fields in periodic structures

8/9/2019 Fields in periodic structures

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T H E S Y S T E M O N A C H I P C O M P A N Y ®

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Effects of Periodic Structures onTransmission Lines


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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


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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


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An equivalent circuit

jb jb jb jb jb

Unit Cell

d


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Some definitions

Propagation constant or wavenumber

Complex propagation constant

Phase constant

Attenuation constant

µε =k

µε ω γ j=

)Im( β =

)Re(=


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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


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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


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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


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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

®


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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 −=+=



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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 −=



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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



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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



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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



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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



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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.



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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

fields in periodic structures

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