Kevin Hao1 and Asher Noel2 1University of Florida - BS Biology ’21, MD ’24 kevin@floridascienceolympiad.org 2Harvard College – BS Physics ’23 ashernoel@college.harvard.edu

2020 Solon High School Invitational

Circuit Lab C

Answer Key

− DO NOT BEGIN UNTIL GIVEN PERMISSION

− You will have 50 minutes to complete the exam || You may separate the exam

− For calculation questions, it is not required that you show your work, however partial credit will be

assigned if correct steps are shown with an incorrect answer.

− Answers must be given with appropriate significant figures and units to receive full credit.

− All final answers must be placed inside the designated box, including multiple choice.

− Lab: You will have up to 20 minutes to complete each lab section (2 total). A proctor will instruct you

when it is your turn. Components may not be replaced between teams, so let us know if you suspect

faulty or damaged components and we will be happy to provide replacements.

− Allowed materials: 3-ring binder, writing utensils, two calculators, basic multimeter

− Tie-breaker order: 9, 16, 21, 23, 35, 39, 41, 44, LAB 2

Competitors: ________________________________________

_________________________________________

School Name: ________________________________________

Team Number: ___________

Rank: ___________

Score: ___________

2

This page is for administrative use only. DO NOT WRITE ON THIS PAGE.

Page Number Possible Score Your Score

3 26

4 26

5 28

6 32

7 30

8 37

9 30

10 22

LAB 1 47

LAB 2 55

Total 333

3

1. Identify the individual that coined the term for item A. 1 (2 pts)

2. Identify the individual that discovered the law

demonstrated in item B. 1 (2 pts)

3. Identify the individual who is credited with

the law demonstrated in item C. 1 (2 pts)

4. Identify the individual who is credited with

the law demonstrated in item D. 1 (2 pts)

5. Identify the inventor of item E. 1 (2 pts)

6. Identify the individual who is credited with

the law demonstrated in item F. 1 (2 pts)

7. Identify item G. 1 (2 pts)

8. A proton is moving rightward between two

parallel charged plates separated by distance

d = 2.80 cm, as shown in item H. The plate

potentials are V1 = 15.0 V and V2 = 35.0 V. If

the initial speed of the proton at the left plate

is 100. km/s, what is its speed just as it

reaches the right plate? 1 (4 pts)

9. What is the net electrostatic force acting on Q1

and Q2 in item I? Indicate a repulsive force

with a (+) and an attractive force with an (-). 1

(4 pts) (TB#1)

10. The parallel plate capacitor in item J has an area (A) of 8.2 cm2 and plate separation (d) of 5.0 mm. If muscovite

(Dielectric of 5.4) is used as the dielectric material, what is the capacitance of the capacitor? 1 (2 pts)

11. If 125 V is applied across the plates of the capacitor in item J, how much energy is stored by the capacitor? Reference

Q10 for additional information. 1 (2 pts)

