DesignCon 2010

Frequency Dependent Material Properties: So What? Eric Bogatin, Bogatin Enterprises eric@beTheSignal.com Don DeGroot, CCNi Don.degroot@ccn-i.com Colin Warwick, Agilent Technologies Colin_warwick@agilent.com Sanjeev Gupta, Agilent Technologies Sanjeev_gupta@agilent.com

Abstract It is well known that the dielectric constant and dissipation factor of real laminate materials are slightly frequency dependent, as is the loop inductance per length of circuit board transmission lines. While the series resistance is often assumed to increase with the square root of frequency, in most simulations, the assumption is made that the other terms are constant with frequency. What do we miss by ignoring their frequency dependence? In this paper, we will use a model and simulation environment that accurately incorporates the frequency dependence of R, L, Dk and Df to explore the impact on typical high speed serial links from low bit rates to above 10 Gbps due to frequency dependent material properties. Author(s) Biography Eric Bogatin received his BS in physics from MIT and MS and PhD in physics from the University of Arizona in Tucson. He has held senior engineering and management positions at Bell Labs, Raychem, Sun Microsystems, Ansoft and Interconnect Devices. Eric has written 6 books on signal integrity and interconnect design and over 300 papers. His latest book, Signal and Power Integrity- Simplified, was published in 2009 by Prentice Hall. He has taught over 5,000 engineers in the last 20 years. Many of his papers and columns are posted on the www.BeTheSignal.com web site. Dr. Don DeGroot operates CCNi, a test and measurement business for high-speed electronic design. Don has 25 years experience in high- frequency measurements and design with industry, government, and academia. He currently focuses on services with value to product development, including multiport TDR and S-parameter measurements, dielectric characterization, and component testing. Colin Warwick is signal integrity product manager at Agilent EEsof EDA, where he is focused on multigigabit per second signal integrity analysis tools. Prior to joining Agilent, Colin was with Royal Signals and Radar Establishment in Malvern, England, Bell Labs in Holmdel, NJ, and The MathWorks in Natick, MA. He completed his bachelor, masters, and doctorate degrees at the University of Oxford, England. He has published over 50 technical articles and holds thirteen patents. Sanjeev Gupta, the Signal Integrity Applications Expert in the EEsof EDA Division of Agilent Technologies, has over eighteen years of experience in high frequency design and simulation. Before joining Hewlett Packard, he worked as a high frequency design engineer/scientist at the Defense Research and Development Organization in India. His background includes the design and development of 100 MHz to 100 GHz active and passive circuits for a wide variety of applications. His most recent activity is focused on influencing the Signal Integrity Design Flow in ADS. He received a Master’s Degree in Microwave Engineering from the University of Delhi, India in 1988. Sanjeev was awarded the Hewlett Packard President’s Award in 1998 for his contributions to the company.

Welcome to the Real World While it is true that the behavior of propagating signals on interconnects is most accurately described by the behavior of the electric and magnetic fields, which are governed by Maxwell’s Equations [1], sometimes, we can approximate this behavior in terms of currents and voltages which are described with equivalent circuit elements in a distributed circuit. A typical equivalent circuit model for a uniform transmission line is shown in Figure 1.

Figure 1. Typical equivalent circuit model to describe an ideal lossy transmission line.

The circuit model for a signal propagating on a transmission line, is really a short hand to describe the differential equations through which the voltage and currents interact with the conductors and dielectrics. The set of coupled differential equations is usually referred to as the Telegrapher’s equations which describe the relationship between the currents and voltages at any location of the line. These are often represented as [2]:

