Van der Pol Oscillator Tuning via Backpropagation through Harmonic Balance¤

import jax
import jax.numpy as jnp
import numpy as np
import optax
import plotly.graph_objects as go
import plotly.io as pio
from plotly.subplots import make_subplots

from circulax import compile_circuit
from circulax.components.base_component import PhysicsReturn, Signals, States, component
from circulax.components.electronic import Capacitor, Inductor, Resistor

# 64-bit precision is important: HB Newton requires accurate Jacobians, and
# gradient-based optimisation accumulates floating-point error across steps.
jax.config.update("jax_enable_x64", True)

pio.templates.default = "plotly_white"
pio.renderers.default = "png"

@component(ports=("p1", "p2"))
def VanDerPolElement(signals: Signals, s: States, mu: float = 2.0, G0: float = 0.01) -> PhysicsReturn:
    """Nonlinear two-terminal element with cubic I-V characteristic.

    I(V) = -mu*G0*V + (G0/3)*V^3

    The first term is a negative conductance (-mu*G0 < 0), which supplies
    energy to the tank when |V| is small.  The cubic term (G0/3)*V^3 saturates
    the gain at large amplitudes, producing a stable oscillation.

    Limit-cycle amplitude (fundamental, from harmonic balance energy balance):
        -mu*G0*A + G0/4*A^3 = 0  -->  A = 2*sqrt(mu)
    With G0=0.01 and mu=2.0: A = 2*sqrt(2) ≈ 2.83 V

    Note: mu appears only in the linear term so that the limit-cycle amplitude
    A = 2*sqrt(mu) is strongly tunable — a factor-of-4 change in mu moves A by
    2×.  If mu were also in the cubic term, amplitude would saturate at ~2 V
    regardless of mu.
    """
    v = signals.p1 - signals.p2
    i = -mu * G0 * v + (G0 / 3.0) * v**3
    return {"p1": i, "p2": -i}, {}


# Quick smoke-test: check that the I-V slope at V=0 is -mu*G0 (negative conductance)
vdp_test = VanDerPolElement(mu=2.0, G0=0.01)
f_test, _ = vdp_test(p1=0.1, p2=0.0)
print(f"VDP I at V=0.1 V : {f_test['p1']:.6f} A  (expected {-2.0*0.01*0.1 + (0.01/3)*0.1**3:.6f} A)")

print(f"Expected limit-cycle amplitude: 2*sqrt(mu) = {2*np.sqrt(2.0):.3f} V  (at mu=2, G0=0.01)")

# ── Circuit parameters ────────────────────────────────────────────────────────
L_val  = 1e-6    # H  (1 µH)
C_val  = 1e-9    # F  (1 nF)
R_damp = 1e4     # Ω  (10 kΩ tank loss — small, Q ≈ R_damp * sqrt(C/L) ≈ 316)

f0 = 1.0 / (2.0 * np.pi * np.sqrt(L_val * C_val))
print(f"Tank resonant frequency : {f0 / 1e6:.4f} MHz")
print(f"Tank Q-factor (Rdamp)   : {R_damp * np.sqrt(C_val / L_val):.1f}")
print(f"VDP negative conductance: {-2.0 * 0.01:.4f} S  (at mu=2, G0=0.01)")
print(f"Tank loss conductance   : {1.0 / R_damp:.6f} S  (1/Rdamp)")
print(f"Net gain margin         : {abs(-2.0 * 0.01) / (1.0 / R_damp):.0f}×  >> 1, oscillation guaranteed")

# ── Netlist ──────────────────────────────────────────────────────────────────
# All four elements are in parallel between top_node and GND.
# GND is declared as the reserved ground instance and assigned node 0.
vdp_net = {
    "instances": {
        "VDP":   {"component": "vdp",       "settings": {"mu": 2.0,    "G0": 0.01}},
        "L1":    {"component": "inductor",   "settings": {"L": L_val}},
        "C1":    {"component": "capacitor",  "settings": {"C": C_val}},
        "Rdamp": {"component": "resistor",   "settings": {"R": R_damp}},
        "GND":   {"component": "ground"},
    },
    "connections": {
        # Connect all negative terminals to GND (node 0)
        "GND,p1":  ("VDP,p2", "L1,p2", "C1,p2", "Rdamp,p2"),
        # Connect all positive terminals to the same top_node
        "VDP,p1":  ("L1,p1", "C1,p1", "Rdamp,p1"),
    },
    "ports": {"osc": "VDP,p1"},
}

models = {
    "vdp":      VanDerPolElement,
    "inductor":  Inductor,
    "capacitor": Capacitor,
    "resistor":  Resistor,
    "ground":    lambda: 0,
}

