r"""
Black's negative-feedback amplifier
===================================

Harold Black (1927), commuting on the Lackawanna ferry, sketched an
amplifier that fed a fraction :math:`\beta` of its output back *out of
phase*. With a huge open-loop gain :math:`A`, the closed-loop gain
:math:`A/(1 + A\beta)` (:func:`~engineeringkit.control_systems.feedback`)
is almost exactly :math:`1/\beta`, set by a passive divider rather than by
drifting vacuum tubes. Distortion and gain variations shrink by the factor
:math:`1 + A\beta`, and the bandwidth grows by the same factor.
"""

# %%
import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.control_systems import bode, dc_gain, feedback, poles, transfer_function

beta = 0.01
wa = 2 * np.pi * 10.0  # open-loop pole [rad/s]

# %%
# Insensitivity to the tube gain
# ------------------------------

for A0 in [5e4, 1e5, 1.5e5]:
    amp = transfer_function([A0], [1 / wa, 1])
    closed = feedback(amp, beta)
    gain = dc_gain(closed)
    print(f"A0 = {A0:8.0f}: closed-loop gain {gain:.4f}, bandwidth {-poles(closed)[0].real / (2 * np.pi):8.0f} Hz")
    assert np.isclose(-poles(closed)[0].real, wa * (1 + A0 * beta))
    assert np.isclose(gain, A0 / (1 + A0 * beta)) and abs(gain * beta - 1) < 2.5e-3

# %%
# Gain traded for bandwidth
# -------------------------

amp = transfer_function([1e5], [1 / wa, 1])
w = np.logspace(0, 8, 400)
fig, ax = plt.subplots()
for model, label in [(amp, "open loop $A$"), (feedback(amp, beta), r"closed loop $A/(1+A\beta)$")]:
    _, mag, _ = bode(model, w)
    ax.semilogx(w / (2 * np.pi), mag, label=label)
ax.set(xlabel="frequency [Hz]", ylabel="gain [dB]", title="Gain-bandwidth exchange")
ax.grid(True, which="both", alpha=0.3)
ax.legend()

plt.show()