A

B

C

D

E

F

G

H

I

Q1 = -3.50 µC

Q2 = +2.00 µC

r = 10.0 cm

J

____ / 26

Ampere

Coulomb

Kirchhoff

Ohm

Tesla

Faraday

Electromagnet

78.5 km/s

−6.29 𝑁

0.886 F

6.92 kJ

4

12. Identify the type of rectified AC current shown right. 1 (2 pts)

A. Half Wave B. Full Wave

C. 3 Phase Half Wave D. 3 Phase Full Wave

13. Identify the type of rectified AC current shown right. 1 (2 pts)

A. Half Wave B. Full Wave

C. 3 Phase Half Wave D. 3 Phase Full Wave

14. Identify the type of rectified AC current shown right. 1 (2 pts)

A. Half Wave B. Full Wave

C. 3 Phase Half Wave D. 3 Phase Full Wave

15. Which of the following devices converts a direct current (DC) input into alternating current (AC)? 1 (2 pts)

A. Rectifier B. Alternator

C. Inverter D. Bigfoot

16. What is the value of Va in Circuit 1? 1 (2 pts) (TB#2)

A. 2.0 V B. 2.5 V

C. 5.0 V D. 50 V

17. What is the value of the power supplied in Circuit 1? 1 (2 pts)

A. 2.0 W B. 2.5 W

C. 5.0 W D. 50 W

18. What is the value of the total power dissipated in Circuit 1? 1 (2 pts)

A. 2.0 W B. 2.5 W

C. 5.0 W D. 50 W

19. What is the value of Va in Circuit 2? 1 (2 pts)

A. 2.0 V B. 3.0 V

C. 6.0 V D. 8.0 V

20. What is the value of Vb in Circuit 2? 1 (2 pts)

A. 2.0 V B. 3.0 V

C. 6.0 V D. 8.0 V

21. What is the value of the current through R1 in Circuit 2? 1 (2 pts) (TB#3)

A. 1.3 A B. 4.0 A

C. 6.0 A D. 8.9 A

22. Determine the voltages V1, V2, and V3. Provide your answer to 3 significant figures. 1 (6 pts; 2, 2, 2)

____ / 26

Circuit 1

Circuit 2

V1 = _______________ V2 = _______________ V3 = _______________

A

B

D

B

A

B

B

D

C

B

60.0 V 86.0 V -13.0 V

V1, V2, V3 Criteria for points (TIER): Grade each answer separately using this rubric (+2) – Answer is correct (+1) – Incorrect significant figures (2 or >3) AND/OR missing units, wrong sign If all answers are wrong (+1) – Reasonable Attempt

5

23. Which of the following is NOT a technique that can be utilized to remove logic redundancy? 1 (2 pts) (TB#4) A. Boolean algebra B. Karnaugh maps

C. Computational derivation D. Quine-McCluskey algorithm

24. Which of the following terms describes the number of logic inputs that can be controlled be a single output? 1 (2 pts) A. Reliability B. Fanout

C. Speed D. Cost

25. Consider the function MAJ: {0, 1}3 -> {0,1} that is defined as follows: MAJ(x) = \piecewise{1 , x0 +x1+x2 >= 2}{0, otherwise}. Which of the following is a formula involving AND, OR, and NOT to compute MAJ? 2 (3 pts)

A. OR( AND(x0, x1), OR(AND(x0, x1), AND(x0, x1)))) B. AND( OR(x0, x1), AND(OR(x0, x1), OR(x0, x1)))

C. OR( AND(x0, x1), AND(AND(x0, x1), AND(x0, x1))) D. AND( AND(x0, x1), AND(OR(x0, x1), AND(x0, x1)))

26. Let MAJ: {0, 1}3 -> {0,1} be the function that on input a, b, c outputs 1 if a + b + c >= 2. What is the size of the minimum Boolean circuit composed of NAND’s for computing MAJ? The size of a Boolean circuit is the number of gates it contains. Gates in a Boolean circuit have at most 2 inputs and 1 output? 2 (3 pts)

A. 8 B. 9

C. 10 D. 11

27. Let XOR3 : {0,1}3 -> {0,1} be the function defined as XOR3(a,b,c) = a + b + c mod 2. Which equals 1? 2 (3 pts) A. XOR3 (0,1,1) B. XOR3 (1,0,1)

C. XOR3 (0,1,0) D. XOR3 (0,0,0)

28. Define ALLEQ: {0,1}4 -> {0,1} to the function that on input x \in {0,1}4 outputs 1 if x0 = x1 = x2 = x3. What is the size of the minimum Boolean circuit composed of NOT, AND, and OR for computing ALLEQ? 2 (3 pts)

A. 8 B. 9

C. 10 D. 11

29. For every Boolean circuit C of s gates, there exists a NAND circuit C’ of at most how many gates that computes the same function as C? 2 (3 pts)

A. 3s B. 4s

C. 2s/s D. 5s

30. A classical system of n bits can be modeled by a string s \in {0,1}n. Which of the following is logically equivalent to writing the NAND of the 3rd and 5th bits to the 17th bit? 2 (3 pts)

A. s17 = 1 – s3s5 B. s17 = s3s5

C. s17 = 1 – (s3 + s5) D. s17 = s3 + s5

31. In quantum computing, the state of an individual bit, called a ‘qubit’, is modeled by a pair of numbers (𝛼, 𝛽) such that | 𝛼 |2 + | 𝛽 |2 = 1. Which of the following models the NOT operation with a map N: ℝ2→ ℝ2? 2 (3 pts)

A. 𝑁(𝛼, 𝛽) = (𝛼, 𝛽) B. 𝑁(𝛼, 𝛽) = (𝛼, 𝛼)

C. 𝑁(𝛼, 𝛽) = (𝛽, 𝛼) D. 𝑁(𝛼, 𝛽) = (𝛽2, 𝛼2)

32. A quantum circuit is analogous to a Boolean circuit and can be described as a directed acyclic graph. One crucial difference, however, is that all gates must be reversible (this ensures that operations can be expressed as unitary matrices). Instead of thinking of NAND as a non-reversible map from {0,1}2 to {0,1}, how could we think about it as a reversible map on three qubits a, b, c? 2 (3 pts)

