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 independentstudy of various topics relating to VLSI modeling. In ourproject, the first part is an extension of the class presentationwhere we introduced the basic idea of signal integrity. Herewe address the philosophy behind frequency domain and timedomain. Then, we give two examples of interconnects, a criticalissue related to signal integrity. The second part is aboutdifferential signaling used in high speed application. Afteranalyzing general principles, we simulate two parallel-tracetransmission line, which has practical meaning in real designcase. Maybe this project is not fancy, but we believe all thefundamentals provided here and those reference list in the endare the best material for everyone who wants to go further inthis area.

We would like to express our deepest gratitude to Prof. He, forhis endless endeavor in and out of class. From him, we learnedhow to be an independent researcher. This experience will beprecious for our future academic life.

I. INTRODUCTION

Recently, high speed and three dimension (3D) are the maintrends of future very large scale integrated (VLSI) circuits.One feature is that the clock frequency in CMOS chip canreach several GHz, which has reached microwave frequencybands, and the corresponding pulses signals’ frequencies canapproach to even higher frequencies. As known to us, levelof integration has been increased due to deep sub-microtechnology, which made system-on-chip (SoC) become possi-ble. Also, to realize complex systems, along with applicationspecific integrated circuits(ASIC), printed circuit board (PCB)method is still used widely. No matter what the methodmentioned above is used for our system, when the operatingfrequency increase, all high speed systems has similar prop-erties. As deep sub-micro technology develops, the size ofsemiconductor devices and basic logic gates is very tiny, sotheir parasitic effects is not obvious as we can still use generalcircuit theory. However, when we consider those interconnectswhich connect different ICs within chip or within one PCB,their length, complexity and density will cause severe parasiticeffects. In the mean time, package, which are used to protectand brace the inner circuits, will cause these effects too. Forexample, bonding wires, ground wires and vias can be problemfor high speed signal propagation. As the working frequencyincrease, these effects can encroach from PCB to inter-chipconnection and even interconnects within chips.

As a result, the high speed theory, is to investigate the elec-trical property of the whole system, which consists of systemsinterconnects, packaging and connector with semiconductordevices, under the action of high speed pulses. Since the

electric mechanism of semiconductor circuit doesn’t changevery much, so the concentration is on the interconnects andpackaging.

The impact of to the system can be classified to signalintegrity, which refers to the contamination of signal’s in-tegrity due to parasitic effects of interconnects and packaging,compared with ideal case. This may result in unsatisfactoryperformance. For example, we design the systems for 10Gbps;however, due to this effect, it can only achieve 5Gbps. Also,this may produce inaccurate logic state and even logicalfailure. This impact can be classified in two ways

1) Delay, distortion and cross-talk directly happening onthe transmission channel;

2) Noise caused by grounding wire, pin connection, powersupply, which can be called simultaneous switchingnoise (SSN).

Traditionally, microwave circuit and high speed circuits arebelonging to different subjects. But they are can be connectedtogether in the sense of “fast varying”. In microwave domain,we study fast varying sine wave, while in high speed circuit,we study fast varying pulse wave. The wave effect becomeobvious when the physical size is comparable to wave lengthin microwave domain; correspondingly, when the physicalsize is becoming comparable to the rising time or pulseduration in digital circuits, it must be studied specifically dueto wave effect. Usually, microwave circuits are analyzed infrequency domain, while high speed circuit is analyzed intime domain. But as the wave effect is obvious, the traditionalcircuit theory are limited. For example, some elements will befrequency or size dependent, which will make time domainsolution complicated and inaccurate. In that case, we have touse electromagnetic simulator, such as momentum simulatorprovided by Agilent ADS to include the field effect to get anaccurate result.

II. GENERAL APPROACH FOR SIGNAL INTEGRITY ISSUE

As we have seen in class, Pade approximation is based onmoment matching. First, it expand both the original networkfunction and Pade approximation as power series around somepoint in s domain. Then, equaling the coefficient with samepower factor in these two series’ can determine the coefficientsin both numerator and denominator. However, there is somesituation where moment matching theory cannot be used.According to Appel’s theorem in complex power series theory,power series are convergent only in the range within their

convergence radius and in this range the function must be an-alytical. But actually, both original network function and Padeapproximation cannot be analytical within whole frequencyrange (they both have poles), so the range of convergent radiusis small. So, by performing Pade approximation, the final resultmay be not very accurate.

For systematic description of a network’s property, we usetwo types of electric parameter.

