Negative-Resitance-CRLH-Calculations/README.md

Active CRLH Transmission Line Modeler

A MATLAB simulation of an active Composite Right/Left-Handed (CRLH) transmission line with a frequency-dependent negative resistance element. The structure achieves controlled gain across a user-defined passband by embedding an active shunt element (modeled as a negative resistance Rn) within a periodic CRLH unit cell topology. This is supplemental code for the paper "Negative Resistance Enabled Amplifying CRLH Transmission Lines With Uniform Insertion Gain" DOI: 10.1109/LMWT.2026.3655243 (https://ieeexplore.ieee.org/document/11366944)

Background

CRLH transmission lines support both left-handed (LH) and right-handed (RH) wave propagation, enabling precise dispersion control. By introducing a frequency-shaped negative conductance into the shunt branch, this design compensates for losses and achieves net gain across the passband.

The unit cell uses a symmetric T-network topology with:

• Series branch: Right-handed inductance LR + left-handed capacitance CL
• Shunt branch: Right-handed capacitance CR + left-handed inductance LL + active negative resistance Rn(f)

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Features

• Calculates CRLH element values (LR, CR, LL, CL) analytically from two frequency/phase design targets
• Models frequency-dependent negative resistance as Rn(f) = -A·exp(α·f), fit to desired shunt conductance at two frequencies
• Cascades n unit cells in a passive–active–passive symmetric arrangement
• Converts ABCD matrices to S-parameters
• Plots:
  • Cascaded S-parameters (S11, S21)
  • Dispersion diagram (βp vs. frequency)
  • Shunt conductance vs. frequency
  • Required Rn(f) vs. exact analytical solutions
  • Single unit cell S-parameters with approximate insertion loss overlay
  • Bloch impedance (real and imaginary)
• Outputs (in command window):
  • Constituent CRLH parameters
  • R_n(\omega) exponential fit parameters
  • S_{21} at desired frequencies (\omega_1 and \omega_2)
  • Maximum gain

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Dependencies

• SPARAMS — provides SPARAMS.abcd2s() for ABCD-to-S-parameter conversion. Add as a submodule or clone separately and add to your MATLAB path.
• closestIdx — a utility function for finding the nearest index in a vector. Included in the bottom of the main file.

After cloning, add dependencies to your MATLAB path

Usage

1. Open Active_NRCRLH_Calcs.m in MATLAB
2. Set your design parameters in the Input parameters section:

Parameter	Description
f1, f2	Lower and upper frequency bounds of the passband (Hz)
theta1, theta2	Desired phase shifts at f1 and f2 (rad)
n	Number of sub-unit cells (must be odd for passive–active–passive symmetry)
Z0	Reference impedance (default: 50 Ω)
G_desired	Target shunt conductance (set as a fraction of G_max)

3. Run the script — six figures will be generated automatically. You may need to tune the figX.Position parameter to fit them on your screen

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Design Equations

CRLH element values are solved analytically from the two phase/frequency constraints in Reference [1].

Then, the required R_n(\omega) is calculated as:

R_n(\omega) = -R_0e^{\alpha\omega}

with

\alpha = \frac{\ln{\left(\frac{R_n(\omega_1)}{R_n(\omega_2)}\right)}}{\omega_1 - \omega_2},

R_0 = -R_n(\omega_1)e^{-\alpha \omega_1}.

This will approximately fit the exact solution of:

R_n(\omega) = \frac{1 \pm \sqrt{1 - 4(G'_{sh} \omega L_L)^2}}{2G'_{sh}}

The required negative resistance at each frequency satisfies:

G_{sh} = \frac{R_n}{R_n^2 + (\omega L_L)^2}

which is solved exactly at f1 and f2, then fit with an exponential model across the full band.

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Output Figures

Figure	Contents
1	Cascaded S11 and S21 (dB)
2	Dispersion: βp (deg) vs. frequency
3	Shunt conductance G(f) vs. frequency
4	Rn(f): exponential fit vs. exact analytical solutions
5	Single unit cell S-parameters + approximate S21
6	Bloch impedance ZB (real and imaginary)

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Notes

• n must be odd to maintain the passive–active–passive cascade symmetry
• Loss-balanced condition (Gsh = Rse/Z0²) is available but commented out by default
• The Rollett (K-Δ) stability analysis block is present in the code but commented out — uncomment to assess stability across frequency
• Maximum achievable gain is estimated and printed to the console on each run

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License

MIT License — see LICENSE for details.