6.776 High Speed Communication Circuits and Systems Lecture 5
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6.776
High Speed Communication Circuits and Systems
Lecture 5
Generalized Reflection Coefficient, Smith Chart
Massachusetts Institute of Technology
February 15, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H.
Perrott
Reflection Coefficient
We defined, at the load
Load and characteristic impedances were related
Alternately
Can we find reflection coefficient at locations other than
the load location?: Generalized reflection coefficient
H.-S. Lee & M.H. Perrott
MIT OCW
Voltage and Current Waveforms
In Lecture 3, we found that in sinusoidal steady-state in a
transmission line,
where the – sign is for the wave traveling in the +z
direction,a nd the + sign is for the wave traveling in the –z
direction.
Thus, the incident wave has the voltage
And the reflected wave
Similarly for currents:
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Determine Voltage and Current At Different Positions
ZL
Incident Wave
Ex
jwt jkz
V+e e
x
y
I+ejwtejkz
Hy
ZL
z
Reflected Wave
Hy
Ex
V-ejwte-jkz
I-ejwte-jkz
z
L
0
Incident and reflected waves must be added together
H.-S. Lee & M.H. Perrott
MIT OCW
Determine Voltage and Current At Different Positions
ZL
Incident Wave
Ex
jwt jkz
V+e e
x
y
I+ejwtejkz
Hy
ZL
z
Reflected Wave
Hy
Ex
V-ejwte-jkz
I-ejwte-jkz
z
L
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0
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Define Generalized Reflection Coefficient
Recall:
Define Generalized Reflection Coefficient:
Since
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Generalized Reflection Coefficient Cont’d
Similarly:
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A Closer Look at Γ(z)
Recall ΓL is
Im{Γ(z)}
Note:
|ΓL|
∆
ΓL
We can view Γ(z)
as a complex
number that
rotates clockwise
as z (distance
from the load)
increases
H.-S. Lee & M.H. Perrott
Γ(z)
0
|ΓL|
= 2kz
ΓL
Re{Γ(z)}
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Calculate |Vmax| and |Vmin| Across The Transmission Line
We found that
So that the max and min of V(z,t) are calculated as
We can calculate this geometrically!
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A Geometric View of |1 + Γ(z)|
Im{1+Γ(z)}
Γ(z)
|1+Γ(z)|
|ΓL|
0
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1
Re{1+Γ(z)}
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Reflections Cause Amplitude to Vary Across Line
Equation:
Graphical representation:
direction
of travel
jwt jkz
V+e e
t
z
λ
|1 + Γ(z)|
max|1+Γ(z)|
|1+Γ(0)|
min|1+Γ(z)|
z
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0
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Voltage Standing Wave Ratio (VSWR)
Definition
For passive load (and line)
We can infer the magnitude of the reflection
coefficient based on VSWR
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Reflections Influence Impedance Across The Line
From Slide 7
- Note: not a function of time! (only of distance from load)
Alternatively
- From Lecture 3:
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Impedance as a Function of Location
We can now express Z(z) as
Also
Note: Z(z) and Γ(z) are periodic in z with a period of λ/2
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Example: Z(λ/4) with Shorted Load
λ/4
x
ZL
Z(λ/4)
y
z
z
L
0
Calculate reflection coefficient
Calculate generalized reflection coefficient
Calculate impedance
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Generalize Relationship Between Z(λ/4) and Z(0)
General formulation
At load (z=0)
At quarter wavelength away (z = λ/4)
- Impedance is inverted (Z
o
is real)
Shorts turn into opens
Capacitors turn into inductors
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Now Look At Z(∆) (Impedance Close to Load)
Impedance formula (∆ very small)
- A useful approximation:
- Recall from Lecture 2:
Overall approximation:
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Example: Look At Z(∆) With Load Shorted
ZL
x
Z(∆)
y
z
z
∆ 0
Reflection coefficient:
Resulting impedance looks inductive!
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Example: Look At Z(∆) With Load Open
ZL
x
Z(∆)
y
z
z
∆ 0
Reflection coefficient:
Resulting impedance looks capacitive!
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Compare to an Ideal LC Tank Circuit
Zin
L
C
Calculate input impedance about resonance
=0
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negligible
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Transmission Line Version: Z(λ0 /4) with Shorted Load
λ0/4
Z(λ0/4)
x
y
z
ZL
z
L
As previously calculated
Impedance calculation
Relate λ to frequency
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0
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Calculate Z(∆ f) – Step 1
λ0/4
Z(λ0/4)
x
y
z
ZL
z
L
Wavelength as a function of ∆ f
Generalized reflection coefficient
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0
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Calculate Z(∆ f) – Step 2
λ0/4
Z(λ0/4)
x
y
z
ZL
z
L
Impedance calculation
Recall
0
Looks like LC tank circuit (but more than one mode)!
