Differential Signaling 1 Introduction to Frequency Domain Analysis (3 Classes) Many thanks to Steve Hall, Intel for the use of his slides Reference Reading:

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1 Differential Signaling 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 Slide content from Stephen Hall Instructor: Richard Mellitz

2 Differential Signaling 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 Differential Signaling 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 Differential Signaling 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 Differential Signaling 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 Differential Signaling 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 Differential Signaling 7 Network characterized with Port Impedance Measuring the port impedance is network is the most simplistic and intuitive method of characterizing a network Port 1 Port 2 Case 1 Case 1: Inject current I 1 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 Case 2: Inject current I 1 into port 1 and measure the voltage at port 1 and calculate the resultant input impedance 2-port Network I 1 I 2 + - V 1 V 2 + - 2- port Networ k I 1 I 2 + - V 1 V 2 + -

8 Differential Signaling 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 Open Circuit Voltage measured at Port i Current Injected at Port j Z ii  the impedance looking into port i Z ij  the impedance between port i and j Or

9 Differential Signaling 9 Impedance Matrix: Example #2 Calculate the impedance matrix for the following circuit: Port 1 Port 2 R1R1 R2R2 R3R3

10 Differential Signaling 10 Impedance Matrix: Example #2 Step 1: Calculate the input impedance R1R1 R2R2 R3R3 I1I1 V1V1 +-+- Step 2: Calculate the impedance across the network R1R1 R2R2 R3R3 I1I1 V2V2 +-+-

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

12 Differential Signaling 12 Measuring the impedance matrix 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?  L probe =0.1nH  C probe =0.3pF Assume F=5 GHz T-line 0.1nH Port 1 Port 2 0.3pF 0.1nH 0.3pF Zo=50 ohms, length=5 in

13 Differential Signaling 13 Measuring the impedance matrix T-line Port 2 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 0.1nH 106 ohms Zo = 50 Without Probe Capacitance Zo = 50 With Probe Capacitance @ 5 GHz Z 21 = 50 ohms Z 21 = 63 ohms Port 1 Port 2 Port 1 Port 2 T-line 0.1nH Port 1 Port 2 0.3pF 0.1nH 0.3pF Zo=50 ohms, length=5 in

14 Differential Signaling 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 Differential Signaling 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 2-port Network 2-port Network Port 1Port 2 a i represents the square root of the power wave injected into port i a1a1 a2a2 b2b2 b1b1 b j represents the power wave coming out of port j R R

16 Differential Signaling 16 Scattering Matrix A set of linear equations can be written to describe the network in terms of injected and transmitted power waves Where: S ii = the ratio of the reflected power to the injected power at port i S ij = the ratio of the power measured at port j to the power injected at port i

17 Differential Signaling 17 Making sense of S-Parameters – Return Loss When there is no reflection from the load, or the line length is zero, S 11 = Reflection coefficient S 11 is measure of the power returned to the source, and is called the “Return Loss” R=Zo Z=-lZ=0 Zo R=50

18 Differential Signaling 18 Making sense of S-Parameters – Return Loss When there is a reflection from the load, S 11 will be composed of multiple reflections due to the standing waves RLRL Z=-l Z=0 Zo If the network is driven with a 50 ohm source, then S11 is calculated using the input impedance instead of Zo 50 ohms S 11 of a transmission line will exhibit periodic effects due to the standing waves

19 Differential Signaling 19 Example #3 – Interpreting the return loss Based on the S11 plot shown below, calculate both the impedance and dielectric constant 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 1.01.52.02.53..03.54.04.55.0 Frequency, GHz S11, Magnitude R=50 L=5 inches Zo R=50

20 Differential Signaling 20 Example – Interpreting the return loss Step 1: Calculate the time delay of the t-line using the peaks Step 2: Calculate Er using the velocity 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 1.01.52.02.53.03.54.04.55.0 Frequency, GHz S11, Magnitude 1.76GHz2.94GHz Peak=0.384

21 Differential Signaling 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 Differential Signaling 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 2-port Network 2-port Network V1V1 a1a1 a 2 =0 b2b2 b1b1 V2V2 Zo S21 is measure of the power transmitted from port 1 to port 2, and is called the “Insertion Loss”

23 Differential Signaling 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 Differential Signaling 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 Differential Signaling 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 Differential Signaling 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 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 ABCE will be discussed later. If S 11 < ~ 0.2 (-14 dB), then the above approximation is valid

27 Differential Signaling 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 Differential Signaling 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 Differential Signaling 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 Differential Signaling 30 Cascading S parameter 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 a1 1 b1 1 a2 1 b2 1 a1 2 b1 2 a2 2 b2 2 a1 3 b1 3 a1 3 b1 3 s11 1 s12 1 s21 1 s22 1 s11 2 s12 2 s21 2 s22 2 s11 3 s12 3 s21 3 s22 3 3 cascaded s parameter blocks

