EE 40 Introduction to Microelectronic Circuits Spring 2008 HW 9 due 4 28 5 pm Venkat Anantharam April 21 2008 Referenced problems from Hambley 4th edition 1 P10 83 2 Design a clipper circuit for a negative limit of 2 1V and a positive limit of 2 2V The input voltage is peak limited to 5V You are allowed to use two diodes two d c voltage sources and a resistor The maximum allowable current through each diode is 2 5mA and the threshold voltage of each diode is 0 7V Use an ideal voltage with threshold ignoring breakdown Hint See Figure 10 32 in textbook 3 A voltage source produces a periodic voltage vin t with period T 2s with waveform as in Figure 1 The positive peaks are of 6V and the negative peaks are at 2V Note that 6 5 4 2 in v t in V 3 1 0 1 2 2 1 0 1 2 3 t in s Figure 1 Waveform Input Voltage vin t vd va t where vd 2V and va t is a periodic signal with zero dc component It is desired to clamp vin t to a positive peak of 10V You are allowed to use at most one of each of the following 1 a an ideal diode b an ideal Zener diode of arbitrary breakdown voltage c a resistor of size 1k d a dc voltage source of arbitrary value e a capacitor of arbitrary value You may assume implicitly that there is a small resistance in series with the voltage source vin which is so small compared to 1k that it can be neglected Design a clamp circuit to perform the desired task giving some guidelines for the choice of values for the elements b d and e 4 In the circuit in Figure 2 the op amp is assumed to be ideal The circuit is called a precision rectifier Here V denotes the supply voltages Take V 12V Note that vout t is defined at the input of V vin t V R vout t Figure 2 Precision Rectifier the op amp a Let vin t Vm sin t Let Vm be significantly smaller than V e g Vm 6V Determine vout t under the assumption that the diode is ideal b Now suppose that the diode is modelled as having a threshold voltage vth 0 7V being an ideal short circuit at threshold voltage and an ideal open circuit below the threshold voltage Again dtermine vout t c Why is the circuit called a precision rectifier 5 Consider the circuit in Figure 3 This circuit is known as an inverting precision rectifier Assume that the op amp has supply voltages at 12V and that vin t 6V sin t Further assume that the diode has a threshold voltage of 0 7V but is otherwise ideal Determine the voltage vout t 2 12V vin t 6V sin t vout t 12V Figure 3 Inverting Precision Rectifier 6 For this problem note that the intrinsic carrier concentration of both electrons and holes in pure Si at room temperature 300K can be taken to be 1010 cm 3 while that in pure Ge can be taken to be 2 1013 cm 3 Identify the majority carrier and find the electron and hole concentrations at room temperature in the following semiconductors a Silicon doped with phosphorus at a doping concentration of 1016 cm 3 b Silicon simultaneously doped with arsenic at a concentration of 5 1017 cm 3 and with boron at a concentration of 5 1 1017 cm 3 c Germanium doped with boron at a concentration of 2 1015 cm 3 Consider a slat of silicon as shown in Figure 4 We focus on the x direction and assume that there is no Figure 4 A slat of Silicon variation in planes transverse to the x direction C Recall that Gauss s law tells us that if there is a charge density profile x in the slat measured in cm 3 V then the associate electric field E measured in cm and with reference as pointing in the x direction is given by dE x x dx Here is the permittivity of the material here the material is silicon for which you can take 11 7 0 F where 0 the permittivity of vacuum is 8 85 10 14 cm Further the associated electrostatic potential x measure in V satisfies E x d x dx which together with Gauss s law gives Poisson s equation x d2 x 2 dx 3 This determines potential differences the actual value of the potential depends on a choice of reference This discussion is relevant to the following three problems 7 Suppose the charge density in a slat of silicon variations only in the x direction is given by the graph in Figure 5 2 1 x in mC cm 3 0 1 2 3 4 120 100 80 60 40 20 x in nm 0 20 40 60 80 Figure 5 Charge Density Profile a Verify that the sample as a whole is electrically neutral b Use Gauss s law to determine the electric field as a function of x c Explain why the direction of the electric field which is determined by its sign does not change throughout the range 120nm x 80nm d Determine the potential x as a function of x assuming as a reference that 0 0 thus solving Poisson s equation 8 The electric field in a slat of silicon variations only in the x direction is given by the graph in Figure 6 a Determine the associated charge density x b Assuming as a reference that 0 0 solve for the corresponding potential x c Verify that the sample as a whole is electrically neutral In addition to the discussion of Gauss s law and Poisson s equation the following two problems refer to the depletion approximation for pn diodes In addition to the supplementary reader you should also recall as discussed in class that in the absence of an eventually applied bias the in the bulk of the n type material away from the depletion potential Nd region can be taken to be VT ln ni and that in the bulk of the p type material away from the depletion ni region can be taken to be VT ln N Here VT kT q denotes the thermal voltage ni the intrinsic carrier a concentration of holes and of electrons in pure silicon and Na and Nd respectively denote the doping densities of acceptors and donors in the p type and n type materials respectively The potentials are thought of as referenced to an intrinsic situation if one solves Poisson s equation exactly with these boundary conditions the potential will be zero precisely when the electron density and the hole density are equal each equal to the intrinsic carrier concentration 4 3 5 3 2 5 E x in kV cm 2 1 5 1 0 5 0 0 5 1 1 5 0 100 200 300 400 500 x in nm 600 700 800 Figure 6 Profile of Electric Field 9 Consider a …
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