# compile_circuit runs once; scalar params passed to circuit.hb(...) update
# instance values without re-compiling the topology.
circuit = compile_circuit(vdp_net, models)
num_vars = circuit.sys_size

print(f"\nSystem size : {num_vars} unknowns")
print(f"Named ports : {list(vdp_net['ports'])}")

# Advanced initialization detail: the custom HB warm starts below need the flat
# state-vector index of the oscillator port.
osc_idx = circuit.port_map["osc"]
print("Oscillator port: 'osc'")

# DC operating point: V=0 is the only fixed point (the VDP element has I(0)=0)
y_dc = circuit.dc()
print(f"DC operating point: max|y_dc| = {float(jnp.max(jnp.abs(y_dc))):.2e} V  (trivially zero)")

N_harm = 7   # 7 harmonics → K = 15 time points per period
K      = 2 * N_harm + 1

# Pass osc_node so circuit.hb automatically tries several sinusoidal initial
# amplitudes and selects the limit-cycle solution.
y_time, y_freq = circuit.hb(freq=f0, harmonics=N_harm, y0=y_dc, osc_node="osc")

print(f"y_time shape : {y_time.shape}  (K={K} time samples × {num_vars} nodes)")
print(f"y_freq shape : {y_freq.shape}  ({N_harm+1} harmonics × {num_vars} nodes)")

y_osc_freq = circuit.port(y_freq, "osc")

# y_freq[0] is the DC component, y_freq[1] is the fundamental, etc.
# Two-sided amplitude at harmonic k>=1 is 2 * |y_freq[k]|  (rfft folds negative freqs)
A_dc   = float(jnp.abs(y_osc_freq[0]))
A_fund = float(2.0 * jnp.abs(y_osc_freq[1]))
A_2nd  = float(2.0 * jnp.abs(y_osc_freq[2]))
A_3rd  = float(2.0 * jnp.abs(y_osc_freq[3]))

print("\nOscillator port 'osc' harmonic amplitudes:")
print(f"  DC (0f0)       : {A_dc:.4f} V")
print(f"  Fundamental f0 : {A_fund:.4f} V")
print(f"  2nd harmonic   : {A_2nd:.4f} V  ({A_2nd/A_fund*100:.1f}% of fundamental)")
print(f"  3rd harmonic   : {A_3rd:.4f} V  ({A_3rd/A_fund*100:.1f}% of fundamental)")
print(f"\nExpected amplitude (VdP limit cycle): ~2*sqrt(mu) = {2*np.sqrt(2.0):.3f} V")

# ── Plot: time-domain waveform and harmonic spectrum ─────────────────────────
T     = 1.0 / f0
t_ns  = np.linspace(0.0, T * 1e9, K, endpoint=False)
v_osc = np.array(circuit.port(y_time, "osc"))

harmonics   = np.arange(N_harm + 1)
scale       = np.where(harmonics == 0, 1.0, 2.0)
spectrum    = scale * np.abs(np.array(circuit.port(y_freq, "osc")))
tick_labels = ["DC" if k == 0 else f"{k}f₀" for k in harmonics]

fig = make_subplots(
    rows=1, cols=2,
    subplot_titles=(
        f"Van der Pol limit cycle — f₀ = {f0/1e6:.3f} MHz",
        "Harmonic spectrum",
    ),
)

fig.add_trace(
    go.Scatter(x=t_ns, y=v_osc, mode="lines+markers",
               marker=dict(size=7), line=dict(width=1.5),
               name=f"V_osc (HB, K={K})"),
    row=1, col=1,
)
fig.add_hline(y=0, line_dash="dash", line_color="grey", line_width=0.7, row=1, col=1)

fig.add_trace(
    go.Bar(x=tick_labels, y=spectrum, marker_color="#636EFA", opacity=0.85, name="spectrum"),
    row=1, col=2,
)

fig.update_xaxes(title_text="Time (ns)", row=1, col=1)
fig.update_yaxes(title_text="Voltage (V)", row=1, col=1)
fig.update_xaxes(title_text="Harmonic", row=1, col=2)
fig.update_yaxes(title_text="Amplitude (V)", row=1, col=2)
fig.update_layout(margin=dict(t=80, b=60, l=60, r=60), height=450, showlegend=True)
fig.show()

print("The odd-harmonic dominance (f0, 3f0, 5f0) is a hallmark of the symmetric cubic nonlinearity.")
print("Even harmonics (2f0, 4f0) are near-zero because I(-V) = -I(V) — the VDP element is odd.")