31 Differential Signaling 31 ABCD Parameters The transmission matrix describes the network in terms of both voltage and current waves 2-port Network 2-port Network V1V1 I1I1 I2I2 V2V2 The coefficients can be defined using superposition

32 Differential Signaling 32 Transmission (ABCD) Matrix Since the ABCD matrix represents the ports in terms of currents and voltages, it is well suited for cascading elements V1V1 I1I1 I2I2 V2V2 The matrices can be cascaded by multiplication I3I3 V3V3 This is the best way to cascade elements in the frequency domain. It is accurate, intuitive and simplistic.

33 Differential Signaling 33 Relating the ABCD Matrix to Common Circuits Z Port 1Port 2 Port 1 Y Port 2 Z1Z1 Port 1Port 2 Z2Z2 Z3Z3 Y1Y1 Port 1 Port 2 Y2Y2 Y3Y3 Port 1Port 2 Assignment 6: Convert these to s-parameters

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

36 Differential Signaling 36 ABCD Matrix – Example #1 The model can be extracted as either a Pi or a T network L1 C VIA L2 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 Port 1 Port 2

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

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

39 Differential Signaling 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 Z1Z1 Port 1Port 2 Z2Z2 Z3Z3

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

41 Differential Signaling 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 Differential Signaling 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 Differential Signaling 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 Differential Signaling 44 Signal Flow Graphs of S Parameters “In a signal flow graph, each port is represented by two nodes. Node a n represents the wave coming into the device from another device at port n, and node b n 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.”* a1 b1 b2 a2 SS LL s21 s12 s11 s22 Example Measurement equipment strives to be match i.e. reflection coefficient is 0 See: http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdfhttp://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf

45 Differential Signaling 45 Mason’s Rule ~ Non-Touching Loop Rule T is the transfer function (often called gain) T k is the transfer function of the k th 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 Differential Signaling 46 Voltage Transfer function 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

47 Differential Signaling 47 Forward Wave Path a1 b1 b2 a2 Vs SS LL s21 s12 s11 s22

48 Differential Signaling 48 Reflected Wave Path a1 b1 b2 a2 Vs SS LL s21 s12 s11 s22

49 Differential Signaling 49 Combine b2 and a2

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

51 Differential Signaling 51 Voltage transfer function using ABCD Let’s see if we can get this results another way

52 Differential Signaling 52 Cascade [ABCD] to determine system [ABCD]

53 Differential Signaling 53 Extract the voltage transfer function Same as with flow graph analysis

54 Differential Signaling 54 Cascading S-Parameter As promised we will now look at how to cascade s- parameters and solve with Mason’s rule The problem we will use is what was presented earlier The assertion is that the loss of cascade channel can be determine just by adding up the losses in dB. We will show how we can gain insight about this assertion from the equation and graphic form of a solution. a1 1 b1 1 a2 1 b2 1 a1 2 b1 2 a2 2 b2 2 a1 3 b1 3 a1 3 b1 3 s11 1 s12 1 s21 1 s22 1 s11 2 s12 2 s21 2 s22 2 s11 3 s12 3 s21 3 s22 3

55 Differential Signaling 55 Creating the signal flow graph We map output a to input b and visa versa. Next we define all the loops Loop “A” and “B” do not touch each other 1 11 1 A1 1 B2 1 A1 2 B2 2 A1 3 B2 3 B1 1 A2 1 B1 2 A2 2 B1 3 A2 3 s12 1 s22 1 s21 1 s12 3 s21 3 s11 3 s22 3 s21 2 s11 2 s22 2 s12 2 a1 1 b1 1 a2 1 b2 1 a1 2 b1 2 a2 2 b2 2 a1 3 b1 3 a1 3 b1 3 s11 1 s12 1 s21 1 s22 1 s11 2 s12 2 s21 2 s22 2 s11 3 s12 3 s21 3 s22 3

56 Differential Signaling 56 Use Mason’s rule There is only one forward path a1 1 to b2 3. There are 2 non touching looks Mason’s Rule 1 11 1 A1 1 B2 1 A1 2 B2 2 A1 3 B2 3 B1 1 A2 1 B1 2 A2 2 B1 3 A2 3 s12 1 s22 1 s21 1 s12 3 s21 3 s11 3 s22 3 s21 2 s11 2 s22 2 s12 2

57 Differential Signaling 57 Evaluate the nature of the transfer function If response is relatively flat and reflection is relatively low –Response through a channel is s21 1 *s21 2 *21 3 … Assumption is that these are ~ 0

