Small-signal model: Difference between revisions

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{{Short description|Electronic circuit analysis method}}

{{unreferenced|date=March 2018}}

'''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".

=== Radioelectronics ===

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 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.

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A large signal is any signal having enough magnitude to reveal a circuit's nonlinear behavior. The signal may be a DC signal or an AC signal or indeed, any signal. How large a signal needs to be (in magnitude) before it is considered a ''large signal'' depends on the circuit and context in which the signal is being used. In some highly nonlinear circuits practically all signals need to be considered as large signals.

A small signal is part of a model consistsof a large signal. To avoid confusion, note that there is such a thing as a ''small signal'' (a part of ana ACmodel) and a ''small-signal model'' (morea technically,model of a large signal).

A small signal model consists of a small signal (having zero average value, for example a sinusoid, but any AC signal could be used) superimposed on a bias signal (or superimposed on a DC constant signal) such that the sum of the ACsmall signal plus the bias signal gives the total signal which is exactly equal to the original (large) signal to be modeled. This resolution of a signal into two components allows the technique of superposition to be used to simplify further analysis. (If superposition applies in the context.)

In analysis of the small signal's contribution to the circuit, the nonlinear components, which would be the DC components, are analyzed separately taking into account nonlinearity.