Logic elements, including ?ip-?ops and gates, are physical devices that have ?niteresponse times to stimuli. Each of these elements exhibits a certain propagation delaybetween the time that an input is presented and the time that an output is generated.As more gates are chained together to create more complex logic functions, theoverall propagation delay of signals between the end points increases. Flip-?ops aretriggered by the rising edge of a clock to load their new state, requiring that the inputto the ?ip-?op is stable prior to the rising edge. Similarly, a ?ip-?ops output stabilizesat a new state some time after the rising edge. In between the output of a ?ip-?opand the input of another ?ip-?op is an arbitrary collection of logic gates, as seen inthe preceding synchronous counter circuit. Synchronous timing analysis is the studyof how the various delays in a synchronous circuit combine to limit the speed at whichthat circuit can operate. As might be expected, circuits with lesser delays areable to run faster.
A clock breaks time into discrete intervals that are each the duration of a singleclock period. From a timing analysis perspective, each clock period is identical tothe last, because each rising clock edge is a new ?op triggering event. Therefore,timing analysis considers a circuits delays over one clock period, betweensuccessive rising (or falling) clock edges. Knowing that a wide range of clockfrequencies can be applied to a circuit, the question of time arises of how fast theclock can go before the circuit stops working reliably. The answer is that the clockmust be slow enough to allow suf?cient time for the output of a ?op to stabilize,for the signal to propagate through the combinatorial logic gates, and for the inputof the destination ?op to stabilize. The clock must also be slow enough for the ?opto reliably detect each edge. Each ?op circuit is characterized by a minimumclock pulse width that must be met. Failing to meet this minimum time can result inthe ?op missing clock events.
Timing analysis revolves around the basic timing parameters of a ?op: input setuptime (tSU), input hold time (tH), and clock-to-out time (tCO). Setup time speci?esthe time immediately preceding the rising edge of the clock by which the input mustbe stable. If the input changes too soon before the clock edge, the electrical circuitrywithin the ?op will not have enough time to properly recognize the state of the input.Hold time places a restriction on how soon after the clock edge the input
may begin to change. Again, if the input changes too soon after the clock edge, itmay not be properly detected by the circuitry. Clock-to-out time speci?es howsoon after the clock edge the output will be updated to the state presented at theinput. These parameters are very brief in duration and are usually measured innanoseconds. One nanosecond, abbreviated ns, is one billionth of a second.In very fast microchips, they may be measured in picoseconds, or one trillionthor a second. Consistent terminology is necessary when conducting timing analysis.Timing is expressed in units of both clock frequency and time. Clock frequency,or speed, is quanti?ed in units of hertz, named after the twentieth century Germanphysicist, Gustav Hertz. One hertz is equivalent to one clock cycle per secondone transition from low to high and a second transition from high to low. Units ofhertz are abbreviated as Hz and are commonly accompanied by pre?xes that denotean order of magnitude. Commonly observed pre?xes used to quantify clockfrequency and their de?ni- tions are listed in Table 1.13. Unlike quantities of bytesthat use binary-based units, clock frequency uses decimal-based units.

Units of time are used to express a clocks period as well as basic logic elementdelays such as the aforementioned tSU, tH, and tCO. As with frequency, standardpre?xes are used to indicate the order of magnitude of a time speci?cation. However,rather than expressing positive powers of ten, the exponents are negative.Table 1.14 lists the common time magnitude pre?xes employed in timing analysis.

Aside from basic ?op timing characteristics, timing analysis must take intoconsideration the ?nite propagation delays of logic gates and wires thatconnect ?op outputs to ?op inputs. All real components have nonzeropropagation delays (the time required for an electrical signal to move from aninput to an output on the same component). Wires have an approximatepropagation delay of 1 ns for every 6 in of length. Logic gates can havepropagation delays ranging from more than 10 ns down to the picosecond range,depending on the technology being used. Newly designed logic circuits shouldbe analyzed for timing to ensure that the inherent propagation delays of the logicgates and interconnect wiring do not cause a ?ops tSU and tH speci?cations tobe violated at a given clock frequency.
Basic timing analysis can be illustrated with the example logic circuit shownFig. 1.16. There are two ?ops connected by two gates. The logic inputs shownunconnected are ignored in this instance, because timing analysis operates on asingle path at a time. In reality, other paths exist through these unconnected inputs,and each path must be individually analyzed. Each gate has a ?nite propagationdelay, tPROP , which is assumed to be 5 ns for the sake of discussion. Each ?ophas tCO = 7 ns, tSU = 3 ns, and tH = 1 ns. For simplicity, it is assumed that thereis zero delay through the wires that connect the gates and ?ops.
The timing analysis must cover one clock period by starting with one rising clockedge and ending with the next rising edge. How fast can the clock run? The ?rstdelay encountered is tCO of the source ?op. This is followed by tPROP of the twologic gates. Finally, tSU of the destination ?op must be met. These parametersmay be summed as follows:
tCLOCK = tCO + 2 ? tPROP + tSU = 20 ns
The frequency and period of a clock are inversely related such that F = 1/t. A 20-nsclock period corresponds to a 50-MHz clock frequency: 1/(20 ? 109) = 50 ? 106.Running at exactly the calculated clock period leaves no room for design margin.Increasing the period by 5 ns reduces the clock to 40 MHz and provides headroomto account for propagation delay through the wires.
Hold time compliance can be veri?ed following setup time analysis. Meeting a ?opshold time is often not a concern, especially in slower circuits as shown above.The 1 ns tH speci?cation is easily met, because the destination ?ops D-input willnot change until tCO + 2 ? tPROP = 17 ns after the rising clock edge. Actualtiming parameters have variance associated with them, and the best-casetCO and tPROP would be somewhat smaller numbers. However, there is somuch margin in this case that tH compliance is not a concern.
Hold-time problems sometimes arise in fast circuits where tCO and tPROPare very small. When there are no logic gates between two ?ops, tPROP canbe nearly zero. If the minimum tCO is nearly equal to the maximum tH, thesituation should be carefully investigated to ensure that the destination?ops input remains stable for a suf?cient time period after the active clock edge.
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