 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

Let's address each part of the problem step by step.

### Given:
- The temperature \( T(t) \) varies sinusoidally from 20°C to 100°C.
- The period \( T \) 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:

\[
T(t) = T_{\text{avg}} + A \cdot \sin(\omega t)
\]

Where:
- \( T_{\text{avg}} \) is the average temperature.
- \( A \) is the amplitude of the sinusoidal part.
- \( \omega \) is the angular frequency.

The average value \( T_{\text{avg}} \) is simply the midpoint between the minimum and maximum temperatures:

\[
T_{\text{avg}} = \frac{T_{\text{max}} + T_{\text{min}}}{2} = \frac{100^\circ C + 20^\circ C}{2} = 60^\circ C
\]

So, the average temperature is **60°C**.

### (b) Find the amplitude of the sinusoidal part of the temperature waveform.
The amplitude \( A \) is the distance from the average value to the maximum (or minimum) value:

\[
A = \frac{T_{\text{max}} - T_{\text{avg}}}{1} = 100^\circ C - 60^\circ C = 40^\circ C
\]

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 \( T \) of the sinusoidal waveform is 30 seconds. The angular frequency \( \omega \) is related to the period by:

\[
\omega = \frac{2\pi}{T} = \frac{2\pi}{30 \, \text{s}} = \frac{\pi}{15} \, \text{rad/s}
\]

So, the angular frequency is **\(\frac{\pi}{15}\) rad/s**.

### (d) Find the average value of the system output voltage waveform.
Assume the system output voltage \( V(t) \) is directly proportional to the temperature waveform \( T(t) \), i.e.,

\[
V(t) = k \cdot T(t)
\]

Where \( k \) is a proportionality constant. The average value of the system output voltage \( V_{\text{avg}} \) would be:

\[
V_{\text{avg}} = k \cdot T_{\text{avg}} = k \cdot 60^\circ C
\]

Thus, the average value of the system output voltage is **\( 60k \)** (in units consistent with the proportionality constant \( k \)).

### (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 \( V_{\text{amp}} \) is directly proportional to the amplitude of the temperature waveform:

\[
V_{\text{amp}} = k \cdot A = k \cdot 40^\circ C
\]

So, the amplitude of the sinusoidal part of the system output voltage is **\( 40k \)**.

### Summary:
- (a) **Average temperature**: 60°C
- (b) **Amplitude of temperature waveform**: 40°C
- (c) **Angular frequency**: \( \frac{\pi}{15} \) rad/s
- (d) **Average system output voltage**: \( 60k \)
- (e) **Amplitude of system output voltage**: \( 40k \)

Let me know if you need any further explanation!

### Questions for Further Exploration:
1. How would the output voltage waveform change if the temperature varied in a non-sinusoidal manner?
2. What role does the proportionality constant \( k \) play in the system output?
3. If the period of the sinusoidal temperature variation were doubled, how would this affect the angular frequency?
4. How could you determine the phase shift of the sinusoidal temperature waveform if additional information were provided?
5. 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.

Explore the analysis of sinusoidal temperature and system output voltage waveforms, including finding average values, amplitudes, and angular frequencies. Understand the role of proportionality in system output. Suitable for advanced high school and college students.

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