A_target = 1.5  # V  (requires mu* = (A_target/2)^2 = 0.5625)

# Fixed warm start for the loss function: sinusoidal at 3.5 V at osc_idx, all
# other variables zero. This advanced initialization detail uses the flat state
# vector because HB solves for all time samples at once.
_phase = 2.0 * jnp.pi * jnp.arange(K, dtype=jnp.float64) / K
y_flat_warmstart = (
    jnp.zeros(K * num_vars, dtype=jnp.float64)
    .at[jnp.arange(K) * num_vars + osc_idx].set(3.5 * jnp.sin(_phase))
)


def loss_fn_mu(mu: jax.Array) -> jax.Array:
    """Squared error between fundamental amplitude and target, as a function of mu."""
    _, y_freq_new = circuit.hb(
        freq=f0,
        harmonics=N_harm,
        y0=y_dc,
        params={"VDP.mu": mu},
        y_flat_init=y_flat_warmstart,
    )
    A_fund = 2.0 * jnp.abs(circuit.port(y_freq_new, "osc")[1])
    return (A_fund - A_target) ** 2


# Test: evaluate loss and gradient at the default mu=2.0
mu_test = jnp.array(2.0)
loss_val, grad_val = jax.value_and_grad(loss_fn_mu)(mu_test)
print(f"At mu=2.0:  loss = {float(loss_val):.4f} V²,  grad = {float(grad_val):.4f} V²/[mu]")
print(f"  Current amplitude ≈ {float(jnp.sqrt(loss_val)) + A_target:.3f} V,  target = {A_target} V")
print("  Positive gradient → decreasing mu reduces amplitude toward target")

# ── Gradient descent — scalar parameter, no need for Adam ───────────────────
mu = jnp.array(3.0)  # start above the solution (A=2*sqrt(3)≈3.46 V)
lr = 0.05

mu_history    = [float(mu)]
loss_history  = []

val_and_grad_jit = jax.jit(jax.value_and_grad(loss_fn_mu))

print(f"\nGradient descent (lr={lr}, starting from mu={float(mu):.1f}):")
for step in range(80):
    loss, g = val_and_grad_jit(mu)
    loss_history.append(float(loss))
    mu = mu - lr * g
    mu_history.append(float(mu))
    if step % 20 == 0 or step == 79:
        _, yf_cur = circuit.hb(freq=f0, harmonics=N_harm, y0=y_dc, params={"VDP.mu": mu}, osc_node="osc")
        A_cur = float(2.0 * jnp.abs(circuit.port(yf_cur, "osc")[1]))
        print(f"  Step {step:3d}: mu = {float(mu):.4f},  loss = {float(loss):.6f},  A_fund = {A_cur:.4f} V")

mu_opt = float(mu)
print(f"\nOptimised mu = {mu_opt:.4f}  (analytical: {(A_target/2)**2:.4f})")

# ── Compute waveforms before and after optimisation ───────────────────────────
_, yf_before = circuit.hb(freq=f0, harmonics=N_harm, y0=y_dc, params={"VDP.mu": jnp.array(2.0)}, osc_node="osc")
yt_after, yf_after = circuit.hb(freq=f0, harmonics=N_harm, y0=y_dc, params={"VDP.mu": jnp.array(mu_opt)}, osc_node="osc")

A_before = float(2.0 * jnp.abs(circuit.port(yf_before, "osc")[1]))
A_after  = float(2.0 * jnp.abs(circuit.port(yf_after, "osc")[1]))

v_before = np.array(circuit.port(y_time, "osc"))
v_after  = np.array(circuit.port(yt_after, "osc"))

fig = make_subplots(
    rows=1, cols=2,
    subplot_titles=(
        "Before and after μ tuning",
        f"Amplitude tuning convergence (target {A_target} V)",
    ),
)

fig.add_trace(
    go.Scatter(x=t_ns, y=v_before, mode="lines", line=dict(width=2),
               name=f"μ=2.00 → A={A_before:.2f} V (initial)"),
    row=1, col=1,
)
fig.add_trace(
    go.Scatter(x=t_ns, y=v_after, mode="lines", line=dict(width=2.5, dash="dash"),
               name=f"μ={mu_opt:.2f} → A={A_after:.2f} V (optimised)"),
    row=1, col=1,
)
fig.add_hline(y= A_target, line_dash="dot", line_color="grey", line_width=0.8, row=1, col=1)
fig.add_hline(y=-A_target, line_dash="dot", line_color="grey", line_width=0.8, row=1, col=1)

