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

2. 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

3. 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

4. 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).

5. 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.

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

7. 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]

8. 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]

9. 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]

10. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.410 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]

11. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.411 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

12. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.412 Frequency Dependence #3 Apply the sinusoid source to the expression for current: [5.4.25] [5.4.26]

13. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.413 Load Impedance [5.4.28] [5.4.27] [5.4.29] Look at the boundary case.

14. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.414 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.

15. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.415 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]

16. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.416 Input Impedance #2 Substituting [5.4.38] into [5.4.35] and doing the algebra: [5.4.39] Use the following relationship: To get:

17. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.417 Input Impedance #3 Alternate expression (for lossless lines): [5.4.40] Use the following relationships: To get:

18. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.418 ExampleLossless Looking into Z 02 : Use Z in2 as the load impedance to get the input impedance looking into Z 01 :

19. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.419 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

20. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.420 Example #2 (2)

21. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.421 Summary We now have the basis for using measurement equipment to characterize interconnect in the frequency domain.

22. Frequency Domain Analysis EE 564 © H. Heck 2008 Section 5.422 References R.E. Matick, Transmission Lines for Digital and Communication Networks, IEEE Press, 1995. 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, 1998. Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, 1985. U. Inan, A. Inan, Engineering Electromagnetics, Addison Wesley, 1999, 1 st edition. Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, 1985.