1) The first electric parameters are voltage and current.The corresponding circuit’s network parameters areimpedance, admittance and their matrix;

2) The second electric parameters are incident wave andreflected wave. The corresponding circuit’s network pa-rameters are reflection coefficients and scattering matrix.

We have used the first type parameter many times as inbasic circuits analysis. Here we give several advantages ofthe second type parameter.

1) It is easy to show the power transmission and matchingcondition in different part within the whole system;

2) For arbitrary circuity, there are the scattering parametersassociated with them. They are more generalized fornetwork properties;

3) When the frequency is higher, the scattering parametersare more easily to measure, especially for active devices;

4) For lossless network, the scattering matrix is unitary,which makes it easy to obtain the relation between thedriving point impedance and the transmission parame-ters, as to make it easy to synthesize.

Wideband amplifier ( )2 Threshold

Decision

Strobe Pulses

Time positionControl

Shift Register

AlgorithmControl

Low-pass filter

LocalPN codes

PhaseControl

ThresholdDecision

ThresholdDecision

Signal out

Syncrhonization Verify

s(t)+n(t)

0()Tdt

Synchronized Acquisition /FPGA

UWB signal in

LosslessLinear

Time invariant

a

bU

I+

-

Fig. 1. Voltage, current, incident and reflected wave of one portnetwork.

The second type parameter is no straight forward relating tothe circuit elements as the first type, which is the drawback.So, as we will see later, these two types parameters are bothused in analysis. Next, we will understand all of these throughan example.

As shown in Fig. 1, the voltage and current of a lossless,linear, time-invariant single port network are U and I . Take areference Z0 with unit of Ohms and get the following lineartransformation of U and I ,

[ab

]=

⎡⎢⎢⎣1

2√Z0

√Z0

21

2√Z0

−√Z0

2

⎤⎥⎥⎦[ UI

](1)

Then, a and b are normalized incident wave and reflectedwave. This transformation can also be interpreted from theperspective of transmission line. Assume U and I are the

voltage and current along the transmission line, then accordingto transmission line theory, we have

U = Uf + Ur (2)

I = If + Ir =UfZ0− UrZ0

(3)

where Z0 is the characteristic impedance. For lossless line,Z0 is pure resistive. Uf and If are the voltage and currentincident wave; Ur and Ir are the voltage and current reflectedwave. For simplicity, we can take the normalized value,

a =Uf√Z0

(4)

b =Ur√Z0

(5)

Then, we haveU =

√Z0(a+ b) (6)

I =1√Z0

(a− b) (7)

Also, a and b can be expressed in terms of U and I .

a =1

2(U√Z0

+√Z0I) (8)

b =1

2(U√Z0

−√Z0I) (9)

As mentioned before, a critical issue is on and off chipinterconnects, which can be treated as transmission line.

III. ONE EXAMPLE USING CHARACTERISTIC METHOD INTIME DOMAIN ANALYSIS

Synchronized Acquisition /FPGA

UWB signal in

LosslessLinear

Time invariant

a

bU

I+

-

Z0Z0

u(l,t)++

--u(0,t)

++

--er(l,t- ) ei(0,t- )

i(0,t) i(l,t)

Fig. 2. Left and right equivalent circuits for characteristic methodin time domain.

Fig. 2 is the equivalent circuit representation for character-istic method [1], which is shown on page 16 of the slides. Weshow an example to illustrate this method. As shown in Fig.3, for a transmission line with impedance Z0, time delay � ,assume the signal resistance and load resistance are R1 andR2, respectively. The driving voltage is e(t). For t < 0, thevalue of e is zero. Also, assume all the initial conditions arezero. We can use the characteristic method to calculate voltageand current along the line in time domain.

At t=0, according to causality, we have

ei(0,−�) = 0, er(l,−�) = 0 (10)

2

Synchronized Acquisition /FPGA

UWB signal in

LosslessLinear

Time invariant

a

bU

I+

-

Z0Z0

u(l,t)++

--u(0,t)

++

--er(l,t- ) ei(0,t- )

i(0,t) i(l,t)

e(t) Z0

R1

R2

Fig. 3. Transmission line terminated with source impedance andload resistance.

