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## Introduction to Frequency Domain Analysis (3 Classes) Many thanks to Steve Hall,

Carrier, Michael, Founder and CEO has reference to this Academic Journal, PHwiki organized this Journal Introduction to Frequency Domain Analysis (3 Classes) Many thanks to Steve Hall, Intel as long as 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 Outline Motivation: Why Use Frequency Domain Analysis 2-Port Network Analysis Theory Impedance in addition to Admittance Matrix Scattering Matrix Transmission (ABCD) Matrix Masons Rule Cascading S-Matrices in addition to Voltage Transfer Function Differential (4-port) Scattering Matrix Motivation: Why Frequency Domain Analysis Time Domain signals on T-lines lines are hard to analyze Many properties, which can dominate per as long as mance, are frequency dependent, in addition to 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 as long as Three reasons Ease in addition to accuracy of measurement at high frequencies Simplified mathematics Allows separation of electrical phenomena (loss, resonance etc)

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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 in addition to characteristic impedance Delay Frequency S-Parameters are used to extract electrical parameters Transmission line parameters (R,L,C,G, TD in addition to 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 Review Important Concepts The impedance looking into a terminated transmission line changes with frequency in addition to line length The input reflection coefficient looking into a terminated transmission line also changes with frequency in addition to 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 in addition to load impedance values 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 in addition to 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 as long as excitation 2 dependent variables as long as response

Network characterized with Port Impedance Measuring the port impedance is network is the most simplistic in addition to intuitive method of characterizing a network Port 1 Port 2 Case 1: Inject current I1 into port 1 in addition to measure the open circuit voltage at port 2 in addition to calculate the resultant impedance from port 1 to port 2 Case 2: Inject current I1 into port 1 in addition to measure the voltage at port 1 in addition to calculate the resultant input impedance 2 – port Network I 1 I 2 + – V 1 V 2 + – 2 – port Network I 1 I 2 + – V 1 V 2 + – 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 as long as any input 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 in addition to j Or Impedance Matrix: Example 2 Calculate the impedance matrix as long as the following circuit: Port 1 Port 2 R1 R2 R3

Impedance Matrix: Example 2 Step 1: Calculate the input impedance R1 R2 R3 I1 V1 + – Step 2: Calculate the impedance across the network R1 R2 R3 I1 V2 + – Impedance Matrix: Example 2 Step 3: Calculate the Impedance matrix Assume: R1 = R2 = 30 ohms R3=150 ohms 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 Lprobe=0.1nH Cprobe=0.3pF Assume F=5 GHz

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 as long as small dimensions Open circuit capacitance = impedance at high frequencies Probe in addition to via impedance not insignificant 0.1nH 106 ohms 106 ohms Zo = 50 Without Probe Capacitance Zo = 50 With Probe Capacitance @ 5 GHz Z21 = 50 ohms Z21 = 63 ohms Port 1 Port 2 Port 1 Port 2 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 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 1 Port 2 ai represents the square root of the power wave injected into port i a1 a2 b2 b1 bj represents the power wave coming out of port j R R

Scattering Matrix A set of linear equations can be written to describe the network in terms of injected in addition to 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 Making sense of S-Parameters Return Loss When there is no reflection from the load, or the line length is zero, S11 = Reflection coefficient S11 is measure of the power returned to the source, in addition to is called the Return Loss R=Zo Z=-l Z=0 Zo R=50 Making sense of S-Parameters Return Loss When there is a reflection from the load, S11 will be composed of multiple reflections due to the st in addition to ing waves RL 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 S11 of a transmission line will exhibit periodic effects due to the st in addition to ing waves

Example 3 Interpreting the return loss Based on the S11 plot shown below, calculate both the impedance in addition to dielectric constant 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 1.0 1.5 2.0 2.5 3 0 3.5 4.0 4.5 5.0 Frequency, GHz S11, Magnitude R=50 L=5 inches Zo R=50 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 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Frequency, GHz S11, Magnitude 1.76GHz 2.94GHz Peak=0.384 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 as long as x=-5 inches Er=1.0 in addition to Zo=75 ohms

Making sense of S-Parameters Insertion Loss When power is injected into Port 1 with source impedance Z0 in addition to measured at Port 2 with measurement load impedance Z0, the power ratio reduces to a voltage ratio 2 – port Network 2 – port Network V1 a1 a2=0 b2 b1 V2 Zo Zo S21 is measure of the power transmitted from port 1 to port 2, in addition to is called the Insertion Loss 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 as long as in the power reflected from the incident port in addition to the power transmitted through network Since s-parameters are the square root of power ratios, the following is true as long as loss-free networks If the above relationship does not equal 1, then there is loss in the network, in addition to the difference is proportional to the power dissipated by the network Insertion loss example Question: What percentage of the total power is dissipated by the transmission line Estimate the magnitude of Zo (bound it)

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 as long as 8 GT/s signal 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 as long as ABCD parameters ABCE will be discussed later. If S11 < ~ 0.2 (-14 dB), then the above approximation is valid 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 in addition to return losses 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 in addition to Return losses include effects such as impedance discontinuities in addition to resonance effects, which are not true losses Loss free networks can still exhibit significant insertion in addition to return losses due to impedance discontinuities Advantages/Disadvantages of S-parameters Advantages: Ease of measurement Much easier to measure power at high frequencies than open/short current in addition to voltage S-parameters can be used to extract the transmission line parameters n parameters in addition to 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 in addition to frequency domain introduces errors 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 a11 b11 a21 b21 a12 b12 a22 b22 a13 b13 a13 b13 3 cascaded s parameter blocks Example 1: Solution (cont.) Since TD in addition to the effective Er is known, the line length can be calculated as in chapter 2 Note the relationship between the peaks in addition to the electrical length This leads to a very useful equation as long as transmission lines 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, in addition to the maximum reflection will occur If the imaginary term is zero, the reflected wave is aligned with the incident wave in addition to 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

## Carrier, Michael Founder and CEO

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