1、Chapter 2SIGNAL INTEGRITYDigital systems rely on signaling from drivers to receivers to pass information be-tween their components. Reliable signaling is achieved when the signaling specifica-tions are met under full adverse noise conditions as well as device behavior variationsdue to both process d
2、eviations in device manufacturing and normal changes overthe operating temperature. A breakdown in communication leads to glitches whereunintended or incorrect data is transferred, a situation called false signaling.Critical data paths within systems often contain safeguards against false signal-ing
3、, and the primary system for this is data redundancy through parity. Parity canenable the system to detect small-scale signaling failures, while an error correctionscheme can be optionally included to correct data faults detected using parity in-formation. However, parity and error correction cannot
4、 be relied upon to make anoisy system stable and reliable. In addition, it is often not practical in terms ofcost or performance to include parity and error correction on every circuit.Superimposed on the desired signals are unwanted waveforms (i.e., noise) gener-ated from many sources. The principa
5、l sources are crosstalk, impedance mismatch,simultaneous switching noise, and multiple reflections. Each can be independentlycharacterized to facilitate an understanding of the mechanisms that degrade signalquality and to help guide design decisions. In real systems, all act simultaneouslyand requir
6、e detailed circuit simulation to obtain good estimates of the total wave-form on each signal line.5556Signal IntegrityChapter 22.1Transmission LinesA transmission line is a two-conductor interconnect (so that it can carry signalfrequency components down to DC) that is long compared to the conductor
7、crosssection and uniform along its length. Because many interconnects are dominatedby long runs over unbroken ground planes (to minimize radiation and EMI suscep-tibility), they can be accurately modeled as transmission lines, and much of signalintegrity analysis is based on them.If a short length o
8、f a transmission line is considered, then the lumped approxi-mation applies and the transmission line can be modeled, as shown in Figure 2.1,with series resistance and inductance and with shunt capacitance and conductance.Applying Kirchhoffs voltage law around the loop, thenv(z + z,t) v(z,t) = Ri(z,
9、t) Ldi(z,t)dt.(2.1)Similarly, Kirchhoffs current law applied at z + z yieldsi(z + z,t) i(z,t) = Gv(z + z,t) Cdv(z + z,t)dt.(2.2)Divide through by z and let z 0, then (2.1) and (2.2) transform from differenceequations to the differential equationsi(z + z,t)v(z + z,t)z + zzLRG+v(z,t)i(z,t)CFigure 2.1.
10、 Lumped model of a short length of a transmission line.Section 2.1.Transmission Lines57v(z,t)z= ri(z,t) )i(z,t)t(2.3)andi(z,t)z= gv(z,t) cv(z,t)t,(2.4)where the lumped component values transition to the per-unit-length quantities r,), c, and g due to normalization by z. Simultaneous solution of the
11、transmissionline equations (2.3) and (2.4) yields the voltage and current at any point on thetransmission line.2.1.1Time-Domain SolutionTransmission lines for digital signaling often have low losses. For example, the effectof half an Ohm of loss on a 50 transmission line driven with a 50 driver is n
12、egligi-ble for most applications. To facilitate investigations of the effects of various systemimperfections on signal integrity, losses will be neglected. The lossless transmissionline equations are recovered from (2.3) and (2.4) by setting r = g = 0 yieldingv(z,t)z= )i(z,t)t(2.5)andi(z,t)z= cv(z,t
13、)t.(2.6)An important property of the lossless transmission line is that pulses propagateundistorted along the length of the line. Consider an arbitrary waveform such as theone in Figure 2.2, where the wave shape is described by the function f() with anindependent variable. Since there are no losses
14、and no frequency dependence to ) orc, the waveform must move down the line unchanged in shape (see section 2.1.3 fora proof) and can be described mathematically by f(z,t) = f() = f(z t), where = z t. A maximum or minimum of the waveform occurs when /t = 0, sot= 0 =dzdt ,58Signal IntegrityChapter 2f(
15、 )Figure 2.2. Arbitrary waveform for propagation down a transmission line.or =dzdt,indicating that the maximum or minimum point is moving in the +z direction withvelocity .The partial derivatives in (2.5) and (2.6) can be rewritten in terms of by notingthatz=z=(z t)z=andt=t= ,thenv()= )i()(2.7)andi(
16、)= cv().(2.8)Eliminatingi()between (2.7) and (2.8) and cancellingv()yields =1)c,(2.9)Section 2.1.Transmission Lines59so the velocity of an arbitrary pulse can be directly computed from the per-unit-length inductance and capacitance of the transmission line. Integrating (2.7) withrespect to while ass
17、uming that no static charge is on the transmission line (sothat the integration constant vanishes), thenv() =?)ci(),where (2.9) is used. Therefore, the voltages and currents of an arbitrary waveformon a lossless transmission line are in phase and related by the impedanceZo=?)c,called the characteris
18、tic impedance of the transmission line.For a transmission line of length d, the time for a wave to travel the length ofthe line is called the delay or the time of flight (TOF) and can be computed asTOF = d)c.(2.10)The lossless transmission line is completely specified by its characteristic impedance
19、and delay. Note that) = ZoTOFd(2.11)andc =1ZoTOFd.(2.12)The analyses can be repeated with = z +t with identical results, except thatthe waveform travels in the z direction with velocity = 1/)c and the voltageand current are related byv() = Zoi().Effective Dielectric ConstantSince the TOF can be foun
20、d given the length of a transmission line and the velocityof a wave on it, the velocity is often the unknown parameter that must be found.60Signal IntegrityChapter 2In a transmission line where the electric and magnetic fields are completely encasedin a dielectric with dielectric constant -r, then t
21、he velocity is =co-r,(2.13)where co= 3108m/s is the velocity of a wave in a vacuum (-r= 1) which is alsocalled the free-space speed of light. For transmission lines such as stripline, dualstripline, embedded microstrip, and coax, the velocity is easily computed once thedielectric constant of the fil
22、ler material is known.When the electric and magnetic fields run through two dielectrics, the wave stillpropagates with some velocity. Generalizing (2.13) yields =co-eff,where -effis an effective dielectric constant. For a transmission line like microstrip,the two dielectrics are air with -r= 1 and t
23、he substrate. The effective dielectricconstant must lie between these two, and since most of the field is below the stripin the substrate, the effective dielectric constant must be closer to the dielectricconstant of the substrate than to that of air.Effective dielectric constants are convenient bec
24、ause they offer a handy shortcutin many situations and can be easily estimated for approximate calculations. Forexample, if -r= 4 for the substrate in microstrip, then -eff 3.For the lossless case, the formulas in (2.11) and (2.12) are easily modified forthe effective dielectric constant to be) =Zo-
25、effco(2.14)andc =-effZoco.(2.15)2.1.2Directional IndependenceThe analysis of lossless transmission lines can be carried further to show that twowaveforms traveling in opposite directions do not interact. Let v+denote a voltageSection 2.1.Transmission Lines61waveform launched in the +z direction, whi
26、le vindicates one traveling in the zdirection. Due to the linearityof Maxwellsequations, and assuming linear materials,the total voltage must be the superposition of these two, sov(z,t) = v+(z t) + v(z + t).(2.16)The total current is theni(z,t) =1Zov+(z t)1Zov(z + t).(2.17)Substituting these express
27、ions into (2.5) and (2.6) yieldsv+z+vz= )Zov+t+)Zovtandv+zvz= cZov+t cZovt.Adding these two results inv+z= 12?)Zo+ cZo?v+t+12?)Zo cZo?vt,while subtracting providesvz= 12?)Zo cZo?v+t+12?)Zo+ cZo?vt.These can be further simplified by noting that?Zo+ cZo?= 2/ while?Zo cZo?= 0, thenv+z= 1v+tandvz= +1vt.
28、Note that these two equations are decoupled, with each equation a function only ofv+or v; therefore, the two waves cannot interact and travel along the transmissionline without influencing each other. The waves interact only at boundaries betweentransmission lines with different impedances or betwee
29、n a transmission line and acomponent. This property is exploited to facilitate analyses and to construct bouncediagrams.62Signal IntegrityChapter 22.1.3Frequency-Domain SolutionWhen losses are significant on transmission lines, the frequency domain is conve-nient for analytic solutions. After Fourie
30、r 4 transformation with (1.4), the lossytransmission line equations in (2.3) and (2.4) becomeV (z,)z= (r + )I(z,)(2.18)andI(z,)z= (g + c)V (z,),(2.19)where V and I are the Fourier transforms of v and i. Eliminating I from (2.18) and(2.19) provides the second-order differential equation2Vz2 (r + )(g
31、+ c)V = 0.(2.20)Define 2= (r + )(g + c), then() =?(r + )(g + c),(2.21)and the solution to (2.20) isV (z,) = A()e()z+ B()e()z.(2.22)Note in particular that A, B, and are functions of . is called the complexpropagation constant.The complex propagation constant can always be written in terms of its rea
32、land imaginary parts as() = () + ();then (2.22) becomesV (z,) = A()e()ze()z+ B()e()ze()z.Section 2.1.Transmission Lines63The time-domain voltage can then be found using the inverse Fourier transformgiven in chapter 1 by (1.5) to obtainf(t) =12?A()e()ze()z+ B()e()ze()z?etd=12?A()e()ze()zt)+ B()e()ze(
33、)z+t)?d,(2.23)where it can be seen in the second form that the total voltage consists of the super-position of many complex exponentials. Each A()e()ze()zt)representsa weighted sinusoidal wave traveling in the +z direction that is attenuated expo-nentially with distance. A fixed point on the exponen
