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**Introduction to Frequency Domain Analysis (3 Classes)**

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 Slide content from Stephen Hall

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**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

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**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)

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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 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

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**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

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**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

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**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 + + 2 2 - - port port + + Port 1 V V V V Port 2 1 1 - - Network Network - - 2 2 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

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**Zii the impedance looking into port i **

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

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**Impedance Matrix: Example #2**

Calculate the impedance matrix for the following circuit: R1 R2 Port 1 R3 Port 2

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**Impedance Matrix: Example #2**

Step 1: Calculate the input impedance R1 R2 + - I1 V1 R3 Step 2: Calculate the impedance across the network R1 R2 + - I1 R3 V2

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**Impedance Matrix: Example #2**

Step 3: Calculate the Impedance matrix Assume: R1 = R2 = 30 ohms R3=150 ohms

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**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 T-line 0.1nH Port 1 Port 2 0.3pF Zo=50 ohms, length=5 in

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**Measuring the impedance matrix**

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 T-line 0.1nH Port 1 Port 2 0.3pF Zo=50 ohms, length=5 in Without Probe Capacitance 0.1nH T-line Zo = 50 Port 2 Port 1 Port 2 Z21 = 50 ohms With Probe 5 GHz Zo = 50 Port 1 Port 2 106 ohms 106 ohms Z21 = 63 ohms

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**Advantages/Disadvantages of Impedance Matrix**

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

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**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 R R Port 2 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

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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

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**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”

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**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 standing waves Zo RL Z=-l Z=0 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

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**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 1.0 1.5 2.0 2.5 3..0 3.5 4.0 4.5 5.0 Frequency, GHz

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**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 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Frequency, GHz Step 1: Calculate the time delay of the t-line using the peaks Step 2: Calculate Er using the velocity

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**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

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**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”

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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

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**Insertion loss example**

Question: What percentage of the total power is dissipated by the transmission line? Estimate the magnitude of Zo (bound it)

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**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

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**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.

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**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

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**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

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**Advantages/Disadvantages of S-parameters**

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

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**3 cascaded s parameter blocks**

Cascading S parameter 3 cascaded s parameter blocks a11 a21 b12 b22 a13 a13 s111 s121 s211 s221 s113 s123 s213 s223 s112 s122 s212 s222 b11 b21 a12 a22 b13 b13 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

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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

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**Transmission (ABCD) Matrix**

Since the ABCD matrix represents the ports in terms of currents and voltages, it is well suited for cascading elements I2 I3 I1 V1 V2 V3 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.

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**Relating the ABCD Matrix to Common Circuits**

Z Assignment 6: Convert these to s-parameters Port 1 Port 2 Port 1 Y Port 2 Z1 Z2 Port 1 Z3 Port 2 Y3 Y1 Y2 Port 1 Port 2 Port 1 Port 2

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**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

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

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

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

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

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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 Z3 Port 2

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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 50 Network X Network Y Network Network Port 1 Port 2

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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

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**Advantages/Disadvantages of ABCD Matrix**

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

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**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

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**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 Measurement equipment strives to be match i.e. reflection coefficient is 0 a1 b1 b2 a2 GS GL s21 s12 s11 s22 See:

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**Mason’s Rule ~ Non-Touching Loop Rule**

T is the transfer function (often called gain) Tk is 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

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**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

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

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

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Combine b2 and a2

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**Convert Wave to Voltage - Multiply by sqrt(Z0)**

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**Voltage transfer function using ABCD**

Let’s see if we can get this results another way

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**Cascade [ABCD] to determine system [ABCD]**

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**Extract the voltage transfer function**

Same as with flow graph analysis

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**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. a11 b11 a21 b21 a12 b12 a22 b22 a13 b13 s111 s121 s211 s221 s112 s122 s212 s222 s113 s123 s213 s223

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**Creating the signal flow graph**

b11 a21 b21 a12 b12 a22 b22 a13 b13 s111 s121 s211 s221 s112 s122 s212 s222 s113 s123 s213 s223 1 A11 B21 A12 B22 A13 B23 B11 A21 B12 A22 B13 A23 s121 s221 s211 s123 s213 s113 s223 s212 s112 s222 s122 B A C 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

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**B A C A B C A B Use Mason’s rule 1 s121 s221 s211 s123 s213 s113 s223**

There is only one forward path a11 to b23. There are 2 non touching looks

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**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 s211*s212*213…

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**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

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**Differential S-Parameters**

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 a 1 2 3 4 = b a a 2 b S S S S 1 1 11 12 13 14 b 4 - port S b 2 21 S S S b 1 2 22 23 24 b 3 S S S S 31 32 33 34 b S 4 S 41 S 43 S 42 44

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**Differential S-Parameters**

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 b a dm1 dm2 cm1 cm2 = dm1 DS DS DCS DCS 11 12 11 12 b dm2 DS 21 DS DCS DCS 22 21 22 b CDS CDS CS CS cm1 11 12 11 12 b cm2 CDS CS 21 CDS 21 CS 22 22 Matrix assumes differential and common mode stimulus

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**Explanation of the Multi-Mode Port**

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 Matrix: Differential Stimulus, differential response i.e., DS21 = differential signal [(D+)-(D-)] inserted at port 1 and diff signal measured at port 2 b a DS DS DCS DCS dm1 dm1 11 12 11 12 b a DS dm2 DS DCS DCS dm2 21 22 21 22 = b a CDS CDS CS CS cm1 cm1 11 12 11 12 b a CDS CS cm2 CDS CS cm2 21 21 22 22 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 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

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**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: 1 2 4 2 - - port port Network Network 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

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**Next class we will develop more differential concepts**

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backup review

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**Advantages/Disadvantages of Multi-Mode Matrix over Traditional 4-port**

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

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**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 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 For typical PCBs at 10 GHz with 5 mil traces (W=0.005”) Y

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**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 (Ex) field and the magnetic (By) are orthogonal

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**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 Ex:

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**High Frequency Electromagnetic Waves**

The 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 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

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**High Frequency Voltage and Current Waves**

The same equation applies to voltage and current waves on a transmission line Incident sinusoid Reflected sinusoid RL z=-l 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 reflecting off the Load and traveling towards the source Voltage wave traveling towards the load

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**High Frequency Voltage and Current Waves**

The 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)

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**High Frequency Voltage and Current Waves**

The voltage wave equation can be put into more intuitive terms by applying the following identity: Subsequently: The amplitude is degraded by The waveform is dependent on the driving function ( ) & the delay of the line

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**Interaction: transmission line and a load**

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 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

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**This is the input impedance looking into a t-line of length l**

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 Zin RL Z=-l Z=0 Note: is dependent on and This is the input impedance looking into a t-line of length l

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**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

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**Line & load interactions**

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 Term 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.

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**Example #1: Periodic Reflections**

Calculate: Line length RL (assume a very low loss line) Er_eff=1.0 RL Zo=75 Z=-l Z=0

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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 1.5E 1.0E 5.0E 02 0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09 Frequency Reflection Coeff . Real Imaginary

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**Example #1: Solution (cont.)**

Note the relationship between the peaks and the electrical length This leads to a very useful equation for transmission lines Since TD and the effective Er is known, the line length can be calculated as in chapter 2

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**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

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