1 Instructor Richard Mellitz Introduction to Frequency Domain

1 Instructor: Richard Mellitz 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 Differential Signaling

Outline 2 üMotivation: Why Use Frequency Domain Analysis ü 2 -Port Network Analysis Theory üImpedance and Admittance Matrix üScattering Matrix üTransmission (ABCD) Matrix üMason’s Rule üCascading S-Matrices and Voltage Transfer Function üDifferential (4 -port) Scattering Matrix Differential Signaling

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

Key Concepts 4 Here are the key concepts that you should retain from this class üThe input impedance & the input reflection coefficient of a transmission line is dependent on: ØTermination and characteristic impedance ØDelay ØFrequency üS-Parameters are used to extract electrical parameters ØTransmission line parameters (R, L, C, G, TD and Zo) can be extracted from S parameters ØVias, connectors, socket s-parameters can be used to create equivalent circuits= üThe behavior of S-parameters can be used to gain intuition of signal integrity problems Differential Signaling

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

Two Port Network Theory üNetwork 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 üNetworks can have any number of ports, however, consideration of a 2 -port network is sufficient to explain theory ØA 2 -port network has 1 input and 1 output port. ØThe ports can be characterized with many parameters, each parameter has a specific advantage üEach Parameter set is related to 4 variables Ø 2 independent variables for excitation Ø 2 dependent variables for response Differential Signaling 6

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

Impedance Matrix 8 ü A set of linear equations can be written to describe the network in terms of its port impedances Or Where: 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 If the impedance matrix is known, the response of the system can be predicted for any input Differential Signaling

Impedance Matrix: Example #2 Calculate the impedance matrix for the following circuit: R 2 R 1 Port 1 R 3 Differential Signaling Port 2 9

Impedance Matrix: Example #2 Step 1: Calculate the input impedance R 2 R 1 + I 1 V 1 R 3 Step 2: Calculate the impedance across the network R 1 R 2 + I 1 R 3 V 2 Differential Signaling 10

Impedance Matrix: Example #2 Step 3: Calculate the Impedance matrix Assume: R 1 = R 2 = 30 ohms R 3=150 ohms Differential Signaling 11

Measuring the impedance matrix Question: ü What obstacles are expected when measuring the impedance matrix of the following transmission line structure assuming that the micro-probes have the following parasitics? Ø Lprobe=0. 1 n. H Ø Cprobe=0. 3 p. F Assume F=5 GHz Port 1 T-line 0. 1 n. H 0. 3 p. F Zo=50 ohms, length=5 in Differential Signaling Port 2 12

Measuring the impedance matrix 13 Answer: ü Open circuit voltages are very hard to measure at high frequencies because they generally do not exist for small dimensions Ø Open circuit capacitance = impedance at high frequencies Ø Probe and via impedance not insignificant Port 1 0. 1 n. H 0. 3 p. F Without Probe Capacitance 0. 1 n. H T-line Zo = 50 Port 1 Port 2 Port 0. 3 p. F Port 2 Z 21 = 50 ohms Zo=50 ohms, length=5 in With Probe Capacitance @ 5 GHz Zo = 50 Port 2 Port 1 106 ohms Differential Signaling 106 ohms Z 21 = 63 ohms

Advantages/Disadvantages of Impedance Matrix 14 Advantages: üThe impedance matrix is very intuitive ØRelates all ports to an impedance ØEasy to calculate Disadvantages: üRequires open circuit voltage measurements ØDifficult to measure ØOpen circuit reflections cause measurement noise ØOpen 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 Differential Signaling

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

Scattering Matrix 16 ü 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 Differential Signaling

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

Making sense of S-Parameters – Return Loss 18 üWhen there is a reflection from the load, S 11 will be composed of multiple reflections due to the standing waves Zo Z=-l RL Z=0 üIf the network is driven with a 50 ohm source, then S 11 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 Differential Signaling

Example #3 – Interpreting the return loss üBased on the S 11 plot shown below, calculate both the impedance and dielectric constant R=50 Zo L=5 inches R=50 0. 45 S 11, Magnitude 0. 4 0. 35 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 1. 5 2. 0 2. 5 3. . 0 3. 5 Differential Signaling Frequency, GHz 4. 0 4. 5 5. 0 19

Example – Interpreting the return loss 0. 45 S 11, Magnitude 0. 4 1. 76 GHz 20 2. 94 GHz Peak=0. 384 0. 35 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0 Frequency, GHz ü Step 1: Calculate the time delay ü Step 2: Calculate Er using the velocity of the t-line using the peaks Differential Signaling