Then, the voltage at two terminals are

u(l, 0) = −Z0i(l, 0) (11)

u(0, 0) = Z0i(0, 0) (12)

Next, according to terminal condition, we have

u(0, 0) = e(0)−R1i(0, 0) (13)

u(l, 0) = R2i(l, 0) (14)

So, we can solve the voltage and current as follows

u(0, 0) =Z0

R1 + Z0e(0), i(0, 0) =

e(0)

R1 + Z0(15)

u(l, 0) = i(l, 0) = 0 (16)

According to the above results, we can write ei(0, 0) ander(l, 0) as

ei(0, 0) = [u(0, 0) + Z0i(0, 0)] =2Z0

R1 + Z0e(0) (17)

er(l, 0) = [u(l, 0)− Z0i(l, 0)] = 0 (18)

Based on the value we get at t=0, we can calculate the nextsample time t= � .

u(l, �) = −Z0i(l, �) + ei(0, 0) = −Z0i(l, �) +2Z0

R1 + Z0e(0)

u(0, �) = Z0i(0, �) + er(l, 0) = Z0i(0, �) (19)

Now, the terminal condition is changed as

u(0, �) = e(�)−R1i(0, �), u(l, �) = R2i(l, �) (20)

Getting these equations together, we can solve it as follows

u(0, �) =Z0

R1 + Z0e(�), i(0, �) =

1

R1 + Z0e(�) (21)

u(l, �) =2Z0R2

(Z0 +R1)(Z0 +R2)e(0) (22)

i(l, �) =2Z0

(Z0 +R1)(Z0 +R2)e(0) (23)

using this method, we can continue calculating the samplingpoint after t = � . Remember that time step size is � ; now ifwe want more precise result in time domain, we can take asmaller step size, such as some fraction of � . For t = kΔt,k=1,2,3,.... These equations can still be used to calculate thetime domain response, just by replacing t with kΔt.

The characteristic method can also be used in the casewhere multiple transmission lines are concatenating, or there

are mixed lumped elements, such as L,C,R at the intersection.The method is to treat each continuous segment separatelyusing the same equations above. At intersection, we can addthe voltage or current continuity equation or the current andvoltage relation of the lumped elements. Combining all theseequation can give the final time response of the whole system.Further, the characteristic method can be used for nonlineartermination load. Those equations are basically the same,except that the terminal condition is described by nonlinearequations, which can solved by iteration method [2].

IV. ONE EXAMPLE USING S PARAMETER IN FREQUENCYDOMAIN ANALYSIS

As we mentioned before, circuit analysis can be performedin time domain and frequency domain. Strictly speaking, a realsignal can only be observed in time domain, which impliesthe time domain analysis is the essence. While frequencydomain is just a integral transform with respect to time t,this makes time variables transformed to imaginary frequencyvariable jw or complex frequency variable s. In jw domainand s domain, differential and integration can be replacedby algebraic operation; convolution becomes multiplication.So, this is the philosophy behind frequency domain analysis.Next, we will use S parameter (also mentioned in the slide)and frequency domain transform to solve the transmission lineproblem [3].

Synchronized Acquisition /FPGA

UWB signal in

LosslessLinear

Time invariant

a

bU

I+

-

Z0Z0

u(l,t)++

--u(0,t)

++

--er(l,t- ) ei(0,t- )

i(0,t) i(l,t)

e(t) Z0

R1

R2

R0SR0

a2b2

a1b1

Fig. 4. Representing interconnects by S parameter.

As shown in Fig. 4, the interconnect transmission line haslength l, characteristic impedance Z0, proportion constant .The reference impedance are both R0 at two terminals. RecallZ0 and has the following form.

Z0 =

√j!L+R

j!C +G, =

√(j!L+R)(j!C +G)

Assume a1, a2 are the incident wave and b1, b2 are thereflected wave at two terminal. Generally, we have

U1 = a1 + b1, U2 = a2 + b2 (24)

I1 =a1

R0− b1R0

, I2 =a2

R0− b2R0

(25)

Γ = Γ(!) =Z0 −R0

Z0 +R0(26)

Here we use reflection coefficient Γ to denote the differencebetween characteristic impedance and reference impedance.The constant propagation coefficient is

� = �(!) = e− t (27)

3

Next, we get the transmission (ABCD) matrix of the segment.

