Final answer:
The total supply voltage is found by adding the sinusoidal voltages for all four loads in series, taking into account their phases, and resulting in a final expression for the supply voltage as a sinusoidal function of time.
Step-by-step explanation:
The student is attempting to determine the total supply voltage for a series circuit with four loads, where each load has a voltage across it represented by a sinusoidal function of time. The supply voltage in a series circuit is the sum of the voltages across each component. When calculating the supply voltage for a series circuit with sinusoidal voltages, phasor addition must be used because voltage signals can be out of phase with each other.
To calculate the supply voltage, we must add all the voltages across the loads while considering their phases. Since v3 is given in terms of cosine, we convert it to a sine function by using the identity cos wt = sin (wt + 90°). Therefore, v3 = 40 sin (wt + 90°). The next step is to sum the sinusoidal functions:
- v1 = 50 sin wt
- v2 = 25 sin (wt + 60°)
- v3 = 40 sin (wt + 90°)
- v4 = 30 sin (wt - 45°)
By adding these voltages, we can find the resulting expression for the supply voltage which will also be a sinusoidal function of time. Finally, we compute the resultant phasor, determine its magnitude and angle, and express our answer as Vs = Vmax sin (wt + φ), where Vmax is the amplitude of the supply voltage and φ is the phase angle.
It's important to note that we can visualize this sum using a phasor diagram representation or perform the addition analytically using complex numbers to facilitate calculations, but these methods are beyond the scope of this question's response.
In conclusion, after appropriate phasor addition, we would mention the correct option answer in the final answer.