Ohmatic™: Resistor Network Synthesizer

Ohmatic™ Synthesizer & Hysteresis Resolver

Normal Synthesis

Resistors (n): 4

Engine Architecture:

Base Resistor Progression (R₁ … R_n): Quick Select:

Target Objective:

Singularity Stress Test:

Pareto States --

Game Fit --%

Fit NRMSE --%

BOM Cost $0.00

Span ΔR --

Min Step ΔR_min --

Max Gap ΔR_max --

Span Ratio S_R --

Base Resistors (1..n): Seed Mode

Boundary Ranges & Switch Fabric

Log₁₀ Bounds

R_min: 10 kΩ

10⁴

R_max: 1.0 MΩ

10⁶

Package:

Tolerance:

C_p (pF):

Switch Fabric Architecture:

Switches:--

R_on:-- Ω

C_off:-- pF

State Sequence Scrubber & Switching Engine

State 1 / 1

Switch Sequentiality: 0% (Parallel)

Switch Toggles:ΔS = 0

Glitch Energy:-- nJ

T_settle:-- μs

Active Circuit Schematic (Crisp Single-Pixel Canvas): --

Resistors: 42×24px [Index Only]

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Pareto Selection Justification:

Canonical optimal selection: Lowest parasitic capacitance C_eq and fewest toggled switches across equivalent resistance paths.

Y-Scale:

• Dark Line: Synthesized Pareto States • Shaded Area: Precision Envelope [±P] • Dashed Line: Ideal Progression Objective

Inverting Op-Amp Feedback Filter Parameters:

R_in: Ω GBW: MHz

Mathematical Models, Circuit Principles & Range Singularity Hysteresis Theorem

Ohmatic™ Synthesis Theory

Singularity Hysteresis & Lockstep

$$\Delta R_{\text{span}} = R_{\text{max}} - R_{\text{min}}, \quad \mu([R_0, R_0]) = 0$$

Lockstep Coupling Theorem: When $n = 1$, the resistor array can only synthesize a single degenerate point. Consequently, $R_{\text{min}}$ and $R_{\text{max}}$ must travel together in lockstep. Reciprocally, when $R_{\text{min}} = R_{\text{max}}$, network degrees of freedom collapse to $n = 1$. The state machine preserves $n_{\text{prior}} \ge 2$ in hysteresis memory, restoring prior state upon range decoupling.

$$0 < \Delta R_{\text{min}} \le R_{k+1} - R_k \le \Delta R_{\text{max}} < R_{\text{max}}$$

Cograph SP-Trees & Pareto Filter

$$R_{\text{series}} = \sum_k R_k, \quad \frac{1}{R_{\text{parallel}}} = \sum_k \frac{1}{R_k}$$

Cographs (complement-reducible graphs) contain no induced $P_4$ paths and decompose into canonical Series-Parallel trees. When redundant topologies yield equivalent resistances ($\vert{}\Delta R / R\vert{} < 0.15\%$), the Ohmatic™ Pareto engine selects:

$$\arg\min \Big[ C_{\text{eq}}(\mathcal{T}) \Big] \to \arg\min \Big[ \Delta S(\mathcal{T}) \Big] \to \arg\max \Big[ I_{\text{max}}(\mathcal{T}) \Big]$$

Parasitic Capacitance & Cutoff

$$\frac{1}{C_{\text{series}}} = \sum_k \frac{1}{C_k}, \quad C_{\text{parallel}} = \sum_k C_k$$

Parasitic capacitance follows the inverse harmonic algebraic dual of resistance. For identical base packaging ($C_p / R_k \approx \text{const}$), the network preserves an inherent time-constant invariance:

$$f_c = \frac{1}{2\pi R_{\text{eq}} C_{\text{eq}}}, \quad \tau = R_{\text{eq}} \cdot C_{\text{eq}} \approx \text{constant}$$

Switch Hamming & Op-Amp Filter

$$\Delta S = \sum_{j=1}^{n} |s_j^{(k)} - s_j^{(k-1)}|, \quad E_{\text{glitch}} \propto \Delta S \cdot (1 + \eta_{\text{seq}})$$

Staggering switch actuation ($\eta_{\text{seq}} > 0$) causes ripple chatter, increasing transient glitch energy. For inverting op-amp feedback loops:

$$A_v(f) = -\frac{Z_f(f)}{R_{\text{in}}} = -\frac{R_{\text{eq}}}{R_{\text{in}} \sqrt{1 + (2\pi f R_{\text{eq}} C_{\text{eq}})^2}}$$