58 Differential Signaling 58 Jitter and dB Budgeting Change s21 into a phasor Insertion loss in db = i.e. For a budget just add up the db’s and jitter =

59 Differential Signaling 59 Differential S-Parameters are derived from a 4-port measurement Traditional 4-port measurements are taken by driving each port, and recording the response at all other ports while terminated in 50 ohms Although, it is perfectly adequate to describe a differential pair with 4-port single ended s-parameters, it is more useful to convert to a multi-mode port Differential S-Parameters 4-port a 1 a 2 b 1 b 2 S 11 S 22 S 21 S 12 S 33 S 44 S 43 S 34 S 31 S 42 S 41 S 32 S 13 S 24 S 23 S 14 b 1 b 2 b 3 b 4 a 1 a 2 a 3 a 4 = a 3 b 3 a 4 b 4

60 Differential Signaling 60 Differential S-Parameters Matrix assumes differential and common mode stimulus It is useful to specify the differential S-parameters in terms of differential and common mode responses  Differential stimulus, differential response  Common mode stimulus, Common mode response  Differential stimulus, common mode response (aka ACCM Noise)  Common mode stimulus, differential response This can be done either by driving the network with differential and common mode stimulus, or by converting the traditional 4-port s-matrix DS 11 DS 22 DS 21 DS 12 CS 11 CS 22 CS 21 CS 12 CDS 11 CDS 22 CDS 21 CDS 12 DCS 11 DCS 22 DCS 21 DCS 12 b dm1 b dm2 b cm1 b cm2 a dm1 a dm2 a cm1 a cm2 =

61 Differential Signaling 61 Explanation of the Multi-Mode Port Differential Matrix: Differential Stimulus, differential response i.e., DS21 = differential signal [(D+)-(D-)] inserted at port 1 and diff signal measured at port 2 Common mode Matrix: Common mode stimulus, common mode Response. i.e., CS21 = Com. mode signal [(D+)+(D-)] inserted at port 1 and Com. mode signal measured at port 2 Common mode conversion Matrix: Differential Stimulus, Common mode response. i.e., DCS21 = differential signal [(D+)-(D-)] inserted at port 1 and common mode signal [(D+)+(D-)] measured at port 2 differential mode conversion Matrix: Common mode Stimulus, differential mode response. i.e., DCS21 = common mode signal [(D+)+(D-)] inserted at port 1 and differential mode signal [(D+)-(D-)] measured at port 2 DS 11 DS 22 DS 21 DS 12 CS 11 CS 22 CS 21 CS 12 CDS 11 CDS 22 CDS 21 CDS 12 DCS 11 DCS 22 DCS 21 DCS 12 b dm1 b dm2 b cm1 b cm2 a dm1 a dm2 a cm1 a cm2 =

62 Differential Signaling 62 Differential S-Parameters Converting the S-parameters into the multi-mode requires just a little algebra Example Calculation, Differential Return Loss The stimulus is equal, but opposite, therefore: 2-port Network 4-port Network 2 1 3 4 Assume a symmetrical network and substitute Other conversions that are useful for a differential bus are shown Differential Insertion Loss: Differential to Common Mode Conversion (ACCM): Similar techniques can be used for all multi-mode Parameters

63 Differential Signaling 63 Next class we will develop more differential concepts

64 Differential Signaling 64 backup review

65 Differential Signaling 65 Advantages/Disadvantages of Multi-Mode Matrix over Traditional 4-port Advantages: Describes 4-port network in terms of 4 two port matrices  Differential  Common mode  Differential to common mode  Common mode to differential Easier to relate to system specifications  ACCM noise, differential impedance Disadvantages: Must convert from measured 4-port scattering matrix

66 Differential Signaling 66 High Frequency Electromagnetic Waves In order to understand the frequency domain analysis, it is necessary to explore how high frequency sinusoid signals behave on transmission lines The equations that govern signals propagating on a transmission line can be derived from Amperes and Faradays laws assumimng a uniform plane wave  The fields are constrained so that there is no variation in the X and Y axis and the propagation is in the Z direction Z X Y Direction of propagation This assumption holds true for transmission lines as long as the wavelength of the signal is much greater than the trace width For typical PCBs at 10 GHz with 5 mil traces (W=0.005”)

67 Differential Signaling 67 High Frequency Electromagnetic Waves For sinusoidal time varying uniform plane waves, Amperes and Faradays laws reduce to: Amperes Law: A magnetic Field will be induced by an electric current or a time varying electric field Faradays Law: An electric field will be generated by a time varying magnetic flux Note that the electric (E x ) field and the magnetic (B y ) are orthogonal

