Small-signal model

Small-signal modeling is a common analysis technique in electronics engineering which is used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC signals, the time-varying currents and voltages in the circuit, have a small magnitude compared to the DC 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 components used in simple electric circuits, such as resistors, inductors, and capacitors are linear, which means the current in them is proportional to the applied voltage. Circuits made with these components, called linear circuits, are governed by linear differential equations, and can be solved easily with powerful mathematical methods such as the Laplace transform.

In contrast, many of the components that make up electronic circuits, such as diodes, transistors, integrated circuits, and vacuum tubes are nonlinear; that is the current through them is not proportional to the voltage, and the output of 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 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 methods or simulation on computers using electronic circuit simulation programs like SPICE.

However in some electronic circuits such as radio receivers, 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 AC equivalent circuit which is linear, allowing the AC behavior of the circuit to be calculated easily. In these circuits a steady DC current or voltage from the power supply, called a bias, is applied to each nonlinear component such as a transistor and vacuum tube to set its operating point, and the time-varying AC current or voltage which represents the 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. These partial derivatives represent the incremental capacitance, resistance inductance and 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 continuous, smooth (differentiable) curve can be approximated by the linear small-signal model. Small-signal models exist for electron tubes, diodes, field-effect transistors (FET) and bipolar transistors, notably the hybrid-pi model and various two-port networks. Manufacturers often list the small-signal characteristics of such components at "typical" bias values on their data sheets.

Variable notation

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 $V_{{\mathrm {IN}}}$ .
Small-signal quantities are denoted using lowercase letters with lowercase subscripts. For example, the input signal of a transistor would be denoted as $v_{{\mathrm {in}}}$ .
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 $v_{{\mathrm {IN}}}(t)=V_{{\mathrm {IN}}}+v_{{\mathrm {in}}}(t)$ .

Example: PN junction diodes

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

Differences between small signal and large signal

A large signal is a DC signal (or an AC signal at a point in time) that is one or more orders of magnitude larger than the small signal and is used to analyse a circuit containing non-linear components and calculate an operating point (bias) of these components.

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-linear components are modeled as linear components.

References

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