Thevenins TheoremEdit

Thevenin's theorem is a central result in circuit theory that greatly simplifies the analysis of linear networks. It says that any linear, bilateral network seen from two terminals can be replaced by an equivalent circuit consisting of a single voltage source in series with a single resistor. The external behavior of the original network, when driving any load connected across those two terminals, is identical to that of the Thevenin equivalent. This idea, along with its dual form known as the Norton equivalent, is foundational for understanding how complex circuitry interacts with loads.

The theorem is named after Léon Thévenin, who introduced the concept in the late 19th century. Its practical value lies in reducing a potentially intricate network to a simple, portable model that can be analyzed using straightforward tools like Ohm's law and voltage division Thevenin's theorem.

Historical background

The Thevenin framework emerged from work in early electrical engineering on how to analyze telegraph and telephone networks. Thévenin's insight was to show that the internal details of a network could be encapsulated by two parameters at the terminals: a voltage that would appear with the load disconnected (the open-circuit voltage) and an equivalent resistance that determines how the network responds to any attached load. This perspective paved the way for a broad class of circuit simplifications used throughout electronics and power engineering. For broader context, see Léon Thévenin and the development of network theorems in circuit theory.

Statement and definitions

The Thevenin equivalent of a network as seen from two terminals consists of:
- a voltage source V_th in series with
- a resistance R_th
Key quantities:
- V_th is the open-circuit voltage across the two terminals (the voltage with no load connected). It depends only on the internal sources and impedances of the network.
- R_th is the equivalent resistance “looking back” into the network from the terminals with all independent sources deactivated (voltage sources replaced by shorts, current sources replaced by opens) while keeping dependent sources active. If the network contains dependent sources, R_th can be found by methods such as applying a test source or computing the ratio V_test / I_test.
The Thevenin form gives the same load current as the original network for any linear load. In mathematical terms, for a load Z_L attached across the terminals, the load voltage V_L and load current I_L satisfy: V_L = V_th × (Z_L / (R_th + Z_L)) and I_L = V_th / (R_th + Z_L) for AC or DC analysis, with Z_L representing the load impedance when generalized to frequency-dependent circuits. See also impedance and AC circuit.
The dual form is the Norton equivalent: a current source in parallel with a resistor, where I_N = V_th / R_th and R_N = R_th. The Norton and Thevenin representations are interchangeable for the same two-terminal network; see Norton’s theorem for details.

How to find the Thevenin equivalent

Open-circuit method:
- Detach the load and measure or compute the voltage across the terminals. This is V_th.
Short-circuit method (for R_th with independent sources deactivated) or resistive-method:
- If there are only independent sources, deactivate them (voltage sources become shorts, current sources become opens) and find the resistance seen from the terminals; that resistance is R_th.
- If dependent sources are present, deactivate independent sources but keep dependent sources active, then determine R_th via a test source (apply a known voltage or current at the terminals and compute the resulting current or voltage) or by evaluating the short-circuit current with the appropriate conditions. See Thevenin's theorem and Norton’s theorem for practical procedures.
A common shortcut is to use the open-circuit voltage V_th and the short-circuit current I_sc:
- R_th = V_th / I_sc
- This formula remains valid as long as the network is linear, even with dependent sources present, provided the short-circuit current is measured with proper conditions in place.

Example (brief illustration)

Consider a network with a mixed set of sources and resistors connected to two terminals. If removing the load yields an open-circuit voltage of 12 V, and shorting the terminals yields a short-circuit current of 3 A, then: - V_th = 12 V - R_th = V_th / I_sc = 12 V / 3 A = 4 Ω

Thus the complex network is equivalent, from the perspective of any connected load, to a 12 V source in series with a 4 Ω resistor. Any load Z_L connected across the terminals will see the same voltage and current as it would in the original network, allowing straightforward analysis via Ohm’s law.

Applications and significance

Simplified circuit analysis: Reducing a complex internal network to a single source and resistor makes it easy to compute voltages and currents for any attached load.
Design and testing: Engineers can model portions of a circuit to understand how variations in sources or impedances affect a connected stage, such as amplifiers, power supplies, or sensor front-ends.
Power transfer considerations: The Thevenin view clarifies how much driving strength a network provides to a load and is related to the maximum power transfer criterion when the load impedance matches R_th.
Education: The theorem is a staple in curricula on electrical engineering and circuit theory, bridging basic concepts like Ohm’s law, voltage division, and source transformation ("source transformation").

Limitations and generalizations

Applicability: The Thevenin equivalent is valid for linear, bilateral networks containing independent and dependent sources, as seen from two terminals. It does not apply to nonlinear or time-varying networks without modification.
Frequency-domain generalization: In AC analysis, R_th extends to impedance Z_th, and the Thevenin model becomes a Thevenin impedance in series with a Thevenin voltage source. See impedance and AC circuit for details.
Multi-terminal extension: For networks with more than two terminals, Thevenin’s idea can be applied locally by selecting a pair of terminals, but a single global Thevenin equivalent for all terminals does not generally exist. See multi-terminal network discussions in modern texts.