Small signal analysis
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1 Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea a tme varyng current ource. We aume that abolute value of at all tme much maller than the D ource. Fndng the operatng pont We et to zero and conder the D crcut drven by actng alone. The D voltage and current n th crcut are denoted ung captal letter. Then, t hold GV, ( ( V g. ( We olve the et of equaton (-( graphcally, a hown n Fg. 3, fndng the operatng V,. pont ( Now we conder the crcut of Fg. ncludng. t decrbed by equaton t ( ( ( l A ω t For each the pont v t, t lne hown n Fg. 3. When Gv, (3 g( v. ( that atfe equaton (3 le on a traght lne parallel to the co, thee traght lne are bounded by the lne k and m depcted n Fg.. The pont of nterecton wth the charactertc g v determne the bounded value of v and. Thu, for all t the pont ( v, le on the arc A of the charactertc. Th arc can be approxmated by a lnear egment. (
2 V V l V Fg. 3 (t t A V m v k l We preent v ( and t a um of two term Fg. V v v ( (6
3 and can be condered a mall dplacement from the operatng pont. Subttutng ( and (6 nto ( yeld where v (. ( g V v Let u expand the rght hand de of equaton ( nto the Taylor ere and neglect the hgher order term Snce g( V, the equaton (8 reduce to dg g( V v vv. (8 G v, (9 where dg G. ( d v v V Next we ubttute ( and (6 nto equaton (3 Snce G( V, we fnd (. ( G V v Gv. ( quaton (9 and ( decrbe the mall gnal equvalent crcut hown n Fg.. G G v Fg. Th crcut lnear and enable u to fnd the mall gnal v and v, (3 G G : 3
4 G. ( G G xample onder the crcut hown n Fg. 6, drven by a D voltage ource V S and a mall voltage v (t. To fnd the mall gnal model of th crcut we determne frt the D operatng pont. For th purpoe we analye the crcut hown n Fg.. k (t - (e 38v - k V S 6V v S (t v(t V S 6V V Fg. 6 Fg. The crcut hown n Fg. decrbed by the equaton whch can be rewrtten n the from 38 ( e V.V.6, 38 e V 9 V 6 9. ( We olve th equaton ung the Newton-aphon method ( j 9 ( j 38V 9 ( j ( j e V 6 V V (, j 38V 9 38e ( ung the ntal gue V (.6. A a reult we obtan: V.9, V (.9, V (3.9. Thu, we aume V o.9 and fnd (6 or d 38.9 G v V 38 e.s,. 8Ω. G The mall gnal model hown n Fg. 8. Ω v (t v.8 Ω Fg. 8
5 . Small gnal analy of bpolar trantor crcut Let u conder a mple amplfer crcut, contanng JT, hown n Fg. 9. v v v S (t Fg. 9 and are D voltage ource, wherea v t a mall tme varyng voltage ource. Fndng the operatng pont. We et to zero v ( t and olve the D crcut drven by the D voltage ource and graphcally, a llutrated n Fg. and. ( ( ( ( ( V V ( V V Fg. Fg. The operatng pont pecfed by ( V, (, ( V, (. Generally the operatng pont the oluton of the et of hybrd equaton decrbng the trantor (, V V vˆ ( (, V and the equaton decrbng the, and, branche î (8, (9 V
6 V. ( Now we conder the crcut hown n Fg. 9 ncludng the mall tme voltage crcut voltage and current are tme varyng: follow: v v v,, v, ( V v v. n th. We preent them a, (, ( ( ( V v, (3, ( ( where v,, v, repreent the mall dplacement from the operatng pont. We et thee equaton nto the hybrd equaton ( ( V v vˆ (, ( V v, ( (, (6 ( î (, ( V v and expand the functon on the rght hand de nto the Taylor ere, about the operatng pont, neglectng the hgher order term ( V v vˆ (, ( V vˆ ( vˆ v ( v ( î (, ( V î ( î v. (8 v Takng nto account equaton ( and (8 we obtan v vˆ vˆ, (9 v v î î. (3 v v The et of equaton can be rewrtten n the form v t h t h v t, (3 ( ( ( h h v, (3 6