A. a, b, c to a, b, c XOR NAND(a, b) [i.e., this flips the last bit if NAND of the first two bits is 1]

B. a, b, c to a, b, c XOR AND(a, b) [i.e., this flips the last bit if AND of the first two bits is 1]

C. a, b, c to a, b, c XOR OR(a, b) [i.e., this flips the last bit if OR of the first two bits is 1]

D. a, b, c to a, b, c XOR NOR(a, b) [i.e., this flips the last bit if XOR of the first two bits is 1]

____ / 28

C

B

A

B

C

D

B

A

C

A

6

33. Determine the current (IAB) thru the resistor R3 and voltage (VAB) across the resistor R3. Indicate direction with

respect to nodes A and B with (+) or (i) sign. Provide your answers to 3 significant figures. 1 (12 pts; 6, 6)

34. Determine the value of X, the resistance of R5, so that the voltage across the resistor R3 is zero in the circuit below.

What is the name of this configuration? Provide numerical answers to 3 significant figures. 1 (8 pts; 6, 2)

35. Draw the Thévenin and Norton equivalent circuits with respect to the terminals A and B in the circuit below. Provide

your answers to 3 significant figures. 1 (12 pts; 6, 6) (TB#5)

IAB = _______________ VAB = _______________

X = _______________ Name = _______________

X

Thévenin Norton

____ / 32

A

B

-51.4 mA -1.28 V

7.00 Wheatstone Bridge (2)

IAB and VAB Criteria for points (TIER): Grade each answer separately using this rubric (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

X Criteria for points (TIER): Grade each answer separately using this rubric (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt Name Criteria for points (TIER) (+2) – Answer is correct

Thevenin and Norton Criteria for points (TIER): Grade each answer separately using this rubric (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

Example Solution 𝐴 − 25

30+

𝐴 − 𝐵

25+

𝐴

5= 0

𝐵 − 25

42+

𝐵 − 𝐴

25+

𝐵

12= 0

IAB = (3.7916-5.0760)/25 = -51.4 mA

Vo = -1.28

Example Solution 𝐴 − 25

30+

𝐴

5= 0

𝐵 − 25

42+

𝐵

𝑋= 0

X = 7.00

7

36. Determine the voltage (V1) across resistor R1 and the current (i) through resistor R3 in the circuit below. Provide

your answers to 3 significant figures. 1 (12 pts; 6, 6)

37. Determine the power dissipated by R6 (PR6) and the power supplied by the source V1 (PV1). Provide your answers to 3

significant figures. 1 (12 pts; 6, 6)

38. Determine the current through the center 5 k resistor (io) Provide your answer to 3 significant figures. 1 (6 pts)

V1 = _______________ i = _______________

____ / 30

PR6 = _______________ PV1 = _______________

io = _______________

50.0 V -1.50 A

1.25 W 21.9 W

2.00 mA

V1 and i Criteria for points (TIER): Grade each answer separately using this rubric (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

PR6 and PV1 Criteria for points (TIER): Grade each answer separately using this rubric (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

io Criteria for points (TIER): (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

Example Solution

−3 +𝑉1

200+

𝑉1 − −5𝑖

10+

𝑉1 − 80

20= 0

𝑖 =𝑉1 − 80

20

V1 = 50.0 V

I = -1.50 A

Example Solution 𝐴 − 20

2000+

𝐴 − 𝐵

5000+

𝐴

30000= 0

𝐵 − 20

5000+

𝐵 − 𝐴

5000+

𝐵

1000= 0

Io = 𝐴−𝐵

5000= 2.00 𝑚𝐴

Example Solution

Two delta-Y conversions (not shown)

PR6 = 502/2000

PV1 = 2302/2421

8

39. State 4 assumptions of the characteristics an ideal operational amplifier. 1 (4 pts) (TB#6)

40. What is the measured output voltage of an operational amplifier if the calculated output voltage based on the input

voltage and the gain exceeds the supply voltage? 1 (3 pts)

41. Determine the output voltage (Vo) and whether an inverting or non-inverting configuration is used in the circuit

below. Assume an ideal operational amplifier. Provide your answer to 3 significant figures. 1 (8 pts; 6, 2) (TB#7)

42. Determine the gain for each stage and the node voltages (V1, V2, V3, V4). Assume an ideal operational amplifier. Provide

your answers to 3 significant figures. 1 (22 pts; 3, 3, 4, 4, 4, 4)

