CEE757 - Homework More senosrs, and intro ADC: Due February 26, 2013 March 20, 2013 Problem - Piezoelectric accelerometer You are interested in measuring vibrations that are induced by an earthquake You decide to accomplish this by building your own piezoelectric accelerometer This is done by sandwiching a ceramic piezo disk between a seismic mass m The sensors base is fastened to a surface you are measuring, so that when an acceleration is applied the seismic mass will move up and down, causing the piezo disk to generate a voltage (think of the capacitor model we derived in class) For this simplified analysis we will assume that the piezo disk is only generating charge if a force is applied in the vertical direction (due to acceleration), proportional to the piezo charge coefficient dyy The circular piezoelectric disk is a 0.5 cm in diameter and mm thick (denote thickness by the variable l) The seismic mass is a steel cylinder cm diameter, cm long Figure shows that the proposed setup can be modeled as a spring mass system, where the piezo disk acts as the spring You also know the following physical properties Parameter E dyy "r "0 Value 71GPa 0.559 Coulombs/meter 450 8.86x10 12 Young’s modulus of the piezo material Piezo charge coefficient dielectric coefficient of the piezo material dielectric coefficient of vaccuum a) What is the mechanical spring constant k of the piezo disk if we treat the sensor as the spring mass model b) What is the capacitance of the piezo disk when undeformed c) Derive a symbolic expression for the acceleration of the mass m that is only a function of the known material properties and dimensions, as well as the output voltage V of the piezoelectric accelerometer To help you simplify your expression you may assume the terms that contain references to y (square of small changes of the displacement) cancel out Figure 1: Piezo electric accelerometer Solutions a)The spring constant of stiffness is given by the applies force divided by the observed displacement k= F l we know that = F l =E =" A l then l= Fl EA and k= F EA EA = = 1.39x109 N/m Fl l b) C = "r "0 A/l = 7.82 ⇥ 10 11 = 78.2pF c) From our spring system we know (no damping) yă + k y=0 m then k y m From our piezoelectric relationship, we know that yă = a = Q = dyy y Then the voltage of the capacitor will be V = Q dyy y l+y dyy yl + dyy y = = dyy y = C C "0 "r A "0 "r A since y is negligible, we get y= V " "r A dyy l plugging into the above a= k V " "r A m dyy l Problem - Loop detectors Loop detectors are used to detect cars on the road When no car is present, the inductance of the loop is a steady, known value When a car passes over the loop, the inductance changes The circuit in figure is a common tool for measuring the change of an unknown inductor Based on known principles of RC and LC impedances,we can obtain a value for L3 (the loop detector of unknown inductance) To obtain this value, a variable capacitor C2 is digitally adjusted until the bridge circuit is balanced (V0 = 0) For the this balanced condition, derive an expression for L3 that only contains the references to known values of R1 , R4 , and C2 Solution The impedance of an RL (resistor inductor) circuit is given by Z = R + j!L you can derive this, or obtain it on the internet Similarly the complex j impedance of an RC circuit is given by Z = R !C Then, at balance we know Z1 Z4 = Z3 Z2 R1 · R4 = (j!L) Then the inductance is given by ✓ j !C L = C · R1 R4 ◆ = L C Figure 2: Inductance measuring circuit Figure 3: Two measuring circuits Problem - Mathematical operations on analog signals For the two circuits in figure derive an expression for the output voltage V0 given the input voltage signals Va and Vb Which mathematical operation does each circuit represent? Assume each op-amp has a nominal gain A, and every resistor has a nominal resistance R Assuming that A is very large may help you reduce to simpler relations (as we’ve done in class) Solution a) The current flowing through the junction of the three resistors must equal zero, then V0 = R Vb Va R+R then V = Va + V b For the inverting amp V0 = Vin Rf = R You can also view it as a paralel voltage going into a known op-amp Slightly different solution, but still works b) This is a summation of the inverting, and non-inverting amplifiers we’ve seen in class, see solution here: http://masteringelectronicsdesign.com/the-differentialamplifier-transfer-function/ We’re adding two sepratat inputs, one of which is a coltage divider ✓ ◆ R R R V0 = Vb0 Va0 = Vb 1+ Va R+R R R Problem - Digital to analog conversion (make new digram, where D is grounded when not used)? An N-bit digital to analog converter (DAC) can be used to provide 2N analog voltage outputs for a number of applications In the case of our SAR ADC, the digital to analog converter was used to source a voltage comparator, helping us isolate the input 0voltage coming into the ADC Digital to analog converter can also be used to drive a number of external peripherals and actuators (more on this later) Needless to say, they are super handy As this problem will show, they are also fairly straight forward to construct using some resistors and an summation op-amp Figure shows a 4-bit DAC A digital controller is used to supply a voltage Vref to four switches (D0 D3 ) When a switch is flipped into the on position, it outputs voltage Vref into a a network of resistors For example, setting D0 to on (logic 1) causes a voltage Vref to be output through the bottom resistors Setting