1. Module 2: Transmission Lines Topic 1: Theory
OGI EE564
Howard Heck
© H. Heck 2008
Section 2.1

2. Where Are We? Introduction Transmission Line Basics Analysis Tools
Transmission Line Theory
Basic I/O Circuits
Reflections
Parasitic Discontinuities
Modeling, Simulation, & Spice
Measurement: Basic Equipment
Measurement: Time Domain Reflectometry
Analysis Tools
Metrics & Methodology
Advanced Transmission Lines
Multi-Gb/s Signaling
Special Topics
© H. Heck 2008
Section 2.1

3. Contents Propagation Velocity Characteristic Impedance
Visualizing Transmission Line Behavior
General Circuit Model
Frequency Dependence
Lossless Transmission Lines
Homogeneous and Non-homogeneous Lines
Impedance Formulae for Transmission Line Structures
Summary
References
© H. Heck 2008
Section 2.1

4. Propagation Velocity Physical example: Circuit:z x y
V, I
Physical example:
Wave propagates in z direction
Circuit:
L = [nH/cm] C = [pF/cm]
Total voltage change across Ldz (use ):
[2.1.1]
Total current change across Cdz (use ):
[2.1.2]
Simplify [2.1.1] & [2.1.2] to get the Telegraphist’s Equations
[2.1.3a]
[2.1.3b]
© H. Heck 2008
Section 2.1

5. Propagation Velocity (2)
Differentiate [2.1.3a] by t:
[2.1.4]
Differentiate [2.1.3b] by z:
[2.1.5]
Equate [2.1.4] & [2.1.5]:
[2.1.6]
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.
Phase velocity definition:
[2.1.7]
Equation in terms of current:
[2.1.8]
An alternate treatment of propagation velocity is contained in the appendix.
© H. Heck 2008
Section 2.1

6. Characteristic Impedance (Lossless)
dz = segment length
C = capacitance per segment
L = inductance per segment
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.
[2.1.9]
© H. Heck 2008
Section 2.1

7. 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]
[2.1.11]
Allow dz to become very small, causing the frequency dependent term to drop out:
[2.1.12]
Solve for Z0:
[2.1.13]
© H. Heck 2008
Section 2.1

8. Visualizing Transmission Line Behaviorf I V
Water flow
Potential = Wave height [m]
Flow = Flow rate [liter/sec]
Transmission Line
Potential = 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.
© H. Heck 2008
Section 2.1

9. Visualizing Transmission Line Behavior #2
Extending the analogy
The diameter of the pipe relates the flow rate and height of the water. This is analogous to electrical impedance.
Ohm’s law and the characteristic impedance define the relationship between current and potential in the transmission line.
Effects of impedance discontinuities
What 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.
© H. Heck 2008
Section 2.1

10. 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.
© H. Heck 2008
Section 2.1

11. General Transmission Line Model #2
Parameters
Parameter
Symbol
Units
Conductor Resistance R
W•cm-1
Self Inductance L
nH•cm-1
Total Capacitance C
pF•cm-1
Dielectric Conductance G
W-1•cm -1
Characteristic Impedance
[2.1.14]
Propagation Constant
[2.1.15]
a = attenuation constant = rate of exponential attenuation
b = phase constant = amount of phase shift per unit length
Phase Velocity
[2.1.16]
In general, a and b are frequency dependent.
© H. Heck 2008
Section 2.1

12. Frequency Dependence From [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.
Output signal from
lossless transmission line
Signal at driven end of
transmission line
Attenuation
Output signal from lossy
transmission line
Edge rate degradation
© H. Heck 2008
Section 2.1

13. Frequency Dependence #2
R and G are sometimes negligible, particularly at low frequencies
Simplifies 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.
© H. Heck 2008
Section 2.1

14. Lossless Transmission Lines
Quasi-TEM Assumption
The electric and magnetic fields are perpendicular to the propagation velocity in the transverse planes. E H x z y
© H. Heck 2008
Section 2.1