34、tial can be identified when()z t = constant. Solving for z and taking the derivative with respect to timeyields the phase velocityp=dzdt=().(2.24)Since is a function of frequency, then the phase velocity is a function of frequency.In a similar fashion, each B()e()ze()z+t)represents a weighted and at
35、tenu-ated wave traveling in the z direction. Due to their roles in wave propagation, is called the attenuation constant and is called the propagation constant.The phase velocity and attenuation of the sinusoidal waves are different at everyfrequency, so the shape of the time-domain waveform must cha
36、nge as it moves downthe line. Attenuation is stronger for higher-frequency components, so the waveformtends to spread, or disperse, with distance as low-frequency components take over.For this reason, the effects of losses on wave shape is called dispersion. Sometimesdispersion is attributed to the
37、source of the losses, such as conductor loss dispersionor dielectric loss dispersion.Because each frequency component has a different phase velocity, the wave ve-locity is not equal to the phase velocity of any given component. For lossy lines,64Signal IntegrityChapter 2the wave velocity is found by
38、 finding the TOF of a pulse between two points anddividing into the distance traveled.1The characteristic impedance can be found by solving for I in (2.18) with thegeneral voltage solution in (2.22) to obtainI(z,) =?g + cr + )?A()e()z B()e()z?.(2.25)The characteristic impedance for a lossy transmiss
39、ion line is thenZo() =?r + )g + c,(2.26)and it is, in general, frequency-dependent.The directional components of voltage and current are clearly apparent with achange of notation in (2.22) toV (z,) = V+()e()z+ V()e+()z,(2.27)then the current from (2.25) isI(z,) =1Zo()?V+()e()z V()e+()z?.(2.28)These
40、two expressions are often a good starting point in solving circuit problemsinvolving transmission lines.Low-Loss Transmission LinesWhen losses on a transmission line are small, additional mathematical manipula-tion can yield good insight into waveform propagation on transmission lines. Thecomplex pr
41、opagation constant can be rearranged as =?(r + )(g + c)=?)(1 +r)c(1 +gc)= )c?1 +r)?1 +gc.(2.29)1. For narrow-band signals, the group velocity can be computed from the frequency dependenceof the phase velocity to find the velocity of a wave packet. Digital signals are baseband, so thegroup velocity i
42、s not applicable since the narrow-band assumption does not apply.Section 2.1.Transmission Lines65For low losses, the approximations?1 +r?1 +12r?,r? 1?1 +gc1 +12gc,gc? 1(2.30)can be applied to (2.29) to obtain )c?1 +12r)+12gc14rg2)c? )c +12?c)r +?)cg?,(2.31)where the term14rg2?cis dropped as a neglig
43、ible second-order small term. The at-tenuation and propagation constants for low losses are then given by =12?c)r +?)cg?(2.32)and = )c,(2.33)and the phase velocity isp=1)c.Note that the attenuation constant and the phase velocity are frequency-independ-ent, so all frequency components of the wavefor
44、m travel together with uniformattenuation. Therefore, the waveform propagates down the transmission line withno change in wave shape except for reduction in amplitude. In this case, the wavevelocity does equal the phase velocity. This propagation is dispersionless.Substituting the attenuation and pr
45、opagation constants in (2.32) and (2.33) forthe low-loss line into (2.23) yieldsf(t) =12ez?A()e?czetd+12e+z?B()e+?czetd= eza(t )cz) + e+zb(t +)cz),(2.34)66Signal IntegrityChapter 2where a(t) = F1A() and b(t) = F1B(). The time-shifting theorem forFourier transforms,Ff(t ) = Ff(t)e,is also utilized. T
46、he result in (2.34) generalizes to low-loss lines the directionalindependence established in section 2.1.2 for lossless lines. In addition, the resultshows that waveforms travel unchanged except for amplitude on low-loss lines.Conductor Loss-Dominated Transmission LinesLosses are often dominated by
47、conductors, so while r is often significant, g is oftennot. Setting g = 0 in the lossy expressions for Zoand yieldsZo=?r + )c(2.35)and =?(r + )c.(2.36)At low frequencies, r ? ), soZo?rce/4and rc e/4.For both Zoand , the real and imaginary parts are equal (ignoring the sign)with a square root depende
48、ncy on frequency. These results are very different fromthose obtained from the lossless transmission line model, yet the lossless model isoften used in signal integrity work where baseband digital signals include significantlow-frequency content.An approximate metric is available for determining if
49、lossless transmission linemodeling is appropriate. At low frequencies, the lumped approximation applies,Section 2.1.Transmission Lines67and the lossy transmission line is simply a loop of wire that can be modeled as aseries RLC circuit. For losses to be negligible, the RLC circuit must be stronglyun
50、derdamped, a condition that occurs whenR ? 2?LC.Using per-unit-length quantities for a transmission line with length d, thend ?2r?)c.(2.37)If the line length is sufficiently short, then lossless transmission line modeling canbe appropriate.To demonstrate the line length dependence, consider a 50 tra
