Small-signal model: Difference between revisions

Revision as of 17:13, 13 April 2005

Revision as of 06:58, 14 April 2005

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

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

==Notational Conventions==

* 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 suscripts.

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 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>

The large signal capacitance of the diode is known to be

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

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.

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

The latter approximation assumes that the bias current I is large enough so that the factor of 1 in the paretheses 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 of the diode, the small signal resistance <math>r</math> is the reciprocal of this quantity:

<math>r=\frac{qV} {kT} I</math>

==References==

[B1] Hu, Chenming. Semiconductor Devices for Integrated Circuits. University of California, Berkeley, Spring 2005.

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