lec.4 smith chart and impedance calculations

RF & Microwave EngineeringBETE-Spring 2009

Department of Electrical EngineeringAir University

Smith Chart and Impedance Calculations

Lecture No. 4

RF & Microwave EngineeringBETE-Fall 2009

Basit Ali ZebDepartment of Electrical Engineering, AU

What is a Smith Chart?

• It is a graphical tool for analyzing and designing transmission line circuits, matching circuits and impedance calculations. It is widely used in antenna design & RF/Microwave engineering

• Transformation from the complex input impedance plane into the complex reflection coefficient plane

• It is a special type of 2-D graph to represent the complex reflection coefficient



The Definition

The Smith Chart is a transformation plot of normalized complex transmission line impedance (resistance and

reactance) in its complex plane into a suitable complex

reflection coefficient plane



The Basics

0Γ

A complex reflection coefficient can be written as:

which is a point in the polar plane with a radius r = |Γ|,

measured from the origin of polar plane and an angle θ,

measured from the positive real axis in counter-clockwise direction.

Transformation of reflection coefficient at load to reflection

coefficient away from the load makes the smith chart useful.

)(lΓ



Complex RepresentationΓ

The load reflection

coefficient in

phasor form in complex plane

0Γ



The Algebra Behind Smith Chart

Multiply the numerator and denominator in the above

equation with the complex conjugate of denominator in above equation. Then rearranging the equation to illustrate the real

and imaginary parts.



Equation of Circle – Resemblance

It resembles with the equation of a circle in xy -plane with radius r and centered at x=a and y=b

By manipulating the REAL PART of the last equation, we can get meaningful graphical representation and resemblance with the equation of a circle.



Constant Resistance Circles

Further modifying the equation by adding

on both sides, and rearrange the resulting equation in the form of circle with :

Center at:

Radius of:

We get:



Constant Resistance Circles

Points of constant resistance form circles on the complex reflection-coefficient plane . Shown here are the circles for various

values of normalized load resistance.



Transformation



Equation of Circle – Resemblance

By modifying the above equation and adding a constant

to make the Γi terms a factorable polynomial:

Similarly, By manipulating the IMAGINARY PART of the last equation, we can get resemblance with the equation

of a circle



Constant Reactance Circles

The resulting equation is in the form of circle with :

Center at:

Radius of:

Values of constant imaginary load impedances xL make up

circles centered at points along the blue vertical line.



Constant Reactance Circles

The segments lying in the top half represent INDUCTIVE Reactances; those lying in the bottom half

represent CAPACITIVE reactances.

All circle centers lie on the blue vertical line. Only the circles segments that lie within the green Γ

r= 1 circle

are relevant for the Smith chart.

Note that xL

= 0 along the horizontal axis, which represents a circle of infinite radius centered at [1, +y] or [1, –y] in the complex Γ plane.



Transformation



The Complete Z Smith Chart



The Complete Smith Chart



Special Points on Z-Smith Chart

Special

Points

+1

-1

-1

+1

Point of Short Cct. ●

Point of Open Cct. ●

Matched Cct. ●

●● ●

For purely real impedance i.e., x=0

For purely imaginary

impedance i.e., r=0



Observations

• Circles of constant normalized resistance having a range of 0 ≤ r < ∞

• Circles of constant normalized reactance (arcs) can represent either positive (i.e. inductive) or negative (i.e. capacitive) in a range of -∞ ≤ x < ∞

• Each full circle describe a transformation over a transmission line length of λ/2 giving an electrical length of 180o

• Rotation on smith chart in clockwise direction is the movement towards generator (or away from load).

• Reflection coefficient does not necessarily satisfy the criteria of |Γ| ≤ 1. For r ≥ 0, we have passive circuits, and for r ≤ 0, leads to the case |Γ| ≥ 1.



Basic Smith Chart Techniques



Given Z(d) Find

)(dΓ

)(dΓ



Given Z(d) Find )(dΓ



Given Find Z(d) )(dΓ



Given & ZR Find & Z(d)RΓ )(dΓ



Graphical Procedure



On Smith Chart



Given & ZR Find d(max) & d(min)R

Γ



On Smith Chart



Given & ZR Find VSWRRΓ



SWR and Smith Chart

For an arbitrary distance l along the transmission line, the SWR is written as:

where

This form of reflection coefficient permits the representation of SWR as circles in the Smith Chart with matched condition

being the origin (SWR =1) or



Find VSWR



On Smith Chart



Impedance & Admittance



Admittance (Y) Smith Chart



Admittances on Smith Chart



Given Z(d) Find Y(d)



Admittances on Smith Chart



Procedure



On Smith Chart



Example-1

Input Impedance of a terminated transmission line

Question:



Example-1



Example-1

Comment:

We note that the reflection coefficient phasor form at

the load is multiplied by a rotator that incorporates twice the electric line length βd. This mathematical

statement thus conveys the idea that voltage/current waves have to travel to the load and return back to

the source to define the input impedance.



Example-2

Calculate the Input Impedance of a transmission

line using Smith Chart



Solution



Example-3

Comments:

As a graphical design tool, Smith Chart allows immediate observation of the degree of mismatch between the line and

the load impedances by plotting the radius of SWR circle.

Question:



SWR circles of Example-3



Study

• Article 2.9 on Smith chart from text book

• Next topic of discussion

–Microstrip Lines

lec.4 smith chart and impedance calculations

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