H. Heck 2008

Section 5.5

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Module 5:Advanced Transmission Lines
Topic 5: 2 Port Networks & S-Parameters

OGI EE564

Howard Heck

Section 5.5

S-Parameters

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Where Are We?

Introduction

Transmission Line Basics

Analysis Tools

Metrics & Methodology

Advanced Transmission Lines

Losses

Intersymbol Interference

Crosstalk

Frequency Domain Analysis

2 Port Networks & S-Parameters

Multi-Gb/s Signaling

Special Topics

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Acknowledgement

Much of the material in this section has been adapted from material developed by Stephen H. Hall and James A. McCall (the authors of our text).

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Contents

Two Port NetworksZ ParametersY ParametersVector Network AnalyzersS Parameters: 2 port, n portsReturn LossInsertion LossTransmission (ABCD) MatrixDifferential S Parameters (MOVE TO 6.2)SummaryReferences Appendices

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Two Port Networks

Linear networks can be completely characterized by parameters measured at the network ports without knowing the content of the networks.Networks can have any number of ports.Analysis of a 2-port network is sufficient to explain the theory and applies to isolated signals (no crosstalk).The ports can be characterized with many parameters (Z, Y, S, ABDC). Each has a specific advantage.Each parameter set is related to 4 variables:2 independent variables for excitation2 dependent variables for response

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Z Parameters

Advantage: Z parameters are intuitive.Relates all ports to an impedance & is easy to calculate.Disadvantage: Requires open circuit voltage measurements, which are difficult to make.Open circuit reflections inject noise into measurements.Open circuit capacitance is non-trivial at high frequencies.

(Open circuit impedance)

Impedance Matrix: Z Parameters

or

[5.5.1]

where

[5.5.2]

2 Port example:

[5.5.4]

[5.5.3]

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Y Parameters

(Short circuit admittance)

Admittance Matrix: Y Parameters

or

[5.5.6]

[5.5.5]

where

2 Port example:

Advantage: Y parameters are also somewhat intuitive.Disadvantage: Requires short circuit voltage measurements, which are difficult to make.Short circuit reflections inject noise into measurements.Short circuit inductance is non-trivial at high frequencies.

[5.5.7]

[5.5.8]

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Example

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Frequency Domain: Vector Network Analyzer (VNA)

VNA offers a means to characterize circuit elements as a function of frequency.VNA is a microwave based instrument that provides the ability to understand frequency dependent effects.The input signal is a frequency swept sinusoid. Characterizes the network by observing transmitted and reflected power waves.Voltage and current are difficult to measure directly.It is also difficult to implement open & short circuit loads at high frequency.Matched load is a unique, repeatable termination, and is insensitive to length, making measurement easier.Incident and reflected waves the key measures.We characterize the device under test using S parameters.

2-Port

Network

+

-

+

-

V

1

V

2

I

1

I

2

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S Parameters

We wish to characterize the network by observing transmitted and reflected power waves.ai represents the square root of the power wave injected into port i.bi represents the square root of the power wave injected into port j.

use

to get

[5.5.9]

[5.5.10]

[5.5.11]

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S Parameters #2

We can use a set of linear equations to describe the behavior of the network in terms of the injected and reflected power waves.For the 2 port case:

where

in matrix form:

[5.5.12]

[5.5.13]

Section 5.5

Howard Heck (HH) - Fix "S55"

S-Parameters

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S Parameters n Ports

[5.5.14]

[5.5.17]

or

[5.5.15]

[5.5.16]

[5.5.18]

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Scattering Matrix Return Loss

S11, the return loss, is a measure of the power returned to the source.When there is no reflection from the load, or the line length is zero, S11 is equal to the reflection coefficient.

[5.5.19]

In general:

[5.5.20]

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Scattering Matrix Return Loss #2

When there is a reflection from the load, S11 will be composed of multiple reflections due to standing waves.Use input impedance to calculate S11 when the line is not perfectly terminated.If the network is driven with a 50 source, S11 is calculated using equation [5.5.22]S11 for a transmission line will exhibit periodic effects due to the standing waves.In this case S11 will be maximum when Zin is real. An imaginary component implies a phase difference between Vinc and Vref. No phase difference means they are perfectly aligned and will constructively add.

[5.5.21]

[5.5.22]

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Scattering Matrix Insertion Loss #1

When power is injected into Port 1 and measured at Port 2, the power ratio reduces to a voltage ratio:S21, the insertion loss, is a measure of the power transmitted from port 1 to port 2.

[5.5.22]

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Comments On 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 other effects, such as impedance discontinuities and resonance, which are not true losses.Loss free networks can still exhibit significant insertion and return losses due to impedance discontinuities.

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Reflection Coefficients

Reflection coefficient at the load:

[5.5.23]

[5.5.24]

[5.5.25]

[5.5.26]

Reflection coefficient at the source:Input reflection coefficient:Output reflection coefficient:

Assuming S12 = S21 and S11 = S22.

