Quick efeence: Amplifie Equations 1.0 Intoduction Befoe deeloping a deep familiaity with the aious chaacteistics of amplifies, it s common to spend a fai amount of time flipping back and foth though you notes and textbook, in seach of aious fomulas. To sae you some touble, hee ae many of the eleant equations in one place. Because this set of notes is not intended to be a textbook on netwok theoy, only the baest sketches of deiations ae usually poided. That said, thee should still be enough that you could econstuct intemediate steps if needed. Pat of what distinguishes an expeienced analog designe fom othes is the ability to apply the simplest appoximation that captues the phenomena of inteest. Thoughout, ecognize that many of the equations in this handout simplify consideably if you ignoe aious factos (e.g. body effect, channel-length modulation, DIBL, etc.. In some cases, we offe those simplifications. In othes, they ae left fo you to finish. In all cases, pause and conside when it may be acceptable to neglect cetain of these phenomena, and how that selectie and conscious neglect simplifies the equations. Thoughout this handout, we use the followinodel fo the tansisto and load/souce esistances: FIGUE 1. Incemental tansisto model, plus teminal esistances C gd G C gs C db gs b bs L C sb As you can see, we ae implicitly assuming that the bulk teminal is tied to an incemental gound. 2.0 Gain, and input/output esistances 2.1 Common-souce input esistance The input esistance is effectiely infinite (not a function of b o CLM/DIBL. In this ea of supethin gate oxides, leakage is moe than measuable, and the input esistance is 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 1 of 9
consequently not tuly infinite. Howee, it still emains high enough fo most puposes that infinity is a useful appoximation. 2.2 Common-souce output esistance Use a test cuent souce, i t. ince gs bs, can simply sum the two tansconductances into a single one (let s call it tot. Compute the esistance looking into the dain, and call it out. The net esistance at the dain will theefoe be out in paallel with any extenal esistance, L. gs i t ; (1 0 ( i t tot gs i t ( 1 tot V test i t i t ( 1 tot i t [ tot ] ; (2. (3 o, at last, the esistance looking into the dain (and theefoe not including L, is out tot (. (4 2.3 Common-souce oltage gain Multiplication of the total esistance at the dain by the effectie tansconductance will gie us the gain. To find the effectie tansconductance, shot the dain to gound (incementally speaking, and measue the atio of shot-cicuit output cuent to input oltage. gs in ; (5 bs ; (6 gx b bs i out - ( in ( i out - ; (7 eff - ------------------------------------------------------- in 1 ( - ------------------------------------------------------ 1 (, (8 whee we hae used g 0 1/. The oltage gain is theefoe A eff ( out L 1 ------------------------------------------------------ ([ ( ( ] L which simplifies initially to (9 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 2 of 9
A L ------------------------------------------------------ ( -------------------------------------------------------------------------- 1 ( ( L. (10 Afte a little moe wok, the equation s complexity educes futhe, leading to a quite easonable expession: A ------------------------------------------------------------------------------ L 1 ( ( L. (11 The eade should eify that this expession educes to the expected appoximations when the body effect may be neglected, when thee is no souce degeneation, etc. 2.4 Common-gate input esistance Use a test oltage souce. Note that gs bs, again, so we can mege the two tansconductances into a single tot. Compute the esistance looking into the souce fist, then accommodate at the end, if desied o necessay. You may use supeposition to simplify the deiation (teat tansconductances as independent cuent souces in this computation, since the contol oltage is fixed thoughout this paticula expeiment. gs test. (12 With tansconductance tot disabled, we compute one contibution to the test cuent: i t1 ----------------- test. (13 L Next, disable (shot out the test oltage souce, taking cae not to zeo out tot, and compute the othe contibutions to the test cuent. The cuent fom the tansconductance is added to the cuent flowing though. i t2 tot gs i L t2 ------ tot test i L g t2 ------ i mtot test t2 ---------------------- 0 1 ------ L tot --------------------------- test L ; (14 i t i t1 i test tot t2 ----------------- --------------------------- test L L test L ----------------- ( 1 tot. (15 o, the esistance looking into the souce, in, is: --------- test i t L -------------------------- 0 ----------------------------------------- L 1 tot 1 (. (16 Note that, if L <<, and if the tansisto s intinsic oltage gain is much geate than unity, then the esistance looking into the souce is appoximately 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 3 of 9
