Download - Lec.4 Smith Chart and Impedance Calculations
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RF & Microwave EngineeringBETE-Spring 2009
Department of Electrical EngineeringAir University
Smith Chart and Impedance Calculations
Lecture No. 4
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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
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Complex RepresentationΓ
The load reflection
coefficient in
phasor form in complex plane
0Γ
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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.
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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.
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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:
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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.
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Transformation
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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.
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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.
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Transformation
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
The Complete Z Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
The Complete Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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.
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Basic Smith Chart Techniques
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Given Z(d) Find
)(dΓ
)(dΓ
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Given Z(d) Find )(dΓ
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Given Find Z(d) )(dΓ
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Given & ZR Find & Z(d)RΓ )(dΓ
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Graphical Procedure
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
On Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Given & ZR Find d(max) & d(min)R
Γ
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
On Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
On Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Given & ZR Find VSWRRΓ
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Find VSWR
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
On Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Impedance & Admittance
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Admittance (Y) Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Admittances on Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Given Z(d) Find Y(d)
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Admittances on Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Procedure
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
On Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Example-1
Input Impedance of a terminated transmission line
Question:
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Example-1
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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.
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Example-2
Calculate the Input Impedance of a transmission
line using Smith Chart
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Solution
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
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:
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
SWR circles of Example-3
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RF & Microwave EngineeringBETE-Fall 2009
Basit Ali ZebDepartment of Electrical Engineering, AU
Study
• Article 2.9 on Smith chart from text book
• Next topic of discussion
–Microstrip Lines