( ) ( )t,xVt

CGt,xIx

⎟⎠⎞

⎜⎝⎛

∂∂+−=

∂∂

( ) ( )t,xIt

LRt,xVx totalac ⎟

⎠⎞

⎜⎝⎛

∂∂+−=

∂∂

The R, L, G and C terms in this model are the per unit length values of the resistance, inductance, conductance and capacitance. In this representation, they can each be frequency dependent, which can arise from material properties. In the case of circuit board transmission lines, the resistance and inductance are determined by conductor properties while the capacitance and conductance are determined by the dielectric properties of the laminate. Since these circuit elements describe real world effects, they must all obey causality. In its simplest form, causality means that “a response can’t happen before the stimulus.” [3] If a stimulus of any sort is applied at t = 0, the response can’t happened before t < 0. On transmission lines, a type of waveguide, this implies that the group velocity must be less than the speed of line in the material. While an obvious condition for real world behavior, describing it mathematically is complicated.

Rac G

Ltotal

C

The impact of causality is that any function describing real world behavior in the frequency domain, usually represented by a complex number to account for amplitude and phase effects, must obey a simple rule for linear systems that are not band limited:

the imaginary part of the normalized transfer function must be the Hilbert transform of the real part of the same complex

function. The consequence of this simple rule is that if the real part of a normalized transfer function is known, the imaginary part can be uniquely determined if the system is not band limited. In other words, “to know the real part is to know the imaginary part.” Any normalized transfer function which has its real and imaginary parts the Hilbert transform of each other is said to be a causal function. Causality is a necessary condition for any real world behavior. [4] This rule can be equally well applied to the relationship between the real and imaginary part of the series impedance and the shunt admittance of a transmission line, when they describe the transfer function between currents and voltages.. A Causal Model for R and L The series impedance of a transmission line can be described as a contribution from the series resistance per length and the series loop inductance pre length. Both terms include the contributions from the signal and the return path conductors. In the series impedance, the real part is from the resistance and the imaginary part is from the inductive reactance. The primary factor affecting the frequency dependence of the resistance is skin depth effects which re-distribute the current distribution in the conductor as frequency increases. From a circuits perspective, the current redistributes to minimize the series impedance. Less current flows through higher inductance paths and a larger fraction flows through the lower inductance path. As frequency increases, the series resistance increases and the series inductance decreases. The inductance is composed of two parts- a contribution from magnetic field lines that are external to the conductors and only depends on the total current through the conductors, and an internal inductance contribution. The internal inductance term will decrease as less current flows through the interior of the conductor. To first order, the skin effect resistance will increase with the square root of frequency and scale inversely with the line width, w, and the dielectric thickness, h. This includes the spreading of the return current in the adjacent planes. The series resistance is approximated as:

⎟⎠⎞

⎜⎝⎛ +

σπμ

h61

w21f~R 0

ac

The resistance and internal inductance must obey causality. When translated in a simple for, they are connected at high frequency as:

ω= ac

intR

L

To know the value of the resistance is to know the value of the internal inductance as well. The external inductance is constant with frequency, but the internal inductance will decrease. Agilent’s Advanced Design System (ADS) uses built in models which relate the resistance and inductance by this causal model. Figure 2 shows the simulated series resistance and total inductance per length of a 50 ohm stripline having a line width of 5 mils.

Figure 2. Simulated inductance per length and resistance per length of a 5 mil wide 50 Ohm stripline

interconnect, simulated with Agilent's ADS. The consequence of the loop inductance decreasing with increasing frequency is that the speed of a signal will be slightly dispersive and the characteristic impedance will vary with frequency. The characteristic impedance of a transmission line, defined as the impedance of a termination which will result in minimal reflections, is given by;

CjGLjRZ0 ω+

ω+=

R is related to the square root of frequency, while the ωL term is directly related to frequency. Surprisingly, this says that the importance of R, even dominated by skin

1E8

1E9

1E10

1E7

2E10

2468

1012141618

0

20

freq, Hz

Tota

l Loo

p In

duct

ance

per

Len

gth,

nH

/inch

1E8

1E9

1E10

1E7

2E10

123456789

0

10

freq, Hz

Res

ista

nce

per L

engt

h, O

hms/

inch

Loop inductance Series resistance

depth, will drop off with frequency. The series resistance of a transmission line will only play a role at low frequency, where it might be larger than the reactance of the series inductance. This behavior is confirmed in Figure 3.