. (17 If you desie the total esistance between the souce node and gound, simply compute the paallel combination of in and. 2.5 Common-gate output esistance This esistance is pecisely the same as the output esistance of the degeneated commonsouce amplifie:. (18 If we die the souce teminal diectly with a oltage souce, then 0, and the equation fo the esistance looking into the dain collapses to simply. That emains a easonably good appoximation as long as is small compaed with, and ( is small compaed with unity. 2.6 Common-gate oltage gain 1 in ( -------------------------- As befoe, let s fist compute the effectie tansconductance of the amplifie, defined hee as the atio of shot-cicuit dain cuent to souce oltage: b out tot ( gs bs in ; (19 gs ; (20 tot in gx ------ in ( tot eff tot (. (21 The oltage gain fom souce to dain is theefoe A 0 eff ( out L ( ( L, (22 whee we hae used the fact that, in this fist computation, we ae diing the souce diectly with a oltage souce. The total dain esistance is thus just L, and the gain in Eqn.22 is theefoe that fom the souce to dain, as stated. The oltage diide fomed at the input must be taken into account to complete the calculation of the oeall gain: o, A A in 0 ------------------ in. (23 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 4 of 9
which simplifies a bit to and futhe to: ----------------------------------------- L 1 ( g A ( ( L m ------------------------------------------------------ 0 L 1 ----------------------------------------- ( 0 A ( ( L 0 -------------------------------------------------------------------------- L L ( ( g A m L ------------------------------------------------------------------------------ 1 ( L (, (24, (25. (26 As expected, the gain expession collapses to moe familia foms when body effect is neglected, the souce esistance is zeo, and if the tansisto s output conductance is zeo, etc. The eade should eify these statements independently. 2.7 ouce followe input esistance Again, we teat it as infinite, independent of body effect and CLM/DIBL. 2.8 ouce followe output esistance The souce followe s output esistance is the same as the input esistance of a commongate stage: L out -------------------------- 1 tot ----------------------------------------- L 1 (. (27 In most souce followes, L is chosen ey small compaed with. In such cases, the output esistance simplifies to: out 1 ----------------------------------------- (. (28 If, in addition, the tansisto s intinsic oltage gain is high, then the unity facto in the denominato can be neglected, simplifying the output esistance expession een futhe: 1 out ---------------------. (29 b 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 5 of 9
2.9 ouce followe oltage gain Again, compute an effectie tansconductance, and multiply by the output esistance. That output esistance needs to include any extenal loading that might appea in paallel with the esistance seen when looking into the tansisto s souce. gs bs in ; (30 gs ; (31 (fo output esistance only; bs 0 fo effectie tansconductance calculations. o, in i L out ------ ; (32 Oeall oltage gain is theefoe: eff - in ---------------- 1 - L. (33 A 0 eff ( out L - ---------------- 1 ----------------------------------------- L 1 ( (34 which, afte some effot, simplifies to A 0 ------------------------------------------------------------------------------- ( ( L. (35 Altenatiely, we may wite: A 0 ------------------------------------------------------------------------------ ( 1 ( L -------------------------------------------------------------------- 1 ( g 1 mb ( 1 g ------------------------- 0 L --------------------------, (36 fom which it is pehaps somewhat easie to see that the oltage gain can only appoach unity, as we would expect fom a souce followe. As usual, the eade should eify that the exact equation simplifies to the numeous known appoximations in the case whee body effect is zeo, tansisto g 0 may be neglected, L is zeo, etc. 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 6 of 9
