# UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences

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1 UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences EECS 40 Spring 2000 Introduction to Microelectronic Devices Prof. King MIDTERM EXAMINATION #2 April 6, 2000 Time allotted: 80 minutes NAME: (print) Last First Signature STUDENT ID#: 1. This is a CLOSED BOOK EXAM. However, you may use 2 pages of notes and a calculator. 2. Show your work on this exam. MAKE YOUR METHODS CLEAR TO THE GRADER. 3. Write your answers clearly in the spaces (lines, boxes or plots) provided. Numerical answers must be accurate to within 10% unless otherwise noted. 4. Remember to specify the units on answers whenever appropriate. 5. Do not unstaple the pages of this exam. SCORE: 1 / 20 2 / 30 3 / 25 4 / 25 Total: / 100 Page 1

2 Physical Constants Description Symbol Value Electronic charge q x C Permittivity of vacuum ε o x F/cm Boltzmann s constant k 8.62 x 10-5 ev/k Thermal voltage at 300K kt/q V Properties of Silicon at 300K Description Symbol Value Thermal velocity v th 10 7 cm/s Relative permittivity ε r 11.7 Intrinsic carrier density n i 1.45 x cm -3 Electron and Hole Mobilities in Silicon at 300K electron mobility hole mobility Conversion Factors 1 ev = x J 1 µm = 10-4 cm = 10-6 m Farad = Coulomb / Volt Henry = Volt / (Ampere/second) Watt = Volt x Ampere Joule = Watt x second Page 2

3 Problem 1 Circuits with Dependent Sources [20 points] a) Find V o. [4 pts] 5 ma 10 kω 40 kω i x 10 i x 10 kω V o V o = V b) In the circuit below, the independent source values and resistances are known. Use the nodal analysis technique to write 3 equations sufficient to solve for V a, V b, and V c. To receive credit, you must write your answer in the box below. [6 pts] DO NOT SOLVE THE EQUATIONS! I AA 10 i a R 2 b c i R1 R 3 V AA R 4 Write the nodal equations here: Page 3

4 Problem 1 (continued) c) Consider the following circuit: 3 kω 20 i 1 1 kω a i a 70 V i 1 7 kω 3 kω b i) Find the voltage V ab. [5 pts] V ab = V ii) What is the current i a when the terminals a and b are shorted together? [3 pts] i a = A iii) Draw the Thevenin Equivalent Circuit. [2 pts] Thevenin Equivalent Circuit: a b Page 4

5 Problem 2: Transient Response [30 points] a) In the circuit below, the switch has been in the closed position for a long time. 5 kω opens at t = 0 5 V 1 µh 1 kω v R i) Find the value of v R just after the switch opens (t = 0 ). [3 pts] v R (0 ) = V ii) How much energy is dissipated in the 1 kω resistor after the switch is opened? [2 pts] Energy dissipated = J b) In the circuit below, the 5 µf capacitor is initially charged to 5 V (v C1 (0 - ) = 5 V). (The 1 µf capacitor is initially uncharged.) The switch is then closed at time t = 0. What is the final value of v C1? [5 pts] closes at t = 0 2 kω v C1 5 µf 1 µf Final value of v C1 = V Page 5

6 Problem 2 (continued) c) The following is a circuit model for an NMOS inverter, in which the transistor is turned on at time t = 0: V DD R NMOS R load t=0 C load i NMOS v C V DD = 5 V R load = 90 kω R NMOS = 10 kω C load = 10 nf i) What is the value of v C at t = 0 -? [3 pts] v C (0 - ) = V ii) What is the value of i NMOS at t = 0? [3 pts] i NMOS (0 ) = A iii) What is the final value of v C? [3 pts] final value of v C = V iv) Neatly sketch the graph of i NMOS for all t, labelling the axes. [5 pts] v) Write an equation for i NMOS as a function of time, for t > 0. [6 pts] Equation for i NMOS : Page 6

7 Problem 3: Op-Amp Circuits [25 points] Assume the op-amps in this problem are ideal. a) Consider the following circuit: 20 kω 10 kω 15 V Va 6 V 20 kω -15 V 40 kω V o i) Find an expression for V o as a function of V a. [6 pts] Expression for V o : ii) Find V o for V a = 2 V. [3 pts] V o = V iii) For what values of V a will the op-amp be saturated? [6 pts] Values of V a for which the op-amp will be saturated: Page 7