fig.add_trace(
    go.Scatter(x=list(range(len(loss_history))), y=loss_history, mode="lines",
               line=dict(width=2), name="loss", showlegend=False),
    row=1, col=2,
)

fig.update_xaxes(title_text="Time (ns)", row=1, col=1)
fig.update_yaxes(title_text="Voltage (V)", row=1, col=1)
fig.update_xaxes(title_text="Gradient descent step", row=1, col=2)
fig.update_yaxes(title_text="Loss (V²)", type="log", row=1, col=2)
fig.update_layout(margin=dict(t=80, b=100, l=60, r=60), height=520, legend=dict(orientation="h", y=-0.25, x=0.5, xanchor="center"))
fig.show()

print(f"Amplitude error after optimisation: {abs(A_after - A_target)*1000:.2f} mV")

f_target = 8e6    # Hz — 8 MHz target (up from ~5 MHz)
A_target_j = 1.0  # V  — 1 V fundamental amplitude

LC_target = 1.0 / (2.0 * np.pi * f_target) ** 2
print(f"Target frequency    : {f_target/1e6:.1f} MHz")
print(f"Required L*C product: {LC_target:.3e} H·F")
print(f"Starting L={L_val*1e6:.2f} µH, C={C_val*1e9:.2f} nF  → f0={f0/1e6:.3f} MHz")

# Fixed sinusoidal warm start for the joint optimisation loss — amplitude 2 V at osc_idx,
# all other nodes zero. This is advanced initialization over the flat HB state vector.
_phase_k = 2.0 * jnp.pi * jnp.arange(K, dtype=jnp.float64) / K
y_flat_hb_init = (
    jnp.zeros(K * num_vars, dtype=jnp.float64)
    .at[jnp.arange(K) * num_vars + osc_idx].set(3.5 * jnp.sin(_phase_k))
)


def loss_joint(log_params: jax.Array) -> jax.Array:
    """Joint loss over (log_L, log_C, log_mu) for frequency + amplitude targets."""
    log_L, log_C, log_mu = log_params
    L   = jnp.exp(log_L)
    C   = jnp.exp(log_C)
    mu  = jnp.exp(log_mu)

    f_resonant = 1.0 / (2.0 * jnp.pi * jnp.sqrt(L * C))

    _, y_freq_j = circuit.hb(
        freq=f_resonant,
        harmonics=N_harm,
        y0=jnp.zeros(num_vars),
        params={"L1.L": L, "C1.C": C, "VDP.mu": mu},
        y_flat_init=y_flat_hb_init,
    )

    A_fund = 2.0 * jnp.abs(circuit.port(y_freq_j, "osc")[1])

    loss_freq = ((f_resonant - f_target) / f_target) ** 2
    loss_amp  = ((A_fund - A_target_j) / A_target_j) ** 2

    return loss_freq + loss_amp


# Sanity check
log_params_0 = jnp.log(jnp.array([L_val, C_val, 2.0]))
loss_0 = loss_joint(log_params_0)
print(f"\nInitial joint loss: {float(loss_0):.4f}  (freq error + amplitude error)")

# ── Adam optimisation ─────────────────────────────────────────────────────────
optimizer   = optax.adam(0.05)
log_params  = log_params_0
opt_state   = optimizer.init(log_params)
losses_joint = []
param_log_hist = [np.array(log_params)]

val_grad_joint = jax.jit(jax.value_and_grad(loss_joint))

print("\nAdam optimisation (200 steps, lr=0.05):")
for i in range(200):
    loss, grads = val_grad_joint(log_params)
    losses_joint.append(float(loss))
    updates, opt_state = optimizer.update(grads, opt_state)
    log_params = optax.apply_updates(log_params, updates)
    param_log_hist.append(np.array(log_params))
    if i % 50 == 0 or i == 199:
        L_c, C_c, mu_c = np.exp(np.array(log_params))
        f_c_cur = 1.0 / (2.0 * np.pi * np.sqrt(L_c * C_c))
        print(f"  Step {i:3d}: loss={float(loss):.5f},  f={f_c_cur/1e6:.3f} MHz,  L={L_c*1e6:.3f} µH,  C={C_c*1e9:.3f} nF,  mu={mu_c:.3f}")

L_opt, C_opt, mu_opt_j = np.exp(np.array(log_params))
f_opt = 1.0 / (2.0 * np.pi * np.sqrt(L_opt * C_opt))
print(f"\nFinal: f={f_opt/1e6:.4f} MHz  (target {f_target/1e6:.1f} MHz),  L={L_opt*1e6:.4f} µH,  C={C_opt*1e9:.4f} nF,  mu={mu_opt_j:.4f}")