Math Problem Statement
If the system is used to measure the temperature of a flowing gas the varies sinusoidally from 20°C to 100° with a period of 30s: a. Find the average value of the temperature waveform. b. Find the amplitude of the sinusoidal part of the temperature waveform. c. Find the angular frequency of the sinusoidal part of the temperature waveform. d. Find the average value of the system output voltage waveform. e. Find the amplitude of the sinusoidal part of the system output voltage waveform
Solution
Let's address each part of the problem step by step.
Given:
- The temperature varies sinusoidally from 20°C to 100°C.
- The period of the sinusoidal waveform is 30 seconds.
(a) Find the average value of the temperature waveform.
The temperature varies between 20°C and 100°C, so it can be described by the equation:
Where:
- is the average temperature.
- is the amplitude of the sinusoidal part.
- is the angular frequency.
The average value is simply the midpoint between the minimum and maximum temperatures:
So, the average temperature is 60°C.
(b) Find the amplitude of the sinusoidal part of the temperature waveform.
The amplitude is the distance from the average value to the maximum (or minimum) value:
So, the amplitude of the sinusoidal temperature waveform is 40°C.
(c) Find the angular frequency of the sinusoidal part of the temperature waveform.
The period of the sinusoidal waveform is 30 seconds. The angular frequency is related to the period by:
So, the angular frequency is rad/s.
(d) Find the average value of the system output voltage waveform.
Assume the system output voltage is directly proportional to the temperature waveform , i.e.,
Where is a proportionality constant. The average value of the system output voltage would be:
Thus, the average value of the system output voltage is (in units consistent with the proportionality constant ).
(e) Find the amplitude of the sinusoidal part of the system output voltage waveform.
The amplitude of the sinusoidal part of the system output voltage is directly proportional to the amplitude of the temperature waveform:
So, the amplitude of the sinusoidal part of the system output voltage is .
Summary:
- (a) Average temperature: 60°C
- (b) Amplitude of temperature waveform: 40°C
- (c) Angular frequency: rad/s
- (d) Average system output voltage:
- (e) Amplitude of system output voltage:
Let me know if you need any further explanation!
Questions for Further Exploration:
- How would the output voltage waveform change if the temperature varied in a non-sinusoidal manner?
- What role does the proportionality constant play in the system output?
- If the period of the sinusoidal temperature variation were doubled, how would this affect the angular frequency?
- How could you determine the phase shift of the sinusoidal temperature waveform if additional information were provided?
- What would be the impact on the system output voltage if the amplitude of the temperature variation increased?
Tip: Understanding how sinusoidal signals are transformed when passing through a system is fundamental in control systems and signal processing. Always ensure you correctly identify the amplitude, frequency, and phase of the signal.
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Math Problem Analysis
Mathematical Concepts
Sinusoidal Functions
Average Value
Amplitude
Angular Frequency
Proportionality
Formulas
Average Value formula
Amplitude formula
Angular Frequency formula
Proportionality formula
Theorems
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Suitable Grade Level
Advanced High School / College
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