# Smith Chart ```The
SMITH
Chart
Review: Reflection Coefficient
Reflection Coefficients Transformation


Reflection Coefficient (at the input)
Complex Number:

Rectangular form
Γ=x+jy

Polar form
Γ=r∟θ

Exponential form
Γ = r ejθ
Γ
• The Smith chart is one of the most useful graphical tools for high
frequency circuit applications.
• Invented by Phillip H. Smith (1905–1987)
• Designed for Electrical and Electronics Engineers specializing in radio
frequency (RF) engineering to assist in solving problems
with transmission lines and matching circuits.
• From a mathematical point of view, the Smith chart is simply a
representation of all possible complex impedances with respect to
coordinates defined by the reflection coefficient.
• The domain of definition of the reflection coefficient is a circle of radius
one (1.0) in the complex plane. This is also the domain of the Smith
chart.
𝑍𝐿 − 𝑍𝑂
Γ=
= 𝑅𝑒(Γ) + 𝐼𝑚(Γ)
𝑍𝐿 + 𝑍𝑂
Possible values for |Γ|
−𝟏 &lt; 𝟎 &lt; +𝟏
“Short” “Matched” “Open”
SWR Circles

From the origin, all impedance points
along a fixed radius circle will have the
same VSWR or |Γ| (magnitude of Γ)
Typical use of a Smith Chart
Find voltage reflection coefficient (Γ∠𝜃 or x+jy)
 Find value of the voltage standing wave ratio (VSWR)
 Find return Loss (dB) / Reflection Loss (dB)
 Find Zin at any given distance along the transmission line
 Find Γ at any given distance along the transmission line
 Find VSWR at any given distance along the transmission line
 Do impedance matching:

◦ Quarter Wave Transformer
◦ Lumped Network
◦ Stubs

Etc.
Special Consideration



The Smith Chart is a very useful tool if we assume the transmission
line to be lossless or distortionless. Since the reflection coefficient
will have a uniform magnitude along the entire length of the line.
If the transmission line is lossy, we can still use the smith chart, but
the graphical analysis become more complicated. It would be
better to use analytical methods for this case.
The spiral pattern will appear since the magnitude of the reflection
coefficient will decrease along the length of the line.
Non-uniform curves forming spiral
patterns becomes evident if the
transmission line is lossy!
Microwave Network Analyzer
An HP8720A network analyzer showing a plot of the Smith Chart
(also known as an RF Vector Analyzer)

Transmission Coefficient (T)
Clarification:
Voltage reflection coefficient ( ΓE = Γ )
 Current reflection coefficient ( ΓI = Γ )
 Power reflection coefficient ( ΓP = Γ2 )

RECALL:
Normalized Impedance (z)
𝑍𝐿 𝑅𝐿 + 𝑗𝑋𝐿
𝑧=
=
= 𝑟 + 𝑗𝑥
𝑍𝑂
𝑍𝑜
𝑅𝐿
𝑟=
𝑍𝑂
Normalized Resistance
Normalized Reactance
𝟎 &lt; 𝒓 &lt; +∞
−∞ &lt; 𝒙 &lt; +∞
𝑋𝐿
𝑥=
𝑍𝑂
In order to use the Smith Chart, impedances must be normalized.
Normalized Resistance
Normalized Reactance
Conversion: Γ(d) &lt;-&gt; Z(d)
Γ(d) = reflection coefficient
Z(d) = input impedance
(d) = distance from the load (towards the generator)

Recall: if ‘d’ is not specified:
Γ(d) = ΓL
Z(d) = ZL
Find the ff:
Case 1 (use analytical method)
Case 1I (use graphical method)
a.) ΓL (polar &amp; rectangular)
b.) TE (polar &amp; rectangular)
b.) VSWR
c.) Return Loss (dB)
d.) Reflection Loss(dB)
Find ZL using smith chart:
Given:
ΓL = 0.665 ∟ 60&deg;
Zo = 100 Ω
```