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8/9/2019 Fields in periodic structures
1/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGICLSI
Effects of Periodic Structures onTransmission Lines
8/9/2019 Fields in periodic structures
2/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Why examine periodic structures?
Periodic structures - a definition• Transmission lines loaded at periodic intervals with identical reactive
elements
The following are examples of periodic structures:• A backplane loaded with SCSI devices• A ribbon cable loaded with SCSI devices• A bare twisted - flat ribbon cable with or without connectors
• Periodic stubs on a microstrip line.The theory of periodic structures:
• Provides insight into the behavior of backplanes and cables
• Explains this behavior quantitatively
8/9/2019 Fields in periodic structures
3/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Periodic structures
Are characterized by a uniform distribution of reactive elementsHave pass-band and stop-band propertiesHave slow wave properties
Can exhibit significant impedance shiftsCan be described mathematically through the use of voltageand current transfer functions
8/9/2019 Fields in periodic structures
4/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
An equivalent circuit
jb jb jb jb jb
Unit Cell
d
8/9/2019 Fields in periodic structures
5/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Some definitions
Propagation constant or wavenumber
Complex propagation constant
Phase constant
Attenuation constant
µε =k
µε ω γ j=
)Im( β =
)Re(=
8/9/2019 Fields in periodic structures
6/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Characteristics
Unloaded line:• Has a propagation constant of k• Has a characteristic impedance of Z0
Structure consists of a number of unit cells• Consist of a length d of transmission line• Have a shunt susceptance b across the midpoint of the cell• The susceptance is normalized to Z0• Can be represented by a cascade of identical 2-port networks
8/9/2019 Fields in periodic structures
7/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Network representation
Structure is a cascade of identical 2 port networksEach cell is represented by an identical ABCD matrixThe ABCD matrix represents
• Section of transmission line of length d/2• A shunt susceptance of b• Another section of transmission line of length d/2
Voltage and currents on either side of each cell are representedby
=+
+
1
1
n
n
n
n
I
V
DC
B A
I
V
8/9/2019 Fields in periodic structures
8/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
ABCD Matrix
θ = k • d where k is the propagation constant of theunloaded line
−
++
−+
−=θ θ θ θ
θ θ θ
θ
sin2
cos2
cos2
sin
2cos2sinsin22cosbbb
j
bb
j
b
DC B A
8/9/2019 Fields in periodic structures
9/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Current and voltage phase
The voltage and current phase only differ by the propagationfactor
Resulting in the following representation
ze γ −
znn
znn
e I I
eV V
γ
γ
−+
−+
=
=
1
1
==+
+
+
+d
n
d n
n
n
n
n
e I
eV
I
V
DC
B A
I
V γ
γ
1
1
1
1
®
8/9/2019 Fields in periodic structures
10/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Solution of matrix
The former equation reduces to the following
Since• We have the following
• And since the right-hand side of the equation must be purely real, either α = 0, and β ≠ 0
or
β = 0 or π , and α ≠ 0
( ) 02 =−+−+ BC e D Ae AD d d γ γ
β j+=
θ θ β α β α γ sin2
cossinsinhcoscoshcosh b
d d jd d d −=+=
T H E S Y S T E M O N A C H I P C O M P A N Y ®
8/9/2019 Fields in periodic structures
11/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Solution case α αα α ==== 0,0,0,0, and β ββ β ≠≠≠≠ 0000
Corresponds to non-attenuating propagating waveDefines a passbandSolved for β if magnitude of right-hand side is less than unity
Infinite number of values of β
θ φ β sin2coscos b
d −=
T H E S Y S T E M O N A C H I P C O M P A N Y ®
8/9/2019 Fields in periodic structures
12/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Solution case α αα α ≠≠≠≠ 0, β ββ β = 0, π ππ π
Wave is attenuatedDefines stopband of structureCauses reflections back to input
Has only one solution for positively traveling waves• α > 0 for positively traveling waves• α < 0 for negatively traveling waves
For β = π then Z0 is same as if β = 0
1sin22
coscosh ≥−= θ θ
α b
d
T H E S Y S T E M O N A C H I P C O M P A N Y ®
8/9/2019 Fields in periodic structures
13/17
T H E S Y S T E M O N A C H I P C O M P A N Y ®
LOGIC
LOGICLSI
Bloch impedance
Defined as the characteristic impedance of waves on thestructure:• impedance is normally less than the impedance of the structure• can be as little as 0.3 Z0
Impedance shift causes attenuation and reflection issuesCan be calculated from the ABCD matrix
120
−
±=±
A
BZ Z
B
T H E S Y S T E M O N A C H I P C O M P A N Y ®
8/9/2019 Fields in periodic structures
14/17
T H E S Y S T E M O N A C H I P C O M P A N Y
LOGIC
LOGICLSI
Attenuation and impedance example Attenuation, Twisted Pair Ribbon, 25 meter sample,
WITHOUT ANY FLATSand
WITH FLATS on 9.85 i nch Cen ters
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0-0.8
-0.6
-0.4
-0.2
0.0
10 100 1000Frequency MHz
G A I N
d B / m
e t e r
NO Flats
9.85 inch Centers
424 MHz
T H E S Y S T E M O N A C H I P C O M P A N Y ®
8/9/2019 Fields in periodic structures
15/17
T H E S Y S T E M O N A C H I P C O M P A N Y
LOGIC
LOGICLSI
Terminated periodic structures
Are similar to SCSI backplanesHave a propagation velocity much slower than light• Velocity can be as little as 0.4 c
Have reflected waves• Γ is a function of the Bloch impedance• Bloch impedance can be as little as 0.3 Z0
For no reflections, Bloch impedance must equal the feedingtransmission line impedance• Achieved through matching sections• Achieved by raising backplane impedance
T H E S Y S T E M O N A C H I P C O M P A N Y ®
8/9/2019 Fields in periodic structures
16/17
T H E S Y S T E M O N A C H I P C O M P A N Y
LOGIC
LOGICLSI
Conclusions
Periodic structure analysis applies to SCSI elements• Backplanes with connectors, w/wo drives• Ribbon cable with connectors, w/wo drives• Flat sections of twisted-flat cable
Period structures have the following properties• Decreased impedance• Comb filter characteristics• Decreased propagation velocity
SCSI backplane and cable performance can be analyzedmathematically through use of periodic structures.
T H E S Y S T E M O N A C H I P C O M P A N Y ®
8/9/2019 Fields in periodic structures
17/17
LOGIC
LOGICLSI
References
D. M. Pozar, Microwave Engineering, Addison-Wesley, N.Y.,1990Ramo, Whinnery, & Van Duzer, Fields and Waves inCommunication Electronics, 2nd Edition, Wiley, N.Y., 1967R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, N.Y., 1966