another comb structure is rigidly fixed to the silicon substrate. In this way the mass can be moved by applying a voltage between the comb structures.
All parts are electrically conducting. The height of the comb fingers (perpendicular to the paper) is h, the number of comb fingers on the moveable comb is N, the overlap between the comb fingers is l0 and the distance between a fixed and a moveable finger is d.
(a) Show that the capacitance between the combs as a function of the position x is given by:
Give an expression for the constant C0.
(b) Derive an expression for the energy stored in the transducer as a function of the position x and the charge q on the comb structures.
The transducer is controlled by a voltage.
(c) Give expressions for the charge q and the force Fext as a function of u and x.
We will now look at the dynamic behavior of the system.
(d) Derive the equation of motion for the system. Give an expression for the resonance frequency. Is the resonance frequency dependent on the voltage u?
(e) Is the resonance frequency at current control dependent on the charge q?
The figure below shows the structure of a capacitive acceleration sensor (or: accelerometer). When the entire system is subject to an acceleration a in vertical direction, a force will be exerted on the spring equal to the mass times the acceleration: Fext = m.a. This results in a change in the length of the spring (or a displacement of the mass with respect to the frame) which can be measured.
For low frequencies, i.e. slow changes in the acceleration, we can easily derive that the change in length is given by:
with Keff the effective spring constant.
When the voltage between the capacitor plates is zero (u=0) the effective spring constant Keff will be equal to the spring constant K.
In this assignment we will study the influence of an applied bias voltage (necessary for measurement of the capacitance value) on the performance of the sensor.
The capacitance as a function of the position of the moving plate is given by C(x) = A/x. The spring is relaxed at x = x0.
(a) Derive the energy function U(q, x) of the transducer.
In order to measure the capacitance a bias voltage is applied. This means that the transducer is voltage controlled.
(b) Derive expressions for Fext and q as a function of u and x.
(c) Derive the characteristic equations of the transducer.
(d1) Derive the effective spring constant Ku of the transducer in the case of voltage control. How does the bias voltage influence the sensitivity of the sensor? That is, how does the voltage influence the relation between the displacement x and the acceleration a?
(d2) Can Ku become equal to 0? Which conclusion(s) can you derive from this?
The useable bandwidth of the sensor is limited by the resonance frequency. Therefore, we need to now how the resonance frequency depends on the bias voltage.
(e) Derive the equation of motion for the mass.
(f) Show that the resonance frequency is given by .
Hint: assume a small harmonic vibration around an equilibrium position xeq and insert this into the equation of motion. Use the following approximation: