深亚微米数字集成电路设计：从电路到系统

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24 INTRODUCTION Chapter 1

expected response. One reason for this aberration are the variations in the manufacturing

process. The dimensions, threshold voltages, and currents of an MOS transistor vary

between runs or even on a single wafer or die. The electrical behavior of a circuit can be

profoundly affected by those variations. The presence of disturbing noise sources on or off

the chip is another source of deviations in circuit response. The word noise in the context

of digital circuits means “unwanted variations of voltages and currents at the logic

nodes.” Noise signals can enter a circuit in many ways. Some examples of digital noise

sources are depicted in Figure 1.10. For instance, two wires placed side by side in an inte-

grated circuit form a coupling capacitor and a mutual inductance. Hence, a voltage or cur-

rent change on one of the wires can influence the signals on the neighboring wire. Noise

on the power and ground rails of a gate also influences the signal levels in the gate.

Most noise in a digital system is internally generated, and the noise value is propor-

tional to the signal swing. Capacitive and inductive cross talk, and the internally-generated

power supply noise are examples of such. Other noise sources such as input power supply

noise are external to the system, and their value is not related to the signal levels. For these

sources, the noise level is directly expressed in Volt or Ampere. Noise sources that are a

function of the signal level are better expressed as a fraction or percentage of the signal

level. Noise is a major concern in the engineering of digital circuits. How to cope with all

these disturbances is one of the main challenges in the design of high-performance digital

circuits and is a recurring topic in this book.

The steady-state parameters (also called the static behavior) of a gate measure how

robust the circuit is with respect to both variations in the manufacturing process and noise

disturbances. The definition and derivation of these parameters requires a prior under-

standing of how digital signals are represented in the world of electronic circuits.

Digital circuits (DC) perform operations on logical (or Boolean) variables. A logical

variable x can only assume two discrete values:

x ∈ {0,1}

As an example, the inversion (i.e., the function that an inverter performs) implements the

following compositional relationship between two Boolean variables x and y:

y = x: {x = 0 ⇒ y = 1; x = 1 ⇒ y = 0} (1.6)

v(t)

i(t)

(a) Inductive coupling (b) Capacitive coupling

Figure 1.10 Noise sources in digital circuits.

noise

chapter1.fm Page 24 Friday, January 18, 2002 8:58 AM

Section 1.3 Quality Metrics of a Digital Design 25

A logical variable is, however, a mathematical abstraction. In a physical implemen-

tation, such a variable is represented by an electrical quantity. This is most often a node

voltage that is not discrete but can adopt a continuous range of values. This electrical volt-

age is turned into a discrete variable by associating a nominal voltage level with each logic

state: 1 ⇔ V

, 0 ⇔ V

, where V

and V

represent the high and the low logic levels,

respectively. Applying V

to the input of an inverter yields V

at the output and vice

versa. The difference between the two is called the logic or signal swing V

(1.7)

The Voltage-Transfer Characteristic

Assume now that a logical variable in serves as the input to an inverting gate that produces

the variable out. The electrical function of a gate is best expressed by its voltage-transfer

characteristic (VTC) (sometimes called the DC transfer characteristic), which plots the

output voltage as a function of the input voltage V

out

= f(V

). An example of an inverter

VTC is shown in Figure 1.11. The high and low nominal voltages, V

and V

, can

readily be identified—V

= f(V

) and V

= f(V

). Another point of interest of the

VTC is the gate or switching threshold voltage V

(not to be confused with the threshold

voltage of a transistor), that is defined as V

= f(V

). V

can also be found graphically at

the intersection of the VTC curve and the line given by V

out

= V

. The gate threshold volt-

age presents the midpoint of the switching characteristics, which is obtained when the out-

put of a gate is short-circuited to the input. This point will prove to be of particular interest

when studying circuits with feedback (also called sequential circuits).

Even if an ideal nominal value is applied at the input of a gate, the output signal

often deviates from the expected nominal value. These deviations can be caused by noise

or by the loading on the output of the gate (i.e., by the number of gates connected to the

output signal). Figure 1.12a illustrates how a logic level is represented in reality by a range

of acceptable voltages, separated by a region of uncertainty, rather than by nominal levels

()=

out

Figure 1.11 Inverter voltage-transfer

characteristic.

out

= V

chapter1.fm Page 25 Friday, January 18, 2002 8:58 AM

26 INTRODUCTION Chapter 1

alone. The regions of acceptable high and low voltages are delimited by the V

and V

voltage levels, respectively. These represent by definition the points where the gain

(= dV

out

/ dV

) of the VTC equals −1 as shown in Figure 1.12b. The region between V

and V

is called the undefined region (sometimes also referred to as transition width, or

TW). Steady-state signals should avoid this region if proper circuit operation is to be

ensured.

Noise Margins

For a gate to be robust and insensitive to noise disturbances, it is essential that the “0” and

“1” intervals be as large as possible. A measure of the sensitivity of a gate to noise is given

by the noise margins NM

(noise margin low) and NM

(noise margin high), which quan-

tize the size of the legal “0” and “1”, respectively, and set a fixed maximum threshold on

the noise value:

(1.8)

The noise margins represent the levels of noise that can be sustained when gates are cas-

caded as illustrated in Figure 1.13. It is obvious that the margins should be larger than 0

for a digital circuit to be functional and by preference should be as large as possible.

Regenerative Property

A large noise margin is a desirable, but not sufficient requirement. Assume that a signal is

disturbed by noise and differs from the nominal voltage levels. As long as the signal is

within the noise margins, the following gate continues to function correctly, although its

output voltage varies from the nominal one. This deviation is added to the noise injected at

the output node and passed to the next gate. The effect of different noise sources may

accumulate and eventually force a signal level into the undefined region. This, fortunately,

does not happen if the gate possesses the regenerative property, which ensures that a dis-

Figure 1.12 Mapping logic levels to the voltage domain.







“1”

“0”

Undefined

Region

(a) Relationship between voltage and logic levels

out

(b) Definition of V

and V

Slope = -1







–=

chapter1.fm Page 26 Friday, January 18, 2002 8:58 AM

28 INTRODUCTION Chapter 1

a degraded amplitude, which could be caused by noise. Instead of swinging from rail to rail,

only extends between 2.1 and 2.9 V. From the simulation, it can be observed that this devi-

ation rapidly disappears, while progressing through the chain; v

, for instance, extends from

0.6 V to 4.45 V. Even further, v

already swings between the nominal V

and V

. The

inverter used in this example clearly possesses the regenerative property.

The conditions under which a gate is regenerative can be intuitively derived by ana-

lyzing a simple case study. Figure 1.15(a) plots the VTC of an inverter V

out

= f(V

) as well

as its inverse function finv(), which reverts the function of the x- and y-axis and is defined

as follows:

(1.9)

Assume that a voltage v

, deviating from the nominal voltages, is applied to the first

inverter in the chain. The output voltage of this inverter equals v

= f(v

) and is applied to

the next inverter. Graphically this corresponds to v

= finv(v

). The signal voltage gradu-

ally converges to the nominal signal after a number of inverter stages, as indicated by the

arrows. In Figure 1.15(b) the signal does not converge to any of the nominal voltage levels

but to an intermediate voltage level. Hence, the characteristic is nonregenerative. The dif-

ference between the two cases is due to the gain characteristics of the gates. To be regener-

ative, the VTC should have a transient region (or undefined region) with a gain greater

than 1 in absolute value, bordered by the two legal zones, where the gain should be

smaller than 1. Such a gate has two stable operating points. This clarifies the definition of

the V

and the V

levels that form the boundaries between the legal and the transient

zones.

Noise Immunity

While the noise margin is a meaningful means for measuring the robustness of a circuit

against noise, it is not sufficient. It expresses the capability of a circuit to “overpower” a

infout

()

⇒

finvout

()

Figure 1.15 Conditions for regeneration.

out

(a) Regenerative gate

f(v)

finv(v)

f(v)

(b) Nonregenerative gate

chapter1.fm Page 28 Friday, January 18, 2002 8:58 AM

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