[A BC D

]=

⎡⎢⎣ e t + e− t

2Z0e t − e− t

2e t − e− t

2Z0

e t + e− t

2

⎤⎥⎦ (28)

The normalized matrix, with respect to R0 is

[a bc d

]=

⎡⎢⎣ e t + e− t

2

Z0

R0

e t − e− t

2R0

Z0

e t − e− t

2

e t + e− t

2

⎤⎥⎦ (29)

Then, the S parameters are

S11 = S22 =(a+ b)− (c+ d)

a+ b+ c+ d

=Z0

R0− R0

Z0(e t − e− t)

e t + e− t + 12 (Z0

R0+ R0

Z0)(e t − e− t)

=12 (Z0

R0− R0

Z0)(1− e−2 t)

1 + e−2 t + 12 (Z0

R0+ R0

Z0)(1− e−2 t)

=12Z2

0−R20

Z0R0(1− e−2 t)

(Z0+R0)2

2Z0R0− (Z0−R0)2

2Z0R0e−2 t

=Γ(1− �2)

1− Γ2�2(30)

S12 = S21 =2

a+ b+ c+ d

=2

e t + e− t + 12 (Z0

R0+ R0

Z0)(e t − e− t)

=2e− t

(Z0+R0)2

2Z0R0− (Z0−R0)2

2Z0R0e−2 t

=

4Z0R0

(Z0+R0)2 e− t

1− Γ2�2=

[1− Z0−R0

Z0+R0]2e− t

1− Γ2�2

=(1− Γ2)�

1− Γ2�2(31)

In special case, where the line is lossless, then Z0 =√

LC is

pure resistive. For R0 = Z0, we have

Γ = 0, = j� = j!√LC, � = e−j�t = e−j!�

S11 = S22 = 0, S12 = S21 = � = e−j!�

So, we can convert the following equations in frequencydomain

b1 = S11a1 + S12a2 (32)b2 = S21a1 + S22a2 (33)

to time domain in lossless condition and get the following,

b1(t) = S12(t) ∗ a2(t) = e−j!� ∗ a2(t) = a2(t− �) (34)b2(t) = S21(t) ∗ a1(t) = e−j!� ∗ a1(t) = a1(t− �) (35)

According to (24) and (25), we can express a1(t), a2(t),b1(t) and b2(t) in terms of u1(t), u2(t), i1(t) and i2(t) and

substitute them into (33) and (34). The result is the same as weuse characteristic method in time domain. For general lossylines, the equations (33) and (34) will contain extra termsS11(t)∗a1(t) and S22(t)∗a2(t), respectively. The time domainS parameter are the inverse Fourier transform of equation (30)and (31) in frequency domain. By fast Fourier transform (FFT)method, we can still get the discrete point in time domain,which means we can still get the algebraic equations and solvethem by numerical method.

Synchronized Acquisition /FPGA

UWB signal in

LosslessLinear

Time invariant

a

bU

I+

-

Z0Z0

u(l,t)++

--u(0,t)

++

--er(l,t- ) ei(0,t- )

i(0,t) i(l,t)

e(t) Z0

R1

R2

R0SR0

a2b2

a1b1

T1(t) T2(t)

a2

b2

a1

b11( )t 2( )t

Fig. 5. Termination condition corresponding to S parameter equa-tions.

Finally, we can add the terminal condition to completediscussion of this part. Assume z1(t), z2(t) are the terminalresistance in time domain representation and e1(t), e2(t) arethe voltage source, then, as shown in Fig. 5, we can associatethen terminal condition with S parameter as follows [2],

Γi(t) =zi(t)−R0

zi(t) +R0(36)

Ti(t) =R0

zi(t) +R0(37)

where i = 1, 2. Next, we can write the terminal condition as

a1(t) = Γ1(t) ∗ b1(t) + T1(t) ∗ e1(t) (38)a2(t) = Γ2(t) ∗ b2(t) + T2(t) ∗ e2(t) (39)

V. BASIC PRINCIPLES AND SIMULATION OF DIFFERENTIALSIGNALING

A. Advantages of Differential Signaling

A differential pair is a simply a pair of transmission linewith some coupling between them. Differential signaling isthe use of two output drivers to drive two independent trans-mission lines, one carrying one bit and the other carryingits complement. The difference signal carries the information.Fig. 6 is an example of using low voltage differential signal(LVDS) in both transmitter and receiver, where current modelogic (CML) is used in transmitter to provide high data rate[4].

Differential signaling has the following main advantages,compared with single-ended version,

1) The total dI/dt noise is reduced in differential signaling,so there is less rail collapse and potentially less ElectroMagnetic Interference(EMI);

2) The differential amplifier at the receiver can have highergain then signal-ended version;

4

Fig. 6. LVDS transmitter(driver) and receiver.

