Cpter 8 Two-Port Circuits 8. Te Terminl Equtions 8. Te Two-Port Prmeters 8.3 Anlysis of te Terminted Two-Port Circuit 8.4 nterconnected Two-Port Circuits
Motivtion Tévenin nd Norton equivlent circuits re used in representin te contribution of circuit to one specific pir of terminls. Usully, sinl is fed into one pir of terminls (input port), processed by te system, ten extrcted t second pir of terminls (output port). t would be convenient to relte te v/i t one port to te v/i t te oter port witout knowin te element vlues nd ow tey re connected inside te blck box.
How to model te blck box? Source (e.. CD plyer) od (e.. speker) We will see tt two-port circuit cn be modeled by mtrix to relte te v/i vribles, were te four mtrix elements cn be obtined by performin experiments. 3
Restrictions of te model No enery stored witin te circuit. No independent source. Ec port is not current source or sink, i.e. i i, i. i No inter-port connection, i.e. between c, d, bc, bd. 4
Key points How to clculte te 6 possible mtrices of two-port circuit? How to find te 4 simultneous equtions in solvin terminted two-port circuit? How to find te totl mtrix of circuit consistin of interconnected two-port circuits? 5
Section 8. Te Terminl Equtions 6
s-domin model Te most enerl description of two-port circuit is crried out in te s-domin. Any out of te 4 vribles {,,, } cn be determined by te oter vribles nd simultneous equtions. 7
8 Six possible sets of terminl equtions () te dmittnce mtrix; is ; te impednce mtrix; is ; - Y y y y y is trnsmission mtrix; ; is trnsmission mtrix; ; - A B b b b b A
9 Six possible sets of terminl equtions () is ybrid mtrix; ; is ybrid mtrix; ; - H G H Wic set is cosen depends on wic vribles re iven. E.. f te source volte nd current {, } re iven, coosin trnsmission mtrix [B] in te nlysis.
Section 8. Te Two-Port Prmeters. Clcultion of mtrix []. Reltions mon 6 mtrixes 0
Exmple 8.: Findin [] () Q: Find te impednce mtrix [] for iven resistive circuit (not blck box ): By definition, =( / ) wen =0, i.e. te input impednce wen port is open. = (0 )//(0 )=0.
Exmple 8.: () By definition, =( / ) wen =0, i.e. te trnsfer impednce wen port is open. Wen port is open: 5 5 5 0, 0.75 (0 ) 0.75,, 0 7.5.
Exmple 8.: (3) By definition, = ( / ) wen =0, i.e. te output impednce wen port is open. = (5 )//(5 ) =9.375. =( / ) wen =0, 0 0 5 9.375, 0.8 9.375 0.8 7.5. (9.375 ),, 3
Comments Wen te circuit is well known, clcultion of [] by circuit nlysis metods sows te pysicl menin of ec mtrix element. Wen te circuit is blck box, we cn perform test experiments to et []: () Open port, pply current to port, mesure te input volte nd output volte. () Open port, pply current to port, mesure te terminl voltes nd. 4
Reltions mon te 6 mtrixes f we know one mtrix, we cn derive ll te oters nlyticlly (Tble 8.). [Y]=[] -, [B]=[A] -, [G]=[H] -, elements between mutully inverse mtrixes cn be esily relted. E.. were y y y y y y dety y y y y. y y y y, 5
Represent [] by elements of [A] () [] nd [A] re not mutully inverse, reltion between teir elements re less explicit. By definitions of [] nd [A],, te independent vribles of [] nd [A] re {, } nd {, }, respectively., Key of mtrix trnsformtion: Representin te distinct independent vrible by {, }. 6
7 Represent [] by elements of [A] () By definitions of [A] nd [], () () det. were, A (3), ) ( (4) (),(3)
Section 8.3 Anlysis of te Terminted Two-Port Circuit. Anlysis in terms of []. Anlysis in terms of [T][] 8
Model of te terminted two-port circuit A two-port circuit is typiclly driven t port nd loded t port, wic cn be modeled s: Te ol is to solve {,,, } s functions of iven prmeters,,, nd mtrix elements of te two-port circuit. 9
0 Anlysis in terms of [] Four equtions re needed to solve te four unknowns {,,, }. equtions due to termintions constrint (4) (3) port equtions two - () (), 0 0 0 0 0 0 0 0 0 {,,, } re derived by inverse mtrix metod.
