Small-signal model

Small signal modelling is a common analysis method used in electrical engineering to describe nonlinear devices in terms of linear equations. The basic idea is to first calculate (possibly by an iterative process if the circuit 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

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, 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 ${\displaystyle V_{IN}}$ .

• Small signal quantities are denoted using lowercase letters with lowercase subscripts.

For example, the input signal of a trasistor would be denoted as ${\displaystyle v_{in}}$ .

• 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 ${\displaystyle v_{IN}(t)=V_{IN}(t)+v_{in}(t)}$ .

Example: PN junction diodes

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

${\displaystyle I=I_{0}(e^{qV/kT}-1)}$ 

The large signal capacitance of the diode is known to be

${\displaystyle Q=I\tau _{s}}$  where ${\displaystyle \tau _{s}}$  is the recombination lifetime of charge carriers ^[1].

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

${\displaystyle {\frac {dI}{dV}}=I_{0}{\frac {q}{kT}}e^{qV/kT}\approx {\frac {q}{kT}}I}$ 

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

Noting that ${\displaystyle {\frac {dI}{dV}}}$  corresponds to the instantaneous conductivity of the diode, the small signal resistance ${\displaystyle r}$  is the reciprocal of this quantity: ${\displaystyle r={\frac {1}{I}}{\frac {kT}{q}}}$ 

References

1. ^[2] Hu, Chenming. Semiconductor Devices for Integrated Circuits [Class Notes]. University of California, Berkeley, Spring 2005.