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2014-08-25T13:15:34Z

Mathematical definition

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	~~A '''linear circuit'''~~ is an [[electronic circuit]] in which, for a [[Sine wave\|sinusoidal]] input voltage of [[frequency]] ''f'', any steady-state output of the circuit (the [[Electric current\|current]] through any component, or the [[voltage]] between any two points) is also sinusoidal with frequency ''f''. ~~Note that the output need not be [[Phase (waves)\|in phase]] with the input~~.~~<ref>{{cite book\|title=Linear circuit design handbook \|first=Hank\|last= Zumbahlen\|publisher= Newnes\|year=2008\|~~
	~~ISBN=0-7506-8703-7}}<~~/~~ref>~~

	~~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 ''ax<sub>1<~~/~~sub>(t) + bx<sub>2<~~/~~sub>(t)'' is applied to it is equal to the linear combination of the outputs due to the signals ''x<sub>1<~~/~~sub>(t)'' and ''x<sub>2<~~/~~sub>(t)'' applied separately:~~

	~~:<math>F(ax_1 + bx_2) = aF(x_1) + bF(x_2)\,<~~/~~math>~~

	~~Informally, a linear circuit is one in which the values of the [[electronic component~~]~~]s, the~~ [~~[Electrical resistance\|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 [[amplifier]]s, [[differentiator]]s,~~ and ~~[[integrator]]s, or any circuit composed exclusively~~ of ~~''ideal'' [[resistor]]s, [[capacitor]]s, [[inductor]]s, [[op~~-~~amp]]s (in the "non~~-~~saturated" regime), and other "linear" [[circuit element]]s~~.

	~~Some examples of nonlinear electronic components are: [[diode]]s, [[transistor]]s, and [[Magnetic core\|iron core]] [[inductor~~]~~]s and [[transformer]]s when the core is saturated. Some examples of circuits~~ that ~~operate in a nonlinear way are [[Frequency mixer\|mixer]]s, [[modulator]]s, and [[digital logic]] circuits~~.

	~~==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 (waves)\|phase shift]], [[resonant frequency]], [[Bandwidth (signal processing)\|bandwidth]], [[Q factor]], [[pole (complex analysis)\|pole]]s, and ~~[[zero (complex analysis)\|zero]]s~~. ~~The analysis of a linear circuit can often be done by hand using~~ a [~~[scientific calculator]]~~.

	~~In contrast, [[Nonlinear element\|nonlinear circuit]]s usually don't have closed form solutions~~. ~~They must be analyzed using approximate [[numerical methods]] by [[electronic circuit simulation]] computer programs such as [[SPICE\|Spice]], if accurate results are desired. The behavior~~ of ~~such [[Linear element\|linear circuit element]~~]~~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]], 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]]s often divide their product lines into 'linear' and 'digital' lines, for example~~. ~~However, in general, the term “Linear” here means “Analog”~~. The “Linear” product line of manufacturers typically includes a variety of non-linear analog devices such as logarithmic amplifiers, comparators, peak detectors and so forth, in addition to devices such as operational amplifiers.

	==~~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, 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~~.

	==~~References~~==
	~~{{reflist}}~~

	~~[[Category:Electronic circuits]~~]

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← Older revision		Revision as of 04:35, 16 March 2013
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			A '''linear circuit''' is an [[electronic circuit]] in which, for a [[Sine wave\|sinusoidal]] input voltage of [[frequency]] ''f'', any steady-state output of the circuit (the [[Electric current\|current]] through any component, or the [[voltage]] between any two points) is also sinusoidal with frequency ''f''. Note that the output need not be [[Phase (waves)\|in phase]] with the input.<ref>{{cite book\|title=Linear circuit design handbook \|first=Hank\|last= Zumbahlen\|publisher= Newnes\|year=2008\|
			ISBN=0-7506-8703-7}}</ref>

			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 ''ax<sub>1</sub>(t) + bx<sub>2</sub>(t)'' is applied to it is equal to the linear combination of the outputs due to the signals ''x<sub>1</sub>(t)'' and ''x<sub>2</sub>(t)'' applied separately:

			:<math>F(ax_1 + bx_2) = aF(x_1) + bF(x_2)\,</math>

			Informally, a linear circuit is one in which the values of the [[electronic component]]s, the [[Electrical resistance\|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 [[amplifier]]s, [[differentiator]]s, and [[integrator]]s, or any circuit composed exclusively of ''ideal'' [[resistor]]s, [[capacitor]]s, [[inductor]]s, [[op-amp]]s (in the "non-saturated" regime), and other "linear" [[circuit element]]s.

			Some examples of nonlinear electronic components are: [[diode]]s, [[transistor]]s, and [[Magnetic core\|iron core]] [[inductor]]s and [[transformer]]s when the core is saturated. Some examples of circuits that operate in a nonlinear way are [[Frequency mixer\|mixer]]s, [[modulator]]s, and [[digital logic]] circuits.

			==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 (waves)\|phase shift]], [[resonant frequency]], [[Bandwidth (signal processing)\|bandwidth]], [[Q factor]], [[pole (complex analysis)\|pole]]s, and [[zero (complex analysis)\|zero]]s. The analysis of a linear circuit can often be done by hand using a [[scientific calculator]].

			In contrast, [[Nonlinear element\|nonlinear circuit]]s usually don't have closed form solutions. They must be analyzed using approximate [[numerical methods]] by [[electronic circuit simulation]] computer programs such as [[SPICE\|Spice]], if accurate results are desired. The behavior of such [[Linear element\|linear circuit element]]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]], 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]]s often divide their product lines into 'linear' and 'digital' lines, for example. However, in general, the term “Linear” here means “Analog”. The “Linear” product line of manufacturers typically includes a variety of non-linear analog devices such as logarithmic amplifiers, comparators, peak detectors and so forth, in addition to devices such as operational amplifiers.

			==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, 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.

			==References==
			{{reflist}}

			[[Category:Electronic circuits]]

141.23.93.126: /* Mathematical definition */ Wavelets are also translated. They aren't simply formed by scaling a function.

2012-05-15T15:20:04Z

Mathematical definition: Wavelets are also translated. They aren't simply formed by scaling a function.

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