mixed-potential-memristor-circuit-convergence

name

mixed-potential-memristor-circuit-convergence

description

Mixed Potential approach for analyzing convergence of nonlinear RLC circuits with memristors using flux-charge analysis method (FCAM). Provides Lyapunov-like stability proofs for circuits with all four basic elements (resistors, inductors, capacitors, memristors). Applications: content addressable memories (CAMs), neuromorphic computing, nonlinear circuit stability analysis. Activation: memristor, circuit convergence, mixed potential, nonlinear RLC, flux-charge analysis, Lyapunov stability, circuit stability.

Context

From arXiv:2606.05851 (June 2026) - "Mixed Potential Approach to Convergence of Nonlinear RLC Circuits with Memristors" by Mauro Di Marco, Mauro Forti, Luca Pancioni, Giacomo Innocenti, Alberto Tesi.

This paper extends classical Brayton-Moser mixed potential theory to circuits with memristors, enabling convergence analysis for the complete set of four basic circuit elements. Uses Flux-Charge Analysis Method (FCAM) to analyze circuits in flux-charge domain rather than voltage-current domain.

Core Methodology

1. RLCM Circuit Framework

• Complete element set: Resistors + Inductors + Capacitors + Memristors
• Flux-charge domain: Transform from voltage-current to flux-charge variables
• Mixed potential function: Generalized Brayton-Moser potential including memristor contributions

2. Convergence Analysis Steps

1. Complete variable set requirement

   • Ensure circuit has complete set of variables in flux-charge domain
   • Verify state variables span full phase space

2. Capacitor-inductor balance

   • Quantitative estimation of balance between capacitors and inductors
   • Required for convergence proof conditions

3. Mixed potential construction

   • Define generalized potential function P(q,φ) where q = charge, φ = flux
   • Include memristor flux-dependent resistance terms
   • Derive from circuit topology and element characteristics

4. Lyapunov-like stability proof

   • Use mixed potential as Lyapunov function candidate
   • Show dP/dt ≤ 0 under balance conditions
   • Prove convergence to equilibrium set

5. Multiple equilibrium handling

   • Characterize stable equilibrium points
   • Identify basins of attraction
   • Relevant for CAM (Content Addressable Memory) implementations

3. Flux-Charge Analysis Method (FCAM)

Key transformation:

• Voltage v → flux φ = ∫v dt
• Current i → charge q = ∫i dt
• State equations in (φ, q) domain instead of (v, i) domain

Advantages:

• Natural treatment of memristor memory
• Cleaner equilibrium analysis
• Direct connection to energy considerations

Implementation Steps

Step 1: Circuit Model Setup

# Define circuit topology
# - Identify all R, L, C, M elements
# - Construct Kirchhoff equations
# - Transform to flux-charge domain

def construct_rlcm_model(topology):
    """
    Input: Circuit topology (nodes, branches, element types)
    Output: Flux-charge domain equations
    
    Components:
    - Capacitor: dq/dt = i, q stored charge
    - Inductor: dφ/dt = v, φ stored flux
    - Memristor: v = R_M(φ) * i, where R_M(φ) is flux-dependent resistance
    """
    pass

Step 2: Mixed Potential Function

def compute_mixed_potential(q, phi, params):
    """
    Mixed potential P(q,φ) = P_C(q) + P_L(φ) + P_M(q,φ)
    
    Components:
    - P_C: Capacitor contribution (charge-based)
    - P_L: Inductor contribution (flux-based)
    - P_M: Memristor contribution (flux-dependent)
    
    Convergence condition: ∂P/∂q · dq/dt + ∂P/∂φ · dφ/dt ≤ 0
    """
    P_C = capacitor_potential(q, params['C'])
    P_L = inductor_potential(phi, params['L'])
    P_M = memristor_potential(q, phi, params['M'])
    return P_C + P_L + P_M

Step 3: Convergence Verification

def verify_convergence_conditions(P, q, phi):
    """
    Check Lyapunov-like conditions:
    1. P bounded below
    2. dP/dt ≤ 0 along trajectories
    3. Convergence to equilibrium set
    """
    dP_dt = compute_time_derivative(P, q, phi)
    return dP_dt <= 0  # Stability indicator

Step 4: Equilibrium Analysis

def find_equilibria(P, params):
    """
    Solve ∂P/∂q = 0, ∂P/∂φ = 0
    Identify stable vs unstable equilibria
    Basin of attraction characterization
    """
    equilibria = solve_gradient_zero(P)
    stability = classify_stability(equilibria, P)
    return equilibria, stability

Pitfalls

1. Incomplete variable set: Circuit must have complete flux-charge state representation. Missing variables lead to degenerate dynamics.

2. Balance violation: Capacitor-inductor balance must be quantitatively satisfied. Violation breaks convergence proof.

3. Domain confusion: Don't mix voltage-current and flux-charge analyses. Use consistent domain throughout.

4. Memristor modeling: Flux-dependent resistance must be properly characterized. Incorrect R_M(φ) leads to wrong potential.

5. Multiple equilibria: Not all equilibria are stable. Must classify stability type for each equilibrium.

6. Parameter sensitivity: Convergence is robust to parameter variations BUT balance conditions may shift. Re-verify after parameter changes.

Verification

1. Flux-charge completeness: Check that state variables span full phase space
2. Mixed potential boundedness: Verify P(q,φ) is bounded below
3. Time derivative sign: Confirm dP/dt ≤ 0 along trajectories
4. Equilibrium stability: Characterize stable/unstable equilibria
5. Simulation validation: Test convergence numerically for specific circuits
6. CAM functionality: For memory applications, verify addressable pattern storage/retrieval

Key Applications

1. Content Addressable Memories (CAMs): Multiple stable equilibria enable pattern matching
2. Neuromorphic computing: Memristor circuits emulate neural plasticity
3. Power electronics: Stability analysis for nonlinear converter circuits
4. Analog computing: Nonlinear dynamics for computation

Connection to Prior Work

• Brayton-Moser (1964): Original mixed potential for RL/RC circuits (no memristors)
• FCAM extensions: Prior work on memristor circuits without inductors
• This paper: Unified treatment of R, L, C, M together

Key Insight

The mixed potential approach generalizes naturally to memristors because:

• Memristor memory (flux) fits naturally in flux-charge domain
• Flux-dependent resistance R_M(φ) is compatible with potential formulation
• Lyapunov theory extends with careful balance conditions

Activation: memristor circuit, mixed potential, convergence analysis, flux-charge method, RLCM circuit, nonlinear stability, Lyapunov circuit analysis, content addressable memory, CAM implementation, neuromorphic hardware