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© H. Heck 2008Section 5.51 Module 5:Advanced Transmission Lines Topic 5: 2 Port Networks & S-Parameters OGI EE564 Howard Heck
S-Parameters EE 564 © H. Heck 2008 Section 5.52 Where Are We? 1.Introduction 2.Transmission Line Basics 3.Analysis Tools 4.Metrics & Methodology 5.Advanced Transmission Lines 1.Losses 2.Intersymbol Interference 3.Crosstalk 4.Frequency Domain Analysis 5.2 Port Networks & S-Parameters 6.Multi-Gb/s Signaling 7.Special Topics
S-Parameters EE 564 © H. Heck 2008 Section 5.53 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).
S-Parameters EE 564 © H. Heck 2008 Section 5.54 Contents Two Port Networks Z Parameters Y Parameters Vector Network Analyzers S Parameters: 2 port, n ports Return Loss Insertion Loss Transmission (ABCD) Matrix Differential S Parameters (MOVE TO 6.2) Summary References Appendices
S-Parameters EE 564 © H. Heck 2008 Section 5.55 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 excitation 2 dependent variables for response
S-Parameters EE 564 © H. Heck 2008 Section 5.56 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]
S-Parameters EE 564 © H. Heck 2008 Section 5.57 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]
S-Parameters EE 564 © H. Heck 2008 Section 5.58 Example
S-Parameters EE 564 © H. Heck 2008 Section 5.59 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
S-Parameters EE 564 © H. Heck 2008 Section S Parameters We wish to characterize the network by observing transmitted and reflected power waves. a i represents the square root of the power wave injected into port i. b i represents the square root of the power wave injected into port j. use to get [5.5.9] [5.5.10] [5.5.11]
S-Parameters EE 564 © H. Heck 2008 Section 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]
S-Parameters EE 564 © H. Heck 2008 Section S Parameters – n Ports [5.5.14] [5.5.17] or [5.5.15] [5.5.16] [5.5.18]
S-Parameters EE 564 © H. Heck 2008 Section Scattering Matrix – Return Loss S 11, 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, S 11 is equal to the reflection coefficient. [5.5.19] In general: [5.5.20]
S-Parameters EE 564 © H. Heck 2008 Section Scattering Matrix – Return Loss #2 When there is a reflection from the load, S 11 will be composed of multiple reflections due to standing waves. Use input impedance to calculate S 11 when the line is not perfectly terminated. If the network is driven with a 50 source, S 11 is calculated using equation [5.5.22] S 11 for a transmission line will exhibit periodic effects due to the standing waves. In this case S 11 will be maximum when Z in is real. An imaginary component implies a phase difference between V inc and V ref. No phase difference means they are perfectly aligned and will constructively add. [5.5.21] [5.5.22]
S-Parameters EE 564 © H. Heck 2008 Section 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: S 21, the insertion loss, is a measure of the power transmitted from port 1 to port 2. [5.5.22]
S-Parameters EE 564 © H. Heck 2008 Section 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.
S-Parameters EE 564 © H. Heck 2008 Section 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 S 12 = S 21 and S 11 = S 22.
S-Parameters EE 564 © H. Heck 2008 Section Transmission Line Velocity Measurements We can calculate the delay per unit length (or velocity) from S 21 : S 21 = b 2 /a 1 Where ( S 21 ) is the phase angle of the S 21 measurement. f is the frequency at which the measurement was taken. l is the length of the line. [5.5.27]
S-Parameters EE 564 © H. Heck 2008 Section Impedance vs. frequency Recall Z in vs f will be a function of delay ( ) and Z L. We can use Z in equations for open and short circuited lossy transmission. Transmission Line Z 0 Measurements [5.5.28] [5.5.29] [5.5.30] Using the equation for Z in, in, and Z 0, we can find the impedance.
S-Parameters EE 564 © H. Heck 2008 Section Transmission Line Z 0 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]:
S-Parameters EE 564 © H. Heck 2008 Section Scattering Matrix Example Using the S 11 plot shown below, calculate Z 0 and estimate r Frequency [GHz] S 11 Magnitude
S-Parameters EE 564 © H. Heck 2008 Section Scattering Matrix Example #2 1.76GHz2.94GHz Step 1: Calculate the d of the transmission line based on the peaks or dips. Step 2: Calculate r based on the velocity (prop delay per unit length). Peak= S 11 Magnitude
S-Parameters EE 564 © H. Heck 2008 Section Example – Scattering Matrix (Cont.) Step 3: Calculate the input impedance to the transmission line based on the peak S 11 at 1.76GHz, assuming a 50 port. Step 4: Calculate Z 0 from Z in at z =0: Solution: r = 1.0 and Z 0 = 75
S-Parameters EE 564 © H. Heck 2008 Section 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.
