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1. Introduction to Frequency Domain Analysis (3 Classes) Many thanks to Steve Hall, Intel for the use of his slides Reference Reading: Posar Ch 4.5 http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf

2. Outline • Motivation: Why Use Frequency Domain Analysis • 2-Port Network Analysis Theory • Impedance and Admittance Matrix • Scattering Matrix • Transmission (ABCD) Matrix • Mason’s Rule • Cascading S-Matrices and Voltage Transfer Function • Differential (4-port) Scattering Matrix

3. Motivation: Why Frequency Domain Analysis? • Time Domain signals on T-lines lines are hard to analyze • Many properties, which can dominate performance, are frequency dependent, and difficult to directly observe in the time domain • Skin effect, Dielectric losses, dispersion, resonance • Frequency Domain Analysis allows discrete characterization of a linear network at each frequency • Characterization at a single frequency is much easier • Frequency Analysis is beneficial for Three reasons • Ease and accuracy of measurement at high frequencies • Simplified mathematics • Allows separation of electrical phenomena (loss, resonance … etc)

4. Key Concepts Here are the key concepts that you should retain from this class • 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) can be extracted from S parameters • Vias, connectors, socket s-parameters can be used to create equivalent circuits= • The behavior of S-parameters can be used to gain intuition of signal integrity problems

5. Review – Important Concepts • The impedance looking into a terminated transmission line changes with frequency and line length • The input reflection coefficient looking into a terminated transmission line also changes with frequency and line length • If the input reflection of a transmission line is known, then the line length can be determined by observing the periodicity of the reflection • The peak of the input reflection can be used to determine line and load impedance values

6. Two Port Network Theory • Network theory is based on the property that a linear system can be completely characterized by parameters measured ONLY at the input & output ports without regard to the content of the system • Networks can have any number of ports, however, consideration of a 2-port network is sufficient to explain the theory • A 2-port network has 1 input and 1 output port. • The ports can be characterized with many parameters, each parameter has a specific advantage • Each Parameter set is related to 4 variables • 2 independent variables for excitation • 2 dependent variables for response

7. Network characterized with Port Impedance • Measuring the port impedance is network is the most simplistic and intuitive method of characterizing a network I I I I 1 1 2 2 port 2 2 - - port + + + + Port 2 V V V V Port 1 1 1 2 2 Network Network - - - - Case 1: Inject current I1 into port 1 and measure the open circuit voltage at port 2 and calculate the resultant impedance from port 1 to port 2 Case 2: Inject current I1 into port 1 and measure the voltage at port 1 and calculate the resultant input impedance

8. Impedance Matrix • A set of linear equations can be written to describe the network in terms of its port impedances Where: If the impedance matrix is known, the response of the system can be predicted for any input Or Open Circuit Voltage measured at Port i Current Injected at Port j Zii the impedance looking into port i Zij the impedance between port i and j

9. Impedance Matrix: Example #2 Calculate the impedance matrix for the following circuit: R2 R1 R3 Port 2 Port 1

10. Impedance Matrix: Example #2 Step 1: Calculate the input impedance R2 R1 + - R3 I1 V1 Step 2: Calculate the impedance across the network R1 R2 + - R3 I1 V2

11. Impedance Matrix: Example #2 Step 3: Calculate the Impedance matrix Assume: R1 = R2 = 30 ohms R3=150 ohms

12. Measuring the impedance matrix 0.1nH Port 1 T-line 0.1nH Port 2 0.3pF 0.3pF Zo=50 ohms, length=5 in Question: • What obstacles are expected when measuring the impedance matrix of the following transmission line structure assuming that the micro-probes have the following parasitics? • Lprobe=0.1nH • Cprobe=0.3pF Assume F=5 GHz

13. Measuring the impedance matrix 0.1nH Port 1 T-line 0.1nH Port 2 0.3pF 0.3pF Zo=50 ohms, length=5 in Answer: • Open circuit voltages are very hard to measure at high frequencies because they generally do not exist for small dimensions • Open circuit  capacitance = impedance at high frequencies • Probe and via impedance not insignificant Without Probe Capacitance 0.1nH T-line Zo = 50 Port 1 Port 2 Port 2 Z21 = 50 ohms With Probe Capacitance @ 5 GHz Zo = 50 Port 2 Port 1 106 ohms 106 ohms Z21 = 63 ohms

14. Advantages/Disadvantages of Impedance Matrix Advantages: • The impedance matrix is very intuitive • Relates all ports to an impedance • Easy to calculate Disadvantages: • Requires open circuit voltage measurements • Difficult to measure • Open circuit reflections cause measurement noise • Open circuit capacitance not trivial at high frequencies Note: The Admittance Matrix is very similar, however, it is characterized with short circuit currents instead of open circuit voltages

