Show that the output voltage vo of the circuit shown below is proportional to the derivative of the input voltage vi, If C = 10−5 F, what value of R is required if vo = -dvi/dt, assume the operational amplifier is ideal.

Here are the solutions to the questions.

4a. Two types of inductive transducers are:

• Self-inductive transducers: These transducers operate by changing their self-inductance due to the displacement of a ferromagnetic core or a change in the magnetic circuit.
  • Examples:
    • Variable reluctance transducer
    • Eddy current transducer
    • Proximity sensor
• Mutual-inductive transducers: These transducers operate by changing the mutual inductance between two or more coils, typically due to the displacement of a core.
  • Examples:
    • Linear Variable Differential Transformer (LVDT)
    • Rotary Variable Differential Transformer (RVDT)
    • Synchros

4b.

• Advantage of a.c. bridges over d.c. bridges used in strain gauges:
  • A.C. bridges are less susceptible to thermoelectric EMFs (stray DC voltages) that can cause errors in D.C. bridge measurements.
  • They can be used to measure reactive components (inductance and capacitance) in addition to resistance, offering more versatility.
  • They can offer higher sensitivity for detecting small changes in resistance, especially when coupled with phase-sensitive detectors.
• Disadvantage of a.c. bridges over d.c. bridges used in strain gauges:
  • A.C. bridges are generally more complex to design and operate, requiring A.C. excitation sources and phase-sensitive null detectors.
  • They are susceptible to errors caused by stray capacitance and inductance, which can be significant at higher frequencies and require careful shielding.
  • Balancing an A.C. bridge requires adjusting both magnitude and phase, making the balancing procedure more intricate than for D.C. bridges.

4c. Given:
Capacitance of crystal $C_{crystal} = 10^9 \, \text{F}$
Capacitance of cable $C_{cable} = 3 \times 10^{-10} \, \text{F}$
Charge constant of crystal $d = 4 \times 10^{-6} \, \text{C/cm}$
Oscilloscope resistance $R_{scope} = 1 \, \text{M}\Omega = 1 \times 10^6 \, \Omega$
Oscilloscope capacitance $C_{scope} = 10^{-10} \, \text{F}$
Harmonic deformation amplitude $x_0 = 10^{-3} \, \text{mm} = 10^{-4} \, \text{cm}$
Frequency $f = 200 \, \text{Hz}$

Step 1: Calculate the total capacitance ($C_{total}$).
The crystal, cable, and oscilloscope capacitances are in parallel.

$$C_{total} = C_{crystal} + C_{cable} + C_{scope}$$ $$C_{total} = 10^9 \, \text{F} + 3 \times 10^{-10} \, \text{F} + 10^{-10} \, \text{F}$$ $$C_{total} = 10^9 \, \text{F} + 4 \times 10^{-10} \, \text{F} \approx 10^9 \, \text{F}$$

(Note: The given crystal capacitance of $10^9 \, \text{F}$ is an extremely large value for a piezoelectric crystal, typically they are in pF or nF. We proceed with the given value.)

Step 2: Calculate the peak charge generated by the crystal ($Q_0$).

$$Q_0 = d \times x_0 = (4 \times 10^{-6} \, \text{C/cm}) \times (10^{-4} \, \text{cm})$$ $$Q_0 = 4 \times 10^{-10} \, \text{C}$$

Step 3: Calculate the output voltage amplitude ($V_o$).
For a piezoelectric transducer connected to a parallel load (resistance and capacitance), the output voltage amplitude is approximately $V_o = \frac{Q_0}{C_{total}}$ when the capacitive impedance dominates the resistive impedance, which is the case here due to the extremely large $C_{total}$.

$$V_o = \frac{4 \times 10^{-10} \, \text{C}}{10^9 \, \text{F}}$$ $$V_o = 4 \times 10^{-19} \, \text{V}$$

The amplitude of the output voltage is $\boxed{\text{4} \times \text{10}^{-19} \, \text{V}}$.

5a. Devices used for measuring power in rotating systems are referred to as dynamometers.