Figure 3. The consequence of frequency dependent inductance and resistance is dispersion and a frequency dependent characteristic impedance for the case of a 5 mil wide, 50 ohm line. Of course, the specific values of the series resistance depends on the specific geometry of the interconnect, and even on the surface texture of the copper. The relationship between the series resistance and copper surface texture is a topic of current research and has not been fully resolved. [5] Dielectric Properties The dielectric properties of an insulator are described by the complex dielectric constant, having a real and imaginary component. For various reasons [6], the imaginary component is negative and the relative complex dielectric constant is usually written as:

rrr jε ′′−ε′=ε We usually refer to the real part of the relative complex dielectric constant as the dielectric constant, Dk, and the dissipation factor of the material, rather than the imaginary part of the dielectric constant. The dissipation factor is given by

DkDf rε ′′

=

Since the dielectric constant represents a material’s response to a broadband electromagnetic wave, the real and imaginary terms must be related in a way that preserves causality. When the Hilbert transform is applied, we find the real and imaginary components must also obey the Kramers-Kronig relationship even over limited bands:

1E8

1E9

1E10

1E7

2E10

1

2

3

4

5

0

6

freq, Hz

Spe

ed, i

nche

s/ns

ec Dispersion: speed is frequency dependent

1E8

1E9

1E10

1E7

2E10

0

10

20

30

40

50

-10

60

OPTSOLNVALS.f_val

Cha

ract

eris

tic Im

peda

nce,

Ohm

s

Real component

Imaginary component

( )ω′

ω−ω′ω′ε ′′ω′

π+=ωε′ ∫

+∞

d)(21)(0

22 ( )

ω′ω−ω′

ω′ε′−πω=ωε ′′ ∫

+∞

d)(12)(0

22

There are still many behaviors for the complex dielectric constant of any material that obey the Kramers-Kronig relationship. A useful approximation is to model a real material as composed of a series of tethered dipoles which have a restoring force and damping due to friction with the polymer backbone. Each dipole is modeled as a harmonic oscillator with a material resonant frequency or a pole. This model was first proposed by Debye and is referred to as the Debye model of dipole motion. [7]. When the model uses one resonant frequency it is called a single pole Debye model. When multiple resonant frequencies are included, it is called a multi-pole Debye model. Independently, Svensson and Djordjevic [8, 9] proposed a wide band model for the behavior of a laminate assuming a large (infinite) distribution of resonant frequencies. In this model, often referred to as an infinite pole Debye model or a wide bandwidth model or the Djordjevic model, a specific behavior model of the dielectric constant can be derived. Based on observations of the measured behavior of the dielectric constant of real materials, the real part of the dielectric constant is seen to be of the form:

( ))flog()flog()flog()flog(

DkDk)f(Dk 212

2 −−

Δ+=

The real part, over some frequency range, varies with the log of frequency. This behavior has been well documented, as for example in [10] and shown in Figure 4.

Figure 4. Measured frequency variation in Dk for an FR4 sample showing the log behavior, taken

from ref [10].

Using the causality rule, that “to know the real part is to know the imaginary part, and applying the Kramers-Kronig relationship, the imaginary part of the dielectric constant can be derived as:

682.0)flog()flog(

Dk)10ln(2/

)flog()flog(Dk)f(

1212 −Δ=π−

−Δ=ε ′′

This shows the imaginary part of the dielectric constant is a direct measure of the slope of the real part, and is constant with frequency over the range where the real part varies as the log of frequency. From this term, the dissipation factor can be estimated as

682.0)flog()flog(

1)f(Dk

Dk)f(Dk)f()f(Df

12 −Δ=ε ′′

=

Frequency variation of Df comes from the very slight frequency variation of Dk. This illustrates that the dissipation factor is a rough measure of the relative frequency variation of Dk. A larger value of Df suggests Dk will vary more with frequency, while a small value of Df suggests that Dk will be very constant with frequency. Having a value of Df effectively defines the slope of Dk and its frequency variation, in the region where the real part varies with the log of frequency. Figure 5 shows an example of how Dk and Df are connected for a causal model compared with a non causal model in which the real part and dissipation factor are constant with frequency. This is for FR4, with the highest typical dissipation factor and hence the largest frequency variation of Dk.