Tévenin equivlent circuit wit respect to port Once {,,, } re solved, { T, T } cn be determined by nd {, }: () () T T T T. ; T T T T
Terminl bevior () Te terminl bevior of te circuit cn be described by mnipultions of {,,, }: nput impednce: Output current: Current in: olte ins: in ; ( )( ) ; ( ; )( ) ; ;
Terminl bevior () Tévenin volte: Tévenin impednce: T T ; ; 3
Anlysis in term of two-port mtrix [T][] f te two-port circuit is modeled by [T][], T={Y, A, B, H, G}, te terminl bevior cn be determined by two metods: Use te two-port equtions of [T] to et new 44 mtrix in solvin {,,, } (Tble 8.); Trnsform [T] into [] by Tble 8., borrow te formuls derived by nlysis in terms of []. 4
Exmple 8.4: Anlysis in terms of [B] () Q: Find () output volte, (,3) vere powers delivered to te lod P nd input port P, for terminted two-port circuit wit known [B]. R R B 0 ms 3 k 0. -b -b 5
Exmple 8.4 () Use te volte in formul of Tble 8.: b b b b b 0 9 b 5000 b b b b ( )(5 k) ( 3 k) ( 0)(0.5 k) ( 0.)(5 k)... 63.60 ; ( 0)( 0.) ( 3 k)( ms). 4 6 0 9,, 6
Exmple 8.4 (3) Te vere power of te lod is formulted by P 63.60 R 5 k 6.93 W. Te vere power delivered to port is formulted by P Re. in b b P in (0.789) in ( 0.)(5 k) (3 k) ( ms)(5 k) 0 5000 (500 ) (33.33) b b (33.33) 4.55 W. 0.7890 33.33 ; A, 7
Section 8.4 nterconnected Two-Port Circuits 8
Wy interconnected? Desin of lre system is simplified by first desinin subsections (usully modeled by two-port circuits), ten interconnectin tese units to complete te system. 9
Five types of interconnections of two-port circuits. Cscde: Better use [A]. b. Series: [] c. Prllel: [Y] d. Series-prllel: [H]. e. Prllel-series: [G]. 30
Anlysis of cscde connection () Gol: Derive te overll mtrix [A] of two cscded two-port circuits wit known trnsmission mtrixes [A'] nd [A"]. A A Overll two-port circuit [A]=? 3
) (, A A 3 Anlysis of cscde connection () ) ( A A. ) ( ) (,, By (),(), A A A A A A
Key points How to clculte te 6 possible mtrices of two-port circuit? How to find te 4 simultneous equtions in solvin terminted two-port circuit? How to find te totl mtrix of circuit consistin of interconnected two-port circuits? 33
Prcticl Perspective Audio Amplifier 34
Appliction of two-port circuits Q: Weter it would be sfe to use iven udio mplifier to connect music plyer modeled by { = (rms), =00 } to speker modeled by lod resistor =3 wit power rtin of 00 W? 35
Find te [H] by test experiments () Definition of ybrid mtrix [H]: ; Test : =.5 ma (rms) = 0 (sort) =.5 (rms) = 3.75 A (rms),, 0 0.5.5 ma 3.75 A.5 ma 500. 500. nput impednce Current in 36
Find te [H] by test experiments () Definition of ybrid mtrix [H]: Test : = 0 (open) =50 m (rms) ; =50 (rms) =.5 A (rms),, 0 0 50 m 50.5 A 50 0 3. (0 ) - olte in. Output dmittnce 37
Find te power dissiption on te lod For terminted two-port circuit: te power dissipted on is P Re * * Re ( ) Re, were is te rms output current psor. 38
Metod : Use terminted -port eqs for [H] By lookin t Tble 8.: ( )( ).98 A (rms), 3 500 0 were ; - 500 (0 ) (rms), 00, 3. P Re (.98) (3) 6 W Not sfe! 00 W. 39
40 Metod : Use system of terminted eqs of [] Trnsform [H] to [] (Tble 8.):. 00 W 6 W (3) (.98) Re P. 0 30,000 0.0 470..98 A 3.4 ma 63.5.66 0 0 0 0 0 0 0 0 0 By system of terminted equtions: