Tesla’s polyphase system (1888)

In May 1888 Nikola Tesla presented “A new system of alternate current motors and transformers” to the American Institute of Electrical Engineers. Currents displaced in phase produce a rotating magnetic field, which drives an induction motor with no commutator. Polyphase supply has a second advantage that Mikhail Dolivo-Dobrovolsky exploited with three phases in 1889-1891: a balanced three-phase load draws constant instantaneous power, \(\sqrt3 V_L I_L \cos\phi\) (three_phase_power()), and the three return currents cancel, so three wires do the work of six. This example checks both, and draws the rotating field.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.electrical_engineering import real_power, three_phase_power

Constant power from three pulsating phases

V_phase, I_line, phi, f = 230.0, 16.0, np.radians(30), 50.0
w = 2 * np.pi * f
t = np.linspace(0, 2 / f, 1000)
shifts = np.array([0, -2 * np.pi / 3, 2 * np.pi / 3])
v = np.sqrt(2) * V_phase * np.cos(w * t[:, None] + shifts)
i = np.sqrt(2) * I_line * np.cos(w * t[:, None] + shifts - phi)
p_phase = v * i
p_total = p_phase.sum(axis=1)
P = three_phase_power(np.sqrt(3) * V_phase, I_line, np.cos(phi))
print(f"three-phase power {P / 1e3:.3f} kW; instantaneous total varies by {np.ptp(p_total):.1e} W")
assert np.allclose(p_total, P)
assert np.isclose(P, 3 * real_power(V_phase, I_line, np.cos(phi)))
assert np.allclose(i.sum(axis=1), 0, atol=1e-9)  # no neutral current

three-phase power 9.561 kW; instantaneous total varies by 2.4e-11 W

The rotating field

Three coils 120 degrees apart, fed by the three currents, add up to a field of constant magnitude turning at the supply frequency.

coil_dirs = np.exp(1j * np.array([0, 2 * np.pi / 3, 4 * np.pi / 3]))
field = (np.cos(w * t[:, None] + shifts) * coil_dirs).sum(axis=1)
assert np.allclose(np.abs(field), 1.5)

fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
axes[0].plot(t * 1e3, p_phase / 1e3, lw=0.8)
axes[0].plot(t * 1e3, p_total / 1e3, "k", label="sum of the three phases")
axes[0].set(xlabel="time [ms]", ylabel="power [kW]", title="Instantaneous power")
axes[0].legend()
axes[1].plot(field.real, field.imag)
axes[1].set_aspect("equal")
axes[1].set(title="Tip of the rotating field", xlabel="x", ylabel="y")
fig.tight_layout()

plt.show()