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Module 2:Transmission LinesTopic 1: TheoryOGI EE564Howard Heck

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Transmission Line TheoryEE 564

Where Are We? IntroductionTransmission Line BasicsTransmission Line TheoryBasic I/O CircuitsReflectionsParasitic DiscontinuitiesModeling, Simulation, & SpiceMeasurement: Basic EquipmentMeasurement: Time Domain ReflectometryAnalysis ToolsMetrics & MethodologyAdvanced Transmission LinesMulti-Gb/s SignalingSpecial Topics

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ContentsPropagation VelocityCharacteristic ImpedanceVisualizing Transmission Line BehaviorGeneral Circuit ModelFrequency DependenceLossless Transmission LinesHomogeneous and Non-homogeneous LinesImpedance Formulae for Transmission Line StructuresSummaryReferences

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Propagation VelocityPhysical example:Wave propagates in z directionCircuit:L = [nH/cm] C = [pF/cm] [2.1.1][2.1.2]Simplify [2.1.1] & [2.1.2] to get the Telegraphists Equations [2.1.3a][2.1.3b]

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Propagation Velocity (2)Phase velocity definition:[2.1.7]Equation in terms of current:[2.1.8]Equate [2.1.4] & [2.1.5]:[2.1.6]Differentiate [2.1.3b] by z: [2.1.5]Differentiate [2.1.3a] by t: [2.1.4]Equation [2.1.6] is a form of the wave equation. The solution to [2.1.6] contains forward and backward traveling wave components, which travel with a phase velocity.An alternate treatment of propagation velocity is contained in the appendix.

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Characteristic Impedance (Lossless)The input impedance (Z1) is the impedance of the first inductor (Ldz) in series with the parallel combination of the impedance of the capacitor (Cdz) and Z2. dz = segment lengthC = capacitance per segmentL = inductance per segment [2.1.9]

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Characteristic Impedance (Lossless)Assuming a uniform line, the input impedance should be the same when looking into node pairs a-d, b-e, c-f, and so forth. So, Z2 = Z1= Z0. [2.1.10]Allow dz to become very small, causing the frequency dependent term to drop out: [2.1.11][2.1.12]Solve for Z0: [2.1.13]

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Visualizing Transmission Line BehaviorWater flowPotential = Wave height [m]Flow = Flow rate [liter/sec] Transmission LinePotential = Voltage [V]Flow = Current [A] = [C/sec] Just as the wave front of the water flows in the pipe, the voltage propagates in the transmission line. The same holds true for current.Voltage and current propagate as waves in the transmission line.

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Visualizing Transmission Line Behavior #2 Extending the analogyThe diameter of the pipe relates the flow rate and height of the water. This is analogous to electrical impedance.Ohms law and the characteristic impedance define the relationship between current and potential in the transmission line.Effects of impedance discontinuitiesWhat happens when the water encounters a ledge or a barrier?What happens to the current and voltage waves when the impedance of the transmission line changes?The answer to this question is a key to understanding transmission line behavior.It is useful to try visualize current/voltage wave propagation on a transmission line system in the same way that we can for water flow in a pipe.

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General Transmission Line Model (No Coupling)Transmission line parameters are distributed (e.g. capacitance per unit length).A transmission line can be modeled using a network of resistances, inductances, and capacitances, where the distributed parameters are broken into small discrete elements.

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General Transmission Line Model #2ParametersCharacteristic Impedance[2.1.14]Propagation Constant[2.1.15]a = attenuation constant = rate of exponential attenuationb = phase constant = amount of phase shift per unit lengthPhase Velocity[2.1.16]In general, a and b are frequency dependent.

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Frequency DependenceFrom [2.1.14] and [2.1.15] note that:Z0 and depend on the frequency content of the signal.Frequency dependence causes attenuation and edge rate degradation.

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Frequency Dependence #2R and G are sometimes negligible, particularly at low frequenciesSimplifies to the lossless case: no attenuation & no dispersion In modules 2 and 3, we will concentrate on lossless transmission lines.Modules 5 and 6 will deal with lossy lines.

