Fitting a Cable from 1 to 10 GHz (Part 1)

This tutorial demonstrates an end-to-end workflow for fitting a physically-motivated model to a high-frequency (1 to 10 GHz) measurement of a 1.5 ft CBN-1.5FT-SMSM cable from Mini-Circuits. The cable’s datasheet is available at https://www.minicircuits.com/pdfs/CBN-xx-SMSM+.pdf.

We will start by fitting an ideal transmission line model, and then add losses and parasitics to cater for effects such as attenuation and connector mismatch. The tutorial can be followed either directly in the documentation, or as port of a Jupyter notebook under docs/tutorials.

Note that this is part 1 of a two-part tutorial. Part 2 (Bayesian inference) will be added soon.

1. Setting up the environment

We only need a few packages for this tutorial. Make sure the following all import correctly:

import matplotlib.pyplot as plt     # for plotting
import parax as prx                 # for parameters
import skrf as rf                   # for touchstone reading
import pmrf as prf                  # ParamRF

2. Preparing the data

We begin by loading the measurement data. For this tutorial, we will download a sample .s2p file directly from the ParamRF repository and load it into a scikit-rf Network. The original file as available at https://www.minicircuits.com/WebStore/dashboard.html?model=CBN-1.5FT-SMSM%2B

import os
import urllib.request
import skrf as rf
import matplotlib.pyplot as plt

base_url = "https://raw.githubusercontent.com/gvcallen/paramrf/main/docs/tutorials/data/"
file_name = "CBN-1.5FT-SMSM.s2p"
full_url = base_url + file_name
local_dir = "data"
local_filename = os.path.join(local_dir, file_name)

os.makedirs(local_dir, exist_ok=True)
if not os.path.exists(local_filename):
    print(f"Downloading {file_name} to {local_dir}/...")
    urllib.request.urlretrieve(full_url, local_filename)

# Load the measured data into skrf using the new local path
measured = rf.Network(local_filename, f_unit='GHz')
measured = measured[measured.frequency.f_scaled < 10]
fig, ax = plt.subplots(figsize=(4, 3))
measured.plot_s_db(m=0, n=0, ax=ax)

3. Selecting a model

ParamRF provides a few transmission line models, such as PhysicalLine, CoaxialLine and DatasheetLine. Looking at the datasheet, we see no mention of physical parameters like loss tangent, conductor loss or detailed geometry dimensions. However, there is a velocity factor, as well as loss factors k1 and k2. This is just what DatasheetLine was meant for.

Let’s try fitting the model with its expected length and default parameters as a quick check.

default_model = prf.models.DatasheetLine(length=prf.math.feet_2_meter(1.5))
default_results = prf.fit(default_model, measured)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
default_results.plot_s_db(m=0, n=0, ax=ax1)
_ = default_results.plot_s_db(m=1, n=0, ax=ax2)

…not horrible, but we can definitely do better by providing the optimizer with a better starting point.

4. Specifying parameters

In ParamRF, parameter bounds and scales are defined using parax. Any parameter not explicitly passed to the model constructor is assumed to be a free parameter with infinite bounds, as in the previous example.

Looking at the datasheet, we find the cable is 50 ohm with a velocity factor vf of 0.74, skin effect loss k1 of around 1.137 dbB/100m, and dielectric loss k2 of around 0.00253 dB/100m. We use this information directly to specify initial parameters for DatasheetLine. We fix the velocity factory (dielectric constant), since there is a known correlation between it and the cable’s length (explored further in part 2 of this tutorial!). Lastly, we add units (which currently are only for metadata purposes).

We fit the model using both the gradient-based L-BFGS-B as well as the robust Nelder-Mead (which can work better when you are unsure of your initial parameters).

# Specify prior model parameters
baseline_model = prf.models.DatasheetLine(
    length=prx.Uniform(prf.math.feet_2_meter(1.4), prf.math.feet_2_meter(1.6), unit='m'),
    zn=prx.Uniform(45.0, 55.0, unit='ohm'),
    vf=prx.Fixed(0.74),
    k1=prx.Uniform(1.0, 1.3, unit='dB/100m'),
    k2=prx.Uniform(2.0, 3.0, scale=1e-3, unit='dB/100m'),
    name='coax',
)
# Fit the model
baseline_lbfgs_results = prf.fit(baseline_model, measured)
baseline_nelder_results = prf.fit(baseline_model, measured, method='Nelder-Mead')