Figure 5. Example of the frequency variation of Dk and Df for FR4, plotted in various scales. Note: the low frequency limit in this plot is 10 MHz. The frequency variation of Dk and the dissipation factor will introduce dispersion and a frequency dependence to the characteristic impedance. The response for the worst case material, FR4 with a dissipation factor of 0.025, is shown in Figure 6.

1E5

1E6

1E7

1E8

1E9

1E10

1E4

2E10

1

2

3

4

5

0

6

freq, Hz

Dk

2 4 6 8 10 12 14 16 180 20

0.01

0.02

0.03

0.04

0.00

0.05

freq, GHz

Df

2 4 6 8 10 12 14 16 180 20

1

2

3

4

5

0

6

freq, GHz

Dk

2 4 6 8 10 12 14 16 180 20

3.5

4.0

4.5

3.0

5.0

freq, GHz

Dk

causal

non-causal

causal

non-causal

causal

non-causal

Figure 6. Impact on impedance and dispersion from frequency dependence of Dk for FR4. Measured Properties This frequency dependence of the dielectric constant and dissipation factor has been well documented and is typically measured An example of a collection of measured material properties is shown in Figure 7.

Figure 7. Typical examples of measured dielectric constant and dissipation factor for laminate samples. While there are a variety of measurement methods used in the industry to extract material properties, the method based on using two different length transmission lines, often called the multi line method, has the widest bandwidth and most easily implemented [11]. A summary of the various methods is shown in Figure 8.

1E8 1E91E7 7E9

46

47

48

49

50

51

45

52

Frequency

Rea

l Par

t Cha

ract

eris

tic Im

peda

nce

1E8 1E91E7 7E9

-2

-1

0

1

2

-3

3

Frequency

Imag

inar

y P

art C

hara

cter

istic

Impe

danc

e

Real component

Imaginary component

2 4 6 8 10 12 14 16 180 20

5.15.25.35.45.55.65.75.85.96.0

5.0

6.1

freq, GHz

Spe

ed, i

nche

s/ns

ec

causal

non-causal

Figure 8. Summary of the various methods to measure the dielectric properties of laminate materials, showing the multi line method as having the widest frequency range. So What? Impact from conductor loss The frequency dependent inductance and resistance of narrow transmission lines will affect the transient response of signals in a complex way. It is difficult to translate from the frequency domain to the time domain without the use of an accurate simulation tool. The Agilent ADS tool integrates causal, frequency dependent effects into a transient simulator. The TDR response from three different line widths is shown in Figure 9. In these examples, the reflected signal is translated into an impedance by the first order approximation that:

rho1rho1Z

−+=

Multi Line Transmission line methods offer widest bandwidth

Figure 9. Simulated impedance profile of lossy lines, with different line widths, but the same impedance, including only the resistive and inductance effects, not the dielectric effects. This shows the impedance creeping up as the TDR incident signal reflects from regions farther down the line. This creep up is primarily due to the series resistance of the line. IT is seen to increase roughly as the square root of the line length, or time delay. This is reasonable as the resistance per length will vary roughly as the square root of frequency, which is roughly as the square root of the time delay, TD, down the line. The result is the series resistance- as shown by the TDR impedance, will vary as TD x 1/sqrt(TD) or roughly as sqrt(TD). The impact from the frequency dependent inductance is very small compared to the series resistance and is masked by this behavior. In fact, it is impossible to separate the impact from the series R and the series L. They are tightly connected by causality as knowing one, defines the other. What is the characteristic impedance of the line? Using TDR, it can only be extracted as the high frequency impedance, roughly at the start of the line [vxx] Transient Response from Dielectric Effects The TDR response from a 50 ohm line, with conductor losses turned off, can be simulated for the case of a causal dielectric model and a non causal model. In the non-casual model, the Dk and Df values are constant with frequency. This response is shown in Figure 10.

