Small-signal model: Difference between revisions

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{{Short description|Electronic circuit analysis method}}

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'''Small-signal modeling''' is a common analysis technique in [[electronics engineering]] which is 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,) have aare small magnitude comparedrelative 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]],.{{Citation which means the [[electric currentneeded|current]]date=May in them is proportional to the applied [[voltage]].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-voltageCurrent–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]].

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Many of the [[electrical component]]s used in simple electric circuits, such as [[resistor]]s, [[inductor]]s, and [[capacitor]]s are [[linear circuit|linear]], which means the [[electric current|current]] in them is proportional to the applied [[voltage]]. Circuits made with these components, called [[linear circuit]]s, are governed by [[linear differential equation]]s, and can be solved easily with powerful mathematical methods such as the [[Laplace transform]].

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".

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 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 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.

Any nonlinear component whose characteristics are given by a [[continuity (mathematics)|continuous]], [[single-valued]], smooth ([[differentiability|differentiable]]) curve can be approximated by a linear small-signal model. Small-signal models exist for [[Vacuum tube|electron tube]]s, [[diode]]s, [[field-effect transistor]]s (FET) and [[Bipolar junction transistor|bipolar transistors]], notably the [[hybrid-pi model]] and various [[two-port network]]s. Manufacturers often list the small-signal characteristics of such components at "typical" bias values on their data sheets.

==Variable notation==

* Large-signal DC quantities (also known as ''bias''), constant values with respect to time, are denoted by uppercase letters with uppercase subscripts. For example, the DC input bias voltage of a transistor would be denoted <math>V_\mathrm{IN}</math>. For example, one might say that <math>V_\mathrm{IN} = 5</math>.

* Small-signal quantities, which have zero average value, are denoted using lowercase letters with lowercase subscripts. Small signals typically used for modeling are sinusoidal, or "AC", signals. For example, the input signal of a transistor would be denoted as <math>v_\mathrm{in}</math>. For example, one might say that <math>v_\mathrm{in}(t) = 0.2\cos (2\pi t)</math>.

* Total quantities, combining both small-signal and large-signal quantities, are denoted using lower case letters and uppercase subscripts. For example, the total input voltage to the aforementioned transistor would be denoted as <math>v_\mathrm{IN}(t)</math>. The small-signal model of the total signal is then the sum of the DC component and the small-signal component of the total signal, or in algebraic notation, <math>v_\mathrm{IN}(t)=V_\mathrm{IN}+v_\mathrm{in}(t)</math>. For example, <math>v_\mathrm{IN}(t)=5 + 0.2\cos (2\pi t)</math>

==Example: PN junction diodes==

{{main|Diode modelling#Small-signal modelling}}

The (large-signal) Shockley equation for a diode can be linearlinearized about the bias point or quiescent point (sometimes called [[Q-point]]) to find the small-signal [[Electrical conductance|conductance]], capacitance and resistance of the diode. This procedure is described in more detail under [[diode modelling#Small-signal modeling|diode modelingsignal_modelling]], which provides an example of the linearlinearization procedure followed in all small-signal models of semiconductor devices.

==Differences between small signal and large signal==

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 ator indeed, any signal. How large a pointsignal needs to be (in timemagnitude) before thatit is oneconsidered ora more''large orderssignal'' ofdepends magnitudeon largerthe thancircuit theand smallcontext signalin andwhich the signal is being used. to analyseIn asome circuithighly containingnonlinear non-linearcircuits componentspractically andall calculatesignals anneed operatingto pointbe (bias)considered ofas theselarge componentssignals.

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

A small signal is an AC signal superimposed on a circuit containing a large signal.

In analysis of the small signal's contribution to the circuit, the non-linearnonlinear components, arewhich modeledwould asbe linearthe DC components, are analyzed separately taking into account nonlinearity.

==See also==

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*[[Hybrid-pi model]]

*[[Early effect]]

*[[SPICE]] -– Simulation Program with Integrated Circuit Emphasis, a general purpose analog electronic circuit simulator capable of solving small signal models.

==References==