Example – Interpreting the return loss ü Step 3: Calculate the input impedance to the transmission line based on the peak S 11 at 1. 76 GHz Note: The phase of the reflection should be either +1 or -1 at 1. 76 GHz because it is aligned with the incident ü Step 4: Calculate the characteristic impedance based on the input impedance for x=-5 inches Er=1. 0 and Zo=75 ohms Differential Signaling 21

Making sense of S-Parameters – Insertion Loss üWhen power is injected into Port 1 with source impedance Z 0 and measured at Port 2 with measurement load impedance Z 0, the power ratio reduces to a voltage ratio a 2=0 a 1 V 1 Zo 2 -port Network Zo V 2 b 1 S 21 is measure of the power transmitted from port 1 to port 2, and is called the “Insertion Loss” Differential Signaling 22

Loss free networks ü For a loss free network, the total power exiting the N ports must equal the total incident power ü 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 ü Since s-parameters are the square root of power ratios, the following is true for loss-free networks ü 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 Differential Signaling 23

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

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

Insertion loss example Answer: ü Since there are minimal reflections on this line, alpha can be estimated directly from the insertion loss Ø S 21~0. 75 at 4 GHz (8 GT/s) When the reflections are minimal, alpha can be estimated ü If S 11 < ~ 0. 2 (-14 d. B), then the above approximation is valid ü If the reflections are NOT small, alpha must be extracted with ABCD parameters (which are reviewed later) ü The loss parameter is “ 1/A” for ABCD parameters ü ABCE will be discussed later. Differential Signaling 26

Important concepts demonstrated ü The impedance can be determined by the magnitude of S 11 ü The electrical delay can be determined by the phase, or periodicity of S 11 ü The magnitude of the signal degradation can be determined by observing S 21 ü The total power dissipated by the network can be determined by adding the square of the insertion and return losses Differential Signaling 27

A note about the term “Loss” 28 üTrue losses come from physical energy losses ØOhmic (I. e. , skin effect) ØField dampening effects (Loss Tangent) ØRadiation (EMI) üInsertion and Return losses include effects such as impedance discontinuities and resonance effects, which are not true losses üLoss free networks can still exhibit significant insertion and return losses due to impedance discontinuities Differential Signaling

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

Cascading S parameter 30 3 cascaded s parameter blocks a 11 s 121 a 21 b 12 s 211 s 221 b 11 b 22 a 13 s 223 s 112 s 122 b 21 a 12 s 222 s 113 s 123 a 13 a 22 b 13 ü While it is possible to cascade s-parameters, it gets messy. ü Graphically we just flip every other matrix. ü Mathematically there is a better way… ABCD parameters ü We will analyzed this later with signal flow graphs Differential Signaling

ABCD Parameters 31 üThe transmission matrix describes the network in terms of both voltage and current waves I 2 I 1 V 1 2 -port Network V 2 üThe coefficients can be defined using superposition Differential Signaling

Transmission (ABCD) Matrix 32 üSince the ABCD matrix represents the ports in terms of currents and voltages, it is well suited for cascading elements I 1 V 1 I 3 I 2 V 3 üThe matrices can be cascaded by multiplication This is the best way to cascade elements in the frequency domain. Differential Signaling It is accurate, intuitive and simplistic.

33 Relating the ABCD Matrix to Common Circuits Z Port 1 Y Port 1 Z 1 Port 1 Assignment 6: Port 2 Convert these to s-parameters Port 2 Z 3 Port 2 Y 3 Port 1 Y 1 Port 1 Y 2 Port 2 Differential Signaling

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

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

ABCD Matrix – Example #1 36 üThe model can be extracted as either a Pi or a T network L 1 L 2 Port 1 CVIA Port 2 üThe 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 üThe capacitance represents the via pads Differential Signaling

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

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

ABCD Matrix – Example #1 üAssume the following s-matrix measured at 5 GHz üConvert to ABCD parameters üRelating the ABCD parameters to the T circuit topology, the capacitance and inductance is extracted from C & A Z 1 Port 1 Z 2 Z 3 Port 2 Differential Signaling 39

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

ABCD Matrix – Example #2 üStep 1: Convert each measured S-Matrix to ABCD Parameters using the conversions presented earlier üStep 2: Multiply the converted T-matrices üStep 3: Convert the resulting Matrix back into Sparameters using thee conversions presented earlier Differential Signaling 41