3) The propagation of a differential signal over a tightlycoupled differential pair is more robust to cross talk anddiscontinuities in the returning path ;

There are several drawbacks in differential signals, too. First,the common mode signal will produce additional EMI. Sec-ond, it need twice signal traces compared with signal-endedversion. Although there are some drawbacks, yet they can beovercome in careful design. Overall speaking, there will stillbe more and more systems using differential signal. Next, wewill see an popular scheme of LVDS [5].

 

Fig. 7. The common and differential signal components in LVDS.

In Fig. 7, two signal traces are used to represent single bitinformation. Each has a swing from 1.125 V to 1.375 V. Hereare the basic principles

Vdiff(t) = V1(t)− V2(t)

Vcomm(t) =1

2(V1(t) + V2(t)) (40)

where Vdiff(t) is the differential mode signal, Vcomm(t) is thecommon mode signal, V1(t) and V2(t) are the signals on line1 and line 2 with respect to the common return path. Here aretwo important observations here

1) If the value of the common signal gets too high, it maysaturate the input amplifier of the differential receiverand prevent it from accurately reading the differentialsignal;

2) If any of the changing common signal makes it outon a twisted-pair cable, it has the potential of causingexcessive EMI.

B. Impedance Viewpoint

As we have mentioned in class, the impedance is veryimport for determining signal and power reflection and hencesignal integrity. Next, we will consider ideal case and non-ideal case for micro-strip lines.

1) No Coupling Condition: By the definition of impedance,the impedance of differential signal is

Zdiff =Vdiff

Ione= 2× Vone

Ione= 2Z0 (41)

which means the differential impedance is twice the charac-teristic impedance of either line.

2) The Impact of Coupling: As we have analyzed in class,there are capacitances and inductances associated with them.As shown in Fig. 8, C11 and C22 are the self conductanceper length, C12 describes the mutual conductance per length.Similarly, we can define those corresponding inductances. As

Synchronized Acquisition /FPGA

UWB signal in

LosslessLinear

Time invariant

a

bU

I+

-

Z0Z0

u(l,t)++

--u(0,t)

++

--er(l,t- ) ei(0,t- )

i(0,t) i(l,t)

e(t) Z0

R1

R2

R0SR0

a2b2

a1b1

T1(t) T2(t)

a2

b2

a1

b11( )t 2( )t

1 2

C11 C22

C12

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 fieldsbetween signal trace 1 and its return path are interceptedby the adjacent trace; C12 will increase according to simplecapacitance equation. So, we can see the sum of C11 and C12,defined as CL, meaning loaded capacitance, will not changevery much. For the case where two traces are far apart, thecharacteristic impedance of each line is independent of theother line. So,

Z0 ∝1

C11(42)

For the case where two lines are near each other, thepresence of the other line will affect the impedance of line1, which is called proximity effect. Simply consider the case,where the voltage on the second line is zero, the couplingcapacitance will be acted as the load of the first line. We canadd these effects up by superposition. In sum, we can find thefollowing relationship.

Z0 ∝1

C11 + C12=

1

CL(43)

Finally, we will see several methods for calculating differentialimpedance to account for the distance between nearby traces.There are five different ways for analysis,

1) Use the direct results from an approximation;2) Use the direct results from a field solver;3) Use an analysis based on modes;

5

4) Use an analysis based on the capacitance and inductancematrix;

5) Use an analysis based on the characteristic impedancematrix.

We have seen the first and fourth method in class. Onereasonable close approximation, which is based on empiricalfitting of measured data, for edge-coupled micro-strip lineusing FR4 material is shown below [5],

Zdiff = 2× Z0[1− 0.48exp(−0.96s

ℎ)] (44)

where s is the edge-to-edge separation between the traces inmils and ℎ is the dielectric thickness between the signal traceand the returning plane.

C. Even-odd Model Analysis for Non-distortion Transmission

Next, we will see two special cases for signal transmission,even mode and odd mode. For two same lines, assume we adda step input from 0V to 1V on line 1 and keep 0V constanton line 2 (shown in Fig. 9), then we will find as we movealong the line, the actual signal will change due to cross talkbetween line 1 and line 2. Noise will be accumulated on line2, which will decrease the signal quality on line 1.

 

Fig. 9. Voltage pattern with only one step input at port 1.

It is found that there are only two conditions where signalscan propagate undistorted. The first case is when exactly thesame signal is applied to each line; for example, the voltagegoes from 0V to 1V in each line. This is also called evenmode excitation, which is similar as the common mode. Thesecond case is when the opposite-transitioning signals areapplied to each line; for example, one of the signals goesfrom 0V to 1V and the other goes from 0V to -1V. This isalso called odd mode excitation, similar as differential mode,except numerically, we have Vdiff = 2Vodd.

