"""Coupler models."""
from functools import partial
from typing import cast
import jax
import jax.numpy as jnp
import sax
from gdsfactory.typings import CrossSectionSpec
from jax.typing import ArrayLike
from sax.models.rf import capacitor, tee
from qpdk.models.constants import DEFAULT_FREQUENCY, ε_0
from qpdk.models.cpw import (
cpw_ep_r_from_cross_section,
cpw_z0_from_cross_section,
get_cpw_dimensions,
)
from qpdk.models.math import (
capacitance_per_length_conformal,
ellipk_ratio,
epsilon_eff,
)
from qpdk.models.waveguides import straight
@partial(jax.jit, inline=True)
def cpw_cpw_coupling_capacitance_per_length_analytical(
gap: float | ArrayLike,
width: float | ArrayLike,
cpw_gap: float | ArrayLike,
ep_r: float | ArrayLike,
) -> float | jax.Array:
r"""Analytical formula for ECCPW mutual capacitance per unit length.
The model follows the edge-coupled coplanar waveguide (ECCPW) formula
using conformal mapping for even and odd modes:
.. math::
\begin{aligned}
x_1 &= s_c / 2 \\
x_2 &= x_1 + W \\
x_3 &= x_2 + G \\
k_e &= \sqrt{\frac{x_2^2 - x_1^2}{x_3^2 - x_1^2}} \\
k_o &= \frac{x_1}{x_2} \sqrt{\frac{x_3^2 - x_2^2}{x_3^2 - x_1^2}} \\
C_{\text{even}} &= 2 \epsilon_0 \epsilon_{\text{eff}} \frac{K(k_e)}{K(k_e')} \\
C_{\text{odd}} &= 2 \epsilon_0 \epsilon_{\text{eff}} \frac{K(k_o')}{K(k_o)} \\
C_m &= \frac{C_{\text{odd}} - C_{\text{even}}}{2}
\end{aligned}
where :math:`s_c` is the separation (gap) between inner edges, :math:`W` is the
center conductor width, and :math:`G` is the gap to the ground plane.
See :cite:`simonsCoplanarWaveguideCircuits2001`.
Args:
gap: The gap (separation) between the two center conductors in µm.
width: Center conductor width in µm.
cpw_gap: Gap between center conductor and ground plane in µm.
ep_r: Relative permittivity of the substrate.
Returns:
The mutual coupling capacitance per unit length in Farads/meter.
"""
# Geometric parameters in m (convert from μm)
s_c = gap * 1e-6
w_m = width * 1e-6
g_m = cpw_gap * 1e-6
x1 = s_c / 2
x2 = x1 + w_m
x3 = x2 + g_m
# Even-mode modulus squared
ke_sq = (x2**2 - x1**2) / (x3**2 - x1**2)
# Odd-mode modulus squared
ko_sq = (x1**2 / x2**2) * ((x3**2 - x2**2) / (x3**2 - x1**2))
# Capacitances per unit length
# Factor is 2.0 since ECCPW formula uses 2 * ε_0 * ε_eff
c_even_pul = 2.0 * capacitance_per_length_conformal(m=ke_sq, ep_r=ep_r)
# c_odd uses K(1-m)/K(m) which is the inverse of ellipk_ratio(m)
c_odd_pul = 2.0 * ε_0 * epsilon_eff(ep_r) / ellipk_ratio(ko_sq)
# Mutual capacitance per unit length
return (c_odd_pul - c_even_pul) / 2
[docs]
def cpw_cpw_coupling_capacitance(
f: sax.FloatArrayLike, # ruff: ignore[unused-function-argument]
length: float | ArrayLike,
gap: float | ArrayLike,
cross_section: CrossSectionSpec,
) -> float | jax.Array:
r"""Calculate the coupling capacitance between two parallel CPWs.
Args:
f: Frequency array in Hz.
length: The coupling length in µm.
gap: The gap between the two center conductors in µm.
cross_section: The cross-section of the CPW.
Returns:
The total coupling capacitance in Farads.
Note:
Raises ``ValueError`` via :func:`~qpdk.models.cpw.get_cpw_dimensions`
if the cross-section has no section with 'etch' in its name, so the
CPW gap cannot be determined, or if the conductor width or etch gap
is not positive.
