Linear circuit

# Linear circuit

Discussion

Encyclopedia
A linear circuit is an electronic circuit
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...

in which, for a sinusoidal
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

input voltage of frequency
Frequency
Frequency 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 current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

through any component, or the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

between any two points) is also sinusoidal with frequency f. Note that the output need not be in phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

with the input.

An equivalent definition of a linear circuit is that it obeys the superposition principle
Superposition principle
In physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...

. 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 component
Electronic component
An electronic component is a basic electronic element and may be available in a discrete form having two or more electrical terminals . These are intended to be connected together, usually by soldering to a printed circuit board, in order to create an electronic circuit with a particular function...

s, the resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

, capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

, inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

, gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

, etc. don't change with the level of voltage or current in the circuit.

## Examples

A linear circuit is one that has no nonlinear electronic components in it. Examples of linear circuits are ideal amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...

s, differentiator
Differentiator
A 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 integrator
Integrator
An 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
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

s, capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s, inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. 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: diode
Diode
In electronics, a diode is a type of two-terminal electronic component with a nonlinear current–voltage characteristic. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material connected to two electrical terminals...

s, transistor
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed 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 terminals changes the current...

s when they are operated in saturation, iron core
Magnetic core
A magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, inductors and magnetic assemblies. It is made of ferromagnetic metal such...

inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s and transformer
Transformer
A 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, mixer
Frequency mixer
In electronics a mixer or frequency mixer is a nonlinear electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals at frequencies f1 and f2 are applied to a mixer, and it produces new signals at the sum f1 + f2 and difference f1 -...

s, modulators, and digital logic circuits.

## Significance

Because they obey the superposition principle
Superposition principle
In physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...

, linear circuits can be analyzed with powerful mathematical frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

techniques, including Fourier analysis and the Laplace transform. These also give an intuitive understanding of the qualitative behavior of the circuit, characterizing it using terms such as gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

, phase shift
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

, resonant frequency, bandwidth, Q factor
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

, poles, and zero
Zero (complex analysis)
In 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 asf=g\,where 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 calculator
Scientific calculator
A 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 simulation
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...

computer programs such as Spice
SPICE
SPICE is a general-purpose, open source analog electronic circuit simulator.It is a powerful program that is used in integrated circuit and board-level design to check the integrity of circuit designs and to predict circuit behavior.- Introduction :Unlike board-level designs composed of discrete...

, 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. The behavior of such linear circuit element
Linear element
In an electric circuit, a linear element is an electrical element with a linear relationship between current and [output[voltage]]. Resistors are the most common example of a linear element; other examples include capacitors, inductors, and transformers....

s as resistors, capacitors, and inductors can be specified by a single number (resistance, capacitance, inductance, respectively). In contrast, a nonlinear element's behavior is specified by its detailed transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant 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 specifying the characteristics of a nonlinear circuit requires more information than is needed for a linear circuit.

Linear circuits and systems form a separate category within electronic manufacturing. Manufacturers of transistors and integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into 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
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

analysis
Analysis
Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle , though analysis as a formal concept is a relatively recent development.The word is...

techniques to be used.

Conversely, many linear circuit elements show nonlinearity as the signal level is increased. If nothing else, the power supply
Power supply
A power supply is a device that supplies electrical energy to one or more electric loads. The term is most commonly applied to devices that convert one form of electrical energy to another, though it may also refer to devices that convert another form of energy to electrical energy...

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.