3.0 esistances fo open-cicuit time constant calculations It s impotant to ecall that a two-pot model is a geneal and complete desciption fo the teminal behaio of any single-input, single-output amplifie. Hence, the effectie esistance facing any capacito may be ealuated diectly fom the two-pot model if the coesponding paametes ae known. We futhe simplify the situation by assuming that we may safely neglect eese tansmission. This assumption of unilateal behaio is usually well satisfied in pactical amplifies. Howee, this obseation does not mean that you ae absoled of eifying whethe you paticula amplifie satisfies this assumption. Always check assumptions in any case whee it mattes! Using a cuent souce in a hybid model, we find that the effectie esistance may be stated in a simple, uniesal, mnemonic fom:, (37 whee left is the esistance seen between the capacito s left teminal and gound, ight is that seen between the ight teminal and gound, and f is the effectie tansconductance, defined as the atio of shot-cicuit output cuent to input oltage. This equation may also be expessed as eq left ight f left ight eq left ight A f left, (38 whee we hae ecognized the poduct of effectie tansconductance and ight as (minus the oltage gain between the two teminals of the capacito. Fom this fom of the equation, we can see diectly the effect of oltage gain between the capacito s teminals, a phenomenon fist explained by John M. Mille of the National Bueau of tandads, in 1918. 3.0.1 esistance facing c gd Fo the dain-gate capacitance, the effectie esistance is whee g G out m L 1 ------------------------------------------------------ ( ( out L G out L [ ( ] L, (39. (40 The expession fo the effectie esistance simplifies consideably in the case of a common-gate connection, whee G is often zeo (o ey small. In that case, Eqn.40 is all you need. In a souce-followe connection, whee L is typically zeo (o ey small, an een geate simplification esults. In that case, G is pehaps a good appoximation to the 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 7 of 9
esistance facing c gd. o, the expession fo the esistance is somewhat complicated only fo the case of the common-souce amplifie. 3.0.2 esistance facing c gs Hee, the same basic two-pot model applies, but now the model paametes hae to be chosen to eflect the fact that the capacito in question connects between the gate and souce teminals. That is, the amplifie unde consideation has its input at the gate, and poides an output at the souce. If we call G. the esistance to the left, the esistance to the ight is simply the total output esistance of a souce-followe (i.e., the esistance looking into the souce teminal, in paallel with any extenal. imilaly, the effectie tansconductance is also meely that of a souce-followe. o, we hae actually deied all of the pieces aleady. We just need to put them togethe: In the geneal case, this esistance is just the esistance looking into the dain of a tansisto, in paallel with any additional esistance connected to that dain (call that exta esis eq L G ----------------------------------------- 1 ( ---------------- 0 L G ----------------------------------------- 1 L 1 ( g - m, (41 which we ty to simplify as follows: eq G -------------------------------------------------------------------------- [( L ( 0 L G ] Afte some cunching, we finally get something that does look simple: G ( L ( L eq -------------------------------------------------------------------------------------------------------- L ( Collecting tems in a slightly diffeent way is also pehaps useful: G G ( G L G L eq ----------------------------------------------------------------------------------------------------------------------- 1 ( ( L. (42. (43. (44 Note that, in the limit of no body effect and infinite, the equialent esistance facing c gs indeed simplifies to a esult we e seen befoe: G eq ---------------------- 1. (45 3.0.3 esistance facing c db 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 8 of 9
tance L to be consistent with ou notation so fa. We e aleady found the esistance looking into the dain: out tot ( o, just place that alue in paallel with any L that happens to be pesent.. (46 3.0.4 esistance facing c sb imilaly, this esistance is just that looking into the souce of a tansisto, in paallel with any othe esistance connected to it (we e been calling it. Thus, the souce-followe output esistance equation is what we need hee: L out ----------------------------------------- 1 (. (47 Just compute the paallel combination of out and to get the final answe. 2002 Thomas H. Lee, e. Decembe 4, 2002; All ights eseed Page 9 of 9