8 Problem 3 (continued) b) In the following circuit, the op-amps are operating linearly. R 1 R 2 R 4 V 1 R 3 V 2 V out Find V out in terms of V 1, V 2, R 1, R 2, R 3, R 4. [10 pts] (Hint: The superposition method might be helpful here.) V out = Page 8

9 Problem 4: Semiconductor properties; p-n diodes [25 points] a) Consider a silicon sample maintained at 300K under equilibrium conditions, uniformly doped with 1x10 16 cm -3 phosphorus atoms. The surface region of the sample is additionally doped uniformly with 5x10 16 cm -3 boron atoms, to a depth of 1 µm, as shown in the figure below. uniformly doped with 1x10 16 cm -3 P and 5x10 16 cm -3 B I 1 µm uniformly doped with 1x10 16 cm -3 P II Schematic cross-sectional view of silicon sample i) In the figure above, indicate the type of the regions (I and II) by labelling them as n or p type. [2 pts] ii) What are the electron and hole concentrations in Region I? [5 pts] n = cm -3 p = cm -3 iii) What is the sheet resistance of Region I? [5 pts] R s = Ω/square iv) Suppose any voltage between 0 V and 5 V can be applied to Region I. What fixed voltage ( bias ) would you apply to Region II, to guarantee that no current would ever flow between Region I and Region II? Briefly explain your answer. [3 pts] Region II bias voltage = V Page 9

10 Problem 4 (continued) b) If a diode is operated only within a small range of forward-bias voltages, its behavior can be accurately modelled by a resistor, whose value is dependent on the bias voltage. Derive an expression for the diode small-signal resistance: R diode = I V in terms of the saturation current I s, the bias voltage V, and the absolute temperature T. [5 pts] R diode = c) Plot v L vs. V IN for -10 V < V IN < 10 V on the axes provided, for the circuit below. Note that the diode is a perfect rectifier. Label the axes. [5 pts] 5 kω V IN 10 kω v L 2 V v L V IN Page 10

11 UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences EECS 40 Spring 2000 Introduction to Microelectronic Devices Prof. King MIDTERM EXAMINATION #2 April 6, 2000 Time allotted: 80 minutes NAME: (print) Last First Signature STUDENT ID#: 1. This is a CLOSED BOOK EXAM. However, you may use 2 pages of notes and a calculator. 2. Show your work on this exam. MAKE YOUR METHODS CLEAR TO THE GRADER. 3. Write your answers clearly in the spaces (lines, boxes or plots) provided. Numerical answers must be accurate to within 10% unless otherwise noted. 4. Remember to specify the units on answers whenever appropriate. 5. Do not unstaple the pages of this exam. SCORE: 1 / 20 2 / 30 3 / 25 4 / 25 Total: / 100 Page 1

12 Physical Constants Description Symbol Value Electronic charge q x C Permittivity of vacuum ε o x F/cm Boltzmann s constant k 8.62 x 10-5 ev/k Thermal voltage at 300K kt/q V Properties of Silicon at 300K Description Symbol Value Thermal velocity v th 10 7 cm/s Relative permittivity ε r 11.7 Intrinsic carrier density n i 1.45 x cm -3 Electron and Hole Mobilities in Silicon at 300K electron mobility hole mobility Conversion Factors 1 ev = x J 1 µm = 10-4 cm = 10-6 m Farad = Coulomb / Volt Henry = Volt / (Ampere/second) Watt = Volt x Ampere Joule = Watt x second Page 2

13 Problem 1 Circuits with Dependent Sources [20 points] a) Find V o. [4 pts] 5 ma 10 kω 40 kω i x 10 i x 10 kω V o V o = V b) In the circuit below, the independent source values and resistances are known. Use the nodal analysis technique to write 3 equations sufficient to solve for V a, V b, and V c. To receive credit, you must write your answer in the box below. [6 pts] DO NOT SOLVE THE EQUATIONS! I AA 10 i a R 2 b c i R1 R 3 V AA R 4 Write the nodal equations here: Page 3

