A
linear circuit is an
electronic circuitAn electronic circuit is a closed path or paths formed by the interconnection of electronic components through which an electric current can flow....
in which, for a
sinusoidalThe sine wave or sinusoid is a function that occurs often in mathematics, music, physics, signal processing, audition, electrical engineering, and many other fields...
input voltage of
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
f, any output of the circuit (the
currentElectric current can mean, depending on the context, a flow of electric charge or the rate of flow of electric charge ....
through any component, or the
voltageVoltage is commonly used as a short name for electrical potential difference. Its corresponding SI unit is the volt...
between any two points) is also sinusoidal with frequency
f. Note that the output need not be
in phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
with the input.
Another equivalent way of defining a linear circuit is that it obeys the
superposition principleIn physics and systems theory, the superposition principle, also known as superposition property, states that, for all linear systems,So that if input A produces response X and input B produces response Y then input produces response .Mathematically, for all linear systems F = y, where x is some...
. 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 of the outputs due to the signals
x1(t) and
x2(t) applied separately:
Informally, a linear circuit is one in which the values of the
electronic componentAn electronic component is a basic electronic element usually packaged in a discrete form with two or more connecting leads or metallic pads....
s, the
resistanceThe electrical resistance of an object is a measure of its opposition to the passage of a steady electric current. An object of uniform cross section will have a resistance proportional to its length and inversely proportional to its cross-sectional area, and proportional to the resistivity of the...
,
capacitanceIn electromagnetism and electronics, capacitance is the ability of a body to hold an electrical charge.Capacitance is also a measure of the amount of electric charge stored for a given electric potential. A common form of charge storage device is a parallel-plate capacitor...
,
inductanceInductance is the property in an electrical circuit where a change in the electric current through that circuit induces an electromotive force that opposes the change in current ....
,
gainGain is a measure of a system's response to feedback. If the gain in a positive feedback loop is less than 1, the feedback is not of itself sufficient to make the system become unstable. For example, water evaporating from the World's oceans causes a positive feedback, as it is a greenhouse gas...
, etc.
A
linear circuit is an
electronic circuitAn electronic circuit is a closed path or paths formed by the interconnection of electronic components through which an electric current can flow....
in which, for a
sinusoidalThe sine wave or sinusoid is a function that occurs often in mathematics, music, physics, signal processing, audition, electrical engineering, and many other fields...
input voltage of
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
f, any output of the circuit (the
currentElectric current can mean, depending on the context, a flow of electric charge or the rate of flow of electric charge ....
through any component, or the
voltageVoltage is commonly used as a short name for electrical potential difference. Its corresponding SI unit is the volt...
between any two points) is also sinusoidal with frequency
f. Note that the output need not be
in phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
with the input.
Another equivalent way of defining a linear circuit is that it obeys the
superposition principleIn physics and systems theory, the superposition principle, also known as superposition property, states that, for all linear systems,So that if input A produces response X and input B produces response Y then input produces response .Mathematically, for all linear systems F = y, where x is some...
. 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 of the outputs due to the signals
x1(t) and
x2(t) applied separately:
Informally, a linear circuit is one in which the values of the
electronic componentAn electronic component is a basic electronic element usually packaged in a discrete form with two or more connecting leads or metallic pads....
s, the
resistanceThe electrical resistance of an object is a measure of its opposition to the passage of a steady electric current. An object of uniform cross section will have a resistance proportional to its length and inversely proportional to its cross-sectional area, and proportional to the resistivity of the...
,
capacitanceIn electromagnetism and electronics, capacitance is the ability of a body to hold an electrical charge.Capacitance is also a measure of the amount of electric charge stored for a given electric potential. A common form of charge storage device is a parallel-plate capacitor...
,
inductanceInductance is the property in an electrical circuit where a change in the electric current through that circuit induces an electromotive force that opposes the change in current ....
,
gainGain is a measure of a system's response to feedback. If the gain in a positive feedback loop is less than 1, the feedback is not of itself sufficient to make the system become unstable. For example, water evaporating from the World's oceans causes a positive feedback, as it is a greenhouse gas...
, etc. don't change with the level of voltage or current in the circuit.
Examples
Examples of linear circuits are
amplifierGenerally, an amplifier or simply amp, is any device that changes, usually increases, the amplitude of a signal. The relationship of the input to the output of an amplifier—usually expressed as a function of the input frequency—is called the transfer function of the amplifier, and the magnitude of...
s,
differentiatorA Differentiator is a circuit that is designed such that the output of the circuit is proportional to the time derivative of the input. There are two types of differentiator circuits, active and passive.- Theory :...
s, and
integratorAn integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus.The integration function is often part of engineering, physics, mechanical, chemical and scientific calculations....
s, or any circuit composed exclusively of
ideal resistor|- align = "center"||width = "25"|| |- align = "center"||| Potentiometer|- align = "center"| || |- align = "top"| Resistor|| Variable
resistor|- align = "center"||width = "25"|| |- align = "center"||| Potentiometer|- align = "center"...
s,
capacitorA capacitor or condenser is a passive electronic component consisting of a pair of conductors separated by a dielectric. When a voltage potential difference exists between the conductors, an electric field is present in the dielectric. This field stores energy and produces a mechanical force...
s,
inductorAn inductor or a reactor is a passive electrical component that can store energy in a magnetic field created by the electric current passing through it. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s, op-amps (in the "non-saturated" regime), and other "linear" circuit elements.
