Section 1: Conceptual Questions
1. Mechatronics is best described as the synergistic integration of which three primary engineering disciplines?
Solution
The correct answer is b) Mechanical, Electrical, Computer Control. Mechatronics is fundamentally defined by the integration of these three fields.
2. True or False: A "systems approach" in mechatronics encourages engineers to design and analyze components in isolation to prevent integration issues.
Solution
The correct answer is False. A systems approach emphasizes considering the interactions between components to optimize the overall system, not isolating them.
3. When two resistors are connected in parallel, their equivalent resistance will always be ________ than the smallest individual resistor.
Solution
The correct answer is less / smaller. Connecting resistors in parallel provides multiple paths for current, effectively reducing the total resistance.
4. In a steady-state DC circuit, a fully charged capacitor behaves like an:
Solution
The correct answer is b) Open circuit. Once fully charged, a capacitor blocks the flow of DC current.
5. An inductor's primary function is to store energy in a magnetic field and to ________ changes in current flowing through it.
Solution
The correct answer is oppose / resist. Inductors resist changes in current, which is why they store energy in a magnetic field.
6. For a silicon P-N junction diode to conduct current easily, it must be:
Solution
The correct answer is b) Forward biased. Forward biasing reduces the depletion region, allowing current to flow.
7. Which of the following is NOT an ideal characteristic of an Operational Amplifier (Op-Amp)?
Solution
The correct answer is c) Finite bandwidth. Ideal Op-Amps are assumed to have infinite bandwidth, meaning they can amplify all frequencies equally.
8. The term that describes the closeness of a measurement to the true value is _________, while the term describing the reproducibility of repeated measurements is ___________.
Solution
The correct answer is accuracy, precision. Accuracy is about correctness, and precision is about consistency.
9. True or False: A microcontroller is a self-contained computer system on a single chip, whereas a microprocessor is primarily just the Central Processing Unit (CPU).
Solution
The correct answer is True. Microcontrollers integrate CPU, memory, and I/O peripherals, making them suitable for embedded applications, while microprocessors are more general-purpose CPUs.
10. In an Arduino sketch, code placed in the `setup()` function executes only once at the beginning, while code in the `loop()` function executes continuously.
Solution
The correct answer is True. This describes the fundamental execution flow of an Arduino program.
11. True or False: An Analog-to-Digital Converter (ADC) produces a continuous output signal from a discrete input.
Solution
The correct answer is False. An ADC converts a continuous (analog) input into a discrete (digital) output. A Digital-to-Analog Converter (DAC) performs the opposite function.
12. Pulse Width Modulation (PWM) is used to achieve an average analog output by varying the:
Solution
The correct answer is c) Duty cycle of a square wave. By changing the proportion of ON time, the average voltage can be controlled.
13. Which programming construct is best suited for executing a block of code a known, fixed number of times?
Solution
The correct answer is c) `for` loop. `for` loops are ideal for definite iteration where the number of repetitions is known beforehand.
14. The Nyquist-Shannon sampling theorem states that the sampling rate must be at least ________ the highest frequency component of the signal to prevent aliasing.
Solution
The correct answer is twice / double. This ensures that enough data points are captured to accurately reconstruct the original signal.
15. What is the main characteristic that distinguishes an open-loop control system from a closed-loop control system?
Solution
The correct answer is b) The presence or absence of feedback from the system's output. Closed-loop systems use feedback to adjust their output, while open-loop systems do not.
16. A potentiometer is typically used to measure:
Solution
The correct answer is b) Rotational or linear position/displacement. Potentiometers are variable resistors whose resistance changes with physical movement.
17. True or False: An incremental optical encoder directly provides information about the absolute position of a shaft from power-up without needing to be homed.
Solution
The correct answer is False. Incremental encoders provide relative position changes (pulses) and require a homing procedure to establish an absolute reference. Absolute encoders provide absolute position immediately upon power-up.
18. Hall-effect sensors are primarily used to detect:
Solution
The correct answer is c) Magnetic fields. Hall-effect sensors produce a voltage proportional to the strength of a magnetic field perpendicular to the sensor.
19. _________ are transducers that convert mechanical force or strain into an electrical signal, commonly used in load cells.
Solution
The correct answer is Strain gauges. Strain gauges change their electrical resistance in response to applied mechanical strain.
20. Which type of sensor generates a voltage directly in response to mechanical stress or pressure?
Solution
The correct answer is c) Piezoelectric sensor. Piezoelectric materials produce an electric charge when mechanical stress is applied.
