ECE Final Solution

Problem 1: Course Fundamentals (20 points - 2 points per question) Answer the following ten questions, which test your conceptual understanding of what we have learned in class.

    (a) Ohm’s law relates the current, I, through a resistor, R, with the voltage across the resistor. What is the power dissipated through the resistor in terms of I & R?








    (b) In the SI system of units, we have base quantities such as length (L), mass (kg) and time (s) – there are seven base quantities. Which of the following is the electrical base quantity? (circle one)

        (i) Volts

        (ii) Coulombs

        (iii) Amperes

        (iv) Hertz





    (c) We introduced the concept of impedance (Z = R + jX), which is a complex quantity. What is the analog of Ohm’s law using complex impedance?
















    (d) What do we mean by phasors? Consider a voltage phasor, V, (phase= 0°) applied to an impedance Z, (Z = R + jX). What is the current, I, in Phasor form? What is the power, P, expended by the voltage source? What is the meaning of the imaginary component of the power?











    (e) Electrical engineering has perhaps one of the more abstract formulations of any branch of engineering. We can’t see voltage but we can feel it when we get an electric shock! Trust me, it’s very real. A commonly used mechanical analogy is water flow in a pipe. Using this analogy, what are the analogues of voltage, current, resistance and capacitance. Here is the difficult one – is there an analogue of inductance? If so, what is it?












    (f) The continuity equation in most fields of engineering states that if there is no source of matter, the mass flux entering a system must equal the mass flux leaving the system. What is the electrical analog of the continuity equation?
















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    (g) Similarly, energy conservation is a thermodynamic requirement. What electrical engineering law reflects energy conservation?








    (h) Why are resistors, capacitors, and inductors linear components, but a diode or transistor is not?








    (i) Way back Charles Babbage proposed computation based on mechanical computers – a complex machine with all kinds of gears and such. They did not make a big impact. What distinguishes electronic computers that mechanical computers did not possess.








    (j) When you shine light on a semiconductor you can increase the conductivity of the semiconductor a 1000-fold through the generation of free carriers by the light. This effect is used in stores to indicate that someone has entered the store. Draw a simple circuit to do this function.














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Problem 2: Zero- order circuits (10 points)

Consider a non-ideal voltage source, V, with an internal resistance, Rint.

    (a) What is the schematic representation of this voltage source?











    (b) If you do a source transformation to a current source what would the circuit representation be? What would the value of the current source be?








    (c) Now reconsider the non-ideal voltage source and let it drive a load resistor RL. You will use a zero-order differential equation (which is really a linear equation) to solve this circuit. What is the power dissipated in the load resistor, RL?






    (d) At what value of RL is the power dissipated in the load maximum? (Express in terms of Rint).








    (e) Is this true when you use the equivalent current source?






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Problem 3: First-order circuits (10 points)

The switch has been in position A for a long time. At t=0ms, the switch is moved to position B, and at t=1ms, the switch is moved back to position A. Determine the values of R1 and R2 such that (   = 1    ) = 8  and
(   = 2    ) = 1  . Why is this a first order circuit?





















Hint: First consider the circuit from 0 <   ≤1     and then consider the circuit for > 1    .
































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Problem 4: 2nd-order circuits (15 points)

Consider the circuit below. Note that there are two switches in this circuit.

At t=0, Switch 1 is closed. For Switch 2, it is initially connected to Point A.

At t=0, Switch 2 is moved from Point A to Point B.



















    (a) What characterizes 2nd order circuits? Why is the above circuit a 2nd order circuit?




    (b) Draw the equivalent circuits for (i)  < 0and (ii) ≥ 0.
























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    (c) Using KCL, write the second order differential equation for the circuit above ( ≥ 0) in terms of current variable, .
Recall:
2  (  )
+ 2ζω
    (  )
+ ω
2
=   (  )

2

























(d) Solve for the initial conditions of the inductor and capacitor:
  (   = 0+),   (   = 0+).








(e) Solve for the damping factor (ζ) and the resonant frequency (ω ).















(f) Is this system underdamped, overdamped, or critically damped?








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    (g) Solve for the characteristic roots:  1and  2.





    (h) Solve for   (  ).







(i) Solve for the current across the resistor,    (  ).

Hint: remember that the voltage across the inductor, capacitor, and resistor are the same voltage ( = = =   ).







(j) Quickly sketch the voltage across capacitor as a function of time,   (  ).













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Problem 5: MOSFET amplifier (15 points)

Consider an NMOS FET used as an amplifier as shown in figure below. The capacitor C1 couples the incremental input voltage vi to the gate. Because it is an open circuit for DC, vi does not affect VGS, the operating point value of the gate-to source voltage. The FET is described by

= 0 for
<










=   (  

−   )2
for

>

and

> (  

−   )



































= 0.2



,

= 2










2

















































    (a) Determine VGS and RL such that the operating (quiescent) point of the amplifier is: VO = 7 V, ID = 0.2 mA























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    (b) Determine the transconductance gm of the FET at the operating point defined in part (a).















    (c) Given that R1 + R2 = 500kΩ, determine R1 and R2 such that the desired DC voltage, VGS is established.
















    (d) Using the small-signal model for the FET, sketch and label a complete small-signal equivalent model of the original circuit.






















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Problem 6: MOS logic (15 points)

Consider the circuit shown below with three NMOS transistors. Assume that a logical one corresponds to voltage, v > 4 volts and a logical zero corresponds to v < 1 volt. The switch-resistor model for the FET is shown alongside.

















    (a) Write a logic expression or describe the logic function implemented by the above circuit. You may consider using a Boolean expression (Output node is vD).








    (b) Using a switch-resistor model (RON = 1kΩ, VT = 1V) for each FET, determine the output voltage vD when vA = vB = vC = 5V.





















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    (c) The output of the above circuit is connected to a FET inverter as shown below. Will this implement an ‘AND’ logic operation? If not, why? (RON=1kΩ, VT=1V)






























































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Problem 7: Operational Amplifiers (20 points)

Consider the op-amp circuit shown in the schematic shown below.






















    (a) Analyze the circuit shown and derive the output voltage in terms of the input voltage. Indicate where the virtual short (or virtual ground). What are the assumptions on the op-amp Voltage gain, AV? What should be the relationship between R1, R2, and RL be to get a voltage gain AV=-5. What are the constraints on RL?










    (b) How would you convert the circuit in (a) to an ‘integrator’? (draw the new circuit)

You may consider swapping R1 or R2 for a different component.















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    (c) How would you convert the circuit in (a) to a ‘differentiator’? (draw the new circuit)

You may consider swapping R1 or R2 for a different component.
















    (d) How would you modify the circuit in (a) to do a summation of three input voltages? (draw the new circuit)
















    (e) Explain qualitatively (with a block diagram) how you would build a PID controller with Op-Amps.

Hint: “PID” = proportional-integral-derivative controller



















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