Chapter 9 - CD compaio CHAPTER NINE CD-9.2 CD-9.2. Stages With Voltage ad Curret Gai A Geeric Implemetatio; The Commo-Merge Amplifier The advaced method preseted i the text for approximatig cutoff frequecies is developed i detail i this compaio sectio. Detailed Developmet of Time Costats This sectio presets a more detailed justificatio for the short- ad ope-circuit time costat methods of aalysis. The fial results of this sectio will be equatios presetig the approximatios give without derivatio i (9.39) ad (9.46) ad repeated here for coveiece: ω cl, (9.39) τ is ω ch τ io, (9.46) Before we begi however, we eed to take a momet to review a result from etwork theory. ASIDE CD-A9. Trasfer Fuctio Deomiators The purpose of this aside is to demostrate that the deomiators of all possible trasfer fuctios for a give liear etwork are the same. Cosider a LTI etwork with three odes as show i Figure CD-A9-. Oly two of these odes ca be idepedet, so let s arbitrarily take ode 3 as a referece ad write the odal equatios for the other two odes.
Chapter 9 - CD compaio 2 2 Y 2 Y 3 Y 23 Figure CD-A9- A three-termial LTI etwork. The matrix form of the odal equatios is (remember that v 3 = 0 is assumed): 3 or y 2 + y 3 y 2 y 2 y 2 + y 23 YV = I = 0. v v 2 = 0 0, (CD-A9.) (CD-A9.2) If we add a idepedet curret source i parallel with ay brach i the circuit, we modify the curret vector I (addig a curret source i series i a brach is the same, the admittace i series with the perfect source makes o differece, so it is as if the source were i parallel with a brach with zero admittace - i.e., a ope circuit). If we add a idepedet voltage source i series with ay brach it will also modify I (covert the Thévei circuit i the brach to a Norto equivalet). If we add a idepedet voltage source i parallel with a brach it reduces the umber of idepedet odes by oe ad we ca simplify the problem. Therefore, CD-A9. represets a geeral situatio. Now cosider some trasfer fuctio for the etwork. Idepedet of where we isert a sigal source (igorig the possibility of puttig a voltage source across a brach) we will still have YV=I with the Y give i (CD-A9.). Also, idepedet of what we desigate as our output variable, it ca always be expressed as a combiatio of ode voltages (if it is a brach curret, it is the correspodig brach admittace times the differece i two ode voltages). Usig Cramer s rule [9.2], we fid the jth ode voltage to be v j = Y with the jth colum = I Y, (CD-A9.3)
Chapter 9 - CD compaio 3 where Y deotes the determiat of Y. Notice that all ode voltages have the same deomiator. Sice all outputs ca be expressed as liear combiatios of ode voltages, they will also have the same deomiator. We fially coclude, therefore, that all possible trasfer fuctios for this liear etwork have the same deomiator polyomial, D(jω), ad that polyomial is equal to the determiat of the admittace matrix for the etwork. We have ot attempted a formal proof of this assertio, if you are iterested you should cosult a good book o liear etwork theory. Also ote that for some trasfer fuctios i a give etwork there may be a term (or terms) i the umerator that ca cacel a term (or terms) i the deomiator (i.e., a zero at the same frequecy as a pole). If we perform these cacellatios, the the deomiator polyomials will ot all look the same. They are all the same however if we do ot perform the cacellatios. We begi by cosiderig a arbitrary th-order high-pass trasfer fuctio, ad assumig that oe of the poles is domiat (i.e., sigificatly higher tha all the rest of the poles). Writig the trasfer fuctio i a form similar to that used i Aside A.6 for sigle-pole high-pass trasfer fuctios we have A L ( jω ) = A L( j )( jω +ω z )(jω +ω z2 )"( jω +ω zm ), (CD-9.) ( jω +ω p )(jω + ω p2 )"( jω +ω p ) where the subscript L idicates that this trasfer fuctio is for a low-frequecy equivalet circuit ad m may be less tha i geeral. If we assume that all of the zeros are at frequecies well below the domiat pole, the for frequecies ear or above the domiat pole we ca write A L ( jω ) A L ( j )(jω) m ( jω +ω p )(jω + ω p2 )"( jω +ω p ). (CD-9.2) We seek to rewrite (CD-9.2) so that the deomiator polyomial looks like a sigle-pole trasfer fuctio. Our first step is to expad the terms i the deomiator ad divide the top ad bottom of (CD-9.2) by (jω) - to get A L ( jω ) A L ( j ) jω ( ) m + jω + ω + ω pk pi jω + ω pk ω pl jω k=i+ 2 k=i+ l=k+ ( ) 2 +". (CD-9.3)