7 where h vˆ, h vˆ v, h î, h î v. The crcut decrbed by equaton (3-(3, called the mall gnal model of the trantor, hown n Fg.. v h h G h v h v Fg. Typcal value of the coeffcent are a follow: h 3 Ω, h, h, h S. Now we conder the retor-ource branche of the crcut hown n Fg. 9 and wrte the equaton v v, (33 v. (3 Subttutng (-( and takng nto account (9 and ( we obtan v, (3 v v. (36 ombnng the equaton (3-(36 wth the equaton (3-(3 decrbng the mall gnal model of the trantor we fnd the mall gnal equvalent crcut of the amplfer crcut, hown n Fg. 3. v v ( t h h v h h v Fg. 3 We fnd the voltage gan of th crcut defned a v v. For th purpoe we wrte the equaton (9-(3 and (3-(36 decrbng the crcut and rearrange them a hown underneath:
8 v h hv, h v, h h ( h h, h v v v h v, ( h h hh h ( h h hh. (3 Subttutng n the voltage gan (3: h S we fnd kω h kω, h, h, v v Small gnal analy a general cae Let u conder a crcut contanng lnear and nonlnear retor, nductor, capactor, lnear controlled ource, drven by a D ource and a mall tme varyng ource. To form the mall gnal equvalent crcut we fnd frt the D operatng pont. For th purpoe we et to zero the mall tme varyng ource and conder the crcut drven by the D ource only. A a reult we obtan a D crcut, by hort-crcutng all the nductor and open-crcut all the capactor. t can be olved ung an arbtrary method for the analy of retve crcut. f the crcut nonlnear we apply the Newton-aphon method. The operatng pont pecfed by branch current ( V. j and branch voltage ( j n the crcut drven by the D ource and the mall tme varyng ource all branch voltage and current depend on tme. We preent each branch voltage a and each branch current a V v v 8
9 where and v and are mall dplacement from the operatng pont. Snce both voltage V atfy KVL n an arbtrary loop we tate that voltage vj alo atfy KVL v j ( j n an arbtrary loop. Smlarly we prove that the current j atfy KL at all node. Now we conder crcut element and decrbe them n term of mall tme-varyng gnal. Nonlnear retor decrbed by the equaton We ubttut, v V v g( v. (38 and expand the functon g ( v( t n the Taylor ere, about the operatng pont, neglectng the hgher order term Snce n the D crcut g( where V we obtan g( V v Gv d g vv. (39, ( d g G d v v V. ( quaton ( decrbe the mall gnal model of the nonlnear retor, beng a lnear retor havng conductance G pecfed by (. f the retor lnear t mall gnal model the ame retor. Smlarly we prove that the mall gnal model of any lnear controlled ource the ame controlled ource. Nonlnear capactor decrbed by the equaton c( v We ubttute v V v, expand the functon ( v operatng pont and neglect the hgher order term q. ( c nto the Taylor ere about the Snce q dc vv c( V v d q d t. (3 ( 9
10 then where d v, ( dt dc d v v V. (6 quaton ( and (6 decrbe a lnear capactor beng the mall gnal model of the nonlnear capactor. f the capactor lnear t mall gnal model the ame capactor. Nonlnear nductor decrbed by the equaton ( t f ( φ. ( Smlarly a n the cae of capactor we tate that the mall gnal model of the nonlnear nductor a lnear nductor pecfed by the equaton where v d L d d L, (8 dt f. (9 f the nductor lnear, t mall gnal model reman the ame nductor. To contruct the mall gnal equvalent crcut of a gven crcut we et to zero D ource and replace all t element by ther mall gnal model. xample For the crcut hown n Fg. contruct the mall gnal equvalent crcut about t operatng pont. 3 3 Data: q v. v, q. 8v,. 8 φ.. L H e 3 Ω Ω 3 v v 6. v v A Fg. Fndng the operatng pont. We et to zero t, remove the capactor and hort crcut the nductor (ee Fg.. v (
11 3 V 6. V A Fg. The crcut can be decrbed by the node equaton. havng the oluton V. 3 Hence, t hold: 8 A, 6. A, A, V V, V 3 V The mall gnal model of the nonlnear element: d v, d q 3. V. 33F, dt V d v, d q. V. F, dt V d ( v t L. 8, L. e. 89H. dt Ung the above reult we contruct the mall gnal equvalent crcut hown n Fg. 6. Ω H Ω.89H v.333f. 6.F v Fg. 6
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