2.SMITH CHART

This is a circular slide rule graph that enables exact calculations of transmission line matching. It was developed around 1939 by P.H Smith and has become very useful to date in it we are able to match transformers for complex load impedance and correct the electronic enemy of attenuations within the transmission lines. It helps in the design of the antennas.
The chart constitutes the following

1.Pure resistance (or Zero reactance) line.
2.Two circles sets for reactance XL and Xc.

3.Two wavelength scales.
4.The reflection co- efficient angle scale.
5.The reflection co- efficient magnitude scale.
6.The dB of loss scale.

I. PURE RESISTANCE LINE.

It lies horizontally across the chart passing through the prime centre of the graph
extending in both directions to the perimeter of the chart called zero reactance where it is
calibrated from zero; on the left, to infinity, on the right. The value of l is at the graph's
center.

II. REACTANCE CIRCLES (ARCS).

The area above horizontal line is reserved for inductive reactance while the area at
the lower half of the chart is reserved for capacitive reactance. This explains why the horizontal line is called the zero reactance line. The reactance circle sets are a series of eccentric cycles above and below the horizontal line and tangential to the point of infinity. Although the reactance are full-circles, values only at the portion of a circle that falls inside the resistance circle R=0 is included on the Smith chart. The point is that reactance values greater than 1= l+(jx or-jx) will only be located in an area smaller than 1/4 of the total smith chart area. The upper right Quadrant is for values of +jx and lower right for values of –jx values. But first the values for resistance are printed on the chart along the zero reactance line and along part of each X= 1 arc. Values of reactance are printed around the perimeter of the R=0 circle and again along part of the R=l circle.

III. WAVE LENGTH SCALES.

These are two scales around the outside edge of the chart. They both start at the left of the
chart on the Zero reactance line. They advance in clockwise direction only to make half
Wavelength in one full revolution as 0 mark represents starting point and ending point.
This is labeled "wavelengths towards the generator". The one half wavelength measures are used because the transmission lines patterns repeat themselves every half wavelength.
The second inner scale labeled "wavelength toward the load" advances in
Anticlockwise direction. The choice of use of a scale depends on the information known

and what information is to be determined. Measure of one-quarter wavelength toward the

generator and that of one-quarter wavelength toward the load are both at the generator and one quarter wavelength toward the same point (extreme right). The two wavelength scales
are on separate slip rings and are freely positioned anywhere around the perimeter of the
chart

IV. REFLECTION COEFFICIENT.

The third scale from the edge is marked Angle of reflection coefficient in degrees starts
from left to right with positive 180° represented at the upper half while negative angles
represented at lower half of the chart.

V. REFLECTION MAGNITUDE.

Identified as radially scaled parameters. On the smith circular slide rule are scales on a clear bar attached to the center of the calculator and is free to rotate around the chart. The two scales of interests for the reflection coefficient magnitude, at the upper right and the dB loss, found on the upper left. The scales are also applied as the need arises and uses of the chart are established.

NORMALIZED IMPEDANCES.

This is a chart that has the R=l circle passing through the primary center and is made to
process all values of impedance values smaller than 50ohms. Charts would have the R=50 circles passing through the prime center. All impedance values smaller than R=50 ohms and would have be to the left of the center. But the normalized reference impedance chart is designed to represent all values of load impedance. The conversion from real-life load impedance is executed by dividing the characteristics impedance.
Zn=Z1Z0 e.g. 60j80 ohms=>60/50=1.2 and 80/50=1.6
Normalized impedance = 1.2 + j1.6ohms
Standing wave Ratio circle
When a circle on the smith Chart where the prime center (R=l, X=0) as a pivot point and
Zn as a point on the circumference, the circle is called the standing wave ratio-circle.
1) Numerical value of the standing wave ratio, Is the point where the circle crosses the
Zero reactance line to the right.
2) Value of the load Zn is located on this circle; SWR circle is a plot of the impedance on the line at every point along the length of the line.
3) SWR circle as the location of the maximum voltage of the standing wave (Emax) at
X=0 in wavelengths from the load (Znmax) on the line, in wavelengths from the load.
4) Where the SWR circle crosses the zero reactance line to the left of center represents
the location along the line where Emin and Zmax are found.
The load line
This is a line drawn from the prime center of the chart and passes through Zn and extends out past the wavelength scales cutting through the wavelength towards the generator.
It moves along the outer wavelength scale to the point where it intersects the Zero reactance line wavelengths. This wavelength difference is the distance from the load to where the standing wave voltage is maximum.
The SWR circle crosses clock wisely the Zero reactance line left of center where the circle (SWR) circle crosses the X=0 line on the right sides gives the SWR.
Radius of the circle is determined by the point Zn and R=l as the center of pivot,
Zn, Yn (admittance) and Bn (susceptance) lie on the SWR line (circle) with Zn being the inverse of Yn. To find Bn move clockwise from Yn to a point where SWR circle first crosses the R=l circle and that is the point Bn.
Matching load impendence
Inductors and capacitors are the best components as far as load impendence matches are concerned. This is because resistors siphon off 33% of power intended for the load.
When put in parallel 30% of load power is dissipated when connected in series.
Step one is the normalization of the load impendence by dividing it by the
characteristics impendence of the line.
Zn = Zl/Zo e.g. If ZL=120=80 ~ and Zo = 100~(Transmission line)
= 120 = j80
100

The load line is changed to a value that can be plotted an the universal graph(chart)
The value read from the graph and renormalized to a real life impendence value by multiplying the chart value by the line impendence.
Step two is plotting of the normalized values onto the graph and drawing the S.W.R circle. The load line joining the R=l and Zn is extended in the inverse direction to find the admittance of the load. That is Zn divided by any convenient Z value. Plot Zn on the chart, draw a circle and a load line, find the normalized admittance and then multiply the normalized admittance by the selected Z value. Practically, a section of the same transmission line to is be used to form a capacitor or inductor and is connected in parallel with line being matched. This section is called a matching stub. The smith chart determines the size and placement of the matching stub that will cause an irregular load impendence to match the line impendence.
Other uses of smith chat include the uses of an instrument called slotted line. Minimum and maximum voltage of the maximum voltage of the standing wave can be measured with great accuracy using a slotted line.
SWR = E max / E min