68 Differential Signaling 68 High Frequency Electromagnetic Waves If Amperes and Faradays laws are differentiated with respect to z and the equations are written in terms of the E field, the transmission line wave equation is derived This differential equation is easily solvable for E x :

69 Differential Signaling 69 High Frequency Electromagnetic Waves The equation describes the sinusoidal E field for a plane wave in free space Portion of wave traveling In the +z direction Portion of wave traveling In the -z direction Note the positive exponent is because the wave is traveling in the opposite direction = permittivity in Farads/meter (8.85 pF/m for free space) (determines the speed of light in a material) = permeability in Henries/meter (1.256 uH/m for free space and non-magnetic materials) Since inductance is proportional to & capacitance is proportional to, then is analogous to in a transmission line, which is the propagation delay

70 Differential Signaling 70 High Frequency Voltage and Current Waves The same equation applies to voltage and current waves on a transmission line RLRL z=-lz=0 If a sinusoid is injected onto a transmission line, the resulting voltage is a function of time and distance from the load (z). It is the sum of the incident and reflected values Voltage wave traveling towards the load Voltage wave reflecting off the Load and traveling towards the source Incident sinusoid Reflected sinusoid Note: is added to specifically represent the time varying Sinusoid, which was implied in the previous derivation

71 Differential Signaling 71 High Frequency Voltage and Current Waves = Attenuation Constant (attenuation of the signal due to transmission line losses) = Phase Constant (related to the propagation delay across the transmission line) = Complex propagation constant – includes all the transmission line parameters (R, L C and G) (For the loss free case) (lossy case) (For good conductors) (For good conductors and good dielectrics) The parameters in this equation completely describe the voltage on a typical transmission line

72 Differential Signaling 72 High Frequency Voltage and Current Waves Subsequently : The voltage wave equation can be put into more intuitive terms by applying the following identity: The amplitude is degraded by The waveform is dependent on the driving function ( ) & the delay of the line

73 Differential Signaling 73 Interaction: transmission line and a load (Assume a line length of l (z=-l)) This is the reflection coefficient looking into a t-line of length l ZlZl Z=-lZ=0 The reflection coefficient is now a function of the Zo discontinuities AND line length  Influenced by constructive & destructive combinations of the forward & reverse waveforms Zo

74 Differential Signaling 74 This is the input impedance looking into a t-line of length l RLRL Z=-lZ=0 Interaction: transmission line and a load If the reflection coefficient is a function of line length, then the input impedance must also be a function of length Note: is dependent on and Z in

75 Differential Signaling 75 Line & load interactions In chapter 2, you learned how to calculate waveforms in a multi-reflective system using lattice diagrams  Period of transmission line “ringing” proportional to the line delay  Remember, the line delay is proportional to the phase constant In frequency domain analysis, the same principles apply, however, it is more useful to calculate the frequency when the reflection coefficient is either maximum or minimum  This will become more evident as the class progresses To demonstrate, lets assume a loss free transmission line

76 Differential Signaling 76 Line & load interactions The frequency where the values of the real & imaginary reflections are zero can be calculated based on the line length Term 1Term 2 Term 1=0 Term 2 = Term 2=0 Term 1 = Note that when the imaginary portion is zero, it means the phase of the incident & reflected waveforms at the input are aligned. Also notice that value of “8” and “4” in the terms. Remember, the input reflection takes the form

77 Differential Signaling 77 Example #1: Periodic Reflections Calculate: 1.Line length 2.R L (assume a very low loss line) RLRL Z=-lZ=0 Zo=75 Er_eff=1.0

78 Differential Signaling 78 Example #1: Solution Step 1: Determine the periodicity zero crossings or peaks & use the relationships on page 15 to calculate the electrical length -2.5E-01 -2.0E-01 -1.5E-01 -1.0E-01 -5.0E-02 0.0E+00 5.0E-02 1.0E-01 1.5E-01 2.0E-01 2.5E-01 0.0E+005.0E+081.0E+091.5E+092.0E+092.5E+093.0E+09Frequency Reflection Coeff. Real Imaginary

79 Differential Signaling 79 Example #1: Solution (cont.) Since TD and the effective Er is known, the line length can be calculated as in chapter 2 Note the relationship between the peaks and the electrical length This leads to a very useful equation for transmission lines

80 Differential Signaling 80 Example #1: Solution (cont.) The load impedance can be calculated by observing the peak values of the reflection  When the imaginary term is zero, the real term will peak, and the maximum reflection will occur  If the imaginary term is zero, the reflected wave is aligned with the incident wave and the phase term = 1 Important Concepts demonstrated The impedance can be determined by the magnitude of the reflection The line length can be determined by the phase, or periodicity of the reflection