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Smith Chart
Define normalized load impedance
Relation between Zn and ΓL
- Consider working in coordinate system based on Γ
Key relationship between Zn and Γ
- Equate real and imaginary parts to get Smith Chart
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Real Impedance in Γ Coordinates (Equate Real Parts)
Im{ΓL}
ΓL=j
ΓL=-1
0.2
ΓL=0
0.5
1
ΓL=1
2
5
Re{ΓL}
Zn=0.5
ΓL=-j
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Imag. Impedance in Γ Coordinates (Equate Imag. Parts)
j1 Im{ΓL}
j0.5
j0.2
ΓL=j
j2
j5
Zn=j0.5
ΓL=-1
ΓL=1
ΓL=0
0
-j0.2
Re{ΓL}
Zn=-j0.5
-j0.5
-j5
ΓL=-j
-j2
-j1
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What Happens When We Invert the Impedance?
Fundamental formulas
Impact of inverting the impedance
- Derivation:
We can invert complex impedances in Γ plane by
simply changing the sign of Γ !
How can we best exploit this?
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The Smith Chart as a Calculator for Matching Networks
Consider constructing both impedance and
admittance curves on Smith chart
- Conductance curves derived from resistance curves
- Susceptance curves derived from reactance curves
For series circuits, work with impedance
For parallel circuits, work with admittance
- Impedances add for series circuits
- Admittances add for parallel circuits
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Resistance and Conductance on the Smith Chart
Im{ΓL}
ΓL=j
Yn=0.5
Zn=0.5
ΓL=-1
Yn=2
0.2
ΓL=1
ΓL=0
Zn=2
0.5
1
2
5
Re{ΓL}
ΓL=-j
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Reactance and Susceptance on the Smith Chart
j1 Im{ΓL}
j0.5
j0.2
ΓL=j
j2
Yn=-j2
Zn=j2
j5
ΓL=0
ΓL=-1
ΓL=1
Re{ΓL}
0
-j0.2
Yn=j2
-j0.5
Zn=-j2
ΓL=-j
-j5
-j2
-j1
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Overall Smith Chart
j1
j0.5
j2
j0.2
0
j5
0.2
0.5
1
2
5
j5
j0.2
j0.5
j2
j1
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Smith Chart element Paths
INCREASING SERIES L
DECREASING SERIES C
DECREASING SHUNT L
INCREASING SHUNT C
H.-S. Lee & M.H. Perrott
Figure by MIT OCW. Adapted from
http://contact.tm.agilent.com/Agilent/tmo/an-95-1/classes/imatch.html
MIT OCW
L-Match Circuits (Matching Load to Source)
π or L match networks
remove forbidden regions
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Figure by MIT OCW. Adapted from
http://contact.tm.agilent.com/Agilent/tmo/an-95-1/classes/imatch.html
MIT OCW
Example – Match RC Network to 50 Ohms at 2.5 GHz
Circuit
Zin
Matching
Network
ZL
Cp=1pF
Rp=200
Step 1: Calculate ZLn
Step 2: Plot ZLn on Smith Chart (use admittance, YLn)
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Plot Starting Impedance (Admittance) on Smith Chart
j1
j0.5
j2
j0.2
0
j5
0.2
0.5
1
2
5
-j0.2
-j5
-j0.5
YLn=0.25+j0.7854
-j2
-j1
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(Note: ZLn=0.37-j1.16)
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Develop Matching “Game Plan” Based on Smith Chart
By inspection, we see that the following matching
network can bring us to Zin = 50 Ohms (center of
Smith chart)-need to step up impedance
Matching Network
Lm
Zin
Cm
ZL
Cp=1pF
Rp=200
Use the Smith chart to come up with component
values
- Inductance L shifts impedance up along reactance
curve
- Capacitance C shifts impedance down along
m
m
susceptance curve
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MIT OCW
Add Reactance of Inductor Lm
j1
j0.5
j2
Z2n=0.37+j0.48
j0.2
0
-j0.2
j5
0.2
normalized
inductor
reactance
= j1.64
-j0.5
0.5
1
2
5
-j5
ZLn=0.37-j1.16
-j2
-j1
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Inductor Value Calculation Using Smith Chart
From Smith chart, we found that the desired
normalized inductor reactance is
Required inductor value is therefore
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Add Susceptance of Capacitor Cm (Achieves Match!)
j1
j0.5
Z2n=0.37+j0.48
(note: Y2n=1.00-j1.31)
normalized
capacitor
susceptance
= j1.31
j0.2
0
j2
0.2
0.5
1.0+j0.0
1
2
j5
5
-j0.2
-j5
-j0.5
ZLn=0.37-j1.16
-j2
-j1
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Capacitor Value Calculation Using Smith Chart
From Smith chart, we found that the desired
normalized capacitor susceptance is
Required capacitor value is therefore
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MIT OCW
Useful Web Resource
Play the “matching game” at
http://contact.tm.agilent.com/Agilent/tmo/an-95-1/classes/imatch.html
- Allows you to graphically tune several matching
networks
- Note: it is set up to match source impedance to load
impedance rather than match the load to the source
impedance
Same results, just different viewpoint
H.-S. Lee & M.H. Perrott
MIT OCW
