Smith Charts and Impedance
Matching
Constant Resistance Circles
ECE357 / Prof. S. V. Hum
ECE357 / Prof. S. V. Hum
Constant Reactance Circles
ECE357 / Prof. S. V. Hum
ECE357 / Prof. S. V. Hum
ECE357 / Prof. S. V. Hum
Smith Chart
•  rL- and xL- circles
are orthogonal to
each other
•  Intersection of rLcircle and xLcircle defines a
normalized load
impedance
ECE357 / Prof. S. V. Hum
ECE357 / Prof. S. V. Hum
Constant |Γ| Circles
ECE357 / Prof. S. V. Hum
Reading VSWR
•  Recall for RL > R0, RL = SR0
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ECE357 / Prof. S. V. Hum
Input Impedance / Input Reflection
Coefficient from a Lossless Line
ECE357 / Prof. S. V. Hum
ECE357 / Prof. S. V. Hum
Example: What is the
input impedance seen
into a 0.2λ line
terminated in ZL?
ECE357 / Prof. S. V. Hum
Translating the Load along a Line
ECE357 / Prof. S. V. Hum
Smith Chart Calculations for Lossy
Lines
ECE357 / Prof. S. V. Hum
Example: Determine the input impedance of a 2 m long line terminated in
a load impedance ZL = 67.5 – j45 Ω. The parameters of the line are as
follows: Z0 = 75 Ω, α = 0.029 Np/m, and β = 0.2π rad/m.
ECE357 / Prof. S. V. Hum
Impedance Matching
ECE357 / Prof. S. V. Hum
Impedance Matching on the Smith Chart
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Quarter-Wave Transformer
ECE357 / Prof. S. V. Hum
Normalized Admittance
ECE357 / Prof. S. V. Hum
ECE357 / Prof. S. V. Hum
Admittance Smith Chart
ECE357 / Prof. S. V. Hum
Single-Stub Matching: Series
Tuning
ECE357 / Prof. S. V. Hum
Single-Stub Matching: Steps
(shunt version in parentheses)
1.  Plot the normalized load impedance on the
Smith Chart (and convert to admittance for
shunt stub tuning)
2.  Draw the |Γ| circle and translate the impedance
along the line to the rL=1 (gL=1) circle (2
solutions) => r’ = 1±jx’ (y’ = 1±jb’)
3.  Determine load-section length d from angles
between point representing zL (yL)and the point
on the rL=1 (gL=1) circle
4.  Determine stub length ℓ between the open /
short circuit point and the points representing
±jx’ (±jb’)
ECE357 / Prof. S. V. Hum
Example: Match a load impedance ZL = 100 + j80 Ω to a 50 Ω line using a single
series open-circuit stub.
ECE357 / Prof. S. V. Hum
Single-Stub Matching: Shunt
Tuning
ECE357 / Prof. S. V. Hum
Example: For a load impedance ZL = 15 + j10 Ω, design two tuning networks
based on shunt short-circuited stubs to match this load to 50 Ω.
ECE357 / Prof. S. V. Hum
Remarks
•  Shorter line sections and stubs give better
performance in terms of bandwidth
–  Shorter transmissions lines have less
variation of electrical parameters with
frequency
•  Be very careful of:
–  If you need to work on an impedance or an
admittance chart
–  Where the open/short circuit locations are on
each chart
ECE357 / Prof. S. V. Hum
(Shunt) Double-Stub Tuners
•  Alternative to
single-stub tuner
•  d0 is fixed
•  lA, lB used to
tune the network
ECE357 / Prof. S. V. Hum
Analysis of Double-Stub Tuner
ECE357 / Prof. S. V. Hum
Analysis of Double-Stub Tuner
ECE357 / Prof. S. V. Hum
Analysis of Double-Stub Tuner
ECE357 / Prof. S. V. Hum
Analysis of Double-Stub Tuner
ECE357 / Prof. S. V. Hum
Double Stub Tuner Steps
1.  Draw the g=1 circle; this is where yB
should be located.
2.  Rotate this circle d0/λ wavelengths
towards the load (CCW); this is the circle
on which yA should be located.
3.  Plot yL on the Smith Chart.
4.  Translate yL along a resistance/
conductance circle to get onto the
rotated g=1 circle
ECE357 / Prof. S. V. Hum
Double Stub Tuner Steps
5.  Determine the stub length lA required to
produce the reactance/susceptance in part 4.
6.  Rotate the modified load impedance/
admittance onto the g=1 circle
7.  Determine the stub length lB required to
produce the reactance/susceptance to move
the rotated impedance/admittance to the origin.
ECE357 / Prof. S. V. Hum
Example: Design a double-stub shunt tuner to match a load impedance ZL = 60 –
j80 Ω to a 50 Ω line. The stubs are to be short-circuited stubs and are spaced λ/8
apart.
ECE357 / Prof. S. V. Hum
ECE357 / Prof. S. V. Hum
Double-Stub Tuner Limitations
•  What happens if
the load point is
inside the red
circle?
ECE357 / Prof. S. V. Hum
Forbidden Region
ECE357 / Prof. S. V. Hum
Escaping from the Forbidden
Region
ECE357 / Prof. S. V. Hum
Closing Remarks on Tuning
Circuits
•  It is always possible to solve for matching
problems analytically
–  A way to check your work!
•  Smith Charts are a quick and intuitive
method for solving transmission line
problems
ECE357 / Prof. S. V. Hum