# Logic Effort Revisited

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1 Logic Effort Revisited Mark This note ill take another look at logical effort, first revieing the asic idea ehind logical effort, and then looking at some of the more sutle issues in sizing transistors..0 Background As a starting point let s revie asic CMOS gate delay. We model the delay of a gate to e in the form: Td = Tparasitic + Tload * Cload/Cin (EQ ) We can get these parameters from any gate y measuring its delay vs. fanout, and finding the slope (Tload) and the intercept (Tparasitic). We can also get the parameters from using our RC model of a MOS transistor. In the terminology of logical effort, the first term is the parasitic delay, and the second term is the effort delay. We ill see that this effort delay is critical for sizing. Given a string of gates e ant to kno A B hat the optimal sizing is for these gates in a chain For a given numer of gates in a 64 chain, it is relatively easy to find the sizes. If e are looking for minimum delay, then it should not e possile to change any of the device sizes and decrease the delay. If I make gate B % larger, the increase in delay of gate A should match the decrease in the delay of gate B. In other ords changing Cload of Gate A y % should cause a delay change that is equal in magnitude to the delay change caused y increase Cin of gate B y %. Given (EQ ), this alance ill occur hen the effort delay of each gate is the same. The alancing of the effort delays for minimum delay is the main result of logical effort.. Logical Effort Rather than orking in ns, e generally ork in a normalize space. Since an inverter is generally the fastest gate one can uild, it is used as the ruler to measure other (more complex) gates. We define the logical effort of an inverter to e. The effort delay is then ritten as the product of three factors -- a technology constant, the logical effort of the gate, and the electrical effort of the gate (hich is just the fanout Cload/Cin). The technology constant is just Tload for an inverter, and the logical effort is the ratio of Tload(gate)/ Tload(inv). This sets the technology constant to e the numer of picoseconds needed for. You should e careful aout finding optimal solutions -- optimal for hat? This note is mostly interested in optimal timing, ut in most circuits you need to orry aout poer and area too. These other factors often set the transistor sizing

2 an inverter to drive an addition load equal to its input capacitance. I call the product of the electrical effort (fanout) and the logical effort the effective fanout of the gate. The logical effort of a gate is the slope of the delay vs. fanout for the gate, divided y the slope for an inverter. It is easy to estimate this numer using the simple RC model e used in EE27. Since delay of the gate is set y a sum of RC time constants, the effort delay comes from the term hich is the effective resistance of the gate driving the load capacitance. For CMOS gates the resistance is invery proportional to input capacitance, so the logical effort of a gate is: LE = C gate R gate C inv R inv (EQ 2) One can find the LE y either making the input capacitance of the gate equal to the inverter, and looking at the resistance ratio, or y making the resistance the same and looking at the capacitance ratio..2 Optimal Numer of Stages and Optimal Fanout Delay vs Fanout 6 5 Delay Delay a=.5 Delay a= Delay a=.5 Delay a= Fanout The question of the optimal fanout and numer of stages in a chain, is oth simple and a little tricky. This anser comes from solving the delay equation for a string of gates: Total delay = N * (f * Tload+Tparasitic) = ln(cl/cin) * Tload * (f + α) / ln(f) (EQ 3) here α is Tparasitic/Tload. From the delay plot, it does seem like 4 is the right anser, since it seems to make the delay close to the minimum. In addition, the curves are flat, so there is not a huge prolem ith using a range of fanouts eteen 2 and 8. So far this all seems to e revie. 2

