Small-signal model: Difference between revisions

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'''Small-signal modeling''' is a common analysis technique in [[electronics engineering]] used to approximate the behavior of [[electronic circuit]]s containing [[nonlinear device]]s, such as [[diode]]s, [[transistor]]s, [[vacuum tube]]s, and [[integrated circuit]]s, with [[linear equations]]. It is applicable to electronic circuits in which the AC [[signal (electrical engineering)|signal]]s (i.e., the time-varying currents and voltages in the circuit) are small relative to the DC [[bias (electrical engineering)|bias]] currents and voltages. A small-signal model is an AC [[equivalent circuit]] in which the nonlinear circuit elements are replaced by linear elements whose values are given by the first-order (linear) approximation of their characteristic curve near the bias point.

== Overview ==

Many of the [[electrical component]]s used in simple electric circuits, such as [[resistor]]s, [[inductor]]s, and [[capacitor]]s are [[linear circuit|linear]].{{Clarify|reason=At which current?|date=May 2022}}{{Citation needed|date=May 2022}} Circuits made with these components, called [[linear circuit]]s, are governed by [[linear differential equation]]s, and can be solved easily with powerful mathematical [[frequency ___domain]] methods such as the [[Laplace transform]].{{Citation needed|date=May 2022}}

In contrast, many of the components that make up ''electronic'' circuits, such as [[diode]]s, [[transistor]]s, [[integrated circuit]]s, and [[vacuum tube]]s are [[linear circuit|nonlinear]]; that is the current through{{Clarify|reason=At which current?|date=May 2022}} them is not proportional to the [[voltage]], and the output of [[two-port network|two-port]] devices like transistors is not proportional to their input. The relationship between current and voltage in them is given by a curved line on a graph, their [[Current–voltage characteristic|characteristic curve]] (I-V curve). In general these circuits don't have simple mathematical solutions. To calculate the current and voltage in them generally requires either [[graphical method]]s or simulation on computers using [[electronic circuit simulation]] programs like [[SPICE]].

However in some electronic circuits such as [[radio receiver]]s, telecommunications, sensors, instrumentation and [[signal processing]] circuits, the AC signals are "small" compared to the DC voltages and currents in the circuit. In these, [[perturbation theory]] can be used to derive an approximate [[equivalent circuit|AC equivalent circuit]] which is linear, allowing the AC behavior of the circuit to be calculated easily. In these circuits a steady [[direct current|DC]] current or voltage from the power supply, called a ''[[bias (electrical engineering)|bias]]'', is applied to each nonlinear component such as a transistor and vacuum tube to set its operating point, and the time-varying [[alternating current|AC]] current or voltage which represents the [[signal (electrical engineering)|signal]] to be processed is added to it. The point on the graph of the characteristic curve representing the bias current and voltage is called the ''[[quiescent point]]'' (Q point). In the above circuits the AC signal is small compared to the bias, representing a small perturbation of the DC voltage or current in the circuit about the Q point. If the characteristic curve of the device is sufficiently flat over the region occupied by the signal, using a [[Taylor series]] expansion the nonlinear function can be approximated near the bias point by its first order [[partial derivative]] (this is equivalent to approximating the characteristic curve by a straight line [[tangent (geometry)|tangent]] to it at the bias point). These partial derivatives represent the incremental [[capacitance]], [[electrical resistance|resistance]], [[inductance]] and [[gain (electronics)|gain]] seen by the signal, and can be used to create a linear [[equivalent circuit]] giving the response of the real circuit to a small AC signal. This is called the "small-signal model".

The small signal model is dependent on the DC bias currents and voltages in the circuit (the [[Q point]]). Changing the bias moves the operating point up or down on the curves, thus changing the equivalent small-signal AC resistance, gain, etc. seen by the signal.