S-Parameters EE 564 © H. Heck 2008 Section Transmission (ABCD) Matrix The transmission matrix describes the network in terms of both voltage and current waves (analagous to a Thévinin 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]
S-Parameters EE 564 © H. Heck 2008 Section 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]
S-Parameters EE 564 © H. Heck 2008 Section ABCD Matrix Values for 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 [5.5.38] [5.5.39] [5.5.40] [5.5.41] [5.5.42]
S-Parameters EE 564 © H. Heck 2008 Section 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
S-Parameters EE 564 © H. Heck 2008 Section 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 network The 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.
S-Parameters EE 564 © H. Heck 2008 Section 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 : Z1Z1 Port 1Port 2 Z2Z2 Z3Z3
S-Parameters EE 564 © H. Heck 2008 Section 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.
S-Parameters EE 564 © H. Heck 2008 Section 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.
S-Parameters EE 564 © H. Heck 2008 Section References D.M. Posar, Microwave Engineering, John Wiley & Sons, Inc. (Wiley Interscience), 1998, 2 nd edition. B. Young, Digital Signal Integrity, Prentice-Hall PTR, 2001, 1 st edition. S. Hall, G. Hall, and J. McCall, High Speed Digital System Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1 st edition. W. Dally and J. Poulton, Digital Systems Engineering, Chapters 4.3 & 11, Cambridge University Press, “Understanding the Fundamental Principles of Vector Network Analysis,” Agilent Technologies application note , “In-Fixture Measurements Using Vector Network Analyzers,” Agilent Technologies application note , “De-embedding and Embedding S-Parameter Networks Using A Vector Network Analyzer,” Agilent Technologies application note , 2001.
S-Parameters EE 564 © H. Heck 2008 Section Appendix More material on S parameters.
S-Parameters EE 564 © H. Heck 2008 Section Lossless Reciprocal
S-Parameters EE 564 © H. Heck 2008 Section S Parameters Scattering Matrix: S Parameters or [5.5.1] where [5.5.2] ????
S-Parameters EE 564 © H. Heck 2008 Section S Parameters #2 [5.5.1] where [5.5.2] Reciprocal
S-Parameters EE 564 © H. Heck 2008 Section S Parameters – n Ports [5.5.1] [5.5.2]
S-Parameters EE 564 © H. Heck 2008 Section S Parameters #4 [5.5.1] [5.5.2] where S ij = ij is the reflection coefficient of the i th port if i = j with all other ports matched S ij = T ij is the forward transmission coefficient of the i th port if I > j with all other ports matched S ij = T ij is the reverse transmission coefficient of the i th port if I < j with all other ports matched
S-Parameters EE 564 © H. Heck 2008 Section VNA Calibration Proper calibration is critical!!! There are two basic calibration methods Short, Open, Load and Thru (SOLT) Calibrated to known standard( Ex: 50 ) Measurement plane at probe tip Thru, Reflect, Line(TRL) Calibrated to line Z 0 – Helps create matched port condition. Measurement plane moved to desired position set by calibration structure design.
S-Parameters EE 564 © H. Heck 2008 Section SOLT Calibration Structures OPENSHORT LOADTHRU Calibration Substrate G G S S G S Signal Ground G S G S
S-Parameters EE 564 © H. Heck 2008 Section TRL Calibration Structures TRL PCB Structures Normalized Z 0 to line De-embed’s launch structure parasitics 6mil wide gap Short 100 mils Open ? Thru ? L1 ? L2 Measurement Planes
S-Parameters EE 564 © H. Heck 2008 Section Calibration- Verification Always check the calibration prior to taking measurements. Verify open, load etc.. Smith Chart: Open & Short should be inside the perimeter. Ideal response is dot at each location when probing the calibration structures. S 11 (Short) S 11 (Open) S 11 (load) S 21/12 (Thru)
S-Parameters EE 564 © H. Heck 2008 Section One Port Measurements Practical sub 2 GHz technique for L & C data. Structure must be electrically shorter than /4 of f max. 1 st order (Low Loss): Z in = jwL (Shorted transmission line) Z in = 1/ jwC (Open transmission line) For an electrically short structure V and I to order are ~constant. At the short, we have I max and V min. Measure L using a shorted transmission line with negligible loss. At the open you have V max and I min. Measure C using an open transmission line with negligible loss. V RSRS = 50 DUT Short Current Z in = j L·I DUT Open V RSRS = 50 Z in = V / j C
S-Parameters EE 564 © H. Heck 2008 Section One Port Measurements – L & C VNA - Format Use Smith chart format to read L & C data
S-Parameters EE 564 © H. Heck 2008 Section Connector L & C Use test board to measure connector inductance and capacitance Measure values relevant to pinout Procedure Measure test board L & C without connector Measure test board with connector Difference = connector parasitics Short Open
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