3 No notice that (EQ 3) did not specify the logical effort of the gate. It turns out that the optimal fanout, here fanout is the electrical effort (Cload/Cin), for any string of homogeneous gates is around 4. So it ould seem at first glance e have a conflict. The theory of logical effort said e should make the effective fanout (fanout * LE) 4. The reason for this confusion is that life gets more interesting hen the gates are not all the same. With different gates you have a choice on ho you are going to drive your load. If the fanout per stage is too high, you could alays add an inverter to the chain to drive the load. When adding uffers, e alays think of using inverters since they are the fastest ay to drive fanout. So the optimization prolem e think aout hen sizing fanout is should I use this pile of gates to do my logic ith n stages, or should I add a uffer (or to) and do it in n+ (or 2) stages. Since the extra stages e are thinking aout are inverters, it is this element that should have a fanout of around 4 for optimal performance. With inverters having a fanout of 4, it means that the effort delay of all other gates should e around 4 as ell (they all have to e the same) and so the fanout of the logic gates ill e less than 4. Understanding that the optimal effective fanout is hen the est uffer element has a fanout of 4ish is important hen there are gates faster than inverters, as e ill see in section on domino logic. In domino logic, the effective fanout (LE * Cload/Cin) is around 3..3 Parasitic Delay The parasitic delay of a gate can e complicated to calculate, and can depend on hich input causes the output to transition. In general e don t need to calculate this preciy, ut you should kno that the parasitic delay of a gate gros more than linearly ith the numer of inputs. This is ecause the parasitic capacitance of parallel transistors gros linearly ith the numer of inputs (n transistors of the same size), hile the parasitic of series devices gros quadratically (n transistors, each n times larger). The large parasitic delay makes large fanin gates rarely useful. In general, gates ith more than 3 stacked devices are rarely used. 2.0 Using Logical Effort Logical effort is a very handy tool, ut there are some situations that ill still require some thought. The first prolem that arises in real designs is that the input and output capacitance are rarely given to you in a nice form so you can calculate logical effort. Even hen this information is availale there are often fixed ire capacitance that needs to e driven. All this makes formulating the prolem more difficult, ut not impossile. It just takes some reasonale approximations. Once the prolem is formed, you often find some circuits that don t fit nicely into the logical effort frameork, so I ill talk aout these next. The issues that I ill discuss are side loads, logical efforts for transmission gates, and dealing ith reconvergent fanout. Finally I ill take a quick look at using logical effort for dynamic gates. 3

5 delay. I ould recalculate the fanouts in gates after A only if I am going to remove a uffer in this chain, or I added one in the prechain. Let s look at each scenario in a little more detail. In all the case here cap A is small compared to the cap of gate y, the initial sizes are correct, and the prolem is solved. Since this is the minimum delay situation, it is good to try to get the side load further don a chain of gates here its effect ill e smaller. The other easy case is here A is much larger than the gate cap. In this case resolve the prolem assuming that A is the output. Once these sizes are found, the optimal y size is easy to find: x --- A+ y = y = Bx x (EQ 4) Since y is small compared to A the interation converges quickly. Also notice that this is going to push the post chain to using small fanouts per stage, and increase the loading on the prechain. We can guess that this change in the output fanout ill have little effect on delay ecause the improvement in delay in going to lo fanout is small (the fixed parasitic loads really hurt), and this gain is partially offset y the large load on the prechain. The net result is that doing this optimization only slightly improves performance at the expense of area. When A is comparale to x the change in the fanouts in the prestage ill e small, since e amortize the change over a numer of stages. If they did not change at all ( x did not change) the fanout in the post stage ould e exactly correct, since y/x=b/y. Since x only changes slightly, e should not e surprised that its effect on the fanout of the post cap gates is small. Said a different ay, our method gets the change in fanouts from the fixed load approximately correct, so the change in delay to get it exactly correct is small. Just to demonstrate that finding the optimal delay is often not the right approach, I solved the for optimal delay for a chain of gates, hen each gate drives a fixed side load. You might guess that the optimal sizes have each of the gates e the same, ut that ould not e right. Since the delay decreases if the fixed load is small compared to the capacitance of the gate, the optimal sizes ramp up in the middle of the chain and then ramp them ack don (making the fanout less than ) The curves are for different numers in the chain, and the y axis is the size of the gate relative to The equation is not much more complicated hen there are chains of gates instead of the single gate x, since the fanout of all the gates in the chain remains the same 5

6 the fixed load. What is interesting is that the overall delay is only slightly etter than a chain here all the gates are the same size and roughly equal to the load capacitance. 2.3 Transmission Gates and other issues ith Gate sizing For all the gates that e have talked aout in class, e make the logical effort of all the inputs the same. This strategy is a good one, since if you try to improve the LE of one input, the LE of the other inputs gets much orse. But hat should you do aout transmission gates? First since e are interested in the effective resistance driving the load capacitance, e can only find the LE for a transmission gate hen it is used in some context. The LE needs to e found for the supergate it is in. No the question is ho to size the devices in the inverter and the TGate. Inv + TGate _ A A B NAND2 + TGate _ To start e need to kno hether e should consider and _ to different signals, or should e add the cap together. The anser is descried in more detail in the next section, and turns out that the logical effort of a splitter (creates A and A_) is only slightly greater than one. So the logic effort should e considered y only looking at the input. No ho do e ant the logical Inv + T-Gate NAND2 + T-gate efforts to turn out? If the ect signal 2 _ is critical e ould ant to make the 2 _ LE of this signal small, and let the LE A for the inverter or NAND gate gro A larger. This is exactly hat happens if B e make the resistance of the TGate LE A=2 LE A = 8/3 2 equal to the resistance of the inverter. LE Sel=2/3 LE Sel = 2/3 What is happening is that the TGate doules the resistance of the gate ithout changing the input capacitance. To get equal logical efforts, the resistance of the transmission gate must e Inv + T-Gate NAND2 + T-gate 2 _ made much smaller than the resistance 2 3 _ of the inverter. Since the resistance of A 3 4 the supergate is the same for oth A inputs, to get equal logical efforts, e B make the input capacitance the same. LE A=4/3 LE A = 5/3 2 This makes the transistors in the LE Sel=4/3 LE Sel = 5/3 TGate 3 for the inverter, and the LE of the oth A and Sel 4/3. Oviously you can set the LE of the input at any point in 6