Advantages/Disadvantages of ABCD Matrix Advantages: üThe ABCD matrix is very intuitive ØDescribes all ports with voltages and currents üAllows easy cascading of networks üEasy conversion to and from S-parameters üEasy to relate to common circuit topologies Disadvantages: üDifficult to directly measure ØMust convert from measured scattering matrix Differential Signaling 42

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 a 1, a 2 on port 1 and port 2 respectively. The reflected waves b 1 and b 2 are on port 1 and port 2. We will use a’s and b’s in the s-parameter follow slides Differential Signaling 43

Signal Flow Graphs of S Parameters “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 a 1 s 21 b 2 GS s 22 s 11 GL s 12 b 1 a 2 Measurement equipment strives to be match i. e. reflection coefficient is 0 See: http: //cp. literature. agilent. com/litweb/pdf/5952 -1087. pdf Differential Signaling 44

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

Voltage Transfer function ü What is really of most relevance to time domain analysis is the voltage transfer function. ü It includes the effect of non-perfect loads. ü We will show the voltage transfer functions for a 2 port network is given by the following equation. ü Notice it is not S 21 Differential Signaling 46

Forward Wave Path Vs a 1 GS 47 s 21 b 2 s 22 s 11 GL s 12 b 1 Differential Signaling a 2

48 Reflected Wave Path Vs a 1 GS s 21 b 2 s 22 s 11 GL s 12 b 1 Differential Signaling a 2

Combine b 2 and a 2 Differential Signaling 49

Convert Wave to Voltage - Multiply by sqrt(Z 0) Differential Signaling 50

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

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

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

Cascading S-Parameter 54 ü As promised we will now look at how to cascade sparameters and solve with Mason’s rule ü The problem we will use is what was presented earlier ü The assertion is that the loss of cascade channel can be determine just by adding up the losses in d. B. ü We will show we can gain insight about this assertion from the equation and graphic form of a solution. a 11 b 11 a 21 b 12 b 22 a 13 s 111 s 121 s 113 s 123 s 211 s 221 s 213 s 223 s 112 s 122 b 21 a 12 s 222 a 22 b 13 Differential Signaling b 13

Creating the signal flow graph a 11 a 21 b 12 a 13 s 123 s 211 s 221 s 213 s 223 s 112 s 122 s 212 s 222 b 21 a 12 s 211 B 21 1 s 221 B 11 b 22 a 13 s 111 s 121 b 11 A 11 55 s 121 A 12 s 212 a 22 b 13 B 22 1 b 13 A 13 s 112 s 222 1 B 12 s 122 A 22 s 213 s 113 1 B 13 s 123 ü We map output a to input b and visa versa. ü Next we define all the loops ü Loop “A” and “B” do not touch each other Differential Signaling B 23 s 223 A 23

Use Mason’s rule A 11 s 211 B 21 1 s 221 B 11 s 121 A 21 56 A 12 s 212 B 22 1 A 13 s 112 s 222 1 B 12 s 122 A 22 s 213 s 113 1 B 13 s 123 Mason’s Rule ü There is only one forward path a 11 to b 23. ü There are 2 non touching looks Differential Signaling B 23 s 223 A 23

Evaluate the nature of the transfer function Assumption is that these are ~ 0 • If response is relatively flat and reflection is relatively low – Response through a channel is s 211*s 212*213… Differential Signaling 57

Jitter and d. B Budgeting 58 ü Change s 21 into a phasor = ü Insertion loss in db = i. e. For a budget just add up the db’s and jitter Differential Signaling

Differential S-Parameters 59 üDifferential S-Parameters are derived from a 4 -port measurement a 2 a 1 b 1 4 -port b 2 b 1 S 12 S 13 S 14 b 2 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44 a 3 a 4 b 3 b 4 = a 1 a 2 a 3 a 4 üTraditional 4 -port measurements are taken by driving each port, and recording the response at all other ports while terminated in 50 ohms üAlthough, 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 Signaling

60 Differential S-Parameters ü It is useful to specify the differential S-parameters in terms of differential and common mode responses ØDifferential stimulus, differential response ØCommon mode stimulus, Common mode response ØDifferential stimulus, common mode response (aka ACCM Noise) ØCommon mode stimulus, differential response ü This can be done either by driving the network with differential and common mode stimulus, or by converting the traditional 4 -port s-matrix bdm 1 DS 12 DCS 11 DCS 12 adm 1 bdm 2 DS 21 DS 22 DCS 21 DCS 22 adm 2 CDS 11 CDS 12 CS 11 CS 12 acm 1 CS 22 acm 2 bcm 1 bcm 2 = CDS 21 CDS 22 Matrix assumes differential and common mode stimulus Differential Signaling