21. True or False: A Thermocouple measures temperature based on the change in its electrical resistance.
Solution
The correct answer is False. Thermocouples measure temperature based on the Seebeck effect, which generates a voltage difference between two dissimilar metals at different temperatures. Resistance Temperature Detectors (RTDs) and thermistors measure temperature based on resistance changes.
22. Which component is typically used as a switch to control a high-power AC appliance with a low-power DC signal from a microcontroller?
Solution
The correct answer is c) Relay. Relays are electromechanical switches that allow a small current to control a much larger current, often used to interface low-power control circuits with high-power loads.
23. Stepper motors are generally preferred over DC motors when the application requires precise _________ position control without continuous feedback.
Solution
The correct answer is open-loop. Stepper motors move in discrete steps and can achieve precise positioning in open-loop systems as long as the load does not cause them to lose steps.
24. Servomotors are known for their:
Solution
The correct answer is b) Precise closed-loop position control. Servomotors use feedback (e.g., from an encoder) to accurately control their position, velocity, or torque.
25. I2C (Inter-Integrated Circuit) protocol requires how many signal wires for data transmission (excluding power and ground)?
Solution
The correct answer is b) 2. I2C uses two signal lines: SDA (Serial Data Line) and SCL (Serial Clock Line).
26. What is the primary advantage of using a microcontroller in a mechatronic system compared to traditional analog or mechanical control?
Solution
The primary advantage is the flexibility and programmability it offers. Microcontrollers allow for complex control algorithms, adaptive behavior, easy modification of system logic (via software updates), and integration of multiple sensors and actuators, leading to more sophisticated and versatile systems.
27. Briefly explain the concept of "aliasing" in digital signal processing and how it can be avoided.
Solution
Aliasing occurs when an analog signal is sampled at a rate lower than twice its highest frequency component (Nyquist rate). This causes higher frequencies in the original signal to appear as lower, incorrect frequencies in the digitized signal, leading to distortion. It can be avoided by ensuring the sampling rate is at least twice the maximum frequency of interest, often by using an anti-aliasing filter before the ADC to remove frequencies above half the sampling rate.
28. What is the main function of an H-bridge circuit, and why is it essential for controlling DC motors in many mechatronic applications?
Solution
The main function of an H-bridge circuit is to allow a DC motor to be driven in both forward and reverse directions, and often to control its speed using PWM. It is essential because DC motors require a change in voltage polarity to reverse direction, and an H-bridge provides this capability using electronic switches, enabling precise and flexible motor control from a microcontroller.
29. Differentiate between a sensor and a transducer, providing an example for each.
Solution
A sensor is a device that detects or measures a physical quantity (e.g., temperature, light, pressure) and responds to it. An example is a thermistor that changes resistance with temperature.
A transducer is a device that converts energy from one form to another. While all sensors are transducers (converting physical phenomena into electrical signals), not all transducers are sensors. An example of a transducer that is also a sensor is a microphone (converts sound energy to electrical energy). An example of a transducer that is not typically considered a sensor is a loudspeaker (converts electrical energy to sound energy).
30. Explain the concept of "feedback" in a control system and why it is fundamental to achieving precise control in mechatronics.
Solution
Feedback in a control system involves measuring the actual output of the system and comparing it to the desired input (setpoint). The difference (error) is then used by the controller to adjust the system's input, thereby reducing the error and bringing the output closer to the desired value. It is fundamental to achieving precise control because it allows the system to self-correct for disturbances, uncertainties, and variations, ensuring accuracy and stability that open-loop systems cannot provide.
31. What is the role of a filter in signal conditioning, and why might it be necessary in a mechatronic system?
Solution
A filter in signal conditioning is used to selectively pass or block certain frequencies from a signal. It is necessary in mechatronic systems to remove unwanted noise (e.g., high-frequency electrical interference) or to isolate specific frequency components of interest from a sensor signal, ensuring that the signal processed by the microcontroller is clean and accurate.
32. Describe the difference between a digital input and an analog input on a microcontroller like an Arduino.
Solution
A digital input can only read two states: HIGH (typically 5V or 3.3V) or LOW (0V). It's used for binary signals like buttons (pressed/not pressed) or switches (on/off). An analog input can read a continuous range of voltage values (e.g., 0V to 5V) and convert them into discrete numerical values (e.g., 0-1023 for a 10-bit ADC). It's used for sensors that provide varying outputs like temperature, light, or potentiometer readings.