7 eteen these points. The est LE depends on hich of the inputs is critical. I don t think it ould make sense to go outside these to points, since oth are getting a little inefficient in terms of making use of the transistor idth consumed. 2.4 Reconvergent Fanout (phase splitting) Sizing gets a little more complex hen e end up ith a situation like the one shon in the figure. There are to paths from the gate to gate. The path through a is longer, ut the other path requires more fanout per stage. This prolem is called reconvergent fanout. a Some 2-input gate It turns out that the short path is the one you ant to consider hile computing gain per stage in a reconvergent fanout prolem ecause a heavy load on the short path ill also increase the delay of the long path. Let the input have unit of cap and the final gate have a size of times the input. What e ant to kno is hat is the effective logical effort of this comination of gates. That is ho does the delay of the path change as e change. Once e kno this, e can just use this effective logical effort hen e come across this situation and quickly find the fanouts at each stage. The critical path is clearly through inverter a, and the delay of this path is: t = (a+)/ + /a + 2* Tparasitic (EQ 5) Since the load on the first inverter is simply a side load for the prolem of sizing a, the size of a is just the geometric mean of gate and, hich is sqrt(). Thus the total delay of this path is t = Tparasitic dt d = (EQ 6) So the logical effort (the slope of the delay ith load) is not really a constant, ut for reasonale total effort (around 4), the fanout ould e around 2.5, and the effective logical effort is.6. This is a reasonale numer to use for stage efforts from 3 to 8. Actually this is not the est solution for this Some 2-input gate prolem since e have not matched the delay -x of the to paths. We have loaded the critical a path ith directly driving the large gate. This kind of prolem often occurs, and you can x speed up the circuit y duplicating some logic and splitting the critical path from the noncritical path. So hat e should have done is to split the first inverter into to inverters, here the sum of the inverter sizes is equal to. In this prolem, e have to find oth x and a hich minimize the total delay of the circuit. This is a little harder to solve, ut it is easy to rite the equations to solve. At the optimal point the delay of the to paths ill e the same, and a ill e the geometric mean of and (-x). This gives 7

8 t = -- + Tpar = 2 x Tpar x (EQ 7) It is easy to solve for as a function of x, so that is hat I did. Given x and, I can find the delay and this gives the plot shon elo Delay Delay no split The lue curves (ith diamonds) are the result of splitting the first inverter. If the inverter is not split (the first technique) you get the pink (squares) curves. The loer curves are the results you get if you ignore the parasitic delay, the upper curves are the results if the f loading is equal to its gate capacitance. Splitting the gate is alays etter than not splitting the gate. When the gate drive is split, the circuit ehaves like a single stage gate ith logical effort of aout (. if you ant to e precise) and a large effective parasitic delay (around 4). 2.5 Domino Gates Working ith domino gates is nearly the same as orking ith static gates. The one issue you need to e careful aout is setting the correct effective fanout per stage. For static gates the est uffer is an inverter, so the optimal effective fanout per stage is around 4. If the logical effort of a gate is larger than, the fanout of the stage should e reduced. In domino gates, the LE is often less than, so to make the effective fanout 4, the true fanout ould need to e greater than 4. This is not 2 correct, since in domino circuits there are etter uffers than inverters. The est uffer /2 ould look like a domino stage folloed y a skeed inverter. The logical effort for the inverter is 5/6, and the domino stage is somehere eteen /3 and 2/3 depending on the size of the clock transistor (let us assume 2/ 8

9 3). The LE for the to stages is.55, and the average LE per stage is If e used a chain of these gates and tried to find the est fan out per stage, the anser ould e that each stage should have an electrical fanout of aout 4, (actually 4.4 for dynamic stage and 3.6 for the static gate). At this fanout, the effective fanout (LE * Fanout) of the domino gate is 3, not 4. Thus in dynamic circuits you should aim for smaller effective fanout per stage than for static logic, hich makes the electrical fanout more similar to hat you ould do ith static gates. 9

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