Assess and predict the performance of an active low pass filter...

In this exercise, the goal is to analyse, assess and predict the performance of an active low pass filter whose circuit diagram is given in Figure 1. The circuit comprises two passive and one active kind of component: the components labelled ’R1’ and ’R2’ are resistors. Their value is measured in Ohms (named after Georg Simon Ohm, unit symbol: ?). As one Ohm is quite small, it is usual to specify resistances in Kiloohms, K? where of course 1K? = 1000?. The components labelled ’C1’ and ’C2’ are capacitors and their value is measured in Farads (named after Michael Faraday, unit symbol: F). As one Farad is a very large capacitance it is more usual to specify capacitances in µF (Microfarads) or even nF (Nanofarads) where 1F = 106µF = 109nF. The third type of component in the circuit is the operational amplifier with whose function and specification we will not concern
ourselves. These components together function as a low pass filter, i.e. signals presented at the input (vin) will get transformed to the output (vout) and the transformation is (to a good approximation) linear. The signal’s amplitude (i.e. voltage) at the output is different from the signal’s amplitude at the input and the relation is given by vout vin (?) = ? 2 c -?2 + i ?c Q ? + ?2 c . (1) In this equation, ? is the frequency of the input signal and i is the complex root of unity, i.e. we have i 2 = -1. Note that the ratio of input and output signal is a complex number where the number’s modulus, i.e. vout vin (?) gives the signal amplitude ratio. The parameters Q and ?c in equation (1) depend on the component values according to the following formulae: ?c = r 1 R1R2C1C2 Q = 1 R1 + R2 · r R1R2C1 C2 . Task 1 Two data files have been made available on Moodle containing measurements (in the unit ?) of two types of resistors all specified to have resistance R = 4700?. The first type (data in film.dat) are film-type resistors where the resistor is initially produced with too low a resistance and then some of the material is abraded (removed) until the resistance is high enough. The second type (data in comp.dat) is the carbon-composition type where the material is simply pressed into shape without further processing. For each of these two data files, find the minimal, maximal value as well as mean, median, upper and lower quartile and standard deviation of the resistance. Perform a t-test to see whether the average resistance of film type resistors is different from that of carbon-composition type resistors. [5] 4 Task 2 Plot a histogram of the distribution for each of the two resistor data sets from above and produce a QQ-plot (using qqnorm) to compare the quantiles to the normal distribution. Do the samples look as though they come from a normal distribution? How would you describe the deviation for film type resistors? In order to fit another distribution for the film type resistors, take the maximum of the film type resistors and for each resistor subtract its value from the maximum value: R˜ i = (max{R1, R2, . . . , Rn}) - Ri Now fit a Gamma distribution to the transformed values, R˜ i . To do this, compute the mean and variance of the transformed values and use the method of moments (see STAT2001) to infer the scale and location parameter of the Gamma distribution. To check your fit, generate a large sample (at least N = 10000 samples) from the gamma distribution with the parameter values you estimated (use e.g. rgamma to do this) and use a qqplot (use qqplot) to check whether your newly fitted distribution fits the data. [5] Task 3 We now wish the investigate the behaviour of the circuit given in figure 1, more precisely its amplitude response given by formula (1). For the values R1 = R2 = 4600? and C1 = 33nF and C2 = 4.7nF, plot the amplitude ratio of the filter for a frequency ranging from ?min = 100 to ?max = 60000. Also plot the attenuation (measured in decibel, dB) as a function of frequency which is given as D(?) = -20 log10  vout vin (?)  (2) over the same frequency range. Can you see why the filter is called a low-pass filter? [3] Task 4 The filter to be constructed is required to have an attenuation of no more than 0.5dB for frequencies below ?1 = 18500, i.e. the amplitude ratio has to be at least vout vin (?) > 10-0.025 when ? < ?1 (this is called the passband). Also, the filter must attenuate frequencies above ?2 = 42000 by at least 12dB, i.e. the amplitude ratio must be no more than vout vin (?) < 10-0.6 when ? > ?2 (this is called the stopband). Using the mean and variances found above for the carbon composition resistors for R1 and R2, and using a statistical model of C1[nF] ~ N (33, 2.5) and C2[nF] ~ N (4.7, 0.05) for the capacitors, simulate the circuit’s amplitude response at the frequencies ?1 and ?2. What percentage of circuits produced will fail to meet the requirements? Provide a confidence interval for your estimate!


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