33. What is the purpose of a "baud rate" setting in serial communication, and why must it match between communicating devices?
Solution
The baud rate defines the rate at which data is transferred over a serial communication line, measured in bits per second (bps). It must match between communicating devices (e.g., microcontroller and computer) so that both devices interpret the incoming data bits at the same speed. If the baud rates don't match, the receiving device will sample the data at the wrong times, leading to corrupted or unintelligible data.
34. Explain the difference between a "pull-up" and a "pull-down" resistor in a digital input circuit.
Solution
A pull-up resistor connects an input pin to the positive supply voltage (VCC), ensuring the pin is HIGH when a switch is open. When the switch closes, it connects the pin to ground (LOW). A pull-down resistor connects an input pin to ground (GND), ensuring the pin is LOW when a switch is open. When the switch closes, it connects the pin to VCC (HIGH). Both prevent the input pin from "floating" and picking up electrical noise.
35. What is the function of a PID (Proportional-Integral-Derivative) controller in a closed-loop system?
Solution
A PID controller is a widely used feedback control loop mechanism that calculates an "error" value as the difference between a desired setpoint and a measured process variable. It then attempts to minimize this error by adjusting the process control inputs based on three components:
Proportional (P): Reacts to the current error.
Integral (I): Accounts for past errors, helping to eliminate steady-state errors.
Derivative (D): Predicts future errors based on the rate of change of the current error, providing damping.
36. What is the primary purpose of an actuator in a mechatronic system? Provide two examples of common actuators.
Solution
The primary purpose of an actuator is to convert an electrical signal (or other control signal) into a physical action or motion. It is the component that performs the work in a mechatronic system.
Examples: DC motor, stepper motor, servomotor, solenoid, pneumatic cylinder, hydraulic cylinder, voice coil motor.
37. Explain the difference between a normally open (NO) and a normally closed (NC) switch.
Solution
A normally open (NO) switch has contacts that are open (no current flow) when the switch is in its default, unactuated state. When actuated, the contacts close, allowing current to flow.
A normally closed (NC) switch has contacts that are closed (allowing current flow) when the switch is in its default, unactuated state. When actuated, the contacts open, breaking the circuit and stopping current flow.
38. What is a "dead zone" in a sensor or control system, and how can it affect system performance?
Solution
A "dead zone" (or deadband) refers to a range of input values for which a sensor or control system produces no output or response. This can lead to a lack of sensitivity around a certain operating point, causing the system to be unresponsive to small changes, potentially leading to oscillations or poor control accuracy.
39. Describe the function of a "debounce" routine in microcontroller programming when reading mechanical switches. Why is it necessary?
Solution
A debounce routine is a software or hardware technique used to eliminate false readings from mechanical switches. When a mechanical switch is pressed or released, its contacts "bounce" for a short period, creating multiple rapid ON/OFF transitions instead of a single clean transition. Debouncing ensures that the microcontroller registers only one stable press or release event, preventing erratic behavior. It's necessary because microcontrollers are fast enough to detect these rapid bounces, which would be misinterpreted as multiple presses.
40. What is the fundamental difference between sequential logic and combinational logic in digital circuits?
Solution
Combinational logic circuits produce an output that depends solely on the current input values. They have no memory.
Sequential logic circuits produce an output that depends on both the current input values and the past sequence of inputs (i.e., they have memory). Flip-flops and registers are examples of sequential logic elements.
41. Explain the concept of "hysteresis" in a sensor or measurement system.
Solution
Hysteresis occurs when the output of a system depends not only on the current input but also on the history of the input. In sensors, it means the output reading for a given input value is different depending on whether the input was increasing or decreasing to reach that value. This can lead to inaccuracies and non-repeatability in measurements.
42. What is the purpose of a voltage regulator in an electronic circuit?
Solution
A voltage regulator is an electronic circuit that maintains a constant output voltage regardless of variations in the input voltage or changes in the load current. This is crucial for sensitive electronic components that require a stable power supply to function correctly.
43. Briefly describe how a photoresistor (LDR) works and what physical quantity it senses.
Solution
A photoresistor (Light Dependent Resistor or LDR) is a passive component whose resistance decreases with increasing light intensity. It works based on the photoconductive effect, where light energy causes electrons in the semiconductor material to move into the conduction band, increasing conductivity. It senses light intensity.
44. What is the main advantage of using a servomotor over a stepper motor for applications requiring smooth, continuous rotation and high torque at varying speeds?
