Circuit Simulation with QPDK

This notebook demonstrates how to perform circuit simulations using the qpdk models and the sax circuit solver. We will showcase individual components and then combine them to create a custom resonator circuit.

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import jax.numpy as jnp
import sax
from matplotlib import pyplot as plt

from qpdk import PDK
from qpdk.models.generic import capacitor, inductor, tee
from qpdk.models.resonator import quarter_wave_resonator_coupled
from qpdk.models.waveguides import straight, straight_shorted
from qpdk.tech import coplanar_waveguide

PDK.activate()

Setup

First, let’s define a frequency range for our simulations and create a coplanar waveguide (CPW) media that defines the transmission line properties.

# Define frequency range
freq = jnp.linspace(2e9, 8e9, 501)
freq_ghz = freq / 1e9

# Define CPW media
cross_section = coplanar_waveguide(width=10, gap=6)

Individual Component Models

Let’s simulate some of the basic components available in qpdk.

Straight Waveguide

Simulate a \(1\,\textrm{mm}\) straight waveguide

straight_wg = straight(f=freq, length=1000, cross_section=cross_section)

# Plot S-parameters
plt.figure()
plt.title("Straight Waveguide S-parameters")
plt.plot(freq_ghz, 20 * jnp.log10(jnp.abs(straight_wg[("o1", "o2")])), label="$S_{21}$")
plt.plot(freq_ghz, 20 * jnp.log10(jnp.abs(straight_wg[("o1", "o1")])), label="$S_{11}$")
plt.xlabel("Frequency [GHz]")
plt.ylabel("Magnitude [dB]")
plt.grid(True)
plt.legend()
plt.show()

Capacitor

Simulate a \(100\,\textrm{fF}\) capacitor

cap_val = 100e-15
cap = capacitor(f=freq, capacitance=cap_val, z0=50)

# Plot S-parameters
plt.figure()
plt.title(f"Capacitor S-parameters (C={cap_val * 1e15:.0f} fF)")
plt.plot(freq_ghz, 20 * jnp.log10(jnp.abs(cap[("o1", "o2")])), label="$S_{21}$")
plt.plot(freq_ghz, 20 * jnp.log10(jnp.abs(cap[("o1", "o1")])), label="$S_{11}$")
plt.xlabel("Frequency [GHz]")
plt.ylabel("Magnitude [dB]")
plt.grid(True)
plt.legend()
plt.show()

Inductor

Simulate a \(5\,\textrm{nH}\) inductor

ind_val = 5e-9
ind = inductor(f=freq, inductance=ind_val, z0=50)

# Plot S-parameters
plt.figure()
plt.title(f"Inductor S-parameters (L={ind_val * 1e9:.0f} nH)")
plt.plot(freq_ghz, 20 * jnp.log10(jnp.abs(ind[("o1", "o2")])), label="$S_{21}$")
plt.plot(freq_ghz, 20 * jnp.log10(jnp.abs(ind[("o1", "o1")])), label="$S_{11}$")
plt.xlabel("Frequency [GHz]")
plt.ylabel("Magnitude [dB]")
plt.grid(True)
plt.legend()
plt.show()

Coupled Resonator Model

Now let’s use a more complex, pre-built model for a coupled resonator.

# Simulate a coupled resonator
res = quarter_wave_resonator_coupled(
    f=freq,
    cross_section=cross_section,
    coupling_gap=0.3,
    coupling_straight_length=200,
    length=5000,
)

# Plot S-parameters
plt.figure()
plt.title("Coupled Resonator S-parameters")
plt.plot(
    freq_ghz,
    20 * jnp.log10(jnp.abs(res[("coupling_o1", "coupling_o2")])),
    label="$S_{21}$",
)
plt.xlabel("Frequency [GHz]")
plt.ylabel("Magnitude [dB]")
plt.grid(True)
plt.legend()
plt.show()

Building a Custom Resonator Circuit

We can use sax to build our own circuits from basic components. Let’s build a quarter-wave resonator capacitively coupled to a feedline.

The circuit is a feedline with a T-junction. A series combination of a capacitor and a shorted transmission line (the resonator) is connected to the T-junction as a shunt element.

# Define component settings
feedline_segment_length = 500  # um
resonator_length = 4000  # um
coupling_cap_val = 20e-15  # F

# Define models for sax circuit
models = {
    "straight": straight,
    "capacitor": capacitor,
    "straight_shorted": straight_shorted,
    "tee": tee,
}

# Define netlist
netlist = {
    "instances": {
        "feedline1": {
            "component": "straight",
            "settings": {"length": feedline_segment_length, "media": cross_section},
        },
        "feedline2": {
            "component": "straight",
            "settings": {"length": feedline_segment_length, "media": cross_section},
        },
        "cap": {
            "component": "capacitor",
            "settings": {"capacitance": coupling_cap_val, "z0": 50},
        },
        "res": {
            "component": "straight_shorted",
            "settings": {"length": resonator_length, "media": cross_section},
        },
        "tee": "tee",
    },
    "connections": {
        "feedline1,o2": "tee,o1",
        "tee,o2": "feedline2,o1",
        "tee,o3": "cap,o1",
        "cap,o2": "res,o1",
    },
    "ports": {
        "o1": "feedline1,o1",
        "o2": "feedline2,o2",
    },
}

# Create and run the circuit
custom_resonator_circuit, _ = sax.circuit(netlist=netlist, models=models)
custom_res_s_params = custom_resonator_circuit(f=freq)

# Plot S-parameters
plt.figure()
plt.title("Custom-built Resonator S-parameters")
plt.plot(
    freq_ghz,
    20 * jnp.log10(jnp.abs(custom_res_s_params[("o1", "o2")])),
    label="$S_{21}$",
)
plt.xlabel("Frequency [GHz]")
plt.ylabel("Magnitude [dB]")
plt.grid(True)
plt.legend()
plt.show()