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Lossless Transmission LinesQuasi-TEM AssumptionThe electric and magnetic fields are perpendicular to the propagation velocity in the transverse planes.

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Lossless Line ParametersLossless line characteristics are frequency independent.As noted before, Z0 defines the relationship between voltage and current for the traveling waves. The units are ohms [W].u defines the propagation velocity of the waves. The units are cm/ns.Sometimes, we use the propagation delay, td (units are ns/cm). Characteristic ImpedancePropagation Velocity[2.1.17][2.1.18]Lossless transmission lines are characterized by the following two parameters:

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Lossless Line Equivalent CircuitThe transmission line equivalent circuit shown on the left is often represented by the coaxial cable symbol.

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Homogeneous MediaA homogeneous dielectric medium is uniform in all directions.All field lines are contained within the dielectric.For a transmission line in a homogeneous medium, the propagation velocity depends only on material properties: er is the relative permittivity or dielectric constant.

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Non-Homogeneous MediaA non-homogenous medium contains multiple materials with different dielectric constants.For a non-homogeneous medium, field lines cut across the boundaries between dielectric materials.In this case the propagation velocity depends on the dielectric constants and the proportions of the materials. Equation [2.1.19] does not hold: In practice, an effective dielectric constant, er,eff is often used, which represents an average dielectric constant.

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Some Typical Transmission Line StructuresAnd useful formulas for Z0

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Coax Cable Impedance[2.1.20][2.1.21][2.1.22]

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Centered Stripline ImpedanceSource: Motorola application note AN1051.[2.1.23]

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Dual Stripline ImpedanceSource: Motorola application note AN1051.OR[2.1.24][2.1.27][2.1.25][2.1.26]

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Surface Microstrip ImpedanceSource: National AN-991. Source: Motorola MECL Design Handbook. [2.1.28][2.1.29][2.1.30][2.1.31]

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Embedded MicrostripOr [2.1.32][2.1.33][2.1.34][2.1.35]

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SummarySystem level interconnects can often be treated as lossless transmission lines.Transmission lines circuit elements are distributed. Voltage and current propagate as waves in transmission lines.Propagation velocity and characteristic impedance characterize the behavior of lossless transmission lines.Coaxial cables, stripline and microstrip printed circuits are the typical transmission line structures in PCs systems.

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ReferencesS. Hall, G. Hall, and J. McCall, High Speed Digital System Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1st edition.H. Johnson and M. Graham, High-Speed Signal Propagation: Advanced Black Magic, Prentice Hall, 2003, 1st edition, ISBN 0-13-084408-X.W. Dally and J. Poulton, Digital Systems Engineering, Cambridge University Press, 1998. R.E. Matick, Transmission Lines for Digital and Communication Networks, IEEE Press, 1995.R. Poon, Computer Circuits Electrical Design, Prentice Hall, 1st edition, 1995. H.B.Bakoglu, Circuits, Interconnections, and Packaging for VLSI, Addison Wesley, 1990, ISBN 0-201-060080-6.B. Young, Digital Signal Integrity, Prentice-Hall PTR, 2001, 1st edition, ISBN 0-13-028904-3.

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Phase Constant (Lossless Case)Recall the basic voltage divider circuit:We want to find the ratio of the input voltage, V1, to the output voltage, V2.Now, we apply it to our transmission line equivalent circuit...

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Phase Constant (Lossless Case) #2The analogous transmission line circuit looks like this:The phase shift is the ratio of V1 to V2:Substitute the expressions for ZC, ZL, and Z0:

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Phase Constant (Lossless Case) #3The amplitude of the phase constant is:The phase angle, denoted as tanbl, is:Now, we make the assumption that dz is small enough that the applied frequency, w, is much smaller than the resonant frequency, , of each subsection, so that:The phase angle becomes:Since , tanbl is, very small. Therefore:

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Phase Constant (Lossless Case) #4The phase shift per unit length is:bl represents the amount by which the input voltage, V1, leads the output voltage, V2.We can simplify the amplitude ratio by using the condition of small bl:So, there is no decrease in the amplitude of the voltage along the line, for the lossless case. Only a shift in phase.From our definition of phase velocity in equation [2.1.16] we get

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