Solution
The main advantage of a servomotor is its ability to provide smooth, continuous rotation, precise speed control, and high torque across a wide range of speeds, especially under varying loads, due to its closed-loop feedback system. Stepper motors are better for precise step-by-step positioning but can lose steps under load and typically offer less smooth motion at high speeds.
45. Explain the concept of "pull-up" resistors in the context of I2C communication lines (SDA and SCL).
Solution
In I2C communication, both the SDA (Serial Data) and SCL (Serial Clock) lines are "open-drain" or "open-collector" outputs, meaning they can only pull the line LOW. To ensure the lines return to a HIGH state when no device is actively pulling them LOW, pull-up resistors are required. These resistors connect the SDA and SCL lines to the positive supply voltage (VCC), providing a default HIGH state and allowing devices to release the line to go HIGH.
46. What is the purpose of a "checksum" or "CRC" (Cyclic Redundancy Check) in digital communication?
Solution
The purpose of a checksum or CRC in digital communication is to detect errors that may occur during data transmission. A calculated value (checksum/CRC) is appended to the data packet by the sender. The receiver then performs the same calculation on the received data. If the calculated checksum/CRC matches the received one, it indicates that the data was likely transmitted without error. If they don't match, it signals data corruption.
47. Describe one advantage and one disadvantage of using pneumatic actuators (e.g., air cylinders) in mechatronic systems.
Solution
Advantage: Pneumatic actuators are generally fast, clean (no oil leaks like hydraulics), relatively inexpensive, and safe to operate (air is non-flammable). They are good for high-speed, repetitive tasks.
Disadvantage: They are difficult to achieve precise position control without complex additional components (due to air compressibility), can be noisy, and require a constant supply of compressed air, which can be inefficient.
48. In the context of microcontrollers, what is the role of the "firmware"?
Solution
Firmware is a specific type of software that is permanently programmed into a hardware device's non-volatile memory (like a microcontroller's flash memory). Its role is to provide the low-level control for the hardware, enabling it to perform its basic functions and interact with other components. For microcontrollers, the Arduino sketch you upload is essentially its firmware.
49. What is the "resolution" of a digital output (e.g., a PWM signal) and how does it relate to the number of bits used?
Solution
The "resolution" of a digital output refers to the number of discrete steps or levels that the output signal can achieve. For a PWM signal, it determines how finely the average voltage or duty cycle can be controlled. It directly relates to the number of bits used: an N-bit PWM output can produce 2N distinct levels (e.g., 8-bit PWM has 256 levels).
50. Why is mechanical advantage often incorporated into mechatronic systems (e.g., using gears or levers)?
Solution
Mechanical advantage is incorporated to amplify force or torque, or to change speed and direction of motion. This allows smaller, less powerful (and often less expensive) motors to drive larger loads or achieve higher precision. For example, a gearbox provides mechanical advantage to increase torque at the expense of speed, enabling a small motor to move a heavy robotic arm.
Section 2: Problem-Solving Questions
51. Consider a circuit with a 12 V DC power supply, a 1.8 kΩ resistor, and a silicon diode (assume 0.7 V forward voltage drop) connected in series.
a) What is the voltage across the 1.8 kΩ resistor?
b) Calculate the current flowing through the circuit.
c) If the diode were replaced with an ideal wire (short circuit), how would the current in the circuit change? Briefly explain.
Solution
a) Voltage across the 1.8 kΩ resistor:
In a series circuit with a forward-biased silicon diode, the voltage drop across the diode is approximately 0.7 V.
The voltage across the resistor (VR) is the supply voltage minus the diode's forward voltage drop:
VR = Vsupply - Vdiode VR = 12 V - 0.7 V VR = 11.3 V
b) Current flowing through the circuit:
Using Ohm's Law (I = V/R) for the resistor:
I = VR / R I = 11.3 V / 1.8 kΩ I = 11.3 V / 1800 Ω I ≈ 0.006278 A or 6.278 mA
c) If the diode were replaced with an ideal wire (short circuit), how would the current in the circuit change? Briefly explain.
If the diode were replaced by an ideal wire, its voltage drop would become 0 V. The entire supply voltage (12 V) would then drop across the 1.8 kΩ resistor.
New Current = Vsupply / R New Current = 12 V / 1800 Ω New Current ≈ 0.006667 A or 6.667 mA
The current in the circuit would increase from approximately 6.278 mA to 6.667 mA. This is because replacing the diode with an ideal wire removes the 0.7 V voltage drop, effectively increasing the voltage available across the resistor, and thus increasing the current according to Ohm's Law.
