Smith chart

Smith chart

Smith chart is a graphical aid for solving transmission line problems very easily which would be tedious to solve by using analytical methods.

1.        Smith chart consist of resistance and reactance circles.

  r=Rl\Zo

  x=Xl\Zo

  where Zl=Rl+jXl

hence in smith chart. This process is called “normalization”.

and the ratio (Zl\Zo) is called “per unit load impedence” or “normalized impedence”.

2.       Plotting of impedence: Any complex impedence can be shown by a single point on the

               Smith chart . this point is intersection of “r=Rl\Zo” and “x=jXl\Zo”.

EXAMPLE : Let Zl = 120+j160 Ω  and Z0=400 Ω

       Normalising we get,

r=0.3

x=0.4

zl=0.3+j0.4Ω  shown on Smith chart.

                                                
                                                                  BLACK MAGIC DESIGN

3.       VSWR Determination:

With “O” as centre and OA as radius a  circle is drawn  which cuts right side of the horizontal axis at “r” value of 3.9 (shown as point B)

This gives value of VSWR = ρ = 3.9

4.       Determination of Γ in magnitude and direction:

The line oa in figure is produced to meet the outer circle at point C

The angle at point C= Ɵ=133.2° is the phase angle of reflection coefficient.

To find magnitude of  Γ , the linear scale at bottom of chart is refered  which is marked

“voltage reflection coefficient”. With O as centre and radius exactly wqual to “OA”,

An arc is cut on left side.

The value of | Γ | is read as 0.6

Therefore Γ=0.6∟133.2°

5.       Location of  Vmax  and  Vmin  : The constant ρ circle intersects the central horizontal axis at

Points D and B.

Vmax = Rmax\Zo  =rmax = 3.91

Vmin = Rmin\Zo  = rmin = 0.26

6.       Open and short circuit line:

At point F on the horizontal axis r=∞ , x=∞ this represents open circuit termination of line.

At point E in the horizontal axis r=0 , x=0  this represents short circuit termination of line.

7.        Movement along periphery :

Movement in clockwise direction from point E results in no of wavelengths towards generator

Movement in anti-clockwise direction from point E results in no of wavelengths towards load.

One full rotation on pheriphery corresponds to λ\2.

8.       Conversion of impedence to admittance:

To find admittance of an impedence at point a, the point a is rotated through constant S-circle

By an amount λ\4  which is equivalent to 180°.

The point yl is diametrically opposite to “A” or (zl) has a value,
yl = 1.2-j1.6.