In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly, that is a simplest form of a more complex circuit in order to aid analysis. In its most common form, an equivalent circuit is made up of linear, passive elements. However, more complex equivalent circuits are used that approximate the nonlinear behavior of the original circuit as well. These more complex circuits often are called ''macromodels'' of the original circuit. An example of a macromodel is the Boyle circuit for the 741 operational amplifier.

Examples

Thévenin and Norton equivalents

One of linear circuit theory's most surprising properties relates to the ability to treat any two-terminal circuit no matter how complex as behaving as only a source and an impedance, which have either of two simple equivalent circuit forms: * Thévenin equivalent - Any linear two-terminal circuit can be replaced by a single

voltage source A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unl ...

and a series impedance. *

Norton equivalent In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of te ...

- Any linear two-terminal circuit can be replaced by a

current source A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it. A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed ...

and a parallel impedance. However, the single impedance can be of arbitrary complexity (as a function of frequency) and may be irreducible to a simpler form.

DC and AC equivalent circuits

In linear circuits, due to the superposition principle, the output of a circuit is equal to the sum of the output due to its DC sources alone, and the output from its AC sources alone. Therefore, the DC and AC response of a circuit is often analyzed independently, using separate DC and AC equivalent circuits which have the same response as the original circuit to DC and AC currents respectively. The composite response is calculated by adding the DC and AC responses: *A DC equivalent of a circuit can be constructed by replacing all capacitances with open circuits, inductances with short circuits, and reducing AC sources to zero (replacing AC voltage sources by short circuits and AC current sources by open circuits.) *An AC equivalent circuit can be constructed by reducing all DC sources to zero (replacing DC voltage sources with short circuits and DC current sources with open circuits) This technique is often extended to

small-signal Small-signal modeling is a common analysis technique in electronics engineering used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC ...

nonlinear circuits like tube and transistor circuits, by linearizing the circuit about the DC bias point

Q-point In electronics, biasing is the setting of DC (direct current) operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processin ...

, using an AC equivalent circuit made by calculating the equivalent ''small signal'' AC resistance of the nonlinear components at the bias point.

Two-port networks

Linear four-terminal circuits in which a signal is applied to one pair of terminals and an output is taken from another, are often modeled as

two-port network A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sat ...

s. These can be represented by simple equivalent circuits of impedances and dependent sources. To be analyzed as a two port network the currents applied to the circuit must satisfy the ''port condition'': the current entering one terminal of a port must be equal to the current leaving the other terminal of the port. By linearizing a nonlinear circuit about its

operating point {{unreferenced, date=April 2013 The operating point is a specific point within the operation characteristic of a technical device. This point will be engaged because of the properties of the system and the outside influences and parameters. In el ...

, such a two-port representation can be made for transistors: see hybrid pi and h-parameter circuits.

Delta and Wye circuits

three phase power Three-phase electric power (abbreviated 3φ) is a common type of alternating current used in electricity generation, transmission, and distribution. It is a type of polyphase system employing three wires (or four including an optional neutral ...

circuits, three phase sources and loads can be connected in two different ways, called a "delta" connection and a "wye" connection. In analyzing circuits, sometimes it simplifies the analysis to convert between equivalent wye and delta circuits. This can be done with the wye-delta transform.

In biology

Equivalent circuits can be used to electrically describe and model either a) continuous materials or biological systems in which current does not actually flow in defined circuits or b) distributed reactances, such as found in electrical lines or windings, that do not represent actual discrete components.{{cite journal , last1=Van Haeverbeke , first1=Maxime , last2=Stock , first2=Michiel , last3=De Baets , first3=Bernard , title=Equivalent Electrical Circuits and Their Use Across Electrochemical Impedance Spectroscopy Application Domains , journal=IEEE Access , date=2022 , volume=10 , pages=51363 - 51379 , doi=10.1109/ACCESS.2022.3174067 , url=https://ieeexplore.ieee.org/document/9771445 , access-date=18 November 2022 For example, a

cell membrane The cell membrane (also known as the plasma membrane (PM) or cytoplasmic membrane, and historically referred to as the plasmalemma) is a biological membrane that separates and protects the interior of all cells from the outside environment ( ...

can be modelled as a

capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...

(i.e. the

lipid bilayer The lipid bilayer (or phospholipid bilayer) is a thin polar membrane made of two layers of lipid molecules. These membranes are flat sheets that form a continuous barrier around all cells. The cell membranes of almost all organisms and many vir ...

) in parallel with resistance-DC

combinations (i.e. ion channels powered by an ion

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

across the

membrane A membrane is a selective barrier; it allows some things to pass through but stops others. Such things may be molecules, ions, or other small particles. Membranes can be generally classified into synthetic membranes and biological membranes. ...

References

Circuit theorems