52. A 220 μF capacitor is charged to 12 V.
a) What is the maximum charge (in Coulombs) the capacitor can store at this voltage?
b) What is the energy (in Joules) stored in the capacitor when it is charged to 12 V?
c) If this charged capacitor is then connected to a light bulb (modeled as a resistor), would the light bulb illuminate instantly and then dim, or gradually brighten? Explain why.
Solution
a) Maximum charge (in Coulombs) the capacitor can store:
Q = C * V Q = (220 * 10-6 F) * 12 V Q = 0.00264 C
b) Energy (in Joules) stored in the capacitor:
E = 0.5 * C * V2 E = 0.5 * (220 * 10-6 F) * (12 V)2 E = 0.5 * 220 * 10-6 * 144 E = 15840 * 10-6 J E = 0.01584 J
c) If this charged capacitor is then connected to a light bulb (modeled as a resistor), would the light bulb illuminate instantly and then dim, or gradually brighten? Explain why.
The light bulb would illuminate instantly and then gradually dim. When the charged capacitor is connected to the light bulb, it immediately begins to discharge its stored energy through the bulb. The current is highest at the beginning of discharge, causing the bulb to be brightest. As the capacitor discharges, its voltage and the current through the bulb decrease exponentially, causing the light to gradually dim until the capacitor is fully discharged.
53. An Arduino Mega uses a 10-bit ADC with a default 5.0 V reference. A linear position sensor connected to an analog input produces a voltage output that ranges from 0.2 V (at 0 mm position) to 4.8 V (at 100 mm position).
a) Calculate the voltage resolution (step size) of the Arduino's ADC.
b) If the Arduino reads a digital value of 410 from this position sensor, what is the analog voltage output by the sensor?
c) What physical position (in mm) does this digital reading (410) correspond to?
d) If the position is known to change at a maximum rate corresponding to a signal frequency of 10 Hz, what is the theoretical minimum sampling rate required for the ADC to accurately capture this position change?
Solution
a) Voltage resolution (step size) of the Arduino's ADC:
Number of Steps = 210 = 1024
Resolution = Vref / Number of Steps Resolution = 5.0 V / 1024 Resolution ≈ 0.0048828 V/step or 4.8828 mV/step
b) If the Arduino reads a digital value of 410, what is the analog voltage output by the sensor?
Analog Voltage = Digital Reading * Resolution Analog Voltage = 410 * (5.0 V / 1024) Analog Voltage ≈ 410 * 0.0048828 V Analog Voltage ≈ 2.002 V
c) What physical position (in mm) does this digital reading (410) correspond to?
The sensor voltage range is 4.8 V - 0.2 V = 4.6 V.
The sensor position range is 100 mm - 0 mm = 100 mm.
The voltage per mm sensitivity of the sensor is 4.6 V / 100 mm = 0.046 V/mm.
The voltage read (2.002 V) is relative to the sensor's minimum output (0.2 V).
Voltage above min = 2.002 V - 0.2 V = 1.802 V
Position = (Voltage above min) / (Voltage per mm) Position = 1.802 V / 0.046 V/mm Position ≈ 39.17 mm
d) Theoretical minimum sampling rate required:
According to Nyquist-Shannon theorem, Sampling Rate = 2 * Fmax
Sampling Rate = 2 * 10 Hz = 20 Hz
Therefore:
a) Voltage resolution: 0.00488 V/step
b) Analog voltage output: 2.002 V
c) Physical position: 39.17 mm
d) Minimum sampling rate: 20 Hz
54. You are designing a robotic arm joint that needs to rotate precisely between specific angles and hold its position against varying loads. The joint should complete a 180° rotation in 2 seconds.
a) Would a standard DC motor with a gearbox or a servomotor generally be more suitable for this application, and why?
b) If you decide to use a MOSFET H-bridge to control a DC motor for this joint, explain the primary function of the H-bridge and how it allows for bidirectional control of the motor.
c) Suppose the robotic arm needs to report its current joint angle to a central controller using Serial Peripheral Interface (SPI) communication. If the microcontroller sends a 16-bit angle value (2 bytes) and the SPI clock speed is 1 MHz, approximately how long would it take to transmit one angle value? (Assume no overhead for chip select or other protocol specific delays, just data bits).
Solution
a) Motor suitability:
A servomotor would generally be more suitable for this application.