____ / 37

Gain (Stage 1) = _________ Gain (Stage 2) = _________

V1 = ____________ V2 = ____________ V3 = ____________ V4 = ____________

Stage 1

Stage 2

Stage 3

1. ____________________________________________________________________

2. ____________________________________________________________________

3. ____________________________________________________________________

4. ____________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Vo = _______________ Type = _______________

The measured output voltage will be equal to the supply (rail) voltage because the op-amp is

saturated (3)

11.8 V Non-Inverting (2)

-2.50 -1.50

-10.0 V -7.50 V -1.88 V 5.63 V

• Infinite Input impendence

• Infinite open loop gain

• Zero output impendence

• Infinite bandwidth

• Zero input offset voltage

• Zero noise

• Infinite output voltage range (when unspecified supply)

• Infinite CMRR + power supply rejection ratio

Example Solution 5 − 0

2200+

5 − 𝑉𝑜

3000= 0

Vo = 11.8 V

Example Solution 0 − 4

4000+

0 − 𝑉1

10000= 0

V1 = -10.0 V, Gain = -10/4 = -2.5 0 − 5

8000+

0 − 𝑉2

12000= 0

V1 = -7.5 V, Gain = -7.5/5 = -1.5

V3 = (V1-V2)(6/(6+2)) = -1.875 −1.875 − −10

2000+

−1.875 − −7.5

6000+

−1.875 − 𝑉1

1500= 0

V4 = 5.625 V

Gain Criteria for points (TIER): Grade each answer separately using this rubric (+3) – Answer is correct (+2) – Correct work shown (+1) – Incorrect work shown V1, V2, V3, V4 Criteria for points (TIER): Grade each answer separately using this rubric (+4) – Answer is correct (+3) – Incorrect significant figures (2 or >3) AND/OR missing units (+2) – Correct work shown (+1) – Incorrect work shown

Vo Criteria for points (TIER): (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

9

43. Using the template shown below, show the distribution of positive and negative charges in the sphere at each step.

Every sphere should contain charges. The number of charges used is irrelevant, but net neutral spheres should be

depicted with equal numbers of positive and negative charges. 1 (10 pts; 2, 2, 2, 2, 2)

44. The figure shows a mass spectrometer in which charged particles of different masses enter a region through a slit and

move perpendicular to a uniform magnetic field permeating the region (the gray dots represent the magnetic field

pointing out of the page). For each particle, the detector measures the distance between the entrance point to the

place where it strikes the bottom of the region. Assume 12C and 16O singly charged ions are accelerated to the same

velocity before entering the spectrometer. If the 16O ions strike the detector 5.0 m from the slit, at what point will the 12C ions hit it? 1 (8 pts) (TB#8)

45. A proton travels through a region with an electric and magnetic force. Name the force being exerted on the proton. If

the magnitude of the electric field (E) directed downwards is 4500 N/m and the magnitude of the magnetic field (B)

into the page is 2.0 T, determine the velocity (v) that allows the proton to travel in a straight trajectory. 1 (8 pts; 2, 6)

46. Describe how AC and DC motors differ in terms of their construction and the way speed is controlled. 1 (4 pts)

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Distance = _______________

Name of Force = _______________ v = _______________

____ / 30

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

5.77 m

2,300 m/s Lorentz (2)

• DC has brushes, AC does not (2)

• DC is controlled by varying current (thru armature windings), AC is controlled by varying

frequency (2)

Example Solution

𝑣 =4500

2= 2250 𝑚/𝑠

V Criteria for points (TIER): (+6) – Answer is correct (+5) – Incorrect significant figures (2 or >3) OR missing units (+4) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

Example Solution

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 5.0 × √16/12 = 5.77 𝑚

Distance Criteria for points (TIER): (+8) – Answer is correct (+7) – Incorrect significant figures (2 or >3) OR missing units (+6) – Incorrect significant figures (2 or >3) AND missing units (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

10

47. Two long solenoids carrying identical current in opposite directions are shown. If there is no magnetic field present

inside the inner solenoid, what can you determine about the number of turns per unit length of the inner solenoid and

the number of turns per unit length of the outer solenoid? 1 (4 pts)

48. Draw wires connecting the round terminals of the switches so that it satisfies the following two conditions: 1.) The

light is currently turned off. 2.) Flipping any one of the four switches turns on the light. 1 (8 pts)

49. A bank wants to install an alarm system with 3 movement sensors (S1, S2, S3). To prevent false alarms produced by a

single sensor activation, the alarm (Alarm) will be triggered only when at least 2 sensors activate simultaneously.

