Linear Circuit Elements

Transcription

1 1/25/2011 inear ircui Elemens.doc 1/6 inear ircui Elemens Mos microwave devices can be described or modeled in erms of he hree sandard circui elemens: 1. ESISTANE () 2. INDUTANE () 3. APAITANE () For he purposes of circui analysis, each of hese hree elemens are defined in erms of he mahemaical relaionship beween he difference in elecric poenial v ( ) beween he wo erminals of he device (i.e., he volage across he device), and he curren i ( ) flowing hrough he device. We find ha for hese hree circui elemens, he relaionship beween v ( ) and i ( ) can be expressed as a linear operaor! ( i ) v ( ) i ( ) = = v( ) i ( ) v ( ) i ( ) = = v( ) Jim Siles The Univ. of Kansas Dep. of EES

2 1/25/2011 inear ircui Elemens.doc 2/6 i ( ) dv( ) v( ) i( ) d = = 1 i( ) = v( ) = i( ) d v ( ) v( ) ( i ) 1 ( ) ( ) ( ) v = i = v d di( ) i( ) v( ) d = = Since he circui behavior of hese devices can be expressed wih linear operaors, hese devices are referred o as linear circui elemens. Q: Well, ha s simple enough, bu wha abou an elemen formed from a composie of hese fundamenal elemens? For example, for example, how are v ( ) and i ( ) relaed in he circui below?? Jim Siles The Univ. of Kansas Dep. of EES

3 1/25/2011 inear ircui Elemens.doc 3/6 i ( ) i ( ) v ( )??? = = v( ) A: I urns ou ha any circui consruced enirely wih linear circui elemens is likewise a linear sysem (i.e., a linear circui). As a resul, we know ha ha here mus be some linear i in your example! operaor ha relaes v ( ) and ( ) i ( ) v ( ) = The circui above provides a good example of a single-por (a.k.a. one-por) nework. We can of course consruc neworks wih wo or more pors; an example of a wo-por nework is shown below: 1( i ) 2( i ) v1( ) v2( ) Jim Siles The Univ. of Kansas Dep. of EES

4 1/25/2011 inear ircui Elemens.doc 4/6 Since his circui is linear, he relaionship beween all volages and currens can likewise be expressed as linear operaors, e.g.: 21 v1 ( ) = v2 ( ) i ( ) v ( ) = i ( ) v ( ) = Q: ikes! Wha would hese linear operaors for his circui be? How can we deermine hem? A: I urns ou ha linear operaors for all linear circuis can all be expressed in precisely he same form! For example, he linear operaors of a single-por nework are: ( ) = ( ) = ( ) ( ) v i g i d ( ) = ( ) = ( ) ( ) i v g v d In oher words, he linear operaor of linear circuis can always be expressed as a convoluion inegral a convoluion wih a circui impulse funcion g ( ). Q: Bu jus wha is his circui impulse response?? Jim Siles The Univ. of Kansas Dep. of EES

5 1/25/2011 inear ircui Elemens.doc 5/6 A: An impulse response is simply he response of one circui v ) due o a specific simulus by funcion (i.e., i ( ) or ( ) anoher. Tha specific simulus is he impulse funcion δ ( ). The impulse funcion can be defined as: δ ( ) π sin 1 τ = lim τ 0 τ π τ Such ha is has he following wo properies: 1. δ ( ) = 0 for 0 2. δ ( ) d = 10. The impulse responses of he one-por example are herefore defined as: and: g( ) v ( ) i ( ) = δ ( ) g( ) i ( ) v ( ) = δ ( ) Jim Siles The Univ. of Kansas Dep. of EES

6 1/25/2011 inear ircui Elemens.doc 6/6 Meaning simply ha g ( ) is equal o he volage funcion v ( ) when he circui is humped wih a impulse curren (i.e., i ( ) = δ ( ) ), and g ( ) is equal o he curren i ( ) when he circui is humped wih a impulse volage (i.e., v ( ) = δ ( ) ). Similarly, he relaionship beween he inpu and he oupu of a wo-por nework can be expressed as: where: v ( ) = v ( ) = g ( ) v ( ) d g ( ) v ( ) 2 v ( ) = δ ( ) 1 Noe ha he circui impulse response mus be causal (nohing can occur a he oupu unil somehing occurs a he inpu), so ha: g ( ) = 0 for < 0 Q: ikes! I recall evaluaing convoluion inegrals o be messy, difficul and sressful. Surely here is an easier way o describe linear circuis!?! A: Nope! The convoluion inegral is all here is. However, we can use our linear sysems heory oolbox o grealy simplify he evaluaion of a convoluion inegral! Jim Siles The Univ. of Kansas Dep. of EES