Justification: Servomotors are designed for precise angular positioning and holding torque, especially against varying loads, due to their integrated feedback mechanism (encoder) and closed-loop control. While a DC motor with a gearbox can provide torque, it lacks the inherent precision and holding capability of a servomotor without additional external feedback and control loops, which would add complexity.
b) Primary function of an H-bridge for DC motor control:
The primary function of an H-bridge is to allow a DC motor to be driven in both forward and reverse directions, and often to control its speed (using PWM). It achieves bidirectional control by having four switching elements (e.g., MOSFETs or transistors) arranged in an "H" configuration. By selectively turning ON pairs of switches, current can be directed through the motor in either direction. For example, turning on the top-left and bottom-right switches allows current to flow one way, while turning on the top-right and bottom-left switches reverses the current flow, thus reversing the motor's direction.
c) Time to transmit one angle value via SPI:
Total bits to transmit = 16 bits (for 2 bytes)
SPI Clock Speed = 1 MHz = 1,000,000 bits per second (bps)
Time = Total Bits / Baud Rate Time = 16 bits / 1,000,000 bps Time = 0.000016 seconds or 16 μs
Therefore:
a) Motor suitability: Servomotor (with justification)
b) H-bridge explanation: (as above)
c) Transmission time: 16 μs
55. A robotic arm needs to pick up an object weighing 200 grams and move it through a vertical distance of 0.5 meters.
a) Calculate the minimum work (in Joules) required to lift the object. (Assume g = 9.81 m/s2).
b) If the motor lifting the object has an efficiency of 75%, how much electrical energy (in Joules) must be supplied to the motor to perform this lift?
c) If the motor operates at 12 V and draws an average current of 0.5 A during the lift, how long (in seconds) would the lift take?
Solution
a) Minimum work required to lift the object:
Work (W) = Force (F) × Distance (d)
Force (F) = mass (m) × gravity (g)
m = 200 grams = 0.2 kg g = 9.81 m/s2 d = 0.5 meters
F = 0.2 kg * 9.81 m/s2 = 1.962 N W = 1.962 N * 0.5 m = 0.981 J
b) Electrical energy supplied to the motor (given 75% efficiency):
Efficiency (η) = Work Output / Energy Input
Energy Input = Work Output / Efficiency
Work Output = 0.981 J Efficiency = 75% = 0.75
Energy Input = 0.981 J / 0.75 = 1.308 J
c) How long would the lift take (in seconds)?
Electrical Power (P) = Voltage (V) × Current (I)
Energy (E) = Power (P) × Time (t)
So, Time (t) = Energy (E) / Power (P)
V = 12 V I = 0.5 A E = 1.308 J (from part b)
P = 12 V * 0.5 A = 6 W t = 1.308 J / 6 W ≈ 0.218 s
Therefore:
a) Minimum work required: 0.981 J
b) Electrical energy supplied: 1.308 J
c) Time taken for the lift: 0.218 s
56. A simple series circuit consists of a 9 V battery and two resistors, R1 = 2.2 kΩ and R2 = 4.7 kΩ.
a) What is the total current flowing through the circuit?
b) Calculate the voltage drop across R1.
c) What is the power dissipated by R2?
Solution
a) Total current flowing through the circuit:
Total Resistance (Rtotal) = R1 + R2 = 2.2 kΩ + 4.7 kΩ = 6.9 kΩ = 6900 Ω
Itotal = Vsupply / Rtotal Itotal = 9 V / 6900 Ω Itotal ≈ 0.001304 A or 1.304 mA
b) Voltage drop across R1:
VR1 = Itotal * R1 VR1 = 0.001304 A * 2200 Ω VR1 ≈ 2.8688 V
c) Power dissipated by R2:
PR2 = Itotal2 * R2 PR2 = (0.001304 A)2 * 4700 Ω PR2 = (1.700416 * 10-6) * 4700 PR2 ≈ 0.00799 W or 7.99 mW
Therefore:
a) Total current: 1.304 mA
b) Voltage across R1: 2.869 V
c) Power dissipated by R2: 7.99 mW
57. A 100 mH inductor has a current flowing through it that changes from 0 A to 2 A in 50 ms.
a) Calculate the average voltage induced across the inductor during this period.
b) What is the energy stored in the inductor when the current reaches 2 A?