D. Electromagnetic Issue and Simulation

As we mentioned earlier, circuit simulator cannot giveyou very accurate because these parasitic effect is gettingcomplicated for high frequencies. As shown in Fig. 10, theresult is obtained by Mentor Graphics Hyperlynx[5].

 

Fig. 10. Electric-field distribution for odd and even modes in micro-strip lines.

Fig. 11 shows the result got by Ansoft Maxwell 2D sim-ulator when the frequency is 50MHz. We can see the skineffect clearly in the signal’s trace; in the returning plane, thecurrents are most concentrated parallel to the signal’s traceand the total current width is about four times of the signal’strace, which is a feature for design.

 

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

Next, we use Agilent Advanced Design System (ADS) tosimulate a pair of parallel transmission lines on the FR4 board.The specified parameters of this board are as follows, thickness= 0.72mm, the real part of permittivity is 4.5, the loss tangentis 0.02. We use Momentum simulator provided in ADS, whichuses Green functions to account for wave effects, hence moreaccurate than lumped circuit simulator.

 

Fig. 12. Two parallel micro-strip lines for even and odd modeexcitation simulation.

Fig. 12 is the actual two parallel micro-strip line imple-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. 13. Current distribution when there is one excitation with 2GHzat port 1.

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

 

Fig. 14. Current distribution when there is one excitation with37MHz at port 1.

Fig. 14 shows the current distribution when there is onesource with 37MHz frequency source at port 1. Now, ascontrast to Fig. 13, here we can see the current is relativeconstant along line 1. Also, there is rarely coupling currenton line 2, which means coupling effect is very small for lowfrequencies.

Fig. 15 shows the current distribution when we add evenmode excitation at port 1 and port 2 with 37MHz frequency.Now, we can see the current is smaller in the near edges thanthat in the far-apart edges

Fig. 16 shows the current distribution when we add oddmode excitation at port 1 and port 2 with 37MHz frequency.As contrast to Fig. 15, we can see the currents is more densein the far-apart edges.

 

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

 

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

Fig. 17. Far field radiation pattern for even mode excitation.

 

Fig. 18. Far field radiation pattern for odd mode excitation.

7

Beyond that, we can use far field analysis to compare thecase with even and odd mode excitation. Fig. 17 shows theelectric field radiation pattern for even mode excitation. Fig.18 is the case for odd mode excitation. The quantities got inMomentum analysis are summarized as follows.

1) For 37 MHz frequency excitation, the total radiationpower is 5.94×10−19W for odd mode; 1.006×10−12Wfor even mode;

2) For 2 GHz frequency excitation, the total radiationpower is 2.311×10−8W for odd mode; 1.178×10−5Wfor even mode ;

These results reveal several general principles. First, evenmode radiates more power than odd mode. Second, as thefrequency goes up, the total radiation power increases. Theseobservations are important, especially for multiple layers intoday’s ICs, where EMI is becoming a critical issue in highfrequencies.

VI. CONCLUSION

In this report, we first use time domain method and fre-quency domain method to get the response of a traditionaltransmission line. Then, we introduce the idea of differentialsignaling, which is quite often used in high speed system.Finally, we perform a series of simulation on an actual trans-mission line used for differential signals, where we can see forhigh speed application, things are much more complicated, sowe need to consider much more than pure circuit analysis.We have to go deeply into the fundamentals, originated fromMaxwell’s equations for coupling and radiation. Instead ofcollecting other’s report and paper, deeply thinking of thefundamentals and essence is the best lesson we get from thisproject.

REFERENCES

[1] F. Branin, “Transient Analysis of Lossless Transmission Lines,” Proceed-ings of the IEEE, vol. 55, no. 11, pp. 2012–2013, Nov. 1967.

[2] Z. F. Li and J. F. Mao, “Microwave and High Speed Circuits Theory,”Shanghai Jiao Tong University, Oct. 2004.

[3] J. E. Schutt-Aine, T. K. Sarker, and R. F. Harrington, “Analysis of LossyTransmission Line with Arbitrary Non-linear Terminal Networks,” IEEETrans. on MTT, p. 36, 1986.

[4] N. Semiconductor, “LVDS owner’s manual,” Including High-Speed CMLand Signal Conditioning, 4th ed, p. 9, 2008.

[5] E. Bogatin, “Signal Integrity, Simplified,” Prentice Hall, Sept. 2003.

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