15. Scattering Matrix (S-parameters) • Measuring the “power” at each port across a well characterized impedance circumvents the problems measuring high frequency “opens” & “shorts” • The scattering matrix, or (S-parameters), characterizes the network by observing transmitted & reflected power waves a2 a1 2 2 - - port port Port 1 Port 2 R R Network Network b2 b1 ai represents the square root of the power wave injected into port i bj represents the power wave coming out of port j

16. Scattering Matrix • A set of linear equations can be written to describe the network in terms of injected and transmitted power waves Where: Sii = the ratio of the reflected power to the injected power at port i Sij= the ratio of the power measured at port j to the power injected at port i

17. Making sense of S-Parameters – Return Loss • When there is no reflection from the load, or the line length is zero, S11 = Reflection coefficient R=50 Zo R=Zo Z=-l Z=0 S11 is measure of the power returned to the source, and is called the “Return Loss”

18. Making sense of S-Parameters – Return Loss • When there is a reflection from the load, S11will be composed of multiple reflections due to the standing waves Zo RL Z=0 Z=-l • If the network is driven with a 50 ohm source, then S11 is calculated using the input impedance instead of Zo 50 ohms S11 of a transmission line will exhibit periodic effects due to the standing waves

19. Example #3 – Interpreting the return loss • Based on the S11 plot shown below, calculate both the impedance and dielectric constant R=50 Zo R=50 L=5 inches 0.45 0.4 0.35 0.3 S11, Magnitude 0.25 0.2 0.15 0.1 0.05 0 1.0 1.5 2.0 2.5 3..0 3.5 4.0 4.5 5.0 Frequency, GHz

20. Example – Interpreting the return loss 0.45 1.76GHz 2.94GHz 0.4 Peak=0.384 0.35 0.3 S11, Magnitude 0.25 0.2 0.15 0.1 0.05 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Frequency, GHz • Step 2: Calculate Er using the velocity • Step 1: Calculate the time delay of the t-line using the peaks

21. Example – Interpreting the return loss • Step 3: Calculate the input impedance to the transmission line based on the peak S11 at 1.76GHz Note: The phase of the reflection should be either +1 or -1 at 1.76 GHz because it is aligned with the incident • Step 4: Calculate the characteristic impedance based on the input impedance for x=-5 inches Er=1.0 and Zo=75 ohms

22. Making sense of S-Parameters – Insertion Loss • When power is injected into Port 1 with source impedance Z0 and measured at Port 2 with measurement load impedance Z0, the power ratio reduces to a voltage ratio a2=0 a1 2 2 - - port port V1 Zo Zo V2 Network Network b2 b1 S21 is measure of the power transmitted from port 1 to port 2, and is called the “Insertion Loss”

23. Loss free networks • For a loss free network, the total power exiting the N ports must equal the total incident power • If there is no loss in the network, the total power leaving the network must be accounted for in the power reflected from the incident port and the power transmitted through network • Since s-parameters are the square root of power ratios, the following is true for loss-free networks • If the above relationship does not equal 1, then there is loss in the network, and the difference is proportional to the power dissipated by the network

24. Insertion loss example Question: • What percentage of the total power is dissipated by the transmission line? • Estimate the magnitude of Zo (bound it)

25. Insertion loss example • What percentage of the total power is dissipated by the transmission line ? • What is the approximate Zo? • How much amplitude degradation will this t-line contribute to a 8 GT/s signal? • If the transmission line is placed in a 28 ohm system (such as Rambus), will the amplitude degradation estimated above remain constant? • Estimate alpha for 8 GT/s signal

26. Insertion loss example Answer: • Since there are minimal reflections on this line, alpha can be estimated directly from the insertion loss • S21~0.75 at 4 GHz (8 GT/s) When the reflections are minimal, alpha can be estimated • If S11 < ~ 0.2 (-14 dB), then the above approximation is valid • If the reflections are NOT small, alpha must be extracted with ABCD parameters (which are reviewed later) • The loss parameter is “1/A” for ABCD parameters • ABCD will be discussed later.

27. Important concepts demonstrated • The impedance can be determined by the magnitude of S11 • The electrical delay can be determined by the phase, or periodicity of S11 • The magnitude of the signal degradation can be determined by observing S21 • The total power dissipated by the network can be determined by adding the square of the insertion and return losses

28. A note about the term “Loss” • True losses come from physical energy losses • Ohmic (I.e., skin effect) • Field dampening effects (Loss Tangent) • Radiation (EMI) • Insertion and Return losses include effects such as impedance discontinuities and resonance effects, which are not true losses • Loss free networks can still exhibit significant insertion and return losses due to impedance discontinuities

29. Advantages/Disadvantages of S-parameters Advantages: • Ease of measurement • Much easier to measure power at high frequencies than open/short current and voltage • S-parameters can be used to extract the transmission line parameters • n parameters and n Unknowns Disadvantages: • Most digital circuit operate using voltage thresholds. This suggest that analysis should ultimately be related to the time domain. • Many silicon loads are non-linear which make the job of converting s-parameters back into time domain non-trivial. • Conversion between time and frequency domain introduces errors