Solution
a) Average voltage induced across the inductor:
dI/dt = ΔI / Δt = (2 A - 0 A) / (50 * 10-3 s) = 2 A / 0.05 s = 40 A/s V = L * (dI/dt) V = (100 * 10-3 H) * (40 A/s) V = 4 V
b) Energy stored in the inductor when the current reaches 2 A:
E = 0.5 * L * I2 E = 0.5 * (100 * 10-3 H) * (2 A)2 E = 0.5 * 0.1 H * 4 A2 E = 0.2 J
Therefore:
a) Average induced voltage: 4 V
b) Energy stored: 0.2 J
58. A 47 μF capacitor is charged through a 10 kΩ resistor by a 20 V DC supply.
a) Calculate the time constant (τ) of this RC circuit.
b) What is the voltage across the capacitor after one time constant?
Solution
a) Time constant (τ) of this RC circuit:
τ = R * C τ = 10 * 103 Ω * 47 * 10-6 F τ = 0.47 s
b) Voltage across the capacitor after one time constant:
For a charging capacitor, the voltage across it after time t is given by:
Vc(t) = Vsupply * (1 - e-t/τ)
After one time constant (t = τ):
Vc(τ) = Vsupply * (1 - e-τ/τ) Vc(τ) = Vsupply * (1 - e-1) Vc(τ) = 20 V * (1 - 0.36788) Vc(τ) = 20 V * 0.63212 Vc(τ) ≈ 12.64 V
Therefore:
a) Time constant: 0.47 s
b) Voltage after one time constant: 12.64 V
59. An 8-bit Analog-to-Digital Converter (ADC) has a full-scale input range of 0 V to 2.55 V.
a) What is the voltage resolution (step size) of this ADC?
b) If the ADC reads a digital value of 150, what is the analog input voltage?
Solution
a) Voltage resolution (step size) of this ADC:
Number of Steps = 28 = 256
Full-scale range = 2.55 V - 0 V = 2.55 V
Resolution = Full-scale Range / Number of Steps Resolution = 2.55 V / 256 Resolution ≈ 0.00996 V/step or 9.96 mV/step
b) If the ADC reads a digital value of 150, what is the analog input voltage?
Analog Voltage = Digital Reading * Resolution Analog Voltage = 150 * (2.55 V / 256) Analog Voltage ≈ 150 * 0.00996 V Analog Voltage ≈ 1.494 V
Therefore:
a) Voltage resolution: 0.00996 V/step
b) Analog input voltage: 1.494 V
60. A DC motor is used to lift a 500 g mass by 1.2 meters. The motor has an efficiency of 80%.
a) Calculate the useful mechanical work done by the motor (in Joules). (Assume g = 9.81 m/s2).
b) How much electrical energy (in Joules) must be supplied to the motor to perform this task?
c) If the motor draws 0.8 A from a 6 V power supply, how long (in seconds) does it take to lift the mass?
Solution
a) Useful mechanical work done by the motor:
Work (W) = Force (F) * Distance (d) Force (F) = mass (m) * gravity (g) m = 500 g = 0.5 kg g = 9.81 m/s2 d = 1.2 meters
F = 0.5 kg * 9.81 m/s2 = 4.905 N W = 4.905 N * 1.2 m = 5.886 J
b) Electrical energy supplied to the motor:
Efficiency (η) = Work Output / Energy Input
Energy Input = Work Output / Efficiency
Work Output = 5.886 J Efficiency = 80% = 0.80
Energy Input = 5.886 J / 0.80 = 7.3575 J
c) How long does it take to lift the mass?
Electrical Power (P) = Voltage (V) * Current (I)
Energy (E) = Power (P) * Time (t)
So, Time (t) = Energy (E) / Power (P)
V = 6 V I = 0.8 A E = 7.3575 J (from part b)
P = 6 V * 0.8 A = 4.8 W t = 7.3575 J / 4.8 W ≈ 1.533 s
Therefore:
a) Useful mechanical work: 5.886 J
b) Electrical energy supplied: 7.3575 J
c) Time taken for the lift: 1.533 s
61. A voltage divider circuit uses a 15 V power supply. If Rtop = 5 kΩ and Rbottom = 10 kΩ, what is the voltage across Rbottom?
Solution
Total Resistance (Rtotal) = Rtop + Rbottom = 5 kΩ + 10 kΩ = 15 kΩ
Vbottom = Vsupply * (Rbottom / Rtotal) Vbottom = 15 V * (10 kΩ / 15 kΩ) Vbottom = 15 V * (10 / 15) Vbottom = 15 V * (2 / 3) Vbottom = 10 V
Therefore, the voltage across Rbottom is 10 V.