Some examples of circuits and components that operate in a nonlinear way:
diodeIn electronics a diode is a two-terminal electronic component which conducts electric current asymmetrically or unidirectionally; that is, it conducts current more easily in one direction than in the opposite direction. The term usually refers to a semiconductor diode, the most common type today,...
s,
transistorA transistor is a semiconductor device commonly used to amplify or switch electronic signals. A transistor is made of a solid piece of a semiconductor material, with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's...
s when they are operated in saturation,
iron coreA magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical and electromechanical devices such as electromagnets, transformers, electric motors, and inductors. It is usually made of ferromagnetic metal such as iron, or...
inductorAn inductor or a reactor is a passive electrical component that can store energy in a magnetic field created by the electric current passing through it. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s and
transformerA transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core, and thus a varying magnetic field...
s when the core is saturated,
mixerIn telecommunications, a mixer is a nonlinear or time-varying circuit or device that accepts as its input two different frequencies and presents at its output a mixture of signals at several frequencies:# the sum of the frequencies of the input signals...
s, modulators, and digital logic circuits.
Importance
Because they obey the
superposition principleIn physics and systems theory, the superposition principle, also known as superposition property, states that, for all linear systems,So that if input A produces response X and input B produces response Y then input produces response .Mathematically, for all linear systems F = y, where x is some...
, linear circuits can be analyzed with powerful mathematical
frequency domainIn electronics, control systems engineering, and statistics frequency domain is a term used to describe the analysis of mathematical functions or signals with respect to frequency, rather than time....
techniques, including Fourier analysis and the
Laplace transformIn mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory....
. These also give an intuitive understanding of the qualitative behavior of the circuit, characterizing it using terms such as
gainGain is a measure of a system's response to feedback. If the gain in a positive feedback loop is less than 1, the feedback is not of itself sufficient to make the system become unstable. For example, water evaporating from the World's oceans causes a positive feedback, as it is a greenhouse gas...
,
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
,
resonanceIn physics, resonance is the tendency of a system to oscillate at larger amplitude at some frequencies than at others. These are known as the system's resonant frequencies . At these frequencies, even small periodic driving forces can produce large amplitude vibrations, because the system...
, bandwidth,
QQ is the seventeenth letter of the basic modern Latin alphabet. Its name in English is spelled cue.- History :The Semitic sound value of Qôp was , a sound common to Semitic languages, but not found in English or most Indo-European ones...
,
poleIn the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of at...
s, and
zeroIn complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written aswhere g is a holomorphic function g such that g is not zero.Generally, the...
s. The analysis of a linear circuit can often be done by hand using a
scientific calculatorA scientific calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science , engineering, and mathematics...
.
In contrast, nonlinear circuits usually don't have exact solutions. They must be analyzed using approximate numerical methods by
electronic circuit simulationElectronic circuit simulation uses mathematical models to replicate the behavior of an actual electronic device or circuit. Simulating a circuit’s behavior before actually building it greatly improves efficiency and provides insights into the behavior of electronics circuit designs...
languages such as Spice, if accurate results are desired. These can give solutions for any specific circuit, but not much insight into the operation of the circuit in general with different component values or inputs. Linear circuit element (resistors, capacitors, inductors) behavior can be specified by a single number (resistance, capacitance, inductance). In comparison, nonlinear elements' behavior is specified by the device's detailed
transfer functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
which may be given as a graph, so more information is needed to specify nonlinear circuits' characteristics.
Linear circuits and systems form a separate category within electronic manufacturing. Manufacturers of transistors and
integrated circuitIn electronics, an integrated circuit is a miniaturized electronic circuit that has been manufactured in the surface of a thin substrate of semiconductor material...
s divide their product lines into 'linear' and 'digital' lines, for example. In their linear components, manufacturers work to reduce nonlinear behavior to a minimum, to make the real component conform as closely as possible to the 'ideal' model used in circuit theory.
Small signal approximation
Nonlinear elements such as transistors tend to behave linearly when small AC signals are applied to them. So in analysing many circuits where the signal levels are small, for example those in TV and radio receivers, nonlinear elements can be replaced with a linear small-signal model, allowing linear analysis techniques to be used.
Conversely, many linear circuit elements show nonlinearity as the signal level is increased. If nothing else, the
power supplyPower supply is a reference to a source of electrical power. A device or system that supplies electrical or other types of energy to an output load or group of loads is called a power supply unit or PSU...
voltage to the circuit usually puts a limit on the magnitude of voltage output from a circuit. Above that limit, the output ceases to scale in magnitude with the input, failing the definition of linearity.