High Speed Signal Integrity Analysis and Differential Signaling

Project Report of EE201C: Modeling of VLSI Circuits and Systems

Tongtong Yu, UID: 904025158 Zhiyuan Shen, UID: 803987916

Abstract— This course project is designated for independent electric mechanism of semiconductor circuit doesn’t change study of various topics relating to VLSI modeling. In our very much, so the concentration is on the interconnects and project, the first part is an extension of the class presentation packaging. where we introduced the basic idea of signal integrity. Here we address the philosophy behind frequency domain and time The impact of to the system can be classified to signal domain. Then, we give two examples of interconnects, a critical integrity, which refers to the contamination of signal’s in- issue related to signal integrity. The second part is about tegrity due to parasitic effects of interconnects and packaging, differential signaling used in high speed application. After compared with ideal case. This may result in unsatisfactory analyzing general principles, we simulate two parallel-trace performance. For example, we design the systems for 10Gbps; transmission line, which has practical meaning in real design case. Maybe this project is not fancy, but we believe all the however, due to this effect, it can only achieve 5Gbps. Also, fundamentals provided here and those reference list in the end this may produce inaccurate logic state and even logical are the best material for everyone who wants to go further in failure. This impact can be classified in two ways this area. 1) Delay, distortion and cross-talk directly happening on We would like to express our deepest gratitude to Prof. He, for the transmission channel; his endless endeavor in and out of class. From him, we learned 2) Noise caused by grounding wire, pin connection, power how to be an independent researcher. This experience will be supply, which can be called simultaneous switching precious for our future academic life. noise (SSN). I.INTRODUCTION Traditionally, microwave circuit and high speed circuits are belonging to different subjects. But they are can be connected Recently, high speed and three dimension (3D) are the main together in the sense of “fast varying”. In microwave domain, trends of future very large scale integrated (VLSI) circuits. we study fast varying sine wave, while in high speed circuit, One feature is that the clock frequency in CMOS chip can we study fast varying pulse wave. The wave effect become reach several GHz, which has reached microwave frequency obvious when the physical size is comparable to wave length bands, and the corresponding pulses signals’ frequencies can in microwave domain; correspondingly, when the physical approach to even higher frequencies. As known to us, level size is becoming comparable to the rising time or pulse of integration has been increased due to deep sub-micro duration in digital circuits, it must be studied specifically due technology, which made system-on-chip (SoC) become possi- to wave effect. Usually, microwave circuits are analyzed in ble. Also, to realize complex systems, along with application frequency domain, while high speed circuit is analyzed in specific integrated circuits(ASIC), printed circuit board (PCB) time domain. But as the wave effect is obvious, the traditional method is still used widely. No matter what the method circuit theory are limited. For example, some elements will be mentioned above is used for our system, when the operating frequency or size dependent, which will make time domain frequency increase, all high speed systems has similar prop- solution complicated and inaccurate. In that case, we have to erties. As deep sub-micro technology develops, the size of use electromagnetic simulator, such as momentum simulator semiconductor devices and basic logic gates is very tiny, so provided by Agilent ADS to include the field effect to get an their parasitic effects is not obvious as we can still use general accurate result. circuit theory. However, when we consider those interconnects which connect different ICs within chip or within one PCB, their length, complexity and density will cause severe parasitic II.GENERAL APPROACHFOR SIGNAL INTEGRITY ISSUE effects. In the mean time, package, which are used to protect and brace the inner circuits, will cause these effects too. For As we have seen in class, Pade approximation is based on example, bonding wires, ground wires and vias can be problem moment matching. First, it expand both the original network for high speed signal propagation. As the working frequency function and Pade approximation as power series around some increase, these effects can encroach from PCB to inter-chip point in s domain. Then, equaling the coefficient with same connection and even interconnects within chips. power factor in these two series’ can determine the coefficients As a result, the high speed theory, is to investigate the elec- in both numerator and denominator. However, there is some trical property of the whole system, which consists of systems situation where moment matching theory cannot be used. interconnects, packaging and connector with semiconductor According to Appel’s theorem in complex power series theory, devices, under the action of high speed pulses. Since the power series are convergent only in the range within their convergence radius and in this range the function must be an- voltage and current along the transmission line, then according alytical. But actually, both original network function and Pade to transmission line theory, we have approximation cannot be analytical within whole frequency U = U + U (2) UWB signal in range (they both have poles), so the range of convergent radius f r is small. So, by performing Pade approximation, the final result Uf Ur I = If + Ir = − (3) may be not very accurate. Z0 Z0 For systematic description of a network’s property, we use where Z0 is the characteristic impedance. For lossless line, Synchronizedtwo types Acquisition of electric parameter. /FPGA Z0 is pure resistive. Uf and If are the voltage and current 1) The first electric parameters are voltage and current. incident wave; Ur and Ir are the voltage and current reflected The corresponding circuit’s network parameters are wave. For simplicity, we can take the normalized value, impedance, admittance and their matrix; U 2) The second electric parameters are incident wave and a = √ f (4) reflected wave. The corresponding circuit’s network pa- Z0 rameters are reflection coefficients and scattering matrix. U b = √ r (5) We have used the first type parameter many times as in Z0 Low-pass filter basic circuits analysis. Here we give several advantages of Then, we have the second type parameter. ThresholdSignal out p Decision U = Z0(a + b) (6) s(t)+n(t) 1) It is easy to show the powerSyncrhonization transmission and matching Verify 1 Wideband 2 condition in different part within the whole system; I = √ (a − b) (7) Threshold T Threshold Z amplifier ( ) 2)Decision For arbitrary circuity,()dt thereDecision are the scattering parameters 0 0 associated with them. They are more generalized for Also, a and b can be expressed in terms of U and I. network properties; Strobe Shift Local Phase 1 U p Pulses 3)Register When the frequencyPN codes isControl higher, the scattering parameters a = (√ + Z I) (8) UWB signal in 2 Z 0 are more easily to measure, especially for active devices; 0 1 U Time position 4)Algorithm For lossless network, the scattering matrix is unitary, p b = (√ − Z0I) (9) Control Control which makes it easy to obtain the relation between the 2 Z0 driving point impedance and the transmission parame- As mentioned before, a critical issue is on and off chip ters, as to make it easy to synthesize. interconnects, which can be treated as transmission line. Synchronized Acquisition a I /FPGA + III. ONEEXAMPLEUSINGCHARACTERISTICMETHODIN b Lossless U Linear TIME DOMAIN ANALYSIS Time invariant - Z0 i(0,t) i(l,t) Z0

