Smith Sphere

The smith chart is a nomogram used frequently in RF/Microwave Engineering. Since its inception it has been recognised that projecting the chart onto the reimen sphere [1].

[1]H. . Wheeler, “Reflection Charts Relating to Impedance Matching,” IEEE Transactions on Microwave Theory and Techniques, vol. 32, no. 9, pp. 1008–1021, Sep. 1984.



In [1]:

    
#from IPython.display import SVG
#SVG('pics/smith_sphere.svg')
from galgebra.printer import Format, Fmt



In [2]:

    
from galgebra import ga
from galgebra.ga import Ga
from sympy import *
Format()

(o3d,er,ex,es) = Ga.build('e_r e_x e_s',g=[1,1,1])
(o2d,zr,zx) = Ga.build('z_r z_x',g=[1,1])

Bz = er^ex # impedance plance 
Bs = es^ex # reflection coefficient plane
Bx = er^es
I = o3d.I()

def down(p, N):
    '''
    stereographically project a vector in G3 downto the bivector N
    '''
    n= -1*N.dual()
    return -(n^p)*(n-n*(n|p)).inv() 

def up(p):
    '''
    stereographically project a vector in G2 upto the space G3
    '''
    if (p^Bz).obj == 0:
        N = Bz
    elif  (p^Bs).obj == 0:
        N = Bs
        
    n = -N.dual()
    
    return   n + 2*(p*p + 1).inv()*(p-n)
    
a,b,c,z,s,n = [o3d.mv(k,'vector') for k in ['a','b','c','z','s' ,'n']]

Starting with an impedance vector $z$, defined by a vector in the impedance plane $B_z$, this vector has two scalar components ( $z^r$, $z^x$) known as resistance and reactance



In [3]:

    
Bz.dual()









    Out[3]:





\begin{equation*} - \boldsymbol{e}_{s} \end{equation*}



In [4]:

    
Bz.is_zero()









    Out[4]:





False



In [5]:

    
z = z.proj([er,ex])
z









    Out[5]:





\begin{equation*} z^{r} \boldsymbol{e}_{r} + z^{x} \boldsymbol{e}_{x} \end{equation*}

stereographically up-projecting this onto the sphere to point $p$,



In [6]:

    
p = up(z)
p









    Out[6]:





\begin{equation*} \frac{2 z^{r}}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{r} + \frac{2 z^{x}}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{x} + \frac{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} - 1}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{s} \end{equation*}



In [7]:

    
simplify(p.norm2())









    Out[7]:





$\displaystyle 1$

If we stereo-project this back onto the impedance plane



In [8]:

    
down(p, Bz)









    Out[8]:





\begin{equation*} z^{r} \boldsymbol{e}_{r} + z^{x} \boldsymbol{e}_{x} \end{equation*}



In [9]:

    
down(p,Bs).simplify()









    Out[9]:





\begin{equation*} \frac{2 z^{x}}{{\left ( z^{r} \right )}^{2} + 2 z^{r} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{x} + \frac{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} - 1}{{\left ( z^{r} \right )}^{2} + 2 z^{r} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{s} \end{equation*}



In [10]:

    
(z-er)*(z+er).inv()









    Out[10]:





\begin{equation*} \frac{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} - 1}{{\left ( z^{r} \right )}^{2} + 2 z^{r} + {\left ( z^{x} \right )}^{2} + 1}  - \frac{2 z^{x}}{{\left ( z^{r} \right )}^{2} + 2 z^{r} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{x} \end{equation*}



In [11]:

    
p









    Out[11]:





\begin{equation*} \frac{2 z^{r}}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{r} + \frac{2 z^{x}}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{x} + \frac{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} - 1}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{s} \end{equation*}



In [12]:

    
R=((-pi/4)*Bx).exp()
R









    Out[12]:





\begin{equation*} \frac{\sqrt{2}}{2}  - \frac{\sqrt{2}}{2} \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{s} \end{equation*}



In [13]:

    
R*p*R.rev()









    Out[13]:





\begin{equation*} \frac{- {\left ( z^{r} \right )}^{2} - {\left ( z^{x} \right )}^{2} + 1}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{r} + \frac{2 z^{x}}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{x} + \frac{2 z^{r}}{{\left ( z^{r} \right )}^{2} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{s} \end{equation*}



In [14]:

    
down(R*p*R.rev(),Bz)









    Out[14]:





\begin{equation*} \frac{- {\left ( z^{r} \right )}^{2} - {\left ( z^{x} \right )}^{2} + 1}{{\left ( z^{r} \right )}^{2} - 2 z^{r} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{r} + \frac{2 z^{x}}{{\left ( z^{r} \right )}^{2} - 2 z^{r} + {\left ( z^{x} \right )}^{2} + 1} \boldsymbol{e}_{x} \end{equation*}