D0 to off (logic 0), causes no voltage to be output into the bottom resistors Different output voltages V0 can be created through different settings of the switches a) Derive an expression for V0 , the output of the DAC, which only depends on Vref , R, and Rf , and D0 through D3 The values for D0 D3 should be binary (0 or 1) b) Use your solution above to derive a general expression for V0 of an N-bit DAC Solution You can this setep by step (http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/dac.html) or use a trick Successive applications of Thevinen’s theorem yield the following: Rf D D1 D2 D3 V0 = Vref + + + R 16 Figure 4: Digital to analog conversion for an N-bit DAC, we get Rf V0 = Vref R D0 D1 DN + N + + 2N 2 Problem - ADC selection a) An analog voltage channel is used to transmit a signal to an analog to digital converter The input voltage can vary over the range of ±2V , and signal noise level corresponds to 1.25 mV How many different code symbols would be required to record all possible signal values without being affected by noise? What ADC accuracy (in bits) does this correspond to? Solution: 4V Range = 3200 V alues 0.00125V We then need a machine that can process log2 (3200) = 11.6bits of information (12 bits really) b) What is the minimum number of binary bits required for an ADC to digitize the output of a pressure sensor whose range is 800 to 1100 kPa, with an accuracy of 0.05 kPa? Solution: Pressure range = 1100kPa - 800kPA =300kPa Least significant bit determines accuracy which should be 0.05kPa Then 300kP a = 6000 values 0.05kP a which means we need log2 (6000)12.55 bits, or 13 bits Problem - SAR ADC approximation algorithm In class we saw how a successive approximation (SAR) ADC uses a digital-toanalog converter to compare an input voltage Vin to a value VDAC produced by the N bit DAC As we know, an N bit DAC produces 2N possible output voltages This number can be quite substantial for large N A simple algorithm aiming to find Vin could start by setting the DAC to volts, and then incrementing it step by step until our comparator matches it to Vin As you can imagine this may take some time, especially when we have high input voltages In such cases, the ADC would have to make almost the full 2N comparisons before isolating the value of the input Such a method is called digital ramp approximation For this problem, come up with a better approximation approach that will make no more than N comparisons before finding Vin Solution Although there are many variations in the implementation of a SAR ADC, the basic architecture is quite simple We start with a guess in the middle of the Vref range, and then keep adjusting our approximation by half, depending on output of our comparator The analog input voltage (VIN) is held on a track/hold To implement the binary search algorithm, the N-bit register is first set to midscale (that is, 100 00, where the MSB is set to ’1’) This forces the DAC output (VDAC) to be VREF /2, where VREF is the reference voltage provided to the ADC A comparison is then performed to determine if VIN is less than or greater than VDAC If VIN is greater than VDAC, the comparator output is a logic high or ’1’ and the MSB of the N-bit register remains at ’1’ Conversely, if VIN is less than VDAC, the comparator output is a logic low and the MSB of the register is cleared to logic ’0’ The SAR control logic then moves to the next bit down, forces that bit high, and does another comparison The sequence continues all the way down to the LSB Once this is done, the conversion is complete, and the N-bit digital word is available in the register See example below Problem - Fiber optical strain measurements Fiberoptic strain gage sensors work by transmitting light across a Bragg grating This Bragg grating forms the sensing component As we mentioned in class, the light that is reflected by the Bragg grating is an indictor of strain (see figure 5) The frequency of the reflected light is given by B = 2nef f ⇤ where nef f is the refraction index (ratio of the speed of light in vacuum divided by the speed of light in a material), and ⇤ is the bragg spacing (distance between etch marks, see figure) inside the sensor It can be shown that BB = Gf LL , where Gf is the gage factor and LL is the strain being measured by the fiberoptic sensor In class it was asked what a realistic value for Gf would be This problem will help you answer that question nef f and ⇤ are independent of each other, but both are sensitive to strain fluctuations The change in the n refraction index nef f is given by nefefff = pe LL , where pe = n2ef f [p12 v(p11 p12 )] For a fiber optic fiber made from germanium silicate, laboratory experiments show that p11 = 0.113, p12 = 0.252, v = 0.16 and nef f = 1.482 a) Use this information to derive the expression for Gf and calculate its value b) For your value of Gf plot the measured realtive (centered at zero) wavelength of reflected light B as a function of strain ranging across ±8000 µm m Figure 5: Fiberoptic Bragg sensor, used to measure strain Solution a) We first derive an expression for B /dl by taking the deviating of by handwaving ✓ ◆ d B dnef f d⇤ =2 ⇤+ nef f dl dl dl B or just From strain relations we know that the change in bragg spacing is given by strain where d⇤ L = ⇤ dl L then for a change dividing by L we get ✓ d B=2 nef f ⇤ + L ⇤nef f L ◆ B B B = ✓ nef f L + nef f L ◆ = (pe + 1) L L Then Gf = pe + Plug in the values for the constants to get the value for Gf =0.7 10