Fig. 1. Voltage, current, incident and reflected wave of one port + + + + network. u(0,t) er(l,t- ) ei(0,t- ) u(l,t) The second type parameter is no straight forward relating to - - - - the circuit elements as the first type, which is the drawback. So, as we will see later, these two types parameters are both Fig. 2. Left and right equivalent circuits for characteristic method used in analysis. Next, we will understand all of these through in time domain. an example. As shown in Fig. 1, the voltage and current of a lossless, Fig. 2 is the equivalent circuit representation for character- linear, time-invariant single port network are U and I. Take a istic method [1], which is shown on page 16 of the slides. We reference Z0 with unit of Ohms and get the following linear show an example to illustrate this method. As shown in Fig. transformation of U and I, 3, for a transmission line with impedance Z0, time delay , ⎡ √ ⎤ 1 Z0 assume the signal resistance and load resistance are R1 and √ a ⎢ 2 Z 2 ⎥ U R2, respectively. The driving voltage is e(t). For t < 0, the = ⎢ 0 √ ⎥ (1) b ⎣ 1 Z0 ⎦ I value of e is zero. Also, assume all the initial conditions are √ − zero. We can use the characteristic method to calculate voltage 2 Z0 2 and current along the line in time domain. Then, a and b are normalized incident wave and reflected At t=0, according to causality, we have wave. This transformation can also be interpreted from the perspective of transmission line. Assume U and I are the ei(0, −) = 0, er(l, −) = 0 (10)