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0

50515253545556575859

49

60

time, nsec

Firs

t Ord

er T

DR

Impe

danc

e

w = 3 mils

w = 5 mils

w = 10 mils

Figure 10. TDR response for a 50 ohm stripline, with conductor loss effects turned off. Note that the vertical scale is greatly expanded to show the effects. For a non-causal model with constant Dk and Df, the impedance creeps down, less than ½ an Ohm down the length of the line. However, for the non causal case, the impedance starts a little high and drops down as the incident signal passes down the line. This behavior has been described as “creep down.” The predominate difference in these two models is the frequency dependence to Dk. In casual model, the Dk will vary with frequency, which causes the characteristic impedance to vary with frequency. The variation for this line, is shown in Figure 11.

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0

49

50

48

51

time, nsec

TDR

Impe

danc

e, O

hms

Df = 0

Df, Dk = constant

Df = causal

Figure 11. Characteristic impedance for the casual model. The frequency variation of Dk causes a frequency variation to the real part of the impedance. At high frequency, or short transients, the impedance is higher, while for long time periods, or low frequency, the impedance is lower. This is what is observed in the TDR response. However it is a very small effect, less than 2 Ohms in this example for FR4 having a dissipation factor of 0.02. This frequency dependence of the dielectric constant, will also create dispersion on transmitted signals. Higher frequency components will travel faster and have a shorter delay than lower frequency components. This will contribute to spreading out of the rising edge of a signal. The attenuation will also contribute to the spreading out of the signal edge. However, since the dissipation factor is nearly the same for the causal and non casual models, the insertion loss will be mostly the same between the two models and the rise time degradation from attenuation should be of similar magnitude. These two effects are shown in Figure 12.

1E8 1E91E7 7E9

46

47

48

49

50

51

45

52

Frequency

Rea

l Par

t Cha

ract

eris

tic Im

peda

nce

Causal Dk

Figure 12. Impact on insertion loss and dispersion in a 10 inch long 50 ohm interconnect for a causal and non causal model of FR4. The transient received signal is a more important indictor of performance in high speed serial link applications than the TDR response. The transmitted signal or the time domain transmitted (TDT) response for the cases of no loss, a non causal model and a causal model is shown in Figure 13.

2 4 6 8 10 12 14 16 180 20

1.65

1.70

1.75

1.60

1.80

freq, GHz

Del

ay, n

sec

2 4 6 8 10 12 14 16 180 20

-15

-10

-5

-20

0

freq, GHz

Inse

rtion

Los

s, d

B

Df = 0Df = constant

Df = causal

Df = constant

Df = causal

Figure 13. Top: TDT response for a 50 ohm line with only dielectric loss, for three different models. Bottom: the same TDT response of the casual model, time shifted to overlay with the non-casual model's response. Since the high frequency dielectric constant of the causal model is less than 4, its time delay to exit the 10 inch transmission lien in this example, is shorter than for the case of a constant Dk of 4. This is why we see the TDT response of the causal model exiting before the other responses. This is not a non-causal response, but due to the dispersion of the signal with frequency. To clarify the comparison of the response from the constant Dk and Df model and the causal model, the TDT response of the causal model is time shifted forward in time so its start matches the non casual model’s start. This clearly shows the important impact from casual material behavior, an increase in the rise time of the transmitted signal, as compared to the non casual interconnect.

