Switched Capacitor Band-P ass Filter0.1 ObjectTheobjectofthisexperimentistodesignandimplementafourth-orderChebyshevband-passfilter using a switched-capacitor realization. The specifications of the filter are: passbandripple 1.8dB, cen ter frequency fC=4kHz, −3dB bandwidth f3dB=2kHz,gainatthecen ter frequency HBP=1. The transfer fu nction is to be realized as the product of asecond-order high-pass filter in cascade with a second-order lo w -pass filter.To obtain the transfer function for the filter, see pa ge 13 of the Filter P otpourri. Followthe examp le for the 8th order band-pass filter to solve for the required value of x and B in thefrequency transformation from a lo w-pass filter function to a band-pass filter function. Thefrequency transformation for the band-pass filter is given on pa ge 6 of the F ilter Potpourri.Use it to solv e for the tran sfer function.After y ou obtain the band-p ass transfer fun ction, use M ath cad or any math program toplot the dB gain v ersu s frequency for the function . If yo u use M a thcad , be sure to use a rangevariable for the frequency that gives equal x-axis spacings on a log scale. Check to see if thecen ter frequency and the −3dB bandwidth are correct. If they are correct, use Mathcad tofactor the denominator of the transfer function in to the product of two second-order transferfunctions.Rewrite the factored transfer function as the product of a high-pass filter function a ndalow-passfilter function. Realize the t wo filter functions using the m ethods and hardwaredescribed below . The parts will be supplied by our laboratory support staff.Theinformationbelow is from the lab manual for ECE 3042. It describes the devices a vailable and how touse them. Other that that, it con tains more information than is needed for this experiment.0.2 Switched Capacitor Theo ryActive filter design with op amps is a robust mature discipline. All of the classical filte r typesthat w ere imp lemented in bygone eras with resistors, capacitors, an d inductors may now beimplem ented solely with resistors, capacitors, and op amps. The accuracy of these filters islimitedbyonlytheprecisionofthecomponentsandthepropertiesofthephysicalopampsemploy ed. If discrete resistors and capa citors are used, variations with in the manufacturer’sstated toleran ce m ay produce unaccepta ble err or in th e design unless extremely expensivecomponents are employed .The resistors and capacitors required for filter design m ay be fabricated on monolith icintegrated circuits along with the op amps but they usually require a large amoun t of area andare subject to temperature drift and other annoying effects such as parasitic capacitance.Resistors fabricated on integrated circuits are usually restricted to values less than 10 kΩOBJECT 1while the upper limit for capacitors is appro x imately 100 pF. Also, it is quite difficult toobtain precise values of passiv e components fabricated on integr ated circuits. Suc h dedicatedanalog filter in tegra ted circuits ar e available but they are rather expensiv e.The problem of component variation may be o v ercome with switc hed capacitor filters.These use small integrated circuit capacitors whose terminals are switc hed b y a high fre-quency clock signal using MOSFET switc hes to simulate large values of resistance. TheMO SFETs are fabricated on the sam e integrated circuit while the clock may be extern al oralso resident on the integrated circuit.Switched capacitor filters are not a panacea. They are digital circuits and are, therefore,subject to aliasing. The N yquist criterion requires that the wa veform be sampled at a rateat least twice its bandwidth to prev ent aliasing. Norma lly the clock frequency is pic ked tobe large compared to the critical frequencies of the filter (50 to 100 times larger) to preven taliasing. Also, the output is a discrete rather than con tin uous waveform. To minimize bothof these defects it is customary to precede the digital switc h ed capacitor filter with an anti-aliasing analo g lo w-pass filter to limit the bandwidth and to follow the digital filter with ananalog deglitc hing filter. The break frequencies of these analog filters are not crucial so thisuse of analog filters at the input and output is not a major impediment.0.2.1 Switched Capacitor IntegratorFigure 0-1 Switched capacitor integrator.A switched capacitor integrator is sho w n in Fig. 0-1. The clock signal c(t) with frequencyfcand period Tc=1/fcis applied to both the gate input of MOSFET M1and the digital2INVERTER. The signal applied to the gate of MOSFET M2is the com plem ent of the cloc k.Hence, excepts for the switching tran sient, one MO SF E T is on wh ile the oth er is o ff.W hen the cloc k is high MOSF E T M1is on and M2is off. Capacitor C1has a c harge∆q = C1viplaced on it b y the input to the filter. If the clock frequency is large compared tothe bandw idth of vithe input ma y be considered to be constant during the sampling intervalTc/2. During the next cloc k half cycle M1is off and M2is on whic h places the top node ofthe capacitor C1at the virtual ground of the op amp which causes the charge on it to betransferred to CF.Theaveragecurrentflowing in to capacitor C1isi(t)=∆qTc=C1Tcvi= C1fcvi(1)which means that it is equivalen t to a resistor Req=1/C1fc. The output of the op amp isthen giv en byvo= −1CFZt−∞i(u) du = −C1fcCFZt−∞vi(u) du = −1CFReqZt−∞vi(u) du (2)which makes this circuit an integrator. In tegrators are the heart of the state variable filterwhich means that any of the classical filters may be realized with this switch ed capacitorarrangement. Other more elaborate topologies are also employ ed in switc h ed capacitor filt ersbut the circuit in F ig. 0-1 illustrates the basic princip le.Because charge is transferredinspurtsfromcapacitorC1to capacitor CFthis makesthe output voltage discrete rather than con tin uous. The voltage increments are reduced toacceptable values b y pickin g the clock frequency to be large which is also required to preventaliasing . A deglitc h in g analog low -p ass filter cascaded with th e output may also be used tosmooth th e output volta ge.Since the output of the switched capacitor in tegrator depends on the ratio of t wo capac-itances, this can easily be fabricated on an in tegrated circuit. Although precise values ofcomponents are difficu lt to co ntrol, maintain in g ratios is relatively sim p le.Anent the equivalent resistance being set as Req=1/C1fc,