# Plot the results. We pass a separate frequency to plot the model at a higher resolution
freq_plot = prf.Frequency(
    measured.frequency.f_scaled[0],
    measured.frequency.f_scaled[-1],
    1001,
    'GHz',
)
fig, axes = plt.subplots(2, 2, figsize=(12, 6))
baseline_lbfgs_results.plot_s_db(m=0, n=0, ax=axes[0,0], model_frequency=freq_plot)
baseline_lbfgs_results.plot_s_db(m=1, n=0, ax=axes[0,1], model_frequency=freq_plot)
baseline_nelder_results.plot_s_db(m=0, n=0, ax=axes[1,0], model_frequency=freq_plot)
_ = baseline_nelder_results.plot_s_db(m=1, n=0, ax=axes[1,1], model_frequency=freq_plot)
axes[0,0].set_title('L-BFGS S11 [dB]')
axes[0,1].set_title('L-BFGS S21 [dB]')
axes[1,0].set_title('Nelder-Mead S21 [dB]')
_ = axes[1,1].set_title('Nelder-Mead S11 [dB]')

While Nelder-Mead appears to fit the S11 better, L-BFGS-B achieves a better S21 fit. This can easily be affected by the type of loss function, parameter starting point, the specifical model, as well as many other factors, so it is important to experiment. However, do note that L-BFGS-B is generally much faster since it obtained gradients directly from JAX, which may be more important for large models.

5. Adding parasitics

Now that we know that we are in the right ballpark with our initial parameters and Nelder-Mead is working for us, it is clear that S11 drifts over frequency. This can be explained by parasitic capacitance in the cable connectors. We should model this!

To polish this tutorial off, we therefore add two shunt capacitors to both sides of the cable model using the ** operator. Note that naming the capacitors is optional, but useful for inspecting/manipulating the combined model’s parameters.

We print out the composite model parameters before continuing.

# Define a parasitic connector capacitor
# We initialize it at 0.1 pF and constrain it to physical (positive) values
shunt_cap1 = prf.models.ShuntCapacitor(
    C=prx.Uniform(0.0, 1.0, value=0.1, scale=1e-12, unit='pF'),
    name='cap1',
)
shunt_cap2 = prf.models.ShuntCapacitor(
    C=prx.Uniform(0.0, 1.0, value=0.1, scale=1e-12, unit='pF'),
    name='cap2',
)

# Create the composite model by cascading Cap -> Coax -> Cap
composite_model = shunt_cap1 ** baseline_model ** shunt_cap2

print("Composite Model Parameters:")
for name in composite_model.param_names():
    print(name)

Composite Model Parameters:
cap1
coax_length
coax_zn
coax_k1
coax_k2
cap2

6. Final optimization

Finally, we optimize the composite model and print out its best parameters. The model captures the high-frequency drift extremely well, while maintaining reasonable parameter values.

# Fit the composite model
results_composite = prf.fit(composite_model, measured, method='Nelder-Mead')
composite_fitted = results_composite.model

# Plot the final comparison
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 3))

measured.plot_s_db(m=0, n=0, ax=ax1, label='Measured S11', color='k', linewidth=2)
baseline_nelder_results.model.plot_s_db(freq_plot, m=0, n=0, ax=ax1, label='Baseline S11', color='gray', linestyle=':')
composite_fitted.plot_s_db(freq_plot, m=0, n=0, ax=ax1, label='Composite S11 (with Parasitics)', color='b', linestyle='--')

measured.plot_s_db(m=1, n=0, ax=ax2, label='Measured S21', color='k', linewidth=2)
baseline_nelder_results.model.plot_s_db(freq_plot, m=1, n=0, ax=ax2, label='Baseline S21', color='gray', linestyle=':')
composite_fitted.plot_s_db(freq_plot, m=1, n=0, ax=ax2, label='Composite S21 (with Parasitics)', color='b', linestyle='--')

ax1.set_title('Return Loss: Baseline vs Composite')
ax1.set_title('Insertion Loss: Baseline vs Composite')
ax1.legend()
plt.show()

# Inspect the final resolved parameters
print("Final Fitted Parameters:")
for name, param in composite_fitted.named_params().items():
    print(f"{name}: {param.value:.3f} {param.metadata['unit']}")

Final Fitted Parameters:
cap1: 0.019 pF
coax_length: 0.444 m
coax_zn: 50.711 ohm
coax_k1: 1.102 dB/100m
coax_k2: 2.714 dB/100m
cap2: 0.018 pF