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**Module 5:. Advanced Transmission Lines Topic 5:**

Module 5: Advanced Transmission Lines Topic 5: 2 Port Networks & S-Parameters OGI EE564 Howard Heck © H. Heck 2008 Section 5.5

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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 © H. Heck 2008 Section 5.5

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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). © H. Heck 2008 Section 5.5

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**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 © H. Heck 2008 Section 5.5

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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 excitation 2 dependent variables for response © H. Heck 2008 Section 5.5

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**Z Parameters Impedance Matrix: Z Parameters or [5.5.1] where**

(Open circuit impedance) [5.5.2] 2 Port example: [5.5.3] [5.5.4] 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. © H. Heck 2008 Section 5.5

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**Y Parameters Admittance Matrix: Y Parameters or [5.5.5] where**

(Short circuit admittance) [5.5.6] 2 Port example: [5.5.7] [5.5.8] 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. © H. Heck 2008 Section 5.5

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Example © H. Heck 2008 Section 5.5

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**Frequency Domain: Vector Network Analyzer (VNA)**

VNA offers a means to characterize circuit elements as a function of frequency. I 1 I 2 + + 2-Port V 1 V 2 Network - - 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. © H. Heck 2008 Section 5.5

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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 [5.5.9] [5.5.10] to get [5.5.11] © H. Heck 2008 Section 5.5

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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: [5.5.12] where in matrix form: [5.5.13] © H. Heck 2008 Section 5.5

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**S Parameters – n Ports [5.5.14] [5.5.15] [5.5.16] [5.5.17] [5.5.18] or**

© H. Heck 2008 Section 5.5

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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] © H. Heck 2008 Section 5.5

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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. [5.5.21] S11 for a transmission line will exhibit periodic effects due to the standing waves. If the network is driven with a 50 source, S11 is calculated using equation [5.5.22] [5.5.22] 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. © H. Heck 2008 Section 5.5

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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: [5.5.22] S21, the insertion loss, is a measure of the power transmitted from port 1 to port 2. © H. Heck 2008 Section 5.5

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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. © H. Heck 2008 Section 5.5

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**Reflection Coefficients**

Reflection coefficient at the load: [5.5.23] Reflection coefficient at the source: [5.5.24] Input reflection coefficient: [5.5.25] Assuming S12 = S21 and S11 = S22. Output reflection coefficient: [5.5.26] © H. Heck 2008 Section 5.5

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**Transmission Line Velocity Measurements**

S21 = b2/a1 We can calculate the delay per unit length (or velocity) from S21: [5.5.27] Where f(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. © H. Heck 2008 Section 5.5

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**Transmission Line Z0 Measurements**

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. [5.5.28] [5.5.29] [5.5.30] Using the equation for Zin, rin, and Z0, we can find the impedance. © H. Heck 2008 Section 5.5

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**Transmission Line Z0 Measurement #2**

Input reflection coefficients for the open and short circuit cases: Input impedance for the open and short circuit cases: [5.5.31] [5.5.32] Now we can apply equation [5.5.30]: © H. Heck 2008 Section 5.5

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**Scattering Matrix Example**

Using the S11 plot shown below, calculate Z0 and estimate er. 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 © H. Heck 2008 Section 5.5

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**Scattering Matrix Example #2**

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 S11 Magnitude 1.76GHz 2.94GHz Step 1: Calculate the td of the transmission line based on the peaks or dips. Peak=0.384 Step 2: Calculate er based on the velocity (prop delay per unit length). © H. Heck 2008 Section 5.5

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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 © H. Heck 2008 Section 5.5

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**Advantages/Disadvantages of S Parameters**

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. © H. Heck 2008 Section 5.5

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**Transmission (ABCD) Matrix**

The transmission matrix describes the network in terms of both voltage and current waves (analagous to a Thévinin Equivalent). [5.5.33] [5.5.34] The coefficients can be defined using superposition: [5.5.29] [5.5.35] [5.5.31] [5.5.36] © H. Heck 2008 Section 5.5

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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: [5.5.37] This is the best way to cascade elements in the frequency domain. It is accurate, intuitive and simple to use. © H. Heck 2008 Section 5.5

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**ABCD Matrix Values for Common Circuits**

Z [5.5.38] Port 1 Port 2 [5.5.39] Port 1 Y Port 2 Z1 Port 1 Port 2 Z2 Z3 [5.5.40] Y3 [5.5.41] Port 1 Y1 Y2 Port 2 Port 1 Port 2 [5.5.42] © H. Heck 2008 Section 5.5

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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 © H. Heck 2008 Section 5.5

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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 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. © H. Heck 2008 Section 5.5

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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 © H. Heck 2008 Section 5.5

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**Advantages/Disadvantages of ABCD Matrix**

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. © H. Heck 2008 Section 5.5

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**We can characterize interconnect networks using n-Port circuits. **

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. © H. Heck 2008 Section 5.5

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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 , 2000. “In-Fixture Measurements Using Vector Network Analyzers,” Agilent Technologies application note , 2000. “De-embedding and Embedding S-Parameter Networks Using A Vector Network Analyzer,” Agilent Technologies application note , 2001. © H. Heck 2008 Section 5.5

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Appendix More material on S parameters. © H. Heck 2008 Section 5.5

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Reciprocal Lossless © H. Heck 2008 Section 5.5

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**S Parameters Scattering Matrix: S Parameters or [5.5.1] [5.5.2] where**

???? © H. Heck 2008 Section 5.5

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**S Parameters #2 [5.5.1] [5.5.2] where Reciprocal © H. Heck 2008**

Section 5.5

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S Parameters – n Ports [5.5.1] [5.5.2] © H. Heck 2008 Section 5.5

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S Parameters #4 [5.5.1] 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<j with all other ports matched [5.5.2] © H. Heck 2008 Section 5.5

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**Proper calibration is critical!!! **

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 Z0 Helps create matched port condition. Measurement plane moved to desired position set by calibration structure design. © H. Heck 2008 Section 5.5

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**SOLT Calibration Structures**

OPEN SHORT LOAD THRU Calibration Substrate G S Signal Ground © H. Heck 2008 Section 5.5

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**TRL Calibration Structures**

TRL PCB Structures Normalized Z0 to line De-embed’s launch structure parasitics Short Open 6mil wide gap 100 mils 100 mils Measurement Planes Thru L1 ? ? L2 ? © H. Heck 2008 Section 5.5

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**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. S11(Short) S11(Open) S21/12(Thru) S11(load) © H. Heck 2008 Section 5.5

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**One Port Measurements Practical sub 2 GHz technique for L & C data.**

Structure must be electrically shorter than /4 of fmax. 1st order (Low Loss): Zin = jwL (Shorted transmission line) Zin = 1/jwC (Open transmission line) For an electrically short structure V and I to order are ~constant. At the short, we have Imax and Vmin. Measure L using a shorted transmission line with negligible loss. At the open you have Vmax and Imin. Measure C using an open transmission line with negligible loss. DUT DUT RS = 50W RS = 50W Short V V Current Open Zin = jwL·I Zin = V/jwC © H. Heck 2008 Section 5.5

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**One Port Measurements – L & C**

VNA - Format Use Smith chart format to read L & C data © H. Heck 2008 Section 5.5

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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 Open Short © H. Heck 2008 Section 5.5

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