Small-signal modeling is a common analysis technique in
electronics engineering
Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current f ...
used to approximate the behavior of
electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. It is a type of electrical ...
s containing
nonlinear device
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electric ...
s with
linear equations
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
. It is applicable to electronic circuits in which the AC
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
s (i.e., the time-varying currents and voltages in the circuit) are small relative to the DC
bias
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
currents and voltages. A small-signal model is an AC
equivalent circuit In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly ...
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
An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not ...
s used in simple electric circuits, such as
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s,
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s, and
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s are
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
. Circuits made with these components, called
linear circuit
A linear circuit is an electronic circuit which obeys the superposition principle. This means that the output of the circuit ''F(x)'' when a linear combination of signals ''ax1(t) + bx2(t)'' is applied to it is equal to the linear combination o ...
s, are governed by
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
s, and can be solved easily with powerful mathematical
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
methods such as the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
.
In contrast, many of the components that make up ''electronic'' circuits, such as
diode
A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other.
A diode ...
s,
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ...
s,
integrated circuit
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
s, and
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied.
The type kn ...
s are
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
; that is the current through them is not proportional to the
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
, and the output of
two-port
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sat ...
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 method
This is a list of graphical methods with a mathematical basis.
Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization.
There is also a list of computer graphics and descriptive geometry topics.
Simple ...
s or simulation on computers using
electronic circuit simulation Electronic circuit simulation uses mathematical models to replicate the behavior of an actual electronic device or circuit.
Simulation software allows for modeling of circuit operation and is an invaluable analysis tool. Due to its highly accurat ...
programs like
SPICE
A spice is a seed, fruit, root, bark, or other plant substance primarily used for flavoring or coloring food. Spices are distinguished from herbs, which are the leaves, flowers, or stems of plants used for flavoring or as a garnish. Spices a ...
.
Radioelectronics
However in some electronic circuits such as
radio receiver
In radio communications, a radio receiver, also known as a receiver, a wireless, or simply a radio, is an electronic device that receives radio waves and converts the information carried by them to a usable form. It is used with an antenna. Th ...
s, telecommunications, sensors, instrumentation and
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
circuits, the AC signals are "small" compared to the DC voltages and currents in the circuit. In these,
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
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
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
'', 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
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
to be processed is added to it. The point on the graph representing the bias current and voltage is called the ''
quiescent point
In electronics, biasing is the setting of DC (direct current) operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processing ...
'' (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
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion the nonlinear function can be approximated near the bias point by its first order
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
(this is equivalent to approximating the characteristic curve by a straight line
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to it at the bias point). These partial derivatives represent the incremental
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
,
resistance,
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
and
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
seen by the signal, and can be used to create a linear
equivalent circuit In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly ...
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
In electronics, biasing is the setting of DC (direct current) operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processing ...
). 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
,
single-valued, smooth (
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
) curve can be approximated by a linear small-signal model. Small-signal models exist for
electron tubes,
diode
A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other.
A diode ...
s,
field-effect transistor
The field-effect transistor (FET) is a type of transistor that uses an electric field to control the flow of current in a semiconductor. FETs (JFETs or MOSFETs) are devices with three terminals: ''source'', ''gate'', and ''drain''. FETs contro ...
s (FET) and
bipolar transistors
A bipolar junction transistor (BJT) is a type of transistor that uses both electrons and electron holes as charge carriers. In contrast, a unipolar transistor, such as a field-effect transistor, uses only one kind of charge carrier. A bipolar t ...
, notably the
hybrid-pi model The hybrid-pi model is a popular circuit model used for analyzing the small signal behavior of bipolar junction and field effect transistors. Sometimes it is also called Giacoletto model because it was introduced by L.J. Giacoletto in 1969. The m ...
and various
two-port network
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
s. Manufacturers often list the small-signal characteristics of such components at "typical" bias values on their data sheets.
Variable notation
* 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
. For example, one might say that
.
* 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
. For example, one might say that
.
* 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
. 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,
. For example,
PN junction diodes
The (large-signal) Shockley equation for a diode can be linearized about the bias point or quiescent point (sometimes called
Q-point
In electronics, biasing is the setting of DC (direct current) operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processing ...
) to find the small-signal
conductance, capacitance and resistance of the diode. This procedure is described in more detail under
diode modelling#Small-signal_modelling, which provides an example of the linearization procedure followed in 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 or indeed, any signal. How large a signal needs to be (in magnitude) before it is considered a ''large signal'' depends on the circuit and context in which the signal is being used. In some highly nonlinear circuits practically all signals need to be considered as large signals.
A small signal is an AC signal (more technically, a signal having zero average value) superimposed on a bias signal (or superimposed on a DC constant signal). 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.)
In analysis of the small signal's contribution to the circuit, the nonlinear components, which would be the DC components, are analyzed separately taking into account nonlinearity.
See also
*
Diode modelling
In electronics, diode modelling refers to the mathematical models used to approximate the actual behaviour of real diodes to enable calculations and circuit analysis. A diode's I- V curve is nonlinear.
A very accurate, but complicated, physical ...
*
Hybrid-pi model The hybrid-pi model is a popular circuit model used for analyzing the small signal behavior of bipolar junction and field effect transistors. Sometimes it is also called Giacoletto model because it was introduced by L.J. Giacoletto in 1969. The m ...
*
Early effect
The Early effect, named after its discoverer James M. Early, is the variation in the effective width of the base in a bipolar junction transistor (BJT) due to a variation in the applied base-to-collector voltage. A greater reverse bias across ...
*
SPICE
A spice is a seed, fruit, root, bark, or other plant substance primarily used for flavoring or coloring food. Spices are distinguished from herbs, which are the leaves, flowers, or stems of plants used for flavoring or as a garnish. Spices a ...
– Simulation Program with Integrated Circuit Emphasis, a general purpose analog electronic circuit simulator capable of solving small signal models.
References
{{reflist
Electronic device modeling