© H. Heck 2008Section 5.41 Module 5:Advanced Transmission Lines Topic 4: Frequency Domain Analysis OGI ECE564 Howard Heck

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.42 Where Are We? 1.Introduction 2.Transmission Line Basics 3.Analysis Tools 4.Metrics & Methodology 5.Advanced Transmission Lines 1.Losses 2.Intersymbol Interference 3.Crosstalk 4.Frequency Domain Analysis 5.2 Port Networks & S-Parameters 6.Multi-Gb/s Signaling 7.Special Topics

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.43 Contents Motivation Wave Equation Revisited Frequency Dependence Reflection Coefficient and Impedance Input Impedance Examples Summary References

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.44 Motivation At high frequencies, losses become significant. This makes time domain analysis difficult, as the properties are frequency dependent.  Skin effect, dielectric loss & dispersion We need to develop the means to understand those effects.  Example: How would we measure R, L, G, C for a PCB trace? Frequency domain analysis allows discrete characterization of a linear network at each frequency.  Characterization at a single frequency is much easier Frequency Analysis has advantages:  Ease and accuracy of measurement at high frequencies  Simplified mathematics  Allows separation of electrical phenomena (loss, resonance … etc).

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.45 Key Concepts The input impedance & the input reflection coefficient of a transmission line is dependent on:  Termination and characteristic impedance  Delay  Frequency S-Parameters are used to extract electrical parameters.  Transmission line parameters (R,L,C,G, TD and Zo)  Vias, connectors, socket … equivalent circuits Periodic behavior of S-parameters can be used to gain intuition of signal integrity problems. We’ll study S-parameters in section 5.5.

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.46 Derive the lossy wave equation Add a sinusoidal stimulus

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.47 Wave Equation Revisited Goal: derive the frequency dependent impedance and reflection coefficients. Method: Starting with the RLGC equivalent circuit, we derive the differential equations. KVL Rearrange Differentiate w.r.t. z [5.4.1] [5.4.2] [5.4.3] KCL Rearrange Differentiate w.r.t. z [5.4.4] [5.4.5] [5.4.6]

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.48 Wave Equation Revisited #2 Use the equations on the previous page to get: [5.4.7] [5.4.8] Which have solutions: [5.4.9] [5.4.10] where [5.4.11]

Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.49 Wave Equation Solution Let’s work with [5.4.3] and [5.4.10] to relate the currents and voltages: [5.4.12] [5.4.13] Differentiate w.r.t. z : Substitute [5.4.14] [5.4.15] [5.4.16b][5.4.16a] [5.4.17b][5.4.17a] Algebra where note so[5.4.19b][5.4.19a] [5.4.18] [5.4.20]

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Including Frequency Dependence If a sinusoid is injected onto a transmission line, the resulting voltage can be expressed as a function of the distance from the load ( z ) and time. Notice:  The first term represents the forward traveling wave (toward the load)  The second term represents the backward traveling wave reflected from the load (toward the source)  The position dependent exponent is positive for the second term because the wave is traveling back toward the source. [5.4.21]

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Frequency Dependence #2 Note that and use to get: [5.4.22] [5.4.24] [5.4.23] Separating the real and imaginary terms: Expressing in terms of sine/cosine functions: Where  is the amplitude loss of the sinusoid  is the phase shift

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Frequency Dependence #3 Apply the sinusoid source to the expression for current: [5.4.25] [5.4.26]

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Load Impedance [5.4.28] [5.4.27] [5.4.29] Look at the boundary case.

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Reflection Coefficients & Impedance Define the reflection coefficients: [5.4.30] [5.4.33] [5.4.34] [5.4.32] [5.4.31] Define the impedance in terms of reflection coefficients: Note: most microwave texts use the gamma (  ) symbol to represent the reflection coefficient. I have chosen to continue to use  in order to remain consistent with our definition from module 2.

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Input Impedance Define the input impedance: [5.4.35] [5.4.37] The impedance at the load is: Solving [5.4.36] for  v, we get the familiar equation for the reflection coefficient at the load: [5.4.38] Substituting [5.4.37] into [5.4.30], we get the equation reflection coefficient as a function of position along the line: [5.4.36]

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Input Impedance #2 Substituting [5.4.38] into [5.4.35] and doing the algebra: [5.4.39] Use the following relationship: To get:

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Input Impedance #3 Alternate expression (for lossless lines): [5.4.40] Use the following relationships: To get:

Frequency Domain Analysis EE 564 © H. Heck 2008 Section ExampleLossless Looking into Z 02 : Use Z in2 as the load impedance to get the input impedance looking into Z 01 :

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Example #2 What is  v as measured at z = 0 for the lossless transmission line system as a function of frequency? Start with [5.4.38]: Which can be rewritten: Notice that the real part is zero when. Solving for f : where The imaginary part is zero when. Solving for f : where

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Example #2 (2)

Frequency Domain Analysis EE 564 © H. Heck 2008 Section Summary We now have the basis for using measurement equipment to characterize interconnect in the frequency domain.

Frequency Domain Analysis EE 564 © H. Heck 2008 Section References R.E. Matick, Transmission Lines for Digital and Communication Networks, IEEE Press, D.M. Posar, Microwave Engineering, John Wiley & Sons, Inc. (Wiley Interscience), 1998, 2 nd edition. B. Young, Digital Signal Integrity, Prentice-Hall PTR, 2001, 1 st edition. W. Dally and J. Poulton, Digital Systems Engineering, Cambridge University Press, Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, U. Inan, A. Inan, Engineering Electromagnetics, Addison Wesley, 1999, 1 st edition. Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, 1985.