{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "nyquist-regeneration-feedback-stability-1932", "title": "Nyquist: when feedback stabilizes or makes a circuit sing", "summary": "Nyquist’s 1932 analysis relates the decay of a small disturbance in an idealized linear feedback circuit to how its frequency-response curve winds around a critical point. It explains why high feedback gain can be stable and why looking only at a steady-state gain can give the wrong answer.\n\nA microphone near a loudspeaker can create a sustained howl: part of the output returns as new input. Feedback can also make amplifiers more useful. Nyquist asks which of these outcomes a small disturbance produces. He studies one trip around the loop across frequencies, turning the resulting complex response into a curve. The curve’s winding about the critical point reveals growing transients that a steady-state calculation can miss. The original includes conditional-stability examples where increasing gain can first destabilize and then stabilize a system.", "boundary": "The exact target is the original linear, constant-gain feedback analysis, not all modern Nyquist generalizations. Its causal network impulse response has the stated bounded-variation and integrability restrictions; the open-loop auxiliary function is analytic in the right half-plane and has the required high-frequency behavior. Original winding is about +1 under its AJ convention, not blindly about modern -1. Net winding and boundary contacts matter; a gain-above-one crossing alone is not sufficient when phase behavior reverses. Stability means a dying small disturbance produces a response that dies out; a sustained oscillator is unstable under this definition. Unlimited-power linearization excludes general nonlinear amplitude stability. Original example 8 explicitly shows that omitting high-frequency decay invalidates the apparent curve conclusion; distributed effects change the curve. The original restricted criterion does not automatically cover unstable open-loop poles, multivariable or nonlinear systems. the account does not independently certify every formula, numerical curve or physical experiment. Later introductory commentary, generalized argument-principle formulations and global engineering adoption are separate.", "area": "Control theory", "topic": "Engineering", "year": 1932, "lane": "HUMAN", "result": "THEOREM", "package_id": "nyquist-regeneration-feedback-stability-1932", "target": { "record_id": "urn:uuid:77b96a62-1671-4a59-af1f-c767844de3c1", "version": 1, "content_hash": "sha256:fab062a96b2d4965b3cb29097bbbe594019f4324115e1aa92dff806b2b41c819" }, "humans": [ "Harry Nyquist" ], "authors_complete": true, "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.", "agents": [], "related": [ "nyquist-telegraph-transmission-theory-1928", "kalman-filter", "bellman-dynamic-programming" ], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "wrs24-nyquist-1932-wiley-reprint", "title": "H. Nyquist, Regeneration Theory (1932), original article reprinted with permission in publisher-hosted control anthology excerpt", "url": "https://media.wiley.com/product_data/excerpt/14/07803602/0780360214.pdf" }, { "id": "wrs24-nyquist-1932-publisher-metadata", "title": "Wiley, Regeneration Theory: original January 1932 bibliographic record", "url": "https://onlinelibrary.wiley.com/doi/abs/10.1002/j.1538-7305.1932.tb02344.x" } ], "points": "270000", "published_grade": { "id": "nyquist-regeneration-feedback-stability-1932", "entry_id": "nyquist-regeneration-feedback-stability-1932", "band": "S", "modifier": null, "target": { "record_id": "urn:uuid:77b96a62-1671-4a59-af1f-c767844de3c1", "version": 1, "content_hash": "sha256:fab062a96b2d4965b3cb29097bbbe594019f4324115e1aa92dff806b2b41c819" }, "what_changed": "Connects transient decay to winding of the one-round-trip response about +1, explains conditional stability and exposes missing high-frequency assumptions.", "scope": "Connects transient decay to winding of the one-round-trip response about +1, explains conditional stability and exposes missing high-frequency assumptions. The exact target is the original linear, constant-gain feedback analysis, not all modern Nyquist generalizations. Its causal network impulse response has the stated bounded-variation and integrability restrictions; the open-loop auxiliary function is analytic in the right half-plane and has the required high-frequency behavior. Original winding is about +1 under its AJ convention, not blindly about modern -1. Net winding and boundary contacts matter; a gain-above-one crossing alone is not sufficient when phase behavior reverses. Stability means a dying small disturbance produces a response that dies out; a sustained oscillator is unstable under this definition. Unlimited-power linearization excludes general nonlinear amplitude stability. Original example 8 explicitly shows that omitting high-frequency decay invalidates the apparent curve conclusion; distributed effects change the curve. The original restricted criterion does not automatically cover unstable open-loop poles, multivariable or nonlinear systems. the account does not independently certify every formula, numerical curve or physical experiment. 