Sinusoidal AC Voltages from Rotating Loops
1. Generation of AC Voltage
- AC voltage is produced by the uniform rotation of a loop in a constant magnetic field. This is the fundamental principle behind AC generators (alternators).
- Electromagnetic Induction: As the loop rotates, the magnetic flux through it changes, inducing an electromotive force (EMF), which drives the current.
- Faraday’s Law: The magnitude of the induced EMF is proportional to the rate of change of magnetic flux (\(E = -N \frac{\Delta \Phi}{\Delta t}\)).
2. Sinusoidal Nature of AC Voltage
- The induced voltage is sinusoidal due to the periodic change in the loop’s orientation relative to the magnetic field.
- Maximum EMF: Occurs when the loop is parallel to the magnetic field (maximum rate of flux change).
- Zero EMF: Occurs when the loop is perpendicular to the magnetic field (no change in flux).
- A graph of voltage vs. time follows a sine wave pattern.
3. Key Parameters of Sinusoidal AC Voltage
3.1. Frequency (f)
- Definition: The number of complete cycles of the AC voltage per second.
- Unit: Hertz (Hz).
- Formula: \(f = \frac{1}{T}\), where \(T\) is the period.
- Australian standard frequency: 50 Hz.
3.2. Period (T)
- Definition: The time taken for one complete cycle of the AC voltage.
- Unit: Seconds (s).
- Formula: \(T = \frac{1}{f}\).
3.3. Amplitude (Vp)
- Definition: The maximum voltage reached during a cycle. Also known as peak voltage.
- Symbol: \(V_p\).
- Represents the maximum displacement of the sine wave from the zero axis.
- Domestic Power in Australia: \(V_p \approx 340V\).
3.4. Peak-to-Peak Voltage (Vp-p)
- Definition: The difference between the maximum positive voltage and the maximum negative voltage in a cycle.
- Formula: \(V_{p-p} = 2V_p\).
- Domestic Power in Australia: \(V_{p-p} \approx 680V\).
3.5. Peak Current (Ip)
- Definition: The maximum current reached during a cycle.
- Symbol: \(I_p\).
- Calculated using Ohm’s Law: \(V_p = I_p R\).
3.6. Peak-to-Peak Current (Ip-p)
- Definition: The difference between the maximum positive current and the maximum negative current in a cycle.
- Formula: \(I_{p-p} = 2I_p\).
4. Mathematical Representation
- Sinusoidal Voltage: \(V(t) = V_p \sin(2\pi ft)\)
- Sinusoidal Current: \(I(t) = I_p \sin(2\pi ft)\)
- Where:
- \(V(t)\) is the instantaneous voltage at time \(t\).
- \(I(t)\) is the instantaneous current at time \(t\).
- \(V_p\) is the peak voltage.
- \(I_p\) is the peak current.
- \(f\) is the frequency.
5. Visualization
- Diagram: A sine wave illustrating \(V_p\), \(V_{p-p}\), and \(T\). (Imagine a sine wave with the x-axis representing time and the y-axis representing voltage or current).
- X-axis: Time (t)
- Y-axis: Voltage (V) or Current (I)
- Amplitude: The distance from the x-axis to the peak of the wave (\(V_p\) or \(I_p\)).
- Peak-to-Peak: The total vertical distance from the highest peak to the lowest trough (\(V_{p-p}\) or \(I_{p-p}\)).
- Period: The length of one complete cycle along the x-axis (T).
6. Comparison Table
| Parameter |
Symbol |
Definition |
Formula |
| Frequency |
\(f\) |
Cycles per second |
\(f = \frac{1}{T}\) |
| Period |
\(T\) |
Time for one cycle |
\(T = \frac{1}{f}\) |
| Peak Voltage |
\(V_p\) |
Maximum voltage in a cycle |
|
| Peak-to-Peak Voltage |
\(V_{p-p}\) |
Difference between max and min voltage |
\(V_{p-p} = 2V_p\) |
| Peak Current |
\(I_p\) |
Maximum current in a cycle |
|
| Peak-to-Peak Current |
\(I_{p-p}\) |
Difference between max and min current |
\(I_{p-p} = 2I_p\) |
KEY TAKEAWAY: Understanding the relationship between frequency, period, peak voltage, and peak-to-peak voltage is crucial for analyzing AC circuits.