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Transmission Line Velocity Measurements

We can calculate the delay per unit length (or velocity) from S21:

S21 = b2/a1

Wheref(S21 ) is the phase angle of the S21 measurement.
f is the frequency at which the measurement was taken.
l is the length of the line.

[5.5.27]

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Impedance vs. frequency Recall Zin vs f will be a function of delay () and ZL.We can use Zin equations for open and short circuited lossy transmission.

Transmission Line Z0 Measurements

[5.5.28]

[5.5.29]

[5.5.30]

Using the equation for Zin, rin, and Z0, we can find the impedance.

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Transmission Line Z0 Measurement #2

[5.5.31]

[5.5.32]

Input reflection coefficients for the open and short circuit cases:Input impedance for the open and short circuit cases:Now we can apply equation [5.5.30]:

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Scattering Matrix Example

Using the S11 plot shown below, calculate Z0 and estimate er.

0

1.0

1.5

2.0

2.5

3..0

3.5

4.0

4.5

5.0

Frequency [GHz]

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

S11 Magnitude

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Scattering Matrix Example #2

1.76GHz

2.94GHz

Step 1: Calculate the td of the transmission line based on the peaks or dips.Step 2: Calculate er based on the velocity (prop delay per unit length).

Peak=0.384

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

S11 Magnitude

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Example Scattering Matrix (Cont.)

Step 3: Calculate the input impedance to the transmission line based on the peak S11 at 1.76GHz, assuming a 50W port.Step 4: Calculate Z0 from Zin at z=0:

Solution: er = 1.0 and Z0 = 75W

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Advantages/Disadvantages of S Parameters

Advantages:

Ease of measurement: It is much easier to measure power at high frequencies than open/short current and voltage.

Disadvantages:

They are more difficult to understand and it is more difficult to interpret measurements.

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Transmission (ABCD) Matrix

The transmission matrix describes the network in terms of both voltage and current waves (analagous to a Thvinin Equivalent).The coefficients can be defined using superposition:

[5.5.33]

[5.5.34]

[5.5.35]

[5.5.36]

[5.5.29]

[5.5.31]

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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.The matrices can be mathematically cascaded by multiplication:This is the best way to cascade elements in the frequency domain. It is accurate, intuitive and simple to use.

[5.5.37]

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ABCD Matrix Values for Common Circuits

Z

Port 1

Port 2

Port 1

Y

Port 2

Y1

Port 1

Port 2

Y2

Y3

Port 1

Port 2

[5.5.38]

[5.5.39]

[5.5.40]

[5.5.41]

[5.5.42]

Z1

Port 1

Port 2

Z2

Z3

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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 MatrixAllows conversion into a more intuitive matrixAllows conversion to ABCD for cascadingABCD matrix can be directly related to several useful circuit topologies

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ABCD Matrix Example

Create a model of a via from the measured s-parameters.The model can be extracted as either a Pi or a T networkThe inductance values will include the L of the trace and the via barrel assumes the test setup minimizes the trace length, so that trace capacitance is minimal.The capacitance represents the via pads.

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ABCD Matrix Example #1

The measured S-parameter matrix at 5 GHz is:Converted to ABCD parameters:Relating the ABCD parameters to the T circuit topology, the capacitance can be extracted from C & inductance from A:

Z1

Port 1

Port 2

Z2

Z3

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Advantages/Disadvantages of ABCD Matrix

Advantages:

The ABCD matrix is intuitive: it 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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Summary

We can characterize interconnect networks using n-Port circuits.The VNA uses S- parameters.From S- parameters we can characterize transmission lines and discrete elements.

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References

D.M. Posar, Microwave Engineering, John Wiley & Sons, Inc. (Wiley Interscience), 1998, 2nd edition.B. Young, Digital Signal Integrity, Prentice-Hall PTR, 2001, 1st edition.S. Hall, G. Hall, and J. McCall, High Speed Digital System Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1st edition.W. Dally and J. Poulton, Digital Systems Engineering, Chapters 4.3 & 11, Cambridge University Press, 1998. Understanding the Fundamental Principles of Vector Network Analysis, Agilent Technologies application note 1287-1, 2000.In-Fixture Measurements Using Vector Network Analyzers, Agilent Technologies application note 1287-9, 2000.De-embedding and Embedding S-Parameter Networks Using A Vector Network Analyzer, Agilent Technologies application note 1364-1, 2001.

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Appendix

More material on S parameters.

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Lossless

Reciprocal

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S Parameters

Scattering Matrix: S Parameters

or

[5.5.1]

where

[5.5.2]

????

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S Parameters #2

[5.5.1]

where

[5.5.2]

Reciprocal

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S Parameters n Ports

[5.5.1]

[5.5.2]

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S Parameters #4

[5.5.1]

[5.5.2]

where

Sij = Gij is the reflection coefficient of the ith
port if i=j with all other ports matched
Sij = Tij is the forward transmission coefficient
of the ith port if I>j with all other ports
matched
Sij = Tij is the reverse transmission coefficient
of the ith port if I