15. Lossless Line Parameters
Lossless transmission lines are characterized by the following two parameters:
Characteristic Impedance
[2.1.17]
Propagation Velocity
[2.1.18]
Lossless 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).
© H. Heck 2008
Section 2.1

16. Lossless Line Equivalent Circuit
The transmission line equivalent circuit shown on the left is often represented by the coaxial cable symbol. L C Z , v
, length
Z0, n, length
© H. Heck 2008
Section 2.1

17. Homogeneous Media
A 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:
[2.1.19]
Note: only er is required to calculate n.
Dielectric Permittivity
Permittivity of free space
Magnetic Permeability
Permeability of free space
er is the relative permittivity or dielectric constant.
© H. Heck 2008
Section 2.1

18. Non-Homogeneous Media
A 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.
© H. Heck 2008
Section 2.1

19. Some Typical Transmission Line Structures
And useful formulas for Z0
© H. Heck 2008
Section 2.1

20. Coax Cable Impedance Z , v , length Z0, u, length [2.1.20] [2.1.21]r R e Z , v
, length
Z0, u, length 2 3 4 5 6 7 8 9 10
R/r 20 40 60 80
100
120
140 Z [ W ] e r
= 1
= 4
= 3.5
= 3
= 2.5
= 2
[2.1.20]
[2.1.21]
[2.1.22]
© H. Heck 2008
Section 2.1

21. Centered Stripline Impedanceh 2 w
Valid for t h 1
[2.1.23]
0.003
0.005
0.007
0.009
0.011
0.013
0.015
w [in] 10 15 20 25 30 35 40 45 50 55 60 Z [ W ]
0.070
0.060
0.050
0.040
0.030
0.025
0.020 h2
t = ”
er = 4.0
Source: Motorola application note AN1051.
© H. Heck 2008
Section 2.1

22. Dual Stripline Impedance
[2.1.24] e r h 1 w
[2.1.25] t h 2
[2.1.26] w OR t h 1
[2.1.27]
0.003
0.005
0.007
0.009
0.011
0.013
0.015 w
[in] 10 20 30 40 50 60 70 80 90
100
110 Z [ W ]
0.020”
0.018”
0.015”
0.012”
0.010”
0.008”
0.005”
2h1 + h2 + 2t = 0.062”
t = ”
er = 4.0 h1
Source: Motorola application note AN1051.
© H. Heck 2008
Section 2.1

23. Surface Microstrip Impedance
[2.1.28] w
[2.1.29] e r t h
[2.1.30]
Source: National AN-991.
[2.1.31]
Source: Motorola MECL Design Handbook.
0.003
0.005
0.007
0.009
0.011
0.013
0.015
w [in] 20 40 60 80
100
120
140
160 Z
[W]
0.025”
0.020”
0.015”
0.012”
0.009”
0.006”
0.004” h
t = ”
er = 4.0
© H. Heck 2008
Section 2.1

24. Embedded Microstrip Or [2.1.32] [2.1.33] [2.1.34] [2.1.35]
[2.1.32] w h2 t Or h1
[2.1.33]
[2.1.34]
[2.1.35]
0.003
0.005
0.007
0.009
0.011
0.013
0.015
w [in] 20 40 60 80
100
120
140 Z [ W ]
0.015”
0.012”
0.010”
0.008”
0.006”
0.005”
0.003”
h2 - h1 = 0.002“
t= ”
er = 4.0 h1
© H. Heck 2008
Section 2.1

25. Summary
System 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.
© H. Heck 2008
Section 2.1

26. References
S. 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 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
B. Young, Digital Signal Integrity, Prentice-Hall PTR, 2001, 1st edition, ISBN
© H. Heck 2008
Section 2.1

27. Phase Constant (Lossless Case)
Recall the basic voltage divider circuit: R1 R2 V1 + V2 - I
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...
© H. Heck 2008
Section 2.1

28. Phase Constant (Lossless Case) #2
The 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:
© H. Heck 2008
Section 2.1

29. Phase Constant (Lossless Case) #3
The 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:
© H. Heck 2008
Section 2.1

30. Phase Constant (Lossless Case) #4
The 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
© H. Heck 2008
Section 2.1