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How to calculate acos with a complex number?

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helio1972

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hi,
I am working on antennas with metamaterials using FDTD method.
So to simulate metamaterials with FDTD i need fristly to calculate the permittivity and the permeabilty using retrieval method. so I have used the smith article namely '' Electromagnetic parameters retieval from inhomogeneous metamaterials'' published in March 2005. However, I have found a problem to calculate refraction index n using the equation given below because the b is a complex number, so how i can calculate acos with a complex number?

n=(1/kd)*acos(b)
with b= (1/2*S21)*(1-S11*S11+S21*S21)
 

Re: metamaterials

tink about the COS as an exponential form:

COS (x) = [exp(jx)+exp(-jx)]/2

Then if x is complex (x=a+jb) you get:

COS(x)= [exp(ja)*exp(-b)+exp(-ja)*exp(b)]/2

clearly a parallel reasoning holds for the SIN function.

Then if you want ACOS you need to solve the equation:

ACOS(y)=x => COS(x)=y =>

[exp(ja)*exp(-b)+exp(-ja)*exp(b)]/2 = y= u+jv

where y=u+jv.

Concluding:

u= 1/2*{exp(-b)*cos(a)+exp(b)*cos(a)}
v= 1/2*{exp(-b)*sin(a)-exp(b)*sin(a)}

check the equations... :)
 

Re: metamaterials

thanks,
but i want to calculate a and b not u and V.
 

Re: metamaterials

You can use Matlab to find those values.
 

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