1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.951.50 2.00

0.00.10.20.30.40.50.60.70.80.91.0

-0.1

1.1

time, nsec

Tran

smitt

ed S

igna

l Df = 0

Df, Dk = constantDf = causal

1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.951.50 2.00

0.00.10.20.30.40.50.60.70.80.91.0

-0.1

1.1

time, nsec

Tran

smitt

ed S

igna

l

causal response shifted in time

This rise time difference can be roughly estimated from the TDT response as about 15 psec difference for the 10 inch long line, with the worse case materials of FR4 having a dissipation factor of 0.02. This suggests a simple rule of thumb to estimate the hit taken by using a non causal, though simple model for dielectric properties:

Worst case dispersion in real laminates can contribute to ~ 1.5 psec/inch additional rise time degradation over non-causal

models. This means that non casual models under estimate the rise time degradation from interconnects and when used in serial link analysis might estimate larger eye openings than might be present. Impact on High Speed Serial Links The roughly 1.5 psec/inch difference between causal and non-casual models would have a significant impact when the rise time is a significant fraction of the unit interval. For example, at 10 Gbps, the unit interval is 100 psec. For a 10 inch long interconnect, the difference in the predictions of the two models is 15 psec. For a 20 inch interconnect, it is 30 psec. This is significant. The comparison of the simulated eyes for 10 Gbps PRBS signals through a 10 and 20 inch long interconnect having these two material behaviors, including just the dielectric loss, is shown in Figure 14.

Figure 14. Simulated 10 Gbps PRBS signals through 10 inch and 20 inch long interconnects with causal and non causal material models. At 10 Gbps and 20 inches, the difference is quite significant. In both of these examples, the conductor loss effects were turned off to show the difference between the dielectric models. The eyes will collapse more from conductor losses and the actual impact will depend on the surface texture of the copper. The rougher the copper surfaces the more the impact. Figure 15 shows the impact on the eye for a 5 Gbps and 10 Gbps PRBS signal through 10 inch long interconnects, including the conductor losses and dielectric effects, for a non causal and causal model.

20 40 60 80 100 120 140 160 1800 200

0.1

0.2

0.3

0.4

0.0

0.5

time, psec

Eye_

Prob

e2.D

ensi

ty20 40 60 80 100 120 140 160 1800 200

0.1

0.2

0.3

0.4

0.0

0.5

time, psec

Eye_

Pro

be3.

Den

sity

Non causal

Causal

20 40 60 80 100 120 140 160 1800 200

0.1

0.2

0.3

0.4

0.0

0.5

time, psec

Eye_

Prob

e2.D

ensi

ty

20 40 60 80 100 120 140 160 1800 200

0.1

0.2

0.3

0.4

0.0

0.5

time, psec

Eye_

Pro

be3.

Den

sity

Non causal

Causal

10 inches 20 inches

Figure 15. Left, 5 Gbps PRBS signals and right, 10 Gbps PRBS signals, passing through 10 inch interconnects with causal and non causal material models, including conductor loss. When the unit interval is 200 psec, the 15 psec difference between the causal and non-causal models is not very significant, compared with the rise time degradation present from the conductor loss and dielectric loss. There is very little difference in the eyes from these two cases for the 5 Gbps signal. However, there is a significant difference in the eye collapse and deterministic jitter for the 10 Gbps signal. Since the dielectric losses are very similar in these two cases, the biggest contributor to the rise time degradation must be the dispersion in the causal material. Conclusions Real interconnects have frequency dependent material properties. While most simulation tools take into account the resistance varying with the square root of frequency, not as many take into account the inductance being frequency dependent. This ignores a source of dispersion and under estimates the degradation of high speed serial data signals. However above about 100 MHz, the impact from inductive dispersion is reduced and the transient response is dominated by the loss from the series resistance. Dielectric materials are more difficult to model and incorporate in transient simulations. Material dielectric constants can be measured over broad bands (over 40 GHz) and the Df values measured with precision resonators at discrete points over equally broad bands.

