9. A voltmeter and frequency meter connected across the terminals of a single-phase alternator running on no load gave readings of 777,7V and 50Hz, respectively. Determine
9.1 the mathematical equation describing this open-circuit voltage
9.2 the average value of this emf
9.3 the times needed for the emf ro reach 975V during the first half coffee rotation.
9.4 the magnitude if the emf 45° after the beginning of q cycle
9.5the magnitude of the emf 1,25ms after the beginning of a cycle
9.1 To determine the mathematical equation describing the open-circuit voltage, we can use the formula:
\[ V(t) = V_{max} \sin(\omega t + \phi) \]
where:
- \( V(t) \) is the instantaneous voltage at time \( t \),
- \( V_{max} \) is the maximum voltage,
- \( \omega \) is the angular frequency (\( 2\pi f \)),
- \( t \) is the time, and
- \( \phi \) is the phase angle.
From the given information, the maximum voltage is 777.7V (as it is the reading of the voltmeter) and the frequency is 50Hz (as it is the reading of the frequency meter).
Therefore, the equation describing the open-circuit voltage is:
\[ V(t) = 777.7 \sin(2\pi \cdot 50 \cdot t + \phi) \]
9.2 To find the average value of the emf, we need to calculate the average value of the voltage waveform. Since the waveform is a sinusoidal function, the average value of a sine wave over a complete period is zero.
Therefore, the average value of the emf is zero.
9.3 To determine the time needed for the emf to reach 975V during the first half cycle, we can use the equation:
\[ V(t) = 777.7 \sin(2\pi \cdot 50 \cdot t + \phi) \]
We need to find the time when \( V(t) = 975V \). Setting up the equation:
\[ 975 = 777.7 \sin(2\pi \cdot 50 \cdot t + \phi) \]
Solving for time \( t \) will give us the answer.
9.4 To find the magnitude of the emf 45° after the beginning of a cycle, we can use the same equation:
\[ V(t) = 777.7 \sin(2\pi \cdot 50 \cdot t + \phi) \]
To find the magnitude 45° after the beginning of a cycle, we need to substitute \( t \) with the appropriate value and calculate \( V(t) \).
9.5 Similar to the previous questions, to find the magnitude of the emf 1.25ms after the beginning of a cycle, we substitute \( t \) with the appropriate value and calculate \( V(t) \) using the equation:
\[ V(t) = 777.7 \sin(2\pi \cdot 50 \cdot t + \phi) \]