EE 350 Problem Set 11 Cover Sheet Fall 2014Last Name (Print):First Name (Print):ID number (Last 4 digits):Section:Submission deadl ines:• Turn in the written solutions by 4:00 pm on Friday December 1 2 in the hom ework slot outside 121 EE East.Problem Weight Score52 2053 2054 2055 2056 20Total 100The solution submitted for grading represents my own analysis of the problem, and not that of another student.Signature:Neatly print the name(s) of the students you coll aborated with on this assignment.Reading assignment:• Lathi Chapter 7: Sections 7.1 and 7.2• Priemer Chapter 12: Section 12.3.7 pages 53 6 through 54 0Problem 52: (20 points)Operational amplifiers are important building blocks in a wide spectrum of electronic systems such a s amplifiersand filters. The concept of feedback control is of central importance in understanding the design of operationalampl ifier circuits. For without feedback, operational ampl ifiers behave as comparators. This problem shows why itis necessary to connect the output of an o perationa l amplifier to its inverting input. Negative feedback produces acircuit that is BIBO stable. Figure 1(A) shows the circuit symbol for an operational amplifier, while Figure 1(B)shows an equivalent circuit model in the s-domain. The transfer function Ad(s)Ad(s) =Aos/ωo+ 1has a sing le real pole at s = −ωoand a DC gain of Ao. Depending on the specific operatio nal am plifier chosen,the DC gain may range from 100,000 to over 1 m illion, while the corner frequency ωocan vary from 100 kHz tomore than 1 G Hz. The maximum output voltage of the operational amplifier is limited by the external power supplysources, that are often called supply rails.Figure 1: Operational amplifier circuit symbol (A) and equivalent circuit (B).1. (4 points) Using the equivalent circuit model in Fig ure 1(B), show that the inverting amplifier configuration inFigure 2(A) can be represented using the block diagram in Figure 2(B). Specify the value o f the gai ns K1andK2in terms of R1and R2.Figure 2: Inverting amplifier (A) and equi valent block diagram representation (B).2. (4 points) Determine the transfer function representation of the block diagram representation in Figure 2(B),and place your answer in the standard formVo(s)Vi(s)=bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.3. (2 points) Determine if the closed-loop system represented by the transfer function in part 2 is BIBO stable.If the closed-loop system is BIBO stable, w hat is the DC and high frequency ga in of the amplifier circuit?4. (4 points) Using the equivalent circuit model in Figure 1(B), show that the circuit in Figure 3 (A) can berepresented using the block diagram in Figure 3(B). Specify the value of the gains K1and K2in terms of R1and R2.Figure 3: Circuit (A) and equivalent block diagram representation (B).5. (4 points) Determine the transfer function representation of the block diagram representation in Figure 3(B),and place your answer in the standard formVo(s)Vi(s)=bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.6. (2 points) Determine if the closed-loop system represented by the transfer function in part 5 is BIBO stable.If the closed-loop system is BIBO stable, w hat is the DC and high frequency ga in of the amplifier circuit?Problem 53: (20 points)1. (6 points) Once gain consider the operational amplifier in Figure 1. Sketch the Bode magnitude and phase plotof the open-loop frequency response f unction Vo(ω)/Vd(ω) of the operational amplifier.2. (1 point) What is the 3 dB bandwidth of the open-loop frequency response function Vo(ω)/Vd(ω)?3. (1 point) The gai n bandwidth (GBW) product is the product of the DC gain and the 3 dB bandwidth. Whatis the GBW product of the open-loop operational amplifier?4. (10 points) Once gain consider the inverting amplifier circuit in Figure 2, whose frequency response functionVo(ω)/Vi(ω) can be o bta ined from the result in part 2 of Problem 58. Sketch the Bode magnitude and phaseplot of the closed-loop frequency response function Vo(ω)/Vi(ω) of the inverting amplifier.5. (1 point) What is the 3 dB bandwidth of the closed-loop frequency response f unction Vo(ω)/Vi(ω)?6. (1 point) What is the GBW product of the inverting amplifier?Problem 54: (20 points)Using the semilog paper posted along with Problem Set 11 on the EE 350 web page, carefully sketch the Bodemagnitude and phase plots for the fol lowing transfer functions. Indicate the slope of each straight-l ine segment aswell as the corner frequencies on your plots. You must be able to sketch and interpret Bode magnitude and phaseplots for Exam #4.1. (6 points) H(ω) =100(ω + 16 00)ω + 16000.2. (6 points) H(ω) =10(ω)(ω + 1000).3. (8 points) H(ω) = 1000ω100(ω + 10)(ω + 1000)Problem 55: (20 points)A linear time-invariant system can be represented by an ordinary differential equation, an impulse response function,a frequency response function, or a transfer function. From one form of representation, you must be able to derivethe others. As an example, a certain linear time invariant system has a frequency response function whose exactmagnitude and phase plots are shown in Figure 4.010203040Magnitude (dB)101102103−90−4504590Phase (deg)Bode DiagramFrequency (rad/sec)Figure 4: Exact magnitude and phase pl ots.1. (2 points) Can the system be described by a first-order differential equation? Justify your answer in one or twosentences to receive credit.2. (6 points) Based on the magnitude and phase plots shown in Figure 4, determine the transfer function H(s)of the system. Check your answer by using the MATLAB command bode(num,den) to generate the exactmagnitude and phase plots, where num and den is the p olynomial representation of H(s). Attach a copy ofthe exact magnitude and phase plot obtained using the bode command, and add your name and section tothe plot using the gtext command.3. (2 points) Based on your result in part 2, find an ordi nary differential equation that describes the response y(t)of the system to the input f(t).4. (5 points) Based on your result in part 2, do you expect the unit-step response of the system to be underdamped,critically dam ped, or overdamped? What is the steady-state response of the system to a unit-step input?Verify your answers by usi ng the MATLAB command step(num,den) to generate the unit-step response ofthe system, where nu m a nd den are the polynomial representation of the
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