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Along with ‘backward waves’ the Magneto-Inductive (MI) waves are becoming popular in this particular field
This happens or forms when two loops of SRR close to each other are ‘coupled’ to one another due to magnetic field of one loop ‘threading’ the other SRR. These coupling leads to waves are called MI waves.The dispersion equation is from.

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Along with ‘backward waves’ the Magneto-Inductive (MI) waves are becoming popular in this particular field

This happens or forms when two loops of SRR close to each other are ‘coupled’ to one another due to

magnetic field of one loop ‘threading’ the other SRR. These coupling leads to waves are called MI waves

ω

The dispersion equation is: ,

0

ω =

is resonant frequency

ω

of SRR,

0

inductance, the mutual

2

1 +

co s(

inductance

Axial coupling

For positive

0

will try to increase due to the induced voltage due to adjacent rings

Thus we write KVL for SRR-

+

= 0

0

−1

[

]

= −

+

)

Mutual impedance between two elements is defined as ratio of the voltages in element-2 to the current in

element-1, that introduced it. Corresponding vector potential is

(

∫

0

)

and the magnetic field is

= ∇ ×

over the area of loop-2. The flux threading the two loop-2

with mutual inductance

is . Note the is ‘com

plex quantity’ if the distance

φ =

2 1

2

2 1 1

between elements becomes comparable to wave-length.

Mutual inductance is positive if magnetic lines cross the two loops in the same direction, and negative if the

magnetic lines are in opposite direction

lo o p − 1

1

lo o p − 1

lo o p - 2

lo o p - 2

Axial coupling

The MI waves considered here are threaded to SRR which does not form a very long line so that

retardation effect and its losses due to radiation are not considered presently. So the MI lines are ‘short’.

From the KVL of the

+

= 0

0

−1

[

]

= −

+

)

−

=

w ave - so lu tio n assu m ed an d su b stitu te

0

⎛

1

⎞

+

+

)

⎜

⎟

⎝

⎠

2

2

−

−

−

= −

+

)

0

0

2

2

2

−

2

+

) = ω 2

T ak e

2

ss; an d reso n an t freq u en cy o f S R R as ω

= (1 /

0

2

2

ω

ω 2

1 −

−

co s

2

2

ω

ω

0

0

2

ω ⎛

2

⎞

1 +

co s

= 1

T h is is d isp ersio n relatio n

⎜

⎟

2

ω ⎝

⎠

0

From previous expansion of KVL for the

2

2

2

−

2

+

) = ω 2

2

2

2

ω

ω

2

ω

+ 1 −

−

κ cos

κ =

p u t

2

2

ω

ω

ω

0

0

2

2

2

ω 1

ω

ω

+ 1 −

−

κ cos( β

w h e re

2

2

2

ω

ω

ω

0

0

0

2

2

2

ω 1

ω

ω

+ 1 −

−

κ (cos β

2

2

2

ω

ω

ω

0

0

0

w ith

c o s

sin

2

2

ω 1

ω

2

2

ω

ω

+ 1 −

−

κ cos β

κ sin β

2

2

ω

ω

2

2

ω

ω

0

0

0

0

s e g re g a tin g re a l & im a g in a ry p a rts , a n d e q u a tin g to z e ro

2

2

ω

ω

1 −

−

κ cos β

2

2

ω

ω

0

0

1 − κ sin β

t h

+

(

+

) =

0

K V L f o r

l o o p

0

1

=

+

+

s e l f i m p e d a n c e

0

Assume wave solution in form:

=

= β − α

0

where

complex quantity with

β as propagation constant,

α as attenuation. The dispersion relation

1

− 2

⎛

2

ω

ω

⎞

=

1 +

co s

2

0 ⎜

⎟

⎝

ω

⎠

1 −

[1 + κ co s( β

may be separated into real and imaginary parts yielding

2

ω 0

1 − κ sin(β

κ = 2

co u p lin g co efficien t

lo sses

1

2

3

4

4

1

3

2

1

2

ω

1 −

[1 + κ co s( β

2

ω 0

an d

1 − κ sin(β

α ≅ 0

co sh α

p u ttin g ab o ve

2

ω

1 −

[1 + κ co s( β

2

ω 0

ω = ω / 1 + κ cos(β

is lo ss - less d isp ersio n !

0

If the losses are small then and thus attenuation thus

α ≅ 0

c o s h ( α

1

s i n h ( α

which means in the dispersion equation for phase change per element remains same, and losses

per element given as:

1

α

It may be expected that losses decline as the coupling coefficient a

κ

nd increases

Similar to part-2 where explanation of dispersion given through TL circuit approach

=

−

=

⎡

⎤

⎡

⎤ ⎡

1 1

1 2

=

⎢

⎥

⎢

⎥ ⎢

⎥

⎣

⎣ 21

2 2 ⎦ ⎣

= 1

= −

= −

= 1 +

1 1

1 2

2 1

2 2

G e n e ra l d is p e rs io n e q u a tio n

2 c o s

+

1 1

2 2

⎡

0

−

⎤

⎢

⎥

> 0

2

ω

1

ω ⎞

⎢

ω

0

0

⎥

−

−

⎜ 1 −

⎟

− κ

2

⎢

1

ω

⎥

⎣

⎝

⎠ ⎦

ω 0

< 0

2

ω

0

2 c o s

+

= −

+

0

1 1

2 2

2

ω

1 + κ

ω 0

ω =

κ = 2

0

π

π

1 + κ c o s (

2

Dispersion lossless case

α = 0

κ = ± 0 . 1

Backward wave for

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