To create a phasor diagram for the sum of $3\sin(\omega t + 30^\circ)$ and $3\sin(\omega t)$, we'll follow these steps:

1. Representation as Phasors:

• $3\sin(\omega t)$ can be represented as a phasor of magnitude 3 along the positive real axis (since its phase angle is $0^\circ$).
• $3\sin(\omega t + 30^\circ)$ can be represented as a phasor of magnitude 3 making a $30^\circ$ angle with the real axis.

2. Addition of Phasors:

The two phasors are added using vector addition. This forms a parallelogram (or triangle) in the phasor diagram.

3. Resultant Phasor:

The resultant phasor represents the sum of the two signals in both magnitude and phase.

I'll generate the phasor diagram now. matplotlib.pyplot as plt import numpy as np

Define the magnitudes and angles

magnitude = 3 angle1 = 0 # in degrees angle2 = 30 # in degrees

Convert angles to radians for calculations

angle1_rad = np.deg2rad(angle1) angle2_rad = np.deg2rad(angle2)

Components of the phasors

phasor1 = (magnitude * np.cos(angle1_rad), magnitude * np.sin(angle1_rad)) phasor2 = (magnitude * np.cos(angle2_rad), magnitude * np.sin(angle2_rad))

Resultant phasor

resultant = (phasor1[0] + phasor2[0], phasor1[1] + phasor2[1])

Plotting the phasors

fig, ax = plt.subplots() ax.quiver(0, 0, phasor1[0], phasor1[1], angles='xy', scale_units='xy', scale=1, color='b', label='3sin(ωt)') ax.quiver(0, 0, phasor2[0], phasor2[1], angles='xy', scale_units='xy', scale=1, color='g', label='3sin(ωt + 30°)') ax.quiver(0, 0, resultant[0], resultant[1], angles='xy', scale_units='xy', scale=1, color='r', label='Resultant')

Setting the plot

ax.set_xlim(-1, 6) ax.set_ylim(-1, 4) ax.set_aspect('equal') ax.grid(True)

Labeling the phasors

ax.text(phasor1[0] + 0.1, phasor1[1], '3∠0°', color='b') ax.text(phasor2[0] + 0.1, phasor2[1], '3∠30°', color='g') ax.text(resultant[0] + 0.1, resultant[1], 'Resultant', color='r')

plt.title('Phasor Diagram for 3sin(ωt + 30°) and 3sin(ωt)') plt.show()Here is the phasor diagram for the sum of $3\sin(\omega t + 30^\circ)$ and $3\sin(\omega t)$:

• Blue vector: Represents $3\sin(\omega t)$ with a phase angle of $0^\circ$.
• Green vector: Represents $3\sin(\omega t + 30^\circ)$ with a phase angle of $30^\circ$.
• Red vector: The resultant of the two phasors, representing the sum of the signals.

Would you like me to explain any part of the diagram or calculation further?

Related Questions:

1. How do you calculate the exact magnitude and phase of the resultant phasor?
2. Can we express the resultant as a single sine function?
3. How would the phasor diagram change if one of the amplitudes was different?
4. What happens if we add a third phasor with a different phase angle?
5. How is this phasor addition related to AC circuit analysis?

Tip: When adding phasors, it's often easier to convert them to rectangular form (real and imaginary components), add them, and then convert back to polar form for the magnitude and phase.