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Long and Short Cables

Electrically Long Cable

Consider an Ocean-Earth section with physical length, \(L\), and propagation constant, \(\gamma\). The section has an adjustment distance \(\frac{1}{\gamma}\). For an electrically-long transmission line, where \(L>\frac{4}{\gamma}\) (i.e. length is greater than four times of adjustment distance) we have the following scenario,

\[ e^{\gamma L}>>1>>e^{-\gamma L} \]

When the transmission line length is considerably shorter than the adjustment distance it is referred to as 'electrically-long' and the equivalent-\(\pi\) components reduce to (Boteler et a, 2013):

\[ Y_E=\frac{2}{Z_0e^{\gamma L}} \]
\[ \frac{Y'}{2}=\frac{1}{Z_0} \]
\[ I_E=\frac{E}{Z} \]
\[ V_i=-U_k=-\frac{E}{\gamma} \]
\[ V(x)=Ve^{-\gamma (L-x)}-Ve^{-\gamma x} \]

where \(V_i=-V_k=V=\frac{E}{\gamma}\).

Example

from pathlib import Path

import pandas as pd

from scubas.datasets import PROFILES
from scubas.models import OceanModel
from scubas.plotlib import cable_potential, plot_transfer_function, update_rc_params
from scubas.cables import Cable, TransmissionLine

figures_dir = Path("docs/tutorial/figures")
figures_dir.mkdir(parents=True, exist_ok=True)

update_rc_params(
    {
        "font.family": "sans-serif",
        "font.sans-serif": ["Tahoma", "DejaVu Sans", "Lucida Grande", "Verdana"],
    },
    science=True,
)

ocean_model = OceanModel(PROFILES.DO_3)
transfer_function = ocean_model.get_TFs()
tf_artifacts = plot_transfer_function(transfer_function)
tf_artifacts.figure.suptitle("Deep Ocean Transfer Function")
tf_artifacts.figure.savefig(
    figures_dir / "electrically_cable_transfer_function.png",
    dpi=300,
    bbox_inches="tight",
)

####################################################################
# Simulating the case: Induced electric field 0.3 V/km on a 
# shallow continental shelf with depth 100 m, length 600 km
####################################################################
induced_e_field = pd.DataFrame(
    {"X": [300.0]},
    index=pd.RangeIndex(1, name="Time"),
)

length = 600.0
transmission_line = TransmissionLine(
    sec_id="CS-long",
    directed_length={"length_north": length},
    elec_params={"site": PROFILES.CS, "width": 1.0, "flim": [1e-6, 1.0]},
)
transmission_line.compute_eqv_pi_circuit(
    Efield=induced_e_field,
    components=["X"],
)

cable = Cable([transmission_line], components=["X"])
potentials, distances = cable._pot_along_cable_(timestamp=0)
potential_plot = cable_potential(potentials, distances, ylim=(-200, 200))
potential_plot.axes.text(
    0.05,
    0.85,
    rf"$L_{{cs}}$={length:.0f} km",
    ha="left",
    va="center",
    transform=potential_plot.axes.transAxes,
)
potential_plot.figure.savefig(
    figures_dir / "electrically_long_cable_potential.png",
    dpi=300,
    bbox_inches="tight",
)

Deep ocean transfer function Electrically long cable potential

Electrically Short Cable

For an electrically-short section, where the physical length is less than the adjustment distance, i.e., \(L<\frac{1}{\gamma}\). When the transmission line length is considerably shorter than the adjustment distance it is referred to as 'electrically-short' and the equivalent-\(\pi\) components reduce to (Boteler et a, 2013):

\[ e^{\pm\gamma L}\approx 1\pm\gamma L \]
\[ Y_E=\frac{1}{Z_0\gamma L} \]
\[ \frac{Y'}{2}=0 \]
\[ I_E=\frac{E}{Z} \]
\[ V_i=-V_k=-\frac{LE}{2}=V \]
\[ V(x)=\frac{V}{2L}(2+\gamma L)(2x-L) \]

Example

####################################################################
# Simulating the case: Induced electric field 0.3 V/km on a
# shallow continental shelf with depth 100 m, length 4000 km
####################################################################
from pathlib import Path

import pandas as pd

from scubas.cables import Cable, TransmissionLine
from scubas.datasets import PROFILES
from scubas.models import OceanModel
from scubas.plotlib import (
    cable_potential,
    plot_transfer_function,
    update_rc_params,
)

figures_dir = Path("docs/tutorial/figures")
figures_dir.mkdir(parents=True, exist_ok=True)

update_rc_params(
    {
        "font.family": "sans-serif",
        "font.sans-serif": ["Tahoma", "DejaVu Sans", "Lucida Grande", "Verdana"],
    },
    science=True,
)

ocean_model = OceanModel(PROFILES.DO_3)
transfer_function = ocean_model.get_TFs()
tf_artifacts = plot_transfer_function(transfer_function)
tf_artifacts.figure.suptitle("Deep Ocean Transfer Function")
tf_artifacts.figure.savefig(
    figures_dir / "electrically_cable_transfer_function.png",
    dpi=300,
    bbox_inches="tight",
)

induced_e_field = pd.DataFrame(
    {"X": [300.0]},
    index=pd.RangeIndex(1, name="Time"),
)

length = 4000.0
transmission_line = TransmissionLine(
    sec_id="CS-short",
    directed_length={"length_north": length},
    elec_params={"site": PROFILES.CS, "width": 1.0, "flim": [1e-6, 1.0]},
)
transmission_line.compute_eqv_pi_circuit(
    Efield=induced_e_field,
    components=["X"],
)

cable = Cable([transmission_line], components=["X"])
potentials, distances = cable._pot_along_cable_(timestamp=0)
potential_plot = cable_potential(potentials, distances, ylim=[-200, 200])
potential_plot.axes.text(
    0.05,
    0.85,
    rf"$L_{{cs}}$={length:.0f} km",
    ha="left",
    va="center",
    transform=potential_plot.axes.transAxes,
)
potential_plot.figure.savefig(
    figures_dir / "electrically_short_cable_potential.png",
    dpi=300,
    bbox_inches="tight",
)

Electrically short cable potential