2 a I + Lossless U b Linear - Time invariant UWB signal in

Synchronized Acquisition /FPGA

Z0 i(0,t) i(l,t) Z0 + + + + u(0,t) er(l,t- ) ei(0,t- ) u(l,t) - - - UWB signal in -

R1  are mixed lumped elements, such as L,C,R at the intersection. The method is to treat each continuous segment separately e(t) Z 0 R2 using the same equations above. At intersection, we can add Synchronizedthe voltage or Acquisition current continuity equation or the current and voltage/FPGA relation of the lumped elements. Combining all these Fig. 3. Transmission line terminated with source impedance and equation can give the ﬁnal time response of the whole system. load resistance. Further, the characteristic method can be used for nonlinear termination load. Those equations are basically the same, except that the terminal condition is described by nonlinear Then, the voltage at two terminals are equations, which can solved by iteration method [2]. Z0 i(0,t) i(l,t) Z0 u(l, 0) = −Z0i(l, 0) (11) IV. ONEEXAMPLEUSING S PARAMETER IN FREQUENCY DOMAIN ANALYSIS u(0, 0) = Z0i(0, 0) (12) + + + + As we mentioned before, circuit analysis can be performed u(0,t) e (l,t- ) ei(0,t- ) Next, according to terminal condition, we have in time domain andr frequency domain. Strictly speaking,u(l, at) real signal can only be observed in time domain, which implies u(0, 0) = e(0) − R1i(0, 0) (13) - - - - a the time domain analysis is the essence. While frequency u(l, 0) = R2i(l, 0) I (14) domain is just a integral transform with respect to time t, So, we can solve the voltage and current as follows+ this makesLossless time variables transformed to imaginary frequency variable jw or complex frequency variable s. In jw domain Z0 e(0) b R  u(0, 0) = e(0), i(0, 0) = U (15) and s Lineardomain,1 differential and integration can be replaced R1 + Z0 R1 + Z0 by algebraic operation; convolution becomes multiplication. u(l, 0) = i(l, 0) = 0 - (16) So,Time this is invariant thee(t) philosophyZ behind0 frequencyR2 domain analysis. Next, we will use S parameter (also mentioned in the slide) According to the above results, we can write ei(0, 0) and and frequency domain transform to solve the transmission line er(l, 0) as problem [3]. 2Z0 ei(0, 0) = [u(0, 0) + Z0i(0, 0)] = e(0) (17) R1 + Z0 a1 a2 er(l, 0) = [u(l, 0) − Z0i(l, 0)] = 0 (18) b b 1 S 2 Based on the value we get at t=0, we can calculate the next R0 R0 sample time t= .

2Z0 u(l, ) = −Z0i(l, ) + ei(0, 0) = −Z0i(l, ) + e(0) R1 + Z0 Fig. 4. Representing interconnects by S parameter.

