Thévenin's theorem

Definition
Thévenin's theorem states that any linear, bilateral electrical network composed of voltage sources, current sources, and resistors can be replaced by an equivalent circuit consisting of a single voltage source in series with a single impedance (or resistance for purely resistive networks), as observed from a pair of terminals.

Overview
The theorem provides a method for simplifying complex circuits to facilitate analysis, particularly when determining the current or voltage across a specific load. By reducing the external portion of the network to its Thévenin equivalent, engineers can more easily predict the behavior of the load under varying conditions. The theorem is applicable only to linear (i.e., obeying superposition) and bilateral (i.e., exhibiting the same behavior in both directions) networks; nonlinear or unilateral components invalidate the equivalence.

Etymology/Origin
Thévenin's theorem is named after Léon Charles Thévenin (1857–1926), a French engineer and physicist. Thévenin first presented the theorem in an 1883 paper titled “Sur un principe d'équivalence entre les réseaux électriques” (“On an equivalence principle for electrical networks”), establishing the foundational concept of circuit equivalence.

Characteristics

• Thévenin Equivalent Voltage (Vₜₕ): The open‑circuit voltage measured across the two terminals when the load is removed.
• Thévenin Equivalent Impedance (Zₜₕ): The impedance seen at the terminals with all independent voltage sources replaced by short circuits and independent current sources replaced by open circuits. For purely resistive networks, Zₜₕ reduces to the Thévenin resistance (Rₜₕ).
• Derivation Steps:
  1. Remove the load from the circuit.
  2. Calculate Vₜₕ as the open‑circuit voltage at the terminals.
  3. Deactivate all independent sources (short voltage sources, open current sources) and compute Zₜₕ (or Rₜₕ) looking back into the circuit.
• Relation to Norton’s Theorem: The Norton equivalent is a current source in parallel with an impedance; it is directly convertible to the Thévenin equivalent via source transformation (Vₜₕ = I_N·Z_N, Zₜₕ = Z_N).
• Limitations: The theorem does not apply to circuits containing dependent sources without additional analysis, nor to networks that are non‑linear, time‑variant, or contain active components that violate linearity.

Related Topics

• Ohm's Law – Fundamental relation between voltage, current, and resistance.
• Kirchhoff's Circuit Laws – Governing equations for current and voltage in electrical networks.
• Norton’s Theorem – Dual of Thévenin’s theorem using a current source and parallel impedance.
• Superposition Theorem – Technique for analyzing linear circuits with multiple independent sources.
• Maximum Power Transfer Theorem – Uses the Thévenin equivalent to determine conditions for optimal power delivery to a load.
• Circuit Analysis Methods – Mesh analysis, nodal analysis, and other systematic approaches that may incorporate Thévenin equivalents.