62. A 10 μF capacitor is charged from 0 V to 5 V in 20 ms. Calculate the average charging current.
Solution
Charge (Q) = C * V
Q = 10 * 10-6 F * 5 V = 50 * 10-6 C = 50 μC
Average Current (I) = Q / t
I = 50 * 10-6 C / (20 * 10-3 s) I = 50 / 20 * 10-3 A I = 2.5 * 10-3 A = 2.5 mA
Therefore, the average charging current is 2.5 mA.
63. An inductor has an inductance of 50 mH. If the voltage induced across it is 10 V, what is the rate of change of current (dI/dt) through the inductor?
Solution
V = L * (dI/dt) dI/dt = V / L dI/dt = 10 V / (50 * 10-3 H) dI/dt = 10 / 0.05 A/s dI/dt = 200 A/s
Therefore, the rate of change of current is 200 A/s.
64. A non-inverting Op-Amp amplifier has an input resistor (R1) of 2 kΩ and a feedback resistor (Rf) of 18 kΩ. If the input voltage (Vin) is 0.5 V, what is the output voltage (Vout)?
Solution
For an ideal non-inverting Op-Amp configuration, the output voltage is given by the formula:
Vout = Vin * (1 + Rf / R1) Vout = 0.5 V * (1 + 18 kΩ / 2 kΩ) Vout = 0.5 V * (1 + 9) Vout = 0.5 V * 10 Vout = 5 V
Therefore, the output voltage is 5 V.
65. A sensor outputs a voltage from 0 V to 3.3 V. If this sensor is connected to a 10-bit ADC with a 3.3 V reference, what is the digital reading when the sensor outputs 1.65 V?
Solution
Number of Steps = 210 = 1024
Resolution = Vref / Number of Steps = 3.3 V / 1024 ≈ 0.0032226 V/step
Digital Reading = Analog Voltage / Resolution Digital Reading = 1.65 V / (3.3 V / 1024) Digital Reading = 1.65 * 1024 / 3.3 Digital Reading = 512
Therefore, the digital reading is 512.
Additional Problem-Solving Questions
66. A DC motor is rated for 24 V and draws 1.5 A when operating at its nominal speed. If it produces 25 W of mechanical power, calculate its efficiency.
Solution
Electrical Power Input (Pin) = V * I
Pin = 24 V * 1.5 A = 36 W
Mechanical Power Output (Pout) = 25 W
Efficiency (η) = (Pout / Pin) * 100% η = (25 W / 36 W) * 100% η ≈ 69.44%
Therefore, the motor's efficiency is approximately 69.44%.
67. A temperature sensor has a transfer function of 10 mV/°C. If the sensor outputs 250 mV, what is the measured temperature?
Solution
Temperature = Output Voltage / Sensitivity Temperature = 250 mV / (10 mV/°C) Temperature = 25 °C
Therefore, the measured temperature is 25 °C.
68. A data acquisition system is sampling a vibration signal with a maximum frequency of 50 Hz. What is the minimum sampling interval (time between samples) required to avoid aliasing?
Solution
Minimum Sampling Rate (fs) = 2 * Fmax = 2 * 50 Hz = 100 Hz
Minimum Sampling Interval (Ts) = 1 / fs Ts = 1 / 100 Hz = 0.01 seconds or 10 ms
Therefore, the minimum sampling interval required is 0.01 seconds (or 10 ms).
69. A 3-bit Digital-to-Analog Converter (DAC) has a reference voltage of 8 V. If the digital input is 1012, what is the analog output voltage?
Solution
Number of Steps = 23 = 8
Resolution = Vref / Number of Steps = 8 V / 8 = 1 V/step
Digital input 1012 in decimal is 1*22 + 0*21 + 1*20 = 4 + 0 + 1 = 5
Analog Output Voltage = Digital Value * Resolution Analog Output Voltage = 5 * 1 V/step = 5 V
Therefore, the analog output voltage is 5 V.
70. A robotic gripper uses a servomotor to close its jaws. The motor needs to apply a torque of 0.2 Nm to close the jaws completely. If the motor's arm length is 5 cm, what force (in Newtons) does the motor apply at the end of its arm?
Solution
Torque (τ) = Force (F) * Radius (r) F = τ / r τ = 0.2 Nm r = 5 cm = 0.05 m
F = 0.2 Nm / 0.05 m F = 4 N
Therefore, the motor applies a force of 4 N at the end of its arm.