Left Handed Maxwell Systems
PART-7
Magneto-Inductive Waves
SAMEER
Shantanu Das
RR&PS
Reactor Control Division, B.A.R.C. Mumbai-400085
shantanu@barc.gov.in

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Magneto Inductive (MI) waves
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,
L is
0
inductance, the mutual
2 M
M
1 +
co s( ka )
inductance
L
n + 1
n
n + 1
n − 1
n
n − 1
Axial coupling M > 0 Planer coupling M < 0
For positive M the central ring’s voltage drop due to its own current Z I
0
n
will try to increase due to the induced voltage due to adjacent rings
jω and
M I
jω M I
n −1
n +1
Thus we write KVL for SRR-n as:
Z I + jω M I
+ jω M I
= 0
0
n
n −1
n +1
−1
[ R + jω L + ( jω C )
]I
= − jω M ( I
+ I
)
n
n −1
n +1

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Magneto Inductive (MI) waves positive and negative mutual inductances
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
A = ( μ / 4π
( J / r ) d v
∫
0
)
and the magnetic field is
H fitted
= ∇ × A
over the area of loop-2. The flux threading the two loop-2
with mutual inductance
M
is . Note the is ‘com
M 21
plex quantity’ if the distance
φ =
2 1
M
I
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
I1
I
lo o p − 1
1
lo o p − 1
lo o p - 2
lo o p - 2
Axial coupling M > 0 Planer coupling M < 0
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’.

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The Dispersion expression
From the KVL of the n-th SRR we have
Z I + jω M I
+ jω M I
= 0
0
n
n −1
n +1
−1
[ R + jω L + ( jω C )
]I
= − jω M ( I
+ I
)
n
n −1
n +1
− jkna
I
= I e
w ave - so lu tio n assu m ed an d su b stitu te
n
0
⎛
1
⎞
R +
+ jω L I = − jω M ( I
+ I
)
⎜
⎟ n
n −1
n +1
⎝
jω C
⎠
2
2
jω R C + 1 + j ω L C
− jkna
− jk ( n −1) a
− jk ( n +1)
I e
= − jω M I (
a
e
+ e
)
0
0
jω C
2
2
2
jk a
− jka
2
jω R C + 1 − ω L C = − j ω M C (e
+ e
) = ω 2 M C co s( ka )
T ak e R as zero lo ssle
2
ss; an d reso n an t freq u en cy o f S R R as ω
= (1 / L C )
0
2
2
ω
ω 2 M
1 −
−
co s ka = 0
2
2
ω
ω
L
0
0
2
ω ⎛
2 M
⎞
1 +
co s ka
= 1
T h is is d isp ersio n relatio n
⎜
⎟
2
ω ⎝
L
⎠
0

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Dispersion R not equal to zero a lossy case with k as complex number (with attenuation)
From previous expansion of KVL for the n-th SRR with loss resistance R we have
2
2
2
jk a
− jka
2
jω R C + 1 − ω L C = − j ω M C ( e
+ e
) = ω 2 M C c o s ( k a )
2
2
2
jω R L C
ω
ω
2 M
ω L
+ 1 −
−
κ cos ka = 0
κ =
p u t
Q =
2
2
ω L
ω
ω
L
R
0
0
2
2
2
ω 1
ω
ω
j
+ 1 −
−
κ cos( β a − jα a ) = 0
w h e re
k = β − jα
2
2
2
ω Q
ω
ω
0
0
0
2
2
2
ω 1
ω
ω
j
+ 1 −
−
κ (cos β a cos jα a + sin β a sin jα a ) = 0
2
2
2
ω Q
ω
ω
0
0
0
w ith
c o s jx = c o sh x
sin jx = j sin h x
2
2
ω 1
ω
2
2
ω
ω
j
+ 1 −
−
κ cos β a cosh α a − j
κ sin β a sinh α a = 0
2
2
ω Q
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 a cosh α a = 0
2
2
ω
ω
0
0
1 − κ sin β a sinh α a = 0
Q

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Circuit coupling
M
M
M
L
L
L
L
C
C
C
C
Vn−2
R
R
V
R
V
n −1
n
Vn+1
R
t h
Z
I
+
j ω M
( I
+ I
) =
0
K V L f o r n
l o o p
0
n
n − 1
n + 1
1
Z
=
j ω L
+
+ R
s e l f i m p e d a n c e
0
j ω C
jk n a
Assume wave solution in form:
I
I e −
=
= β − α
n
0
where
k

j
complex quantity with
β as propagation constant,
α as attenuation. The dispersion relation
1
− 2
⎛
2 M
ω
ω
⎞
=
1 +
co s ka
2
0 ⎜
⎟
⎝
ω
L
⎠
1 −
[1 + κ co s( β a ) co sh (α a )] = 0
may be separated into real and imaginary parts yielding
2
ω 0
1 − κ sin(β a) sinh(α a) = 0
Q
κ = 2 M / L
co u p lin g co efficien t
Q = ω L / R
lo sses

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Planer and Axial coupling of SRR with excitation and termination to form ‘backward’
and ‘forward’ MI waves
y
a
N
1
2
3
4
x
z
N
V
4
1
Z t
3
2
Z
1
t
r0
y
a
x
z
V 1

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Small losses dispersion expression
2
ω
1 −
[1 + κ co s( β a ) co sh (α a )] = 0
2
ω 0
an d
1 − κ sin(β a) sinh(α a) = 0
Q
α ≅ 0
co sh α a = 1
p u ttin g ab o ve
2
ω
1 −
[1 + κ co s( β a )] = 0
2
ω 0
ω = ω / 1 + κ cos(β a )
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 a ) ≅ and
1

s i n h ( α a ) ≅ α a
which means in the dispersion equation for phase change per element remains same, and losses
per element given as:
1
α a = κ Q s i n ( β α )
It may be expected that losses decline as the coupling coefficient a
κ
nd increases
Q

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Waves on four pole
Similar to part-2 where explanation of dispersion given through TL circuit approach
I
I
in
out
Z
V
V
Y
out
in
V
= I Z + V
i n
i n
o u t
I
− I
i n
o u t
V
=
o u t
Y
⎡V
⎤
⎡ b
b
⎤ ⎡V ⎤
o u t
1 1
1 2
in
=
⎢
⎥
⎢
⎥ ⎢
⎥
I
b
b
I
⎣ out ⎦
⎣ 21
2 2 ⎦ ⎣
in ⎦
b
= 1
b
= − Z
b
= −Y
b
= 1 + YZ
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 k a = b
+ b
1 1
2 2

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Coupled SRR circuit and dispersion
I
I
in
o u t
L
C
V
M
V
i n
o u t
⎡
0
− jω M
⎤
⎢
⎥
M
> 0
2
B =
ω
1
L ⎛
ω ⎞
⎢
ω
0
0
⎥
−
−
⎜ 1 −
⎟
− κ
2
⎢
1
jω M
M
ω
⎥
⎣
⎝
⎠ ⎦
ω 0
M
< 0
2
L
L ω
ω
0
2 c o s k a = b
+ b
= −
+
0
1 1
2 2
2
M
M
ω
1 + κ
ω 0
ω =
κ = 2 M / L
0
π
π
1 + κ c o s ( k a )
k a
2
Dispersion lossless case
α = 0
Q = ∞
κ = ± 0 . 1
Backward wave for M < 0

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