Small-signal model: Difference between revisions

Revision as of 00:31, 16 January 2006

Revision as of 21:24, 9 February 2006

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'''Small signal modelling''' is ana common analysis method used in [[electrical engineering]], to describe [[nonlinear device]]s in terms of [[linear deviceequations]]s. The basic idea is to first calculate (possibllypossibly by an [[iterative]] process if the cuircuitcircuit is complex) the levels that will be present when no signal is applied, then form linear approximations for the deviations from that base state.

==Motivation==

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Electronic circuits generally involve small time-varying signals carried over a constant [[bias]]. This suggests using an method akin to approximation by [[differentials]] to analyze relatively small perturbations about the [[bias point]].

Any nonlinear device which can be described quantitatively using a formula can then be 'linearized' about a bias point by taking partial derivatives of the formula with respect to all governing variables. These partial derivatives can be associated with physical quantities (such as [[capacitance]], [[electrical resistance|resistance]] and [[inductance]]), and a circuit diagram relating them can be formulated.

Small signal models exist for [[diodes]], [[field effect transistors]] and [[bipolar transistors]].

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* Large signal DC quantities are denoted by uppercase letters with uppercase subscripts. For example, the DC input bias voltage of a transistor would be denoted <math>V_{IN}</math>.

* Small signal quantities are denoted using lowercase letters with lowercase suscriptssubscripts.

For example, the input signal of a trasistor would be denoted as <math>v_{in}</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 <math>v_{IN}(t)=V_{IN}(t)+v_{in}(t)</math>.

==Example: PN Junctionjunction Diodesdiodes==

The large signal I-V characteristicscharacteristic of the PN junction diode under forward bias is described by the Shockley Equation:

==Example: PN Junction Diodes==

The large signal I-V characteristics of the PN junction diode under forward bias is described by the Shockley Equation:

<math>I = I_0(e^{qV/kT}-1)</math>

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<math>Q=I\tau_s</math>

where <math>\tau_s</math> is the recombination lifetime of charge carriers [Hu 36]{{ref|Hu36}}.

where <math>\tau_s</math> is the recombination lifetime of charge carriers [Hu 36].

Given these two relations, the small signal resistance and capacitance of the diode can be derived about some operating point P.

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<math>\frac {dI} {dV} = I_0 \frac{q} {kT} e^{qV/kT} \approx \frac{q} {kT} I</math>

The latter approximation assumes that the bias current I is large enough so that the factor of 1 in the parethesesparentheses of the Shockley Equation can be ignored. This approximation is fairly common in nonlinear circuit analysis.

Noting that <math>\frac {dI} {dV}</math> corresponds to the instantaneous conductivity of the diode, the small signal resistance <math>r</math> is the reciprocal of this quantity:

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==References==

[B1]#{{ref|Hu36}} Hu, Chenming. Semiconductor Devices for Integrated Circuits [Class Notes]. University of California, Berkeley, Spring 2005.

[[Category:Electrical engineering]]