u(0, ) = Z0i(0, ) + er(l, 0) = Z0i(0, ) (19) As shown in Fig. 4, the interconnect transmission line has Now, the terminal condition is changed as length l, characteristic impedance Z0, proportion constant . The reference impedance are both R0 at two terminals. Recall u(0, ) = e() − R1i(0, ), u(l, ) = R2i(l, ) (20) Z0 and has the following form. a Getting these equations together, we can solve it as follows s I j!L + R p + Z0 1 Z = , = (j!L + R)(j!C + G) Lossless u(0, ) = e(), i(0, ) = e() (21) 0 j!C + G R + Z R + Z b 1 0 1 0 U Linear 2Z0R2 Assume a1, a2 are the incident wave and b1, b2 are the u(l, ) = e(0) (22) reflected wave at two terminal. Generally, we have (Z0 + R1)(Z0 + R2) - Time invariant 2Z U1 = a1 + b1,U2 = a2 + b2 (24) i(l, ) = 0 e(0) (23) (Z0 + R1)(Z0 + R2) a1 b1 a2 b2 I1 = − ,I2 = − (25) using this method, we can continue calculating the sampling R0 R0 R0 R0 point after t = . Remember that time step size is ; now if Z − R Γ = Γ(!) = 0 0 (26) we want more precise result in time domain, we can take a Z0 + R0 smaller step size, such as some fraction of . For t = kΔt, Here we use reflection coefficient Γ to denote the difference k=1,2,3,.... These equations can still be used to calculate the between characteristic impedance and reference impedance. time domain response, just by replacing t with kΔt. The constant propagation coefficient is The characteristic method can also be used in the case where multiple transmission lines are concatenating, or there = (!) = e− t (27)

3 UWB signal in

Synchronized Acquisition /FPGA

i(0,t) Z0 i(l,t) Z0 + + + + u(0,t)er(l,t- ) ei(0,t- ) u(l,t) Next, we get the transmission (ABCD) matrix of the segment. substitute them into (33) and (34). The result is the same as we - - - - use characteristic method in time domain. For general lossy ⎡ e t + e− t e t − e− t ⎤ AB Z0 lines, the equations (33) and (34) will contain extra terms = ⎢ 2 2 ⎥ (28) CD ⎣ e t − e− t e t + e− t ⎦ S11(t)∗a1(t) and S22(t)∗a2(t), respectively. The time domain R1  S parameter are the inverse Fourier transform of equation (30) 2Z0 2 and (31) in frequency domain. By fast Fourier transform (FFT) e(t) 0Z R2 The normalized matrix, with respect to R0 is method, we can still get the discrete point in time domain, t − t t − t ⎡ e + e Z0 e − e ⎤ which means we can still get the algebraic equations and solve a b 2 R0 2 them by numerical method. = ⎢ t − t t − t ⎥ (29) c d ⎣ Ra01 e − e e a+2 e ⎦ b1 b2 a a Z0 2 S 2 1 2 R 0 R0  ()t b1 b2  Then, the S parameters are 1 2()t (a + b) − (c + d) S = S = 11 22 a + b + c + d T1(t) T2(t) Z0 − R0 (e t − e− t) = R0 Z0 e t + e− t + 1 ( Z0 + R0 )(e ta − e− t) 2 R0 ZI 0 Fig. 5. Termination condition corresponding to S parameter equa- 1 ( Z0 − R0 )(1+ − e−2 t) Losslesstions. = 2 R0 Z0 b 1 + e−2 t + 1 ( Z0 +U R0 )(1 − e−2 t) Linear 2 R0 Z0 Finally, we can add the terminal condition to complete Z2−R2 Time invariant 1 0 0 (1 − e−2 t- ) discussion of this part. Assume z1(t), z2(t) are the terminal 2 Z0R0 = 2 2 resistance in time domain representation and e1(t), e2(t) are (Z0+R0) − (Z0−R0) e−2 t 2Z0R0 2Z0R0 the voltage source, then, as shown in Fig. 5, we can associate Γ(1 − 2) then terminal condition with S parameter as follows [2], = (30) 1 − Γ2 2 zi(t) − R0 Γi(t) = (36) 2 zi(t) + R0 S12 = S21 = a + b + c + d R0 2 Ti(t) = (37) = zi(t) + R0 e t + e− t + 1 ( Z0 + R0 )(e t − e− t) 2 R0 Z0 where i = 1, 2. Next, we can write the terminal condition as 2e− t = 2 2 a (t) = Γ (t) ∗ b (t) + T (t) ∗ e (t) (38) (Z0+R0) − (Z0−R0) e−2 t 1 1 1 1 1 2Z0R0 2Z0R0 a2(t) = Γ2(t) ∗ b2(t) + T2(t) ∗ e2(t) (39) 4Z0R0 − t Z0−R0 2 − t 2 e [1 − ] e = (Z0+R0) = Z0+R0 1 − Γ2 2 1 − Γ2 2 (1 − Γ2) = 2 2 (31) 1 − Γ V. BASICPRINCIPLESANDSIMULATIONOF DIFFERENTIAL

