en>Mark viking: Added a little more explanation o 2n dimensional rep

2013-11-20T19:52:23Z

Added a little more explanation o 2n dimensional rep

← Older revision		Revision as of 21:52, 20 November 2013
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	~~Greetings! I am Myrtle Shroyer~~. ~~To collect cash~~ is ~~what his family members and him enjoy~~. ~~California~~ is ~~our beginning location~~. ~~Hiring~~ is ~~her working day job now but she~~'s ~~always needed her personal business~~.<br><br>~~Visit my webpage;~~ [~~http~~://~~www~~.~~buzzbit~~.~~net/blog/333162 over~~ the ~~counter std test~~]		{{Unreferenced\|date=December 2006}}

			[[Image:Passive_integrator_circuit_1.png\|thumb\|150px\|Circuit 1]]
			[[Image:Passive_integrator_circuit_2.png\|thumb\|150px\|Circuit 2]]

			'''Passive integrator circuit''' is a simple four-terminal network consisting of two passive elements. It is also the simplest (first-order) [[low-pass filter]].

			We'll analyze only the first circuit; the second is very similar.

			== Transfer function ==

			A [[transfer ratio]] is a gain factor for the sinusoidal input signal with given frequency.

			A [[transfer function]] shows the dependence of the transfer ratio from the signal frequency, given that the input signal is sinusoidal.

			According to [[Ohm's law]],{{br}}
			<center><math>Y=X\frac{Z_C}{Z_C+Z_R}=X\frac{\frac{1}{j \omega C}}{\frac{1}{j \omega C}+R}=X\frac{1}{1+j \omega RC},</math></center>{{br}}
			where <math>X</math> and <math>Y</math> are input and output signals' amplitudes respectively, and <math>Z_R</math> and <math>Z_C</math> are the [[resistor\|resistor's]] and [[capacitor\|capacitor's]] [[Electrical impedance\|impedance]]s.{{br}}
			Therefore, the complex transfer function is{{br}}
			<center><math>K(j \omega)=\frac{1}{1+j \omega RC}=\frac{1}{1+\frac{j \omega}{\omega_0}},</math></center>{{br}}
			where
			<center><math>\omega_0=\frac{1}{RC}.</math>{{br}}</center>

			Amplitude transfer function
			<center><math>H(\omega)=\|K(j \omega)\|=\frac{1}{\sqrt{1+\left(\frac{\omega}{\omega_0}\right)^2}}.</math></center>{{br}}
			Phase transfer function
			<center><math>\phi (\omega)=\arg K(j \omega)=-\arctan \frac{\omega}{\omega_0}.</math></center>{{br}}

			[[Image:Passive_integrator_circuit_transfer_function.png\|center\|thumb\|350px\|Amplitude and phase transfer functions for a passive integrator circuit]]

			Transfer functions for the second circuit are the same (with <math>\omega_0=\frac{R}{L}</math>).

			== Impulse response ==

			The circuit's [[Impulse response]] can be derived as an inverse [[Laplace transform]] of the complex transfer function:{{br}}
			<center><math>h(t)=\mathcal{L}^{-1} \left \{K(p) \right \}=\int_{\beta-j \infty}^{\beta+j \infty} K(p)e^{pt} \, dp=\omega_0 e^{-\omega_0 t}=\frac{1}{\tau} e^{-\frac{t}{\tau}},</math></center>{{br}}
			where <math>\tau=\frac{1}{\omega_0}</math> is a time constant.

			[[Image:Passive_integrator_circuit_impulse_response_1.png\|center\|thumb\|350px\|An impulse response of a passive integrator circuit]]

			== Applications ==

			A passive integrator circuit can serve as a simple [[integrator]]. It is also one of the basic [[electronic circuit]]s, being widely used in circuit analysis based on the [[equivalent circuit]] method.

			It is often used in cheap digital audio systems (i.e. cheap [[soundcard]]s) as a [[reconstruction filter]].

			== See also ==

			[[Passive differentiator circuit]]{{br}}
			[[RC circuit]]{{br}}
			[[Electronic filter]]

			[[Category:Analog circuits]]

			[[id:Sirkuit integrator pasif]]
			[[pl:Układ_całkujący]]

en>Brad7777: /* See also */ removed category geometry, added category complex manifolds (complex geometry would be better)

2011-11-16T18:52:56Z

See also: removed category geometry, added category complex manifolds (complex geometry would be better)

New page

Greetings! I am Myrtle Shroyer. To collect cash is what his family members and him enjoy. California is our beginning location. Hiring is her working day job now but she's always needed her personal business.<br><br>Visit my webpage; [http://www.buzzbit.net/blog/333162 over the counter std test]

Complex space - Revision history

en>Mark viking: Added a little more explanation o 2n dimensional rep

en>Brad7777: /* See also */ removed category geometry, added category complex manifolds (complex geometry would be better)