"""
ep_r = cpw_ep_r_from_cross_section(cross_section)
width, cpw_gap = get_cpw_dimensions(cross_section)
c_pul = cpw_cpw_coupling_capacitance_per_length_analytical(
gap=gap,
width=width,
cpw_gap=cpw_gap,
ep_r=ep_r,
)
return c_pul * length * 1e-6
[docs]
def coupler_straight(
f: ArrayLike = DEFAULT_FREQUENCY,
length: int | float = 20.0,
gap: int | float = 0.27,
cross_section: CrossSectionSpec = "cpw",
) -> sax.SDict:
"""S-parameter model for two coupled coplanar waveguides, :func:`~qpdk.cells.waveguides.coupler_straight`.
Args:
f: Array of frequency points in Hz
length: Physical length of coupling section in µm
gap: Gap between the coupled waveguides in µm
cross_section: The cross-section of the CPW.
Returns:
sax.SDict: S-parameters dictionary
.. code::
o2──────▲───────o3
│gap
o1──────▼───────o4
"""
f = jnp.asarray(f)
straight_settings = {"length": length / 2, "cross_section": cross_section}
capacitor_settings = {
"capacitance": cpw_cpw_coupling_capacitance(f, length, gap, cross_section),
"z0": cpw_z0_from_cross_section(cross_section, f),
}
# Create straight instances with shared settings
straight_instances = {
f"straight_{i}_{j}": straight(f=f, **straight_settings)
for i in [1, 2]
for j in [1, 2]
}
tee_instances = {f"tee_{i}": tee(f=f) for i in [1, 2]}
instances = {
**straight_instances,
**tee_instances,
"capacitor": capacitor(f=f, **capacitor_settings),
}
connections = {
"straight_1_1,o1": "tee_1,o1",
"straight_1_2,o1": "tee_1,o2",
"straight_2_1,o1": "tee_2,o1",
"straight_2_2,o1": "tee_2,o2",
"tee_1,o3": "capacitor,o1",
"tee_2,o3": "capacitor,o2",
}
ports = {
"o2": "straight_1_1,o2",
"o3": "straight_1_2,o2",
"o1": "straight_2_1,o2",
"o4": "straight_2_2,o2",
}
return sax.evaluate_circuit_fg((connections, ports), instances)
[docs]
def coupler_ring(
f: ArrayLike = DEFAULT_FREQUENCY,
length: int | float = 20.0,
gap: int | float = 0.27,
cross_section: CrossSectionSpec = "cpw",
) -> sax.SDict:
"""S-parameter model for two coupled coplanar waveguides in a ring configuration.
The implementation is the same as straight coupler for now.
TODO: Fetch coupling capacitance from a curved simulation library.
Args:
f: Array of frequency points in Hz
length: Physical length of coupling section in µm
gap: Gap between the coupled waveguides in µm
cross_section: The cross-section of the CPW.
Returns:
sax.SDict: S-parameters dictionary
"""
return coupler_straight(f=f, length=length, gap=gap, cross_section=cross_section)
if __name__ == "__main__":
import matplotlib.pyplot as plt
lengths = jnp.linspace(10, 1000, 10)
gaps = jnp.geomspace(0.1, 5.0, 6)
width = 10.0
cpw_gap = 6.0
ep_r = 11.7
plt.figure(figsize=(10, 6))
# Calculate capacitance per unit length for all gaps simultaneously (shape: (6,))
c_pul = cast(
jax.Array,
cpw_cpw_coupling_capacitance_per_length_analytical(
gap=gaps, width=width, cpw_gap=cpw_gap, ep_r=ep_r
),
)
# Broadcast to compute total capacitance for all lengths and gaps (shape: (6, 1000))
capacitances = c_pul[:, None] * lengths[None, :] * 1e-6 * 1e15 # Convert to fF
for i, gap in enumerate(gaps):
plt.plot(lengths, capacitances[i], label=f"gap = {gap:.1f} µm")
plt.xlabel("Coupling Length (µm)")
plt.ylabel("Mutual Capacitance (fF)")
plt.title(
rf"CPW-CPW Coupling Capacitance ($\mathtt{{width}}=${width} µm, $\mathtt{{cpw\_gap}}=${cpw_gap} µm, $\epsilon_r={ep_r}$)"
)
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()