30. Cascading S parameter s112 s122 s212 s222 s113 s123 s213 s223 s111 s121 s211 s221 3 cascaded s parameter blocks • While it is possible to cascade s-parameters, it gets messy. • Graphically we just flip every other matrix. • Mathematically there is a better way… ABCD parameters • We will analyzed this later with signal flow graphs a11 a21 b12 b22 a13 a13 b11 b21 a12 a22 b13 b13

31. ABCD Parameters • The transmission matrix describes the network in terms of both voltage and current waves I2 I1 2 2 - - port port V1 V2 Network Network • The coefficients can be defined using superposition

32. Transmission (ABCD) Matrix • Since the ABCD matrix represents the ports in terms of currents and voltages, it is well suited for cascading elements I3 I2 I1 V3 V1 V2 • The matrices can be cascaded by multiplication This is the best way to cascade elements in the frequency domain. It is accurate, intuitive and simplistic.

33. Relating the ABCD Matrix to Common Circuits Z Assignment 6: Convert these to s-parameters Port 1 Port 2 Y Port 1 Port 2 Z1 Z2 Port 1 Port 2 Z3 Y3 Y1 Y2 Port 2 Port 1 Port 1 Port 2

34. Converting to and from the S-Matrix • The S-parameters can be measured with a VNA, and converted back and forth into ABCD the Matrix • Allows conversion into a more intuitive matrix • Allows conversion to ABCD for cascading • ABCD matrix can be directly related to several useful circuit topologies

35. ABCD Matrix – Example #1 Port 1 Port 2 • Create a model of a via from the measured s-parameters

36. ABCD Matrix – Example #1 Port 1 Port 2 • The model can be extracted as either a Pi or a T network L2 L1 CVIA • The inductance values will include the L of the trace and the via barrel (it is assumed that the test setup minimizes the trace length, and subsequently the trace capacitance is minimal • The capacitance represents the via pads

37. ABCD Matrix – Example #1 • Assume the following s-matrix measured at 5 GHz

38. ABCD Matrix – Example #1 • Assume the following s-matrix measured at 5 GHz • Convert to ABCD parameters

39. ABCD Matrix – Example #1 • Assume the following s-matrix measured at 5 GHz • Convert to ABCD parameters • Relating the ABCD parameters to the T circuit topology, the capacitance and inductance is extracted from C & A Z1 Z2 Port 1 Port 2 Z3

40. ABCD Matrix – Example #2 • Calculate the resulting s-parameter matrix if the two circuits shown below are cascaded Port 1 Port 2 2 - port 50 50 Network X Network Port 1 Port 2 2 - port 50 50 Network Y Network 2 - port 2 - port 50 Network Y 50 Network X Network Network Port 2 Port 1

41. ABCD Matrix – Example #2 • Step 1: Convert each measured S-Matrix to ABCD Parameters using the conversions presented earlier • Step 2: Multiply the converted T-matrices • Step 3: Convert the resulting Matrix back into S-parameters using thee conversions presented earlier

42. Advantages/Disadvantages of ABCD Matrix Advantages: • The ABCD matrix is very intuitive • Describes all ports with voltages and currents • Allows easy cascading of networks • Easy conversion to and from S-parameters • Easy to relate to common circuit topologies Disadvantages: • Difficult to directly measure • Must convert from measured scattering matrix

43. Signal flow graphs – Start with 2 port first The wave functions (a,b) used to define s-parameters for a two-port network are shown below. The incident waves is a1, a2 on port 1 and port 2 respectively. The reflected waves b1 and b2 are on port 1 and port 2. We will use a’s and b’s in the s-parameter follow slides

44. Signal Flow Graphs of S Parameters a1 s21 b2 s22 s11 GS GL s12 b1 a2 “In a signal flow graph, each port is represented by two nodes. Node an represents the wave coming into the device from another device at port n, and node bn represents the wave leaving the device at port n. The complex scattering coefficients are then represented as multipliers (gains) on branches connecting the nodes within the network and in adjacent networks.”* Example Measurement equipment strives to be match i.e. reflection coefficient is 0 See: http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf

45. Mason’s Rule ~ Non-Touching Loop Rule • T is the transfer function (often called gain) • Tkis the transfer function of the kth forward path • L(mk) is the product of non touching loop gains on path k taken mk at time. • L(mk)|(k) is the product of non touching loop gains on path k taken mk at a time but not touching path k. • mk=1 means all individual loops

46. What is really of most relevance to time domain analysis is the voltage transfer function. It includes the effect of non-perfect loads. We will show how the voltage transfer functions for a 2 port network is given by the following equation. Notice it is not S21 Voltage Transfer function

47. Forward Wave Path Vs a1 s21 b2 s22 s11 GS GL s12 b1 a2

48. Reflected Wave Path Vs a1 s21 b2 s22 s11 GS GL s12 b1 a2

49. Combine b2 and a2

50. Convert Wave to Voltage - Multiply by sqrt(Z0)