Draw a logic diagram and provide the simplified Boolean expression. Use the bolded terms above to label your

diagram. 1 (10 pts; 6, 4)

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

Alarm = _______________

____ / 22

Turns per unit length are equal (4)

S1S2+S2S3+S1S3

Diagram Criteria for points (ADDITIVE): (+4; +1 each) – Each logic gate is constructed and connected correctly (+2; +0.5 each) – S1, S2, S3, and Alarm are labeled Alarm Expression Criteria for points (TIER): (+4) – Answer is correct (+3) – Correct work shown (+2) – Incorrect work shown (+1) – Reasonable attempt

Diagram Criteria for points (ADDITIVE): (+4) – light is currently OFF (+4) – flipping and switch turns light ON

11

LAB 1 (Circuit Analysis): At this station, you will analyze a circuit. Perform the following tasks on the circuit built on the

breadboard provided. 1 (47 pts; 15, 12, 20)

50. Draw the circuit constructed on the breadboard. You may represent the power supply as a battery with a labeled

voltage. Represent the potentiometer (trimpot) as a variable resistor. (15 pts)

51. Calculate the theoretical resistance of the variable resistor needed to balance the circuit such that zero current flows

through resistor R3. (12 pts)

52. Part 1: Measure the resistance of the variable resistor needed to balance the circuit such that zero current flows

through resistor R3. Part 2: Explain how you manipulated the circuit to make that measurement. (20 pts; 8; 12)

Measured R = ___________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

____ / 47

3-bit D/A converter diagram Criteria for points (ADDITIVE): (+2) – 5V source present (and ground) (+8) – Correct resistor configuration (+5: +1 each) – Correct resistor values (R5 is variable resistor)

Criteria for points (TIER): (+12) – Answer is correct (+6) – Correct work shown (+3) – Incorrect work shown (+1) – Reasonable attempt

~5 k

Example Solution 20𝑘

10𝑘=

10𝑘

𝑅5

R5 = 5 k

• Placed ammeter in series with R3 (4)

• Varied trimpot resistance (4)

• Until no current was measured (4)

12

LAB 2 (Digital Logic): At this station, you will draw and construct a circuit. One way to achieve Digital-to-Analog (D/A)

conversion is to build an R-2R ladder. 1 (55 pts; 30, 25) (TB#9)

53. Part 1: Draw a 3-bit D/A converter circuit using an R-2R ladder that obeys the provided output table. The component

values and quantities MUST be representative of the physical components provided. The following criteria should be

met:

• All sources and components include values labeled where appropriate.

• Three digital inputs representing each bit of data drawn as SPDT switches that provide 5 V or 0 V in the ON

and OFF positions, respectively. Label the inputs “B0, B1, and B2”.

• The analog output voltage obeys the output table provided below.

Part 2: Derive a formula for the analog output (Vout) in terms of the digital inputs (B2, B1, B0). (30 pts; 18, 12)

54. Construct a 3-bit D/A converter circuit that uses an R-2R ladder from Q53. The component values and quantities

MUST be representative of the physical components provided. When constructed, call over a proctor to demonstrate

the circuit. You will only receive credit if the proctor initials below indicating the circuit functions properly. There is

NO PARTIAL CREDIT. The following criteria should be met: (25 pts)

• Three digital inputs representing each bit of data are constructed using SPDT switches that provide 5 V or 0 V

in the ON and OFF positions, respectively.

• The analog output obeys the output table given in Q53.

Output Table

B2 B1 B0 Digital Vout

0 0 0 0 0.0 V

0 0 1 1 0.625 V

0 1 0 2 1.25 V

0 1 1 3 1.875 V

1 0 0 4 2.5 V

1 0 1 5 3.125 V

1 1 0 6 3.75 V

1 1 1 7 4.375 V

Initials = _______________

____ / 55

𝑉𝑜𝑢𝑡 =𝑉𝐵0

8+

𝑉𝐵1

4+

𝑉𝐵2

2

3-bit D/A converter diagram Criteria for points (ADDITIVE): (+3) – Three SPDT switches labeled B0, B1, and B2 (+3) – Ground is present and connected to SPDT terminals (+3) – 5V source is present and connected to SPDT terminals (+6; +1 each) – Correct resistor values and positions (+15) – Circuit obeys output table Vout Criteria for points (TIER): Grade each answer separately using this rubric (+12) – Answer is correct (+6) – Correct work shown (+3) – Incorrect work shown (+1) – Reasonable attempt

Varies