Question

2. The electromechanical system shown in the figure below represents a simplified model of a capacitor microphone. The system consists of a parallel plate capacitor connected into an electric circuit. Capacitor plate a is rigidly fixed to the microphone frame. Sound waves passing through the mouthpiece exert a force fs(t) on plate b, which has mass M and is connected to the frame by a set of springs and dampers The capacitance C is a function of the distance x between the plates, as follows: C(x) = where is the dielectric constant of the material between the plates, and A is the surface area of the plates The charge q and the voltage e across the plates are related by: 4 = C(x) The electric field produces a force fe on the movable plate, opposite to its motion: for min M a) Write the differential equations that describe the operation of this system Hint: Derive an equation for the mechanical system (the motion of plate b), and another equation for the electrical system (the electric circuit). The inputs are the force fot) and voltage v. b) At equilibrium, f. = fso. V = V., X = Xo, and q = 90- Find the equilibrium conditions c) Linearize the equations using a Taylor series expansion

Answer 1

Answer page ① Given Data: Here Capacitor plate 'al is regedly fixed to The Microphone frame. fs (4) - force on a Plate. Where The Capacitance c is a function of The distance at "it" between the plates (*) = SA x Where A = Surface Area of plate E = Die lec fost constant Relation between charge cq , and voltage a 19 = cca) e where The plate Electric field : opposite of a to force its the fe on motion. 2 EA

page ② fe _qe 2EA 2EA Already we know q = clase fe - (ccx)) we know ((x) = EA fe: (SADY 2EA Ite - EA eV 2xr Required Data (a) Derive Equations for the Mechanical System and Electrecal System (b) find The Equilibrium Condition fs : fsc : V = V. , *=*0,9590 (c) Line vize the Equation using a Taylor series Expansion

(a) cin Stephen In This System The free Body page 3 Draw we first step diagram -KRL - Båt factie Body 22 Step 2: (1) Equations Taking from The free Body diagram Mãe & Bå + Kx + fe = felt) we know de = ERR Måt Bă + ka + ERC = fsct) / This is The Mechanical System. Step 3: Electrical system. Apply Koi to The System vo Rilt) +2 diftite. Le filt)dt - Riff) + d:ft) + 4 J1L43d + 1 ceea = Rilt) + L dilt + en Silidt dt

page (4) v= Rect) + Lil J aude we know olt) = V = Related (da) + Salat r = RG + Lala a Tv = Rů + Lg + 22 a 7 This is The Electrical System -- - (b) At Equilibrium condition - fs = fso ,u=vo, x= xo . and q=% Here it is impossible to get a Linear moder because of the ar sgncä). and qcu)! Here we observe The Linear Model Cannot be obtained. (c) using Taylor Series - The output of a Capacitor Microphone is voice signal. . Where it is corrent. output of System is current :(t) = ģ