Explanation of the Multi-Mode Port Differential Matrix: Differential Stimulus, differential response i. e. , DS 21 = differential signal [(D+)-(D-)] inserted at port 1 and diff signal measured at port 2 Common mode conversion Matrix: Differential Stimulus, Common mode response. i. e. , DCS 21 = differential signal [(D+)-(D-)] inserted at port 1 and common mode signal [(D+)+(D-)] measured at port 2 bdm 1 DS 12 DCS 11 DCS 12 adm 1 bdm 2 DS 21 DS 22 DCS 21 DCS 22 adm 2 CDS 11 CDS 12 CS 11 CS 12 acm 1 CS 22 acm 2 bcm 1 bcm 2 = CDS 21 CDS 22 61 differential mode conversion Matrix: Common mode Stimulus, differential Common mode stimulus, common mode response. i. e. , DCS 21 = common Response. i. e. , CS 21 = Com. mode signal [(D+)+(D-)] inserted at port 1 and Com. mode 1 and differential mode signal [(D+)-(D-)] Differential Signaling signal measured at port 2

Differential S-Parameters 62 ü 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: 1 3 4 -port 2 -port Network 2 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. Differential can be used for all multi-mode Parameters Signaling

Next class we will develop more differential concepts Differential Signaling 63

backup review 64 Differential Signaling

Advantages/Disadvantages of Multi-Mode Matrix over Traditional 4 -port Advantages: üDescribes 4 -port network in terms of 4 two port matrices ØDifferential ØCommon mode ØDifferential to common mode ØCommon mode to differential üEasier to relate to system specifications ØACCM noise, differential impedance Disadvantages: üMust convert from measured 4 -port scattering matrix Differential Signaling 65

High Frequency Electromagnetic Waves ü In order to understand the frequency domain analysis, it is necessary to explore how high frequency sinusoid signals behave on transmission lines ü The equations that govern signals propagating on a transmission line can be derived from Amperes and Faradays laws assumimng a uniform plane wave ØThe fields are constrained so that there is no variation in the X and Y axis and the propagation is in the Z direction ü This assumption holds true for transmission lines as long as the wavelength of the signal is much greater than the trace width X Direction of propagation Z Y For typical PCBs at 10 GHz with 5 mil traces (W=0. 005”) Differential Signaling 66

High Frequency Electromagnetic Waves ü 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 ü Note that the electric (Ex) field and the magnetic (By) are orthogonal Differential Signaling 67

High Frequency Electromagnetic Waves ü 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 Ex: Differential Signaling 68

High Frequency Electromagnetic Waves 69 üThe equation describes the sinusoidal E field for a plane wave in free space Note the positive exponent is because the wave is traveling in the opposite direction Portion of wave traveling In the +z direction Portion of wave traveling In the -z direction = permittivity in Farads/meter (8. 85 p. F/m for free space) (determines the speed of light in a material) = permeability in Henries/meter (1. 256 u. H/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 Differential Signaling

High Frequency Voltage and Current Waves 70 ü The same equation applies to voltage and current waves on a transmission line Incident sinusoid Reflected sinusoid z=-l RL z=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 Note: is added to specifically represent the time varying Sinusoid, which was implied in the previous derivation Voltage wave traveling towards the load Voltage wave reflecting off the Load and traveling towards the source Differential Signaling

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

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

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

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

Line & load interactions ü In chapter 2, you learned how to calculate waveforms in a multi-reflective system using lattice diagrams Ø Period of transmission line “ringing” proportional to the line delay Ø Remember, the line delay is proportional to the phase constant ü 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 Ø This will become more evident as the class progresses To demonstrate, lets assume a loss free transmission line Differential Signaling 75

Line & load interactions 76 Remember, the input reflection takes the form The frequency where the values of the real & imaginary reflections are zero can be calculated based on the line length 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. Differential Signaling

Example #1: Periodic Reflections Calculate: 1. Line length 2. RL 77 Er_eff=1. 0 Zo=75 3. (assume a very low loss line) Z=-l Differential Signaling RL Z=0

Example #1: Solution Step 1: Determine the periodicity zero crossings or peaks & use the relationships on page 15 to calculate the electrical length Imaginary Differential Signaling 78

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

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