14.11 Determine the rms value of the voltage waveform shown in Figure 14-27. v(t) 3. 3. 2.
14.11 Determine the rms value of the voltage waveform shown in Figure 14-27. v(t) 3. 3. 2.
Introductory Circuit Analysis (13th Edition)
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Author:Robert L. Boylestad
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![**Title:**
Calculating the RMS Value of a Voltage Waveform
**Introduction:**
Understanding the root mean square (RMS) value of voltage waveforms is essential in electrical engineering, especially when analyzing AC circuits. Here, we examine the waveform depicted in Figure 14-27 and aim to calculate its RMS value.
**Problem Statement from Figure 14-27:**
Determine the RMS value of the voltage waveform shown.
**Graph Description:**
The waveform graph, labeled \( v(t) \) for voltage over time versus \( t \), illustrates a periodic signal with varying voltage values. Below is a detailed description of the graph:
- **Voltage \( v(t) \) Axis:**
- The vertical axis represents the voltage in unspecified units.
- It ranges from -3 to 3.
- **Time \( t \) Axis:**
- The horizontal axis represents time.
- It is marked with increments from 0 to 8.
- **Waveform Characteristics:**
- The waveform begins at the origin (0, 0).
- From \( t = 0 \) to \( t = 1 \), the voltage rises linearly from 0 to 3.
- From \( t = 1 \) to \( t = 3 \), the voltage stays constant at 3.
- From \( t = 3 \) to \( t = 4 \), the voltage drops linearly from 3 to -3.
- From \( t = 4 \) to \( t = 5 \), the voltage rises linearly back to 0.
- From \( t = 5 \) to \( t = 6 \), the voltage again rises linearly from 0 to 3.
- From \( t = 6 \) to \( t = 8 \), the voltage remains constant at 3.
**Conclusion:**
To find the RMS value of this waveform, use the formula:
\[
V_{\text{rms}} = \sqrt{\frac{1}{T} \int_{0}^{T} [v(t)]^2 \, dt}
\]
where \( T \) is the period of the waveform. By performing this calculation, you can determine the effective voltage value of the waveform as observed in a typical AC system.
**Further Exploration:**
This exercise provides a foundational understanding of RMS calculations, useful in various applications such as](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffcfa310c-2c98-4461-a46a-c39d6b0c5cb5%2F8b296036-77f5-4748-b956-3e37903f2f31%2Fm2uoi8_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:**
Calculating the RMS Value of a Voltage Waveform
**Introduction:**
Understanding the root mean square (RMS) value of voltage waveforms is essential in electrical engineering, especially when analyzing AC circuits. Here, we examine the waveform depicted in Figure 14-27 and aim to calculate its RMS value.
**Problem Statement from Figure 14-27:**
Determine the RMS value of the voltage waveform shown.
**Graph Description:**
The waveform graph, labeled \( v(t) \) for voltage over time versus \( t \), illustrates a periodic signal with varying voltage values. Below is a detailed description of the graph:
- **Voltage \( v(t) \) Axis:**
- The vertical axis represents the voltage in unspecified units.
- It ranges from -3 to 3.
- **Time \( t \) Axis:**
- The horizontal axis represents time.
- It is marked with increments from 0 to 8.
- **Waveform Characteristics:**
- The waveform begins at the origin (0, 0).
- From \( t = 0 \) to \( t = 1 \), the voltage rises linearly from 0 to 3.
- From \( t = 1 \) to \( t = 3 \), the voltage stays constant at 3.
- From \( t = 3 \) to \( t = 4 \), the voltage drops linearly from 3 to -3.
- From \( t = 4 \) to \( t = 5 \), the voltage rises linearly back to 0.
- From \( t = 5 \) to \( t = 6 \), the voltage again rises linearly from 0 to 3.
- From \( t = 6 \) to \( t = 8 \), the voltage remains constant at 3.
**Conclusion:**
To find the RMS value of this waveform, use the formula:
\[
V_{\text{rms}} = \sqrt{\frac{1}{T} \int_{0}^{T} [v(t)]^2 \, dt}
\]
where \( T \) is the period of the waveform. By performing this calculation, you can determine the effective voltage value of the waveform as observed in a typical AC system.
**Further Exploration:**
This exercise provides a foundational understanding of RMS calculations, useful in various applications such as
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