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A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ω is used to measure the amplitude of a step force. The voltage output is measured using a digital voltmeter with input impedance 1013 Ω connected across the crystal. All other capacitances and resistances may be neglected. A step force of 2 N is applied from direction “3” on the crystal. The time in milliseconds within which the voltmeter should sample the crystal output voltage so that the drop from the peak value is no more than 0.12V is ________.
(Important - Enter only the numerical value in the answer)
Correct answer is '2.48'. Can you explain this answer?
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A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = ...
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A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = ...
Understanding the Problem
To solve the problem, we need to analyze the voltage response of the piezoelectric crystal when a step force is applied.
Given Parameters
- d33 = 150 pC/N (piezoelectric coefficient)
- Ccrystal = 25 pF (capacitance of the crystal)
- Rcrystal = 10^10 Ω (resistance of the crystal)
- Input impedance of voltmeter = 10^13 Ω
- Step force = 2 N
- Allowable voltage drop = 0.12 V
Calculating the Peak Voltage
1. The peak voltage (V_peak) generated by the applied force can be calculated using the formula:
V_peak = d33 × Force / Ccrystal
Substituting the values, we have:
V_peak = (150 × 10^-12 C/N) × (2 N) / (25 × 10^-12 F) = 12 V
Time Constant Calculation
2. The effective resistance (R_effective) seen by the crystal can be calculated by considering the resistances in parallel:
R_effective = (Rcrystal × Input impedance) / (Rcrystal + Input impedance)
Now substituting the values:
R_effective = (10^10 × 10^13) / (10^10 + 10^13) ≈ 9.09 × 10^9 Ω
3. The time constant (τ) for the RC circuit formed by the crystal capacitance and the effective resistance is given by:
τ = R_effective × Ccrystal
So, τ ≈ (9.09 × 10^9 Ω) × (25 × 10^-12 F) = 0.22725 seconds
Voltage Drop Calculation
4. The voltage across the capacitor as a function of time is given by:
V(t) = V_peak × exp(-t/τ)
To find the time at which the voltage drops by 0.12 V, we set up the equation:
12 V - 0.12 V = 12 V × exp(-t/0.22725)
Simplifying gives us:
0.01 = exp(-t/0.22725)
Taking the natural logarithm of both sides:
-t/0.22725 = ln(0.01)
Thus:
t = -0.22725 × ln(0.01) ≈ 2.48 seconds
Final Result
The time in milliseconds within which the voltmeter should sample the crystal output voltage to ensure the drop from the peak value is no more than 0.12 V is 2.48 ms.
To solve the problem, we need to analyze the voltage response of the piezoelectric crystal when a step force is applied.
Given Parameters
- d33 = 150 pC/N (piezoelectric coefficient)
- Ccrystal = 25 pF (capacitance of the crystal)
- Rcrystal = 10^10 Ω (resistance of the crystal)
- Input impedance of voltmeter = 10^13 Ω
- Step force = 2 N
- Allowable voltage drop = 0.12 V
Calculating the Peak Voltage
1. The peak voltage (V_peak) generated by the applied force can be calculated using the formula:
V_peak = d33 × Force / Ccrystal
Substituting the values, we have:
V_peak = (150 × 10^-12 C/N) × (2 N) / (25 × 10^-12 F) = 12 V
Time Constant Calculation
2. The effective resistance (R_effective) seen by the crystal can be calculated by considering the resistances in parallel:
R_effective = (Rcrystal × Input impedance) / (Rcrystal + Input impedance)
Now substituting the values:
R_effective = (10^10 × 10^13) / (10^10 + 10^13) ≈ 9.09 × 10^9 Ω
3. The time constant (τ) for the RC circuit formed by the crystal capacitance and the effective resistance is given by:
τ = R_effective × Ccrystal
So, τ ≈ (9.09 × 10^9 Ω) × (25 × 10^-12 F) = 0.22725 seconds
Voltage Drop Calculation
4. The voltage across the capacitor as a function of time is given by:
V(t) = V_peak × exp(-t/τ)
To find the time at which the voltage drops by 0.12 V, we set up the equation:
12 V - 0.12 V = 12 V × exp(-t/0.22725)
Simplifying gives us:
0.01 = exp(-t/0.22725)
Taking the natural logarithm of both sides:
-t/0.22725 = ln(0.01)
Thus:
t = -0.22725 × ln(0.01) ≈ 2.48 seconds
Final Result
The time in milliseconds within which the voltmeter should sample the crystal output voltage to ensure the drop from the peak value is no more than 0.12 V is 2.48 ms.
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A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer?
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A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer?.
A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer?.
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ample number of questions to practice A barium titanate piezoelectric crystal with d33=150 pC/N, Ccrystal = 25 pF and Rcrystal = 1010 Ωis used to measure the amplitude of a step force. The voltage output is measured using adigital voltmeter with input impedance 1013 Ω connected across the crystal. All othercapacitances and resistances may be neglected. A step force of 2 N is applied from direction“3” on the crystal. The time in milliseconds within which the voltmeter should sample thecrystal output voltage so that the drop from the peak value is no more than 0.12V is________.(Important - Enter only the numerical value in the answer)Correct answer is '2.48'. Can you explain this answer? tests, examples and also practice GATE tests.
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