q L SIGNALING In special case, where the line is lossless, then Z0 = is C A. Advantages of Differential Signaling pure resistive. For R0 = Z0, we have √ A differential pair is a simply a pair of transmission line Γ = 0, = j = j! LC, = e−j t = e−j! with some coupling between them. Differential signaling is −j! S11 = S22 = 0,S12 = S21 = = e the use of two output drivers to drive two independent trans- So, we can convert the following equations in frequency mission lines, one carrying one bit and the other carrying domain its complement. The difference signal carries the information. Fig. 6 is an example of using low voltage differential signal b1 = S11a1 + S12a2 (32) (LVDS) in both transmitter and receiver, where current mode

b2 = S21a1 + S22a2 (33) logic (CML) is used in transmitter to provide high data rate[4]. Differential signaling has the following main advantages, to time domain in lossless condition and get the following, compared with single-ended version, −j! b1(t) = S12(t) ∗ a2(t) = e ∗ a2(t) = a2(t − ) (34) 1) The total dI/dt noise is reduced in differential signaling, so there is less rail collapse and potentially less Electro b (t) = S (t) ∗ a (t) = e−j! ∗ a (t) = a (t − ) (35) 2 21 1 1 1 Magnetic Interference(EMI); According to (24) and (25), we can express a1(t), a2(t), 2) The differential ampliﬁer at the receiver can have higher b1(t) and b2(t) in terms of u1(t), u2(t), i1(t) and i2(t) and gain then signal-ended version;

4 UWB signal in

2) If any of the changing common signal makes it out on a twisted-pair cable, it has the potential of causing Synchronized Acquisition excessive EMI. /FPGA B. Impedance Viewpoint As we have mentioned in class, the impedance is very Z 0 i(0,t) i(l,t) Z 0 import for determining signal and power reﬂection and hence + + + + signal integrity. Next, we will consider ideal case and non- u(0,t) e r(l,t- ) ei(0,t- ) u(l,t) ideal case for micro-strip lines. - - - - 1) No Coupling Condition: By the deﬁnition of impedance, Fig. 6. LVDS transmitter(driver) and receiver. the impedance of differential signal is

R  Vdiff Vone 1 Z = = 2 × = 2Z (41) diff I I 0 3) Thee(t) propagationZ 0 of aR 2 differential signal over a tightly one one coupled differential pair is more robust to cross talk and which means the differential impedance is twice the charac- discontinuities in the returning path ; teristic impedance of either line. 2) The Impact of Coupling: As we have analyzed in class, There area 1 several drawbacks ina differential2 signals, too. First, b b 1 S 2 there are capacitances anda 1 inductancesa 2 associated with them. the common0R mode signal willR 0 produce additional EMI. Sec- As shown in Fig. 8, C b 1 andb 2 C are the self conductance ond, it need twice signal traces compared with signal-ended 1()t 11 22 2()t per length, C describes the mutual conductance per length. version. Although there are some drawbacks, yet they can be 12 Similarly, we can define those corresponding inductances. As overcome in careful design. Overall speaking, there will still T 1(t) T 2(t) be more and more systems using differential signal. Next, we C 12 will see an popular scheme of LVDS [5].a 1 I 2 + Lossless C11 C 22 U b Linear - Time invariant Fig. 8. Coupling capacitance modeling.