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Nyquist, Regeneration Theory (1932), original article reprinted with permission in publisher-hosted control anthology excerpt", "url": "https://media.wiley.com/product_data/excerpt/14/07803602/0780360214.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "H. Nyquist, Regeneration Theory (1932), original article reprinted with permission in publisher-hosted control anthology excerpt", "url": "https://media.wiley.com/product_data/excerpt/14/07803602/0780360214.pdf" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "Wiley, Regeneration Theory: original January 1932 bibliographic record", "url": "https://onlinelibrary.wiley.com/doi/abs/10.1002/j.1538-7305.1932.tb02344.x" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "The exact target is the original linear, constant-gain feedback analysis, not all modern Nyquist generalizations. Its causal network impulse response has the stated bounded-variation and integrability restrictions; the open-loop auxiliary function is analytic in the right half-plane and has the required high-frequency behavior. Original winding is about +1 under its AJ convention, not blindly about modern -1. Net winding and boundary contacts matter; a gain-above-one crossing alone is not sufficient when phase behavior reverses. Stability means a dying small disturbance produces a response that dies out; a sustained oscillator is unstable under this definition. Unlimited-power linearization excludes general nonlinear amplitude stability. Original example 8 explicitly shows that omitting high-frequency decay invalidates the apparent curve conclusion; distributed effects change the curve. The original restricted criterion does not automatically cover unstable open-loop poles, multivariable or nonlinear systems. the account does not independently certify every formula, numerical curve or physical experiment. Later introductory commentary, generalized argument-principle formulations and global engineering adoption are separate.", "domain": "Control theory" }, "sources": [ { "citation": "H. Nyquist, Regeneration Theory (1932), original article reprinted with permission in publisher-hosted control anthology excerpt", "url": "https://media.wiley.com/product_data/excerpt/14/07803602/0780360214.pdf" }, { "citation": "Wiley, Regeneration Theory: original January 1932 bibliographic record", "url": "https://onlinelibrary.wiley.com/doi/abs/10.1002/j.1538-7305.1932.tb02344.x" } ], "statement": "# Nyquist: when feedback stabilizes or makes a circuit sing\n\nNyquist’s 1932 analysis relates the decay of a small disturbance in an idealized linear feedback circuit to how its frequency-response curve winds around a critical point. It explains why high feedback gain can be stable and why looking only at a steady-state gain can give the wrong answer.\n\nA microphone near a loudspeaker can create a sustained howl: part of the output returns as new input. Feedback can also make amplifiers more useful. Nyquist asks which of these outcomes a small disturbance produces. He studies one trip around the loop across frequencies, turning the resulting complex response into a curve. The curve’s winding about the critical point reveals growing transients that a steady-state calculation can miss. The original includes conditional-stability examples where increasing gain can first destabilize and then stabilize a system.\n\n## Scientific scope\n\nThe exact target is the original linear, constant-gain feedback analysis, not all modern Nyquist generalizations. Its causal network impulse response has the stated bounded-variation and integrability restrictions; the open-loop auxiliary function is analytic in the right half-plane and has the required high-frequency behavior. Original winding is about +1 under its AJ convention, not blindly about modern -1. Net winding and boundary contacts matter; a gain-above-one crossing alone is not sufficient when phase behavior reverses. Stability means a dying small disturbance produces a response that dies out; a sustained oscillator is unstable under this definition. Unlimited-power linearization excludes general nonlinear amplitude stability. Original example 8 explicitly shows that omitting high-frequency decay invalidates the apparent curve conclusion; distributed effects change the curve. The original restricted criterion does not automatically cover unstable open-loop poles, multivariable or nonlinear systems. the account does not independently certify every formula, numerical curve or physical experiment. Later introductory commentary, generalized argument-principle formulations and global engineering adoption are separate.