{
  "$schema": "https://siliconverify.com/schemas/tool-v1.json",
  "slug": "emft-sparam",
  "title": "S-Parameter Workbench — 2-Port Networks — EMFT",
  "category": "electromagnetics",
  "category_label": "Electromagnetics",
  "urls": {
    "interactive": "https://siliconverify.com/emft-sparam.html",
    "static_sheet": "https://siliconverify.com/ref-emft-sparam.html",
    "api_doc": "https://siliconverify.com/api/emft-sparam.json",
    "blog_index": "https://siliconverify.com/llms-electromagnetics.txt"
  },
  "description": "The S-Parameter workbench takes a 2×2 complex S-matrix and computes the four S_ij on polar plots and on a Smith chart (for S₁₁ and S₂₂). It auto-computes stability circles (input/output), the Rollett stability factor K, the maximum stable gain, and the unilateral figure of merit. Use it to characterize transistors, filters, and any 2-port network.",
  "keywords": [
    "emft",
    "s-parameters",
    "2-port network",
    "scattering parameters",
    "abcd",
    "z-matrix",
    "y-matrix",
    "rollett",
    "stability circle",
    "gain",
    "isolation",
    "reciprocity",
    "envelope correlation",
    "smith chart",
    "microwave",
    "GATE ECE"
  ],
  "pricing": {
    "class": "workbench-tool",
    "protocol": "x402",
    "network": "base",
    "currency": "USDC",
    "price_atomic": 1000,
    "price_usd": 0.001,
    "payTo": "0x97DAA5649Fd7Dfe3e46fCd9f75516e36E750eed7",
    "static_sheet_price_atomic": 1000,
    "export_price_atomic": 5000
  },
  "content": {
    "formulas": [
      "S-parameter definition: b_i = Σ S_ij·a_j (waves, not V/I)",
      "S₁₁ = input reflection (port 2 matched)",
      "S₂₁ = forward transmission (port 2 matched)",
      "Rollett stability: K = (1 − |S₁₁|² − |S₂₂|² + |Δ|²) / (2·|S₁₂·S₂₁|) > 1 (unconditional stability)",
      "Δ = S₁₁·S₂₂ − S₁₂·S₂₁",
      "Maximum stable gain: MSG = |S₂₁|/|S₁₂|",
      "Maximum available gain: MAG = (|S₂₁|/|S₁₂|)·(K − √(K² − 1)) for K > 1"
    ],
    "citations": [
      "Pozar, D. M., \"Microwave Engineering,\" 4th ed., 2011, Ch. 12.",
      "Gonzalez, G., \"Microwave Transistor Amplifiers,\" 2nd ed., Prentice Hall, 1997."
    ],
    "faq": [
      {
        "q": "My K = 0.5 — what does that mean?",
        "a": "K < 1 means potentially unstable (oscillation possible). The output stability circle tells you which load impedances are stable."
      },
      {
        "q": "Why is S₁₁ not the input impedance?",
        "a": "S₁₁ is the input reflection coefficient. The input impedance is Z_in = Z₀·(1 + S₁₁)/(1 − S₁₁)."
      }
    ]
  },
  "accessibility": {
    "crawlable": true,
    "javascript_required": true,
    "static_alternative": "/ref-emft-sparam.html",
    "structured_data": [
      "TechArticle",
      "SoftwareApplication",
      "FAQPage",
      "BreadcrumbList"
    ]
  },
  "x402_challenge_example": {
    "request": "curl -A 'GPTBot/1.0' -i https://siliconverify.com/emft-sparam.html",
    "response_headers": [
      "HTTP/1.1 402 Payment Required",
      "X-Siliconverify-Resource-Class: workbench-tool",
      "X-Siliconverify-Price-Atomic: 1000",
      "X-Siliconverify-Price-USD: 0.001"
    ],
    "response_body": "{\n  \"x402Version\": 1,\n  \"accepts\": [\n    {\n      \"scheme\": \"exact\",\n      \"network\": \"base\",\n      \"maxAmountRequired\": \"1000\",\n      \"resource\": \"https://siliconverify.com/emft-sparam.html\",\n      \"description\": \"SiliconVerify workbench tool — interactive browser-based engineering calculator. Client-side computation, no server cost.\",\n      \"mimeType\": \"text/html\",\n      \"payTo\": \"0x97DAA5649Fd7Dfe3e46fCd9f75516e36E750eed7\",\n      \"validAfter\": 1783274656,\n      \"validBefore\": 1783274956\n    }\n  ]\n}",
    "paid_request": "curl -A 'GPTBot/1.0' -H 'X-PAYMENT: 0x<txhash>' -i https://siliconverify.com/emft-sparam.html"
  },
  "last_updated": "2026-07-05"
}