# Linear circuit

A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any steady-state output of the circuit (the current through any component, or the voltage between any two points) is also sinusoidal with frequency f. Note that the output need not be in phase with the input.[1]

An equivalent definition of a linear circuit is that it 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 of the outputs due to the signals x1(t) and x2(t) applied separately:

$F(ax_1 + bx_2) = aF(x_1) + bF(x_2)\,$

Informally, a linear circuit is one in which the values of the electronic components, the resistance, capacitance, inductance, gain, 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 small-signal amplifiers, differentiators, and integrators, or any circuit composed exclusively of ideal resistors, capacitors, inductors, op-amps (in the "non-saturated" regime), and other "linear" circuit elements.

Some examples of nonlinear electronic components are: diodes, transistors when they are operated in saturation, and iron core inductors and transformers when the core is saturated. Some examples of circuits that operate in a nonlinear way are mixers, modulators, and digital logic circuits.

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## Significance

Because they obey the superposition principle, linear circuits can be analyzed with powerful mathematical frequency domain 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, phase shift, resonant frequency, bandwidth, Q factor, poles, and zeros. The analysis of a linear circuit can often be done by hand using a scientific calculator.

In contrast, nonlinear circuits usually don't have exact solutions. They must be analyzed using approximate numerical methods by electronic circuit simulation computer programs 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. The behavior of such linear circuit elements 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, 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 circuits 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.

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## 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 supply 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.

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## References

1. ^ Zumbahlen, Hank (2008). Linear circuit design handbook. Newnes. ISBN 0-7506-8703-7.
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