14 Problem 1 (continued) c) Consider the following circuit: 3 kω 20 i 1 1 kω a i a 70 V i 1 7 kω 3 kω b i) Find the voltage V ab. [5 pts] V ab = V ii) What is the current i a when the terminals a and b are shorted together? [3 pts] i a = A iii) Draw the Thevenin Equivalent Circuit. [2 pts] Thevenin Equivalent Circuit: a b Page 4

15 Problem 2: Transient Response [30 points] a) In the circuit below, the switch has been in the closed position for a long time. 5 kω opens at t = 0 5 V 1 µh 1 kω v R i) Find the value of v R just after the switch opens (t = 0 ). [3 pts] v R (0 ) = V ii) How much energy is dissipated in the 1 kω resistor after the switch is opened? [2 pts] Energy dissipated = J b) In the circuit below, the 5 µf capacitor is initially charged to 5 V (v C1 (0 - ) = 5 V). (The 1 µf capacitor is initially uncharged.) The switch is then closed at time t = 0. What is the final value of v C1? [5 pts] closes at t = 0 2 kω v C1 5 µf 1 µf Final value of v C1 = V Page 5

16 Problem 2 (continued) c) The following is a circuit model for an NMOS inverter, in which the transistor is turned on at time t = 0: V DD R NMOS R load t=0 C load i NMOS v C V DD = 5 V R load = 90 kω R NMOS = 10 kω C load = 10 nf i) What is the value of v C at t = 0 -? [3 pts] v C (0 - ) = V ii) What is the value of i NMOS at t = 0? [3 pts] i NMOS (0 ) = A iii) What is the final value of v C? [3 pts] final value of v C = V iv) Neatly sketch the graph of i NMOS for all t, labelling the axes. [5 pts] v) Write an equation for i NMOS as a function of time, for t > 0. [6 pts] Equation for i NMOS : Page 6

17 Problem 3: Op-Amp Circuits [25 points] Assume the op-amps in this problem are ideal. a) Consider the following circuit: 20 kω 10 kω 15 V Va 6 V 20 kω -15 V 40 kω V o i) Find an expression for V o as a function of V a. [6 pts] Expression for V o : ii) Find V o for V a = 2 V. [3 pts] V o = V iii) For what values of V a will the op-amp be saturated? [6 pts] Values of V a for which the op-amp will be saturated: Page 7

18 Problem 3 (continued) b) In the following circuit, the op-amps are operating linearly. R 1 R 2 R 4 V 1 R 3 V 2 V out Find V out in terms of V 1, V 2, R 1, R 2, R 3, R 4. [10 pts] (Hint: The superposition method might be helpful here.) V out = Page 8

19 Problem 4: Semiconductor properties; p-n diodes [25 points] a) Consider a silicon sample maintained at 300K under equilibrium conditions, uniformly doped with 1x10 16 cm -3 phosphorus atoms. The surface region of the sample is additionally doped uniformly with 5x10 16 cm -3 boron atoms, to a depth of 1 µm, as shown in the figure below. uniformly doped with 1x10 16 cm -3 P and 5x10 16 cm -3 B I 1 µm uniformly doped with 1x10 16 cm -3 P II Schematic cross-sectional view of silicon sample i) In the figure above, indicate the type of the regions (I and II) by labelling them as n or p type. [2 pts] ii) What are the electron and hole concentrations in Region I? [5 pts] n = cm -3 p = cm -3 iii) What is the sheet resistance of Region I? [5 pts] R s = Ω/square iv) Suppose any voltage between 0 V and 5 V can be applied to Region I. What fixed voltage ( bias ) would you apply to Region II, to guarantee that no current would ever flow between Region I and Region II? Briefly explain your answer. [3 pts] Region II bias voltage = V Page 9

20 Problem 4 (continued) b) If a diode is operated only within a small range of forward-bias voltages, its behavior can be accurately modelled by a resistor, whose value is dependent on the bias voltage. Derive an expression for the diode small-signal resistance: R diode = I V in terms of the saturation current I s, the bias voltage V, and the absolute temperature T. [5 pts] R diode = c) Plot v L vs. V IN for -10 V < V IN < 10 V on the axes provided, for the circuit below. Note that the diode is a perfect rectifier. Label the axes. [5 pts] 5 kω V IN 10 kω v L 2 V v L V IN Page 10

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