20 40 60 80 100 120 140 160 1800 200

0.1

0.2

0.3

0.4

0.0

0.5

time, psec

Eye

_Pro

be2.

Den

sity

20 40 60 80 100 120 140 160 1800 200

0.1

0.2

0.3

0.4

0.0

0.5

time, psecE

ye_P

robe

3.D

ensi

ty

Non causal

Causal

10 Gbps

50 100 150 200 250 300 3500 400

0.1

0.2

0.3

0.4

0.0

0.5

time, psec

Eye_

Prob

e2.D

ensi

ty

50 100 150 200 250 300 3500 400

0.1

0.2

0.3

0.4

0.0

0.5

time, psec

Eye

_Pro

be3.

Den

sity

Non causal

Causal

5 Gbps

These measured functions coupled with measured transmission line response can be incorporated into simulations. In all cases transmission lines composed of real materials obey causality, which implies a definite relationship between the frequency dependence of the dielectric constant and dissipation factor. In the wide bandwidth Debye model for dielectrics, the dissipation factor is a good measure of the slope of the dielectric constant with the log of frequency. A higher dissipation factor suggests the material’s dielectric constant will vary more with frequency. This creates dispersion. When a material is modeled with a constant dissipation factor and a constant dielectric constant, this contribution to dispersion is not included. It can contribute an additional rise time degradation on the order of 1.5 psec/inch for the worst performance laminate such as FR4. This is the additional impact from the dispersion from the dielectric constant which is not included in models assuming a constant dielectric constant. When the additional rise time degradation is significant compared with a bit’s unit interval, the actual impact from the serial link on the data signal might be worse than that predicted by a dielectric constant which is constant with frequency. For example, a 10 Gbps signal through a 10 inch length of FR4 will suffer a significantly worse eye using a causal model than a constant Dk model for the laminate. This effect may be a potential issue for all FR4 interconnects at 5 Gbps and above. It is recommended that for accurate simulations causal models of the materials should be used, and an accurate measure of the laminate’s dielectric properties should be included. Acknowledgements We would like to acknowledge the assistance of Don Degroot’s student from Andrews University, Joel Kitchen, and assistance from Dr. Mike Janezic of NIST. References

1. Huray, Paul, The Foundations of Signal Integrity, John Wiley and Sons, 2010

2. Bogatin, Eric, as shown in Essential Principles of Signal Integrity, and online and live lecture series at www.beTheSignal.com

3. Hall, Stephen and Heck, Howard, Advanced Signal Integrity for High-Speed

Digital Designs, John Wiley and Sons, 2009

4. Morgan, Chad, “Solutions for Causal Modeling and a Technique for Measuring Causal, Broadband Dielectric Properties,” DesignCon 2008.

5. Huray, Paul, et.al., “Impact of Copper Surface Texture on Loss: A Model that

Works,” DesignCon 2010.

6. Bogatin, Eric, Signal and Power Integrity- Simplified, Prentice Hall, 2010.

7. Debye, P., Polar Molecules, Dover Publications, 1929.

8. Djordjevic, A. R., et. Al., “Wide Band Frequency Domain Characterization of FR4 and Time Domain Causality,” IEEE Trans EMC, vol 43, no. 4, Nov 2001, pp 662-667.

9. Svensson, C., and Dermer, G. E., “Time Domain Modeling of Lossy

Interconnects,”, IEEE Trans on Adv Pkg, vol 24, no. 2, May 2001.

10. Bogatin, Eric, “A Closed Form Analytical Model For The Electrical Properties of Microstrip Interconnects,” IEEE Trans. CHMT, June, 1990 p. 258

11. DeGroot, D. C., Jargon, J. A., and Marks, R. B., "Multiline TRL

Revealed," 60th ARFTG Conference Digest, pp. 131-155, Dec. 5-6, 2002.