Linear circuit

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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 of the outputs due to the signals x₁(t) and x₂(t) applied separately:

${\displaystyle F(ax_{1}+bx_{2})=aF(x_{1})+bF(x_{2})\,}$

It is called a linear circuit because the output voltage and current of such a circuit are linear functions of its input voltage and current.^[1][2][3] This kind of linearity is not the same as that of straight-line graphs.

In the common case of a circuit in which the components' values are constant and don't change with time, an alternate definition of linearity is that when a sinusoidal input voltage or current of frequency f is applied, 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.^[1][4] A linear circuit with constant component values is called linear time-invariant (LTI).

Informally, a linear circuit is one in which the

Alternate definition

The superposition principle, the defining equation of linearity, is equivalent to two properties, additivity and homogeneity, which are sometimes used as an alternate definition

• ${\displaystyle F(x_{1}+x_{2})=F(x_{1})+F(x_{2})\qquad }$ Additivity
• ${\displaystyle F(hx)=hF(x)\qquad \qquad \qquad \qquad }$ Homogeneity

That is, a linear circuit is a circuit in which (1) the output when a sum of two signals is applied is equal to the sum of the outputs when the two signals are applied separately, and (2) scaling the input signal ${\displaystyle x(t)}$ by a factor ${\displaystyle h}$ scales the output signal ${\displaystyle F(x(t))}$ by the same factor.

Linear and nonlinear components

A linear circuit is one that has no

Some examples of nonlinear electronic components are:

Significance

Linear time-invariant circuits are important because they can process

Nonlinear elements such as transistors tend to behave linearly when small AC signals are applied to them. So in analyzing many circuits where the signal levels are small, for example those in TV and radio receivers, nonlinear elements can be replaced with a linear

Conversely, all circuit elements, even "linear" 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.

See also

• Network analysis (electrical circuits)

References