Explaining square root 3 in power systems

The root 3 factor is a relationship between the line-to-line voltage and the line-to-neutral voltage in three-phase power systems.

But what does that mean?

In a balanced three-phase system, three sinusoidal voltages are displaced 120 degrees from each other. (Remember, the phasor diagram is depicted in steady-state, but it actually spins as the sine waves move). At 60 Hz, one cycle is roughly 0.016 seconds. Meaning in a three-phase system, all three phasors spin one rotation in 16 ms. This leads to a more continuous and higher average power transfer compared to single-phase systems. How do we account for this higher power? Using the square root three.

The three phasors from a common neutral point (in a WYE system).

If we take phasor A, which we will denote as V_an (voltage of phase a to neutral) and subtract it from V_bn, we get the line-to-line voltage. (Think of high-school math vector adding and subtracting). As the math works out, the magnitude of the line-to-line voltage is 1.73 times the phasor. In other words, the distance from phase A vector to the phase B line-to-neutral vector is 1.73 times the line-to-neutral vector.

Here is a proof in python to demostrate:

import matplotlib.pyplot as plt
import numpy as np

# Define line-to-neutral voltage phasor angles
phasor_angles = [0, 120, 240]
phasors = [np.exp(1j * np.deg2rad(angle)) for angle in phasor_angles] # Convert degrees to radians, then calculate phasor polar values

# Calculate line-to-line voltage
v_ll = phasors[0] - phasors[1] # V_AB = V_A - V_B
v_ll_magnitude = np.abs(v_ll) # Calculate magnitude of V_AB

# Plot the phasors
plt.figure(figsize=(6, 6))
plt.axhline(0, color='black', linewidth=0.5)
plt.axvline(0, color='black', linewidth=0.5)
plt.quiver(0, 0, phasors[0].real, phasors[0].imag, angles='xy', scale_units='xy', scale=1, color='blue', label="V_LN Phase A")
plt.text(phasors[0].real / 2, phasors[0].imag / 2-.1, "V_an", fontsize=12, color='blue')

plt.quiver(0, 0, phasors[1].real, phasors[1].imag, angles='xy', scale_units='xy', scale=1, color='green', label="V_LN Phase B")
plt.text(phasors[1].real + phasors[1].real / 2, phasors[0].imag + phasors[1].imag / 2, "V_bn", fontsize=12, color='green')

plt.quiver(phasors[0].real, phasors[0].imag, -phasors[0].real + phasors[1].real, -phasors[0].imag + phasors[1].imag, angles='xy', scale_units='xy', scale=1, color='red', label="V_LN Phase B")
plt.text(0.5,0.5, "V_ab", fontsize=12, color='red')
plt.xlim(-1.5, 1.5)
plt.ylim(-1.5, 1.5)
plt.legend()
plt.title("Phasor Diagram")
plt.xlabel("Real")
plt.ylabel("Imaginary")
plt.grid(True)
plt.gca().set_aspect('equal', adjustable='box')
plt.show()

# Return the magnitude of line-to-line voltage
print("The phase 1 (V_ln) magnitude is:", np.abs(phasors[0]))
print("The phase 2 (V_ln) magnitude is:", np.abs(phasors[1]))
print("The phase 3 (V_ln) magnitude is:", np.abs(phasors[2]))
print("The V_ll magnitude is:", v_ll_magnitude)

# result
# The phase 1 (V_ln) magnitude is: 1.0
# The phase 2 (V_ln) magnitude is: 1.0
# The phase 3 (V_ln) magnitude is: 1.0
# The V_ll magnitude is: 1.7320508075688772 # square root three

There you have it! The square root of three is the ratio between the line-to-line voltage and the line-to-neutral voltage in a three-phase system.