these two traces become near each other, both C11 and C12 will change. C11 will decrease as some of the fringe ﬁelds between signal trace 1 and its return path are intercepted by the adjacent trace; C12 will increase according to simple capacitance equation. So, we can see the sum of C11 and C12, deﬁned as CL, meaning loaded capacitance, will not change very much. For the case where two traces are far apart, the characteristic impedance of each line is independent of the other line. So, 1 Z0 ∝ (42) Fig. 7. The common and differential signal components in LVDS. C11

For the case where two lines are near each other, the

In Fig. 7, two signal traces are used to represent single bit presence of the other line will affect the impedance of line information. Each has a swing from 1.125 V to 1.375 V. Here 1, which is called proximity effect. Simply consider the case, are the basic principles where the voltage on the second line is zero, the coupling capacitance will be acted as the load of the first line. We can Vdiff (t) = V1(t) − V2(t) 1 add these effects up by superposition. In sum, we can find the Vcomm(t) = (V1(t) + V2(t)) (40) following relationship. 2 1 1 where Vdiff (t) is the differential mode signal, Vcomm(t) is the Z ∝ = (43) 0 C + C C common mode signal, V1(t) and V2(t) are the signals on line 11 12 L 1 and line 2 with respect to the common return path. Here are Finally, we will see several methods for calculating differential two important observations here impedance to account for the distance between nearby traces. 1) If the value of the common signal gets too high, it may There are five different ways for analysis, saturate the input amplifier of the differential receiver 1) Use the direct results from an approximation; and prevent it from accurately reading the differential 2) Use the direct results from a field solver; signal; 3) Use an analysis based on modes;

5 4) Use an analysis based on the capacitance and inductance matrix; 5) Use an analysis based on the characteristic impedance matrix. We have seen the first and fourth method in class. One reasonable close approximation, which is based on empirical fitting of measured data, for edge-coupled micro-strip line Fig. 10. Electric-field distribution for odd and even modes in micro- using FR4 material is shown below [5], strip lines. s Zdiff = 2 × Z0[1 − 0.48exp(−0.96 )] (44) ℎ Fig. 11 shows the result got by Ansoft Maxwell 2D sim- where s is the edge-to-edge separation between the traces in mils and ℎ is the dielectric thickness between the signal trace ulator when the frequency is 50MHz. We can see the skin and the returning plane. effect clearly in the signal’s trace; in the returning plane, the currents are most concentrated parallel to the signal’s trace C. Even-odd Model Analysis for Non-distortion Transmission and the total current width is about four times of the signal’s Next, we will see two special cases for signal transmission, trace, which is a feature for design. even mode and odd mode. For two same lines, assume we add a step input from 0V to 1V on line 1 and keep 0V constant on line 2 (shown in Fig. 9), then we will find as we move along the line, the actual signal will change due to cross talk between line 1 and line 2. Noise will be accumulated on line 2, which will decrease the signal quality on line 1.

Fig. 11. Current distribution in the cross section of a micro-strip

line.

Next, we use Agilent Advanced Design System (ADS) to simulate a pair of parallel transmission lines on the FR4 board.

The speciﬁed parameters of this board are as follows, thickness Fig. 9. Voltage pattern with only one step input at port 1. = 0.72mm, the real part of permittivity is 4.5, the loss tangent is 0.02. We use Momentum simulator provided in ADS, which

It is found that there are only two conditions where signals uses Green functions to account for wave effects, hence more can propagate undistorted. The ﬁrst case is when exactly the accurate than lumped circuit simulator. same signal is applied to each line; for example, the voltage goes from 0V to 1V in each line. This is also called even mode excitation, which is similar as the common mode. The second case is when the opposite-transitioning signals are applied to each line; for example, one of the signals goes from 0V to 1V and the other goes from 0V to -1V. This is also called odd mode excitation, similar as differential mode, except numerically, we have V = 2V . diﬀ odd

D. Electromagnetic Issue and Simulation Fig. 12. Two parallel micro-strip lines for even and odd mode As we mentioned earlier, circuit simulator cannot give excitation simulation. you very accurate because these parasitic effect is getting complicated for high frequencies. As shown in Fig. 10, the Fig. 12 is the actual two parallel micro-strip line imple- result is obtained by Mentor Graphics Hyperlynx[5]. mented on a substrate. The width is 3.2mm and the thickness is

6 0.04mm, which provide 50 Ω as the characteristic impedance. All four ports are terminated with 50 Ω.

