The Hopkinsons’ dynamo and the DC machine (1886)

John Hopkinson introduced the characteristic curve of a dynamo, its EMF against field current at constant speed, in 1879, and with his brother Edward showed in 1886 how to predict it from the magnetic circuit of the iron. Underneath both lies the armature equation of every DC machine: an EMF \(E = k\Phi\omega\) proportional to flux and speed, behind the armature resistance \(R_a\), with torque \(T = k\Phi I_a\). As a motor on a fixed supply the torque-speed curve is then a straight line from the stall torque to the no-load speed (dc_motor_torque(), dc_motor_speed()); driven faster than that the same machine generates. This example draws the torque-speed lines for three armature voltages and checks the power balance at an operating point.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.electrical_engineering import dc_motor_speed, dc_motor_torque

Torque-speed lines

A 240 V, 0.4 ohm armature with \(k\Phi = 2.2\) V s/rad.

Ra, kphi = 0.4, 2.2
omega = np.linspace(0, 130, 200)
fig, ax = plt.subplots(figsize=(7, 3.8))
for V in (120.0, 180.0, 240.0):
    T = dc_motor_torque(V, Ra, kphi, omega)
    ax.plot(omega, T, label=f"V = {V:.0f} V")
    print(f"V = {V:.0f} V: no-load speed {V / kphi:.1f} rad/s, stall torque {kphi * V / Ra:.0f} N m")
assert dc_motor_torque(240.0, Ra, kphi, 240.0 / kphi) == 0

V = 120 V: no-load speed 54.5 rad/s, stall torque 660 N m
V = 180 V: no-load speed 81.8 rad/s, stall torque 990 N m
V = 240 V: no-load speed 109.1 rad/s, stall torque 1320 N m

Power balance at a load

Electrical input = copper loss + mechanical output, since \(VI_a = I_a^2R_a + EI_a\) and \(EI_a = T\omega\).

V, T_load = 240.0, 150.0
w = dc_motor_speed(V, Ra, kphi, T_load)
Ia = T_load / kphi
p_in, p_copper, p_out = V * Ia, Ia**2 * Ra, T_load * w
print(f"at {T_load:.0f} N m: speed {w:.2f} rad/s ({w * 60 / (2 * np.pi):.0f} rpm), efficiency {p_out / p_in:.1%}")
assert np.isclose(p_in, p_copper + p_out)

at 150 N m: speed 96.69 rad/s (923 rpm), efficiency 88.6%

Generating

Above the no-load speed the EMF exceeds the supply, the current and torque reverse, and the machine returns power to the supply.

assert dc_motor_torque(V, Ra, kphi, 1.1 * V / kphi) < 0

ax.axhline(0, color="gray", lw=0.6)
ax.plot(w, T_load, "ko")
ax.set(xlabel="speed [rad/s]", ylabel="torque [N m]", title="Separately excited DC machine")
ax.legend(fontsize="small")
fig.tight_layout()

plt.show()