Chapter 9 - CD compaio 4 You are ecouraged to write out the first few terms to see that the sums i (CD- 9.3) are correct. We ow otice that if oe of the poles is domiat ad we are iterested i frequecies ear this pole, the higher-order sums i the deomiator of (CD-9.3) ca be eglected ad we get A L ( jω ) A L (j ) jω jω + ( )m +. (CD-9.4) Settig the deomiator equal to our stadard form for sigle-pole high-pass deomiators, ad recogizig that the resultig pole is a approximatio to the low cutoff frequecy, we get D( jω ) jω +ω cl jω +. (CD-9.5) We ow eed to set this result aside for a few momets ad show that there is a alterate way to fid the sum of the pole frequecies i our circuit. Cosider a liear time ivariat (LTI) etwork with idepedet capacitors ad o iductors. We pull the capacitors out of the etwork ad show the result as i Figure CD-9-. i + v C - LTI etwork with o eergy storage i + C v - i 2 C 2 i - C - v 2 + - v - + - Figure CD-9- A LTI etwork with capacitors ad o iductors. The capacitors have bee pulled out to leave behid a LTI etwork with o eergy storage. Let s first cosider the LTI etwork with o eergy storage. We ca write the port equatios as
Chapter 9 - CD compaio 5 i g " g v # = # $ # #, (CD-9.6) i g " g v or I = GV, (CD-9.7) where g ii is the drivig-poit coductace see by C i whe all other ports are shorted (i.e., their voltages are set to zero). This coductace is called the shortcircuit drivig-poit coductace. The o-diagoal coductaces i the matrix are trascoductaces. Rewritig (CD-9.6) with the capacitors icluded, the coductace matrix becomes a admittace matrix. Chagig our otatio to reflect the fact that we are ow dealig with trasforms (i.e., istead of i(t) we have I(jω), but to simplify the otatio we drop the explicit argumet sice o cofusio is likely) we have I g + jωc " g V # = # $ # #, (CD-9.8) I g " g + jωc V where the capacitors oly show up o the diagoal elemets of the admittace matrix. This equatio ca also be writte as I=YV. (CD-9.9) Usig the results of Aside CD-A9. we kow that the deomiator of ay trasfer fuctio for this etwork will be give by the determiat of the admittace matrix; that is, D( jω ) = G + jω C i G ii +" + jω ( ) g ii C k k= k i ( ) C i + jω, (CD-9.0) The otatio x i, meas the product of all the terms; that is, x x 2 x 3 x.
Chapter 9 - CD compaio 6 where G is the determiat of the coductace matrix i (CD-9.6) or, i other words, the determiat of Y whe ω = 0, ad G ij is the cofactor of the term i the ith row ad jth colum 2. Although it is a bit tedious, you are agai strogly ecouraged to write out a few terms ad covice yourself that (CD-9.0) is correct. Now, otice that for frequecies ear the largest of ay of the pole frequecies, ad still assumig a domiat pole, all but the two highest-order terms o the right had side of (CD-9.0) will go to zero ad we ca write D( jω ) ( jω) g ii C k k= k i ( ) C i + jω. (CD-9.) We fid the domiat pole frequecy by settig (CD-9.) equal to zero. If we do that ad divide both sides by the product of all the capacitors ad (jω) -, we get Comparig (CD-9.5) ad (CD-9.2) we fid g ii C i 0= D( jω) jω +. (CD-9.2) ω cl g ii C i. (CD-9.3) Rememberig that g ii is the short-circuit drivig-poit coductace for C i we recogize R is = /g ii as the short-circuit drivig-poit resistace see by C i ad write ω cl =, (CD-9.4) R is C i τ is where we have used the short-circuit time costat associated with C i, τ is = R is C i. This result was first preseted i (9.39). 2 The cofactor of the i,j term is defied to be the determiat of the matrix formed by deletig the ith row ad the jth colum [].