\n\n## People and systems\n\nHarry Nyquist\n\n\n\n## Original sources\n\n- [H. Nyquist, Regeneration Theory (1932), original article reprinted with permission in publisher-hosted control anthology excerpt](https://media.wiley.com/product_data/excerpt/14/07803602/0780360214.pdf)\n- [Wiley, Regeneration Theory: original January 1932 bibliographic record](https://onlinelibrary.wiley.com/doi/abs/10.1002/j.1538-7305.1932.tb02344.x)\n", "summary": "Nyquist’s 1932 analysis relates the decay of a small disturbance in an idealized linear feedback circuit to how its frequency-response curve winds around a critical point. It explains why high feedback gain can be stable and why looking only at a steady-state gain can give the wrong answer.\n\nA microphone near a loudspeaker can create a sustained howl: part of the output returns as new input. Feedback can also make amplifiers more useful. Nyquist asks which of these outcomes a small disturbance produces. He studies one trip around the loop across frequencies, turning the resulting complex response into a curve. The curve’s winding about the critical point reveals growing transients that a steady-state calculation can miss. The original includes conditional-stability examples where increasing gain can first destabilize and then stabilize a system.", "title": "Nyquist: when feedback stabilizes or makes a circuit sing" }, "record_id": "urn:uuid:77b96a62-1671-4a59-af1f-c767844de3c1", "version": 1, "source_ref": { "record_id": "urn:uuid:77b96a62-1671-4a59-af1f-c767844de3c1", "version": 1, "content_hash": "sha256:fab062a96b2d4965b3cb29097bbbe594019f4324115e1aa92dff806b2b41c819" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:17568aebdb09cbc40da970338a0e663ba2ffb452be567e7f44587ebd8f64751d" }, "target": { "record_id": "urn:uuid:77b96a62-1671-4a59-af1f-c767844de3c1", "version": 1, "content_hash": "sha256:fab062a96b2d4965b3cb29097bbbe594019f4324115e1aa92dff806b2b41c819" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "nyquist-regeneration-feedback-stability-1932", "entry_id": "nyquist-regeneration-feedback-stability-1932", "revision": 1, "title": "Nyquist: when feedback stabilizes or makes a circuit sing", "context": "Control theory", "track": "Control theory", "band": "S", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-04", "evidence_cutoff": "2026-10-04", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Connects transient decay to winding of the one-round-trip response about +1, explains conditional stability and exposes missing high-frequency assumptions.", "scope": "Connects transient decay to winding of the one-round-trip response about +1, explains conditional stability and exposes missing high-frequency assumptions. The exact target is the original linear, constant-gain feedback analysis, not all modern Nyquist generalizations. Its causal network impulse response has the stated bounded-variation and integrability restrictions; the open-loop auxiliary function is analytic in the right half-plane and has the required high-frequency behavior. Original winding is about +1 under its AJ convention, not blindly about modern -1. Net winding and boundary contacts matter; a gain-above-one crossing alone is not sufficient when phase behavior reverses. Stability means a dying small disturbance produces a response that dies out; a sustained oscillator is unstable under this definition. Unlimited-power linearization excludes general nonlinear amplitude stability. Original example 8 explicitly shows that omitting high-frequency decay invalidates the apparent curve conclusion; distributed effects change the curve. The original restricted criterion does not automatically cover unstable open-loop poles, multivariable or nonlinear systems. the account does not independently certify every formula, numerical curve or physical experiment. Later introductory commentary, generalized argument-principle formulations and global engineering adoption are separate.", "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.", "references": [ { "title": "H. Nyquist, Regeneration Theory (1932), original article reprinted with permission in publisher-hosted control anthology excerpt", "url": "https://media.wiley.com/product_data/excerpt/14/07803602/0780360214.pdf", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." }, { "title": "Wiley, Regeneration Theory: original January 1932 bibliographic record", "url": "https://onlinelibrary.wiley.com/doi/abs/10.1002/j.1538-7305.1932.tb02344.x", "locator": "Original linked work. 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