Fig. 15. Current distribution with even mode excitation at 37MHz.

Fig. 13. Current distribution when there is one excitation with 2GHz at port 1.

Fig. 13 shows the current distribution when there is only one source with 2GHz frequency excitation at port 1. There are two important observations here. First, we can see the current density distribution (according to color and arrows) along one line is varying, which shows the wave effect at high frequencies. Second, there are some small currents on the edge of line 2 with the same direction as on line 1, which accounts for the coupling effect.

Fig. 16. Current distribution with odd mode excitation at 37MHz.

Fig. 14. Current distribution when there is one excitation with

37MHz at port 1. Fig. 17. Far ﬁeld radiation pattern for even mode excitation. Fig. 14 shows the current distribution when there is one source with 37MHz frequency source at port 1. Now, as contrast to Fig. 13, here we can see the current is relative constant along line 1. Also, there is rarely coupling current on line 2, which means coupling effect is very small for low frequencies. Fig. 15 shows the current distribution when we add even mode excitation at port 1 and port 2 with 37MHz frequency. Now, we can see the current is smaller in the near edges than that in the far-apart edges

Fig. 16 shows the current distribution when we add odd mode excitation at port 1 and port 2 with 37MHz frequency.

As contrast to Fig. 15, we can see the currents is more dense Fig. 18. Far ﬁeld radiation pattern for odd mode excitation. in the far-apart edges.

7 Beyond that, we can use far field analysis to compare the REFERENCES case with even and odd mode excitation. Fig. 17 shows the [1] F. Branin, “Transient Analysis of Lossless Transmission Lines,” Proceed- electric field radiation pattern for even mode excitation. Fig. ings of the IEEE, vol. 55, no. 11, pp. 2012–2013, Nov. 1967. 18 is the case for odd mode excitation. The quantities got in [2] Z. F. Li and J. F. Mao, “Microwave and High Speed Circuits Theory,” Shanghai Jiao Tong University, Oct. 2004. Momentum analysis are summarized as follows. [3] J. E. Schutt-Aine, T. K. Sarker, and R. F. Harrington, “Analysis of Lossy 1) For 37 MHz frequency excitation, the total radiation Transmission Line with Arbitrary Non-linear Terminal Networks,” IEEE −19 −12 Trans. on MTT, p. 36, 1986. power is 5.94×10 W for odd mode; 1.006×10 W [4] N. Semiconductor, “LVDS owner’s manual,” Including High-Speed CML for even mode; and Signal Conditioning, 4th ed, p. 9, 2008. 2) For 2 GHz frequency excitation, the total radiation [5] E. Bogatin, “Signal Integrity, Simplified,” Prentice Hall, Sept. 2003. power is 2.311×10−8W for odd mode; 1.178×10−5W for even mode ; These results reveal several general principles. First, even mode radiates more power than odd mode. Second, as the frequency goes up, the total radiation power increases. These observations are important, especially for multiple layers in today’s ICs, where EMI is becoming a critical issue in high frequencies.

VI.CONCLUSION In this report, we ﬁrst use time domain method and frequency domain method to get the response of a traditional transmission line. Then, we introduce the idea of differential signaling, which is quite often used in high speed system. Finally, we perform a series of simulation on an actual transmission line used for differential signals, where we can see for high speed application, things are much more complicated, so we need to consider much more than pure circuit analysis. We have to go deeply into the fundamentals, originated from Maxwell’s equations for coupling and radiation. Instead of collecting other’s report and paper, deeply thinking of the fundamentals and essence is the best lesson we get from this project.