Chapter 9 - CD compaio 7 The result show i (CD-9.4) is extremely importat. We have discovered that if a high-pass trasfer fuctio has a domiat pole, the the domiat pole is approximately the sum of the pole frequecies as show by (CD-9.5). Furthermore, the domiat pole is also approximately the sum of the reciprocal short-circuit time costats as foud i (CD-9.4). We should be very careful at this poit to recogize that while the two sums are equal, as show i (CD-9.3), we caot associate a pole with the reciprocal of each short-circuit time costat. Let s ow cosider the ope-circuit time costat method for approximatig ω ch. Start by cosiderig a arbitrary th-order low-pass trasfer fuctio, (we write it i a form similar to that used i Aside A.6 for sigle-pole low-pass trasfer fuctios) A H ( jω ) = A H ( j0) + jω + jω " + jω ω z ω z2 ω zm + jω + jω " + jω, (CD-9.5) ω p ω p2 ω p where the subscript H idicates that this trasfer fuctio is for a high-frequecy equivalet circuit ad m may be less tha i geeral. If we assume that oe of the poles is domiat (i.e., sigificatly lower tha all of the other poles) ad that all of the zeros are at frequecies well above the domiat pole, the for frequecies ear or below the domiat pole we ca write ( ) A A H ( jω ) H j0 + jω + jω " + jω. (CD-9.6) ω p ω p2 We seek to rewrite (CD-9.6) so that the deomiator polyomial looks like a sigle-pole trasfer fuctio. Our first step is to expad the terms i the deomiator to get A H ( jω ) + jω ω p A H ( j0) + (. (CD-9.7) jω )2 +" k=i+ ω pk You are ecouraged to write out the first few terms to see that the sums i (CD- 9.7) are correct. We ow otice that if oe of the poles is domiat ad we are
Chapter 9 - CD compaio 8 iterested i frequecies ear this pole, the higher-order sums i the deomiator of (CD-9.7) ca be eglected ad we get ( ) A A H ( jω ) H j0 + jω. (CD-9.8) Settig the deomiator equal to our stadard form for sigle-pole low-pass deomiators, ad recogizig that the resultig pole is a approximatio to the high cutoff frequecy, we get + jω + jω. (CD-9.9) ω ch We ow set this result aside for a few momets ad show that there is a alterate way to fid the sum of the reciprocal pole frequecies i our circuit. Cosider agai the LTI etwork with idepedet capacitors ad o iductors show i Figure CD-9-. The deomiator polyomial of the etwork is give by (CD- 9.0). If a domiat low-frequecy pole exists ad we cosider frequecies ear or below this domiat pole, the all of the higher-order terms i (CD-9.0) ca be safely igored ad we are left with D( jω ) G + jω C i G ii. (CD-9.20) We fid the domiat pole frequecy by settig (CD-9.20) equal to zero. If we do that ad divide both sides by the determiat of G, we get 0= D( jω) + jω G C i G ii. (CD-9.2) Comparig (CD-9.9) ad (CD-9.2) we fid + jω + jω + jω ω ch G C ig ii. (CD-9.22) We ow diverge for a momet ad fid the resistace see by C i if all the capacitors are ope circuited i the LTI system of Figure CD-9-. This resistace is the ope-circuit drivig-poit resistace see by C i ad is deoted by R io as
Chapter 9 - CD compaio 9 oted earlier. With the capacitors ope circuited, their correspodig currets are all zero. To fid the resistace we apply a test curret at its port, I i, ad see what the voltage is. The LTI system without capacitors is described by (CD-9.6). Applyig Cramer s rule to this system yields V i = G with the ith colum = I G. (CD-9.23) With all of the currets except I i equal to zero we get R io = V i = G ii I i I k =0 G, for all k i (CD-9.24) ad usig this result i (CD-9.22) we fid that ω ch R io C i = τ io, (CD-9.25) where we have used the ope-circuit time costat associated with C i, τ io = R io C i, as previously defied. This result is idetical to the approximatio preseted i (9.46). Equatio (CD-9.25) is extremely importat. If a low-pass trasfer fuctio has a domiat pole, the the cutoff frequecy is approximately equal to the reciprocal of the sum of the ope-circuit time costats. Usig (CD-9.22) we also see that ω CH. (CD-9.26) Comparig (CD-9.26) ad (CD-9.25) it is temptig, but completely wrog, to associate a pole with the reciprocal of each time costat. All we ca say is that the sum of the ope-circuit time costats is equal to the sum of the reciprocal pole frequecies.