5b. Three (3) categories of dynamometers are:

• Absorption dynamometers (e.g., Prony brake, hydraulic dynamometer)
• Transmission dynamometers (e.g., torsion dynamometer, strain gauge dynamometer)
• Driving dynamometers (or Engine dynamometers, which can both absorb and drive)

5c. Given:
Strain gauge resistance $R = 120 \, \Omega$
Bridge supply voltage $V_s = 9 \, \text{V}$
Power transmitted $P = 5 \, \text{hp}$
Speed of shaft $N = 900 \, \text{rpm}$
Diameter of shaft $d = 2 \, \text{cm} = 0.02 \, \text{m}$
Modulus of rigidity $G = 8 \times 10^{10} \, \text{N/m}^2$
Gauge factor $GF = 2.5$

Step 1: Convert power to Watts and speed to rad/s.

$$P = 5 \, \text{hp} \times 745.7 \, \text{W/hp} = 3728.5 \, \text{W}$$ $$\omega = N \times \frac{2\pi}{60} = 900 \, \text{rpm} \times \frac{2\pi}{60} \, \text{rad/(s}\cdot\text{rpm)} = 30\pi \, \text{rad/s} \approx 94.248 \, \text{rad/s}$$

Step 2: Calculate the torque ($T$) transmitted by the shaft.

$$T = \frac{P}{\omega} = \frac{3728.5 \, \text{W}}{94.248 \, \text{rad/s}} \approx 39.56 \, \text{N}\cdot\text{m}$$

Step 3: Calculate the polar moment of inertia ($J$) of the shaft.
For a solid circular shaft, $J = \frac{\pi d^4}{32}$.

$$J = \frac{\pi (0.02 \, \text{m})^4}{32} = \frac{\pi \times 16 \times 10^{-8}}{32} = \frac{\pi}{2} \times 10^{-8} \, \text{m}^4 \approx 1.5708 \times 10^{-8} \, \text{m}^4$$

Step 4: Calculate the maximum shear stress ($\tau_{max}$) in the shaft.

$$\tau_{max} = \frac{T r}{J} = \frac{(39.56 \, \text{N}\cdot\text{m}) \times (0.01 \, \text{m})}{1.5708 \times 10^{-8} \, \text{m}^4} \approx 2.518 \times 10^7 \, \text{N/m}^2$$

Step 5: Calculate the shear strain ($\gamma$) and longitudinal strain ($\epsilon$).

$$\gamma = \frac{\tau_{max}}{G} = \frac{2.518 \times 10^7 \, \text{N/m}^2}{8 \times 10^{10} \, \text{N/m}^2} \approx 3.1475 \times 10^{-4}$$

For strain gauges placed at $45^\circ$ to the shaft axis, the longitudinal strain is $\epsilon = \frac{\gamma}{2}$.

$$\epsilon = \frac{3.1475 \times 10^{-4}}{2} = 1.57375 \times 10^{-4}$$

Step 6: Calculate the output voltage of the Wheatstone bridge ($V_o$).
For a full Wheatstone bridge with four active strain gauges (two in tension, two in compression), the output voltage is given by:

$$V_o = V_s \times GF \times \epsilon$$ $$V_o = 9 \, \text{V} \times 2.5 \times 1.57375 \times 10^{-4}$$ $$V_o \approx 0.003541 \, \text{V} = 3.541 \, \text{mV}$$

The output voltage is $\boxed{\text{3.541 mV}}$.

6a. A thermistor is a type of resistor whose electrical resistance changes significantly with temperature. It is typically made from semiconductor materials and is highly sensitive, exhibiting a large change in resistance for a small change in temperature, making it suitable for precise temperature measurement and control.

6b. The function of a thermostat as a controlling device is to sense the temperature of a system and automatically maintain it within a desired range. It does this by comparing the measured temperature to a set point and then activating or deactivating a heating or cooling mechanism to correct any deviation from the set point.

6c. Given:
Copper-constantan thermocouple, linear calibration.
EMF at $400 \, ^\circ C$ (with $0 \, ^\circ C$ reference) = $20.68 \, \text{mV}$.

Step 1: Determine the Seebeck coefficient ($S$) of the thermocouple.

$$S = \frac{\text{EMF}}{\text{Temperature difference}} = \frac{20.68 \, \text{mV}}{400 \, ^\circ C - 0 \, ^\circ C} = 0.0517 \, \text{mV/}^\circ C$$