6.776 high speed communication circuits and systems ... · h.-s. lee & m.h. perrott mit ocw...
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6.776High Speed Communication Circuits and Systems
Lecture 5Generalized Reflection Coefficient, Smith Chart
Massachusetts Institute of TechnologyFebruary 15, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H. Perrott
H.-S. Lee & M.H. Perrott MIT OCW
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
x
z
Ex
Hyy
ZL
ExHy
ZL
Incident Wave
Reflected Wave
0Lz
V+ejwtejkz
I+ejwtejkz
V-ejwte-jkz
I-ejwte-jkz
Incident and reflected waves must be added together
H.-S. Lee & M.H. Perrott MIT OCW
Determine Voltage and Current At Different Positions
x
z
Ex
Hyy
ZL
ExHy
ZL
Incident Wave
Reflected Wave
0Lz
V+ejwtejkz
I+ejwtejkz
V-ejwte-jkz
I-ejwte-jkz
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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
We can view Γ(z) as a complex number that rotates clockwise as z (distance from the load) increases
Note: |ΓL|
|ΓL| Re{Γ(z)}
Im{Γ(z)}
ΓL
∆ = 2kzΓL
Γ(z)
0
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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!
H.-S. Lee & M.H. Perrott MIT OCW
A Geometric View of |1 + Γ(z)|
|ΓL|
Re{1+Γ(z)}
Im{1+Γ(z)}
Γ(z)
10
|1+Γ(z)|
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Reflections Cause Amplitude to Vary Across Line
Equation:Graphical representation:
directionof travel
z
tV+ejwtejkz
z
|1 + Γ(z)|
max|1+Γ(z)|
λ
0
min|1+Γ(z)|
|1+Γ(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
Also
We can now express Z(z) as
Note: Z(z) and Γ(z) are periodic in z with a period of λ/2
H.-S. Lee & M.H. Perrott MIT OCW
Example: Z(λ/4) with Shorted Load
x
z
y
ZL
Calculate reflection coefficient
Calculate generalized reflection coefficient
Calculate impedance
0Lz
λ/4
Z(λ/4)
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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 (Zo is real)Shorts turn into opensCapacitors 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
x
z
y
ZL
0∆z
Z(∆)
Reflection coefficient:
Resulting impedance looks inductive!
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Example: Look At Z(∆) With Load Open
Reflection coefficient:
Resulting impedance looks capacitive!
x
z
ZL
y
0∆z
Z(∆)
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Compare to an Ideal LC Tank Circuit
Calculate input impedance about resonance
L CZin
= 0 negligible
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Transmission Line Version: Z(λ0 /4) with Shorted Load
As previously calculated
Impedance calculation
Relate λ to frequency
x
z
y
ZL
0Lz
λ0/4
Z(λ0/4)
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Calculate Z(∆ f) – Step 1
Wavelength as a function of ∆ f
Generalized reflection coefficient
x
z
y
ZL
0Lz
λ0/4
Z(λ0/4)
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Calculate Z(∆ f) – Step 2
x
z
y
ZLλ0/4
Impedance calculation
Recall
0Lz
Z(λ0/4)
- Looks like LC tank circuit (but more than one mode)!
H.-S. Lee & M.H. Perrott MIT OCW
Smith Chart
Define normalized load impedance
Relation between Zn and ΓL
- Consider working in coordinate system based on ΓKey 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)
0.2 0.5 1 2 5
ΓL=0
Im{ΓL}
Re{ΓL}
ΓL=1ΓL=-1
ΓL=j
ΓL=-j
Zn=0.5
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Imag. Impedance in Γ Coordinates (Equate Imag. Parts)
j0.2
-j0.2
0
j0.5
-j0.5
0
j1
-j1
0
j2
-j2
0
j5
-j5
0
Im{ΓL}
Re{ΓL}
ΓL=0 ΓL=1ΓL=-1
ΓL=j
ΓL=-j
Zn=j0.5
Zn=-j0.5
H.-S. Lee & M.H. Perrott MIT OCW
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- Impedances add for series circuits
For parallel circuits, work with admittance- Admittances add for parallel circuits
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Resistance and Conductance on the Smith Chart
0.2 0.5 1 2 5
Im{ΓL}
Re{ΓL}
ΓL=0 ΓL=1ΓL=-1
ΓL=j
ΓL=-j
Yn=2
Yn=0.5 Zn=0.5
Zn=2
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Reactance and Susceptance on the Smith Chart
j0.2
-j0.2
0
j0.5
-j0.5
0
j1
-j1
0
j2
-j2
0
j5
-j5
0
Im{ΓL}
Re{ΓL}
ΓL=0 ΓL=1ΓL=-1
ΓL=j
ΓL=-j
Yn=-j2
Yn=j2
Zn=j2
Zn=-j2
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Overall Smith Chart
0.2 0.5 1 2 5
j0.2
j0.2
0
j0.5
j0.5
0
j1
j1
0
j2
j2
0
j5
j 5
0
H.-S. Lee & M.H. Perrott MIT OCW
Smith Chart element Paths
INCREASING SERIES L DECREASING SERIES C
DECREASING SHUNT L INCREASING SHUNT C
Figure by MIT OCW. Adapted fromhttp://contact.tm.agilent.com/Agilent/tmo/an-95-1/classes/imatch.html
H.-S. Lee & M.H. Perrott MIT OCW
L-Match Circuits (Matching Load to Source)
π or L match networksremove forbidden regions
Figure by MIT OCW. Adapted fromhttp://contact.tm.agilent.com/Agilent/tmo/an-95-1/classes/imatch.html
H.-S. Lee & M.H. Perrott MIT OCW
Example – Match RC Network to 50 Ohms at 2.5 GHz
Circuit
Step 1: Calculate ZLn
Step 2: Plot ZLn on Smith Chart (use admittance, YLn)
Zin Rp=200Cp=1pFMatchingNetwork ZL
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Plot Starting Impedance (Admittance) on Smith Chart
0.2 0.5 1 2 5
j0.2
-j0.2
0
j0.5
-j0.5
0
j1
-j1
0
j2
-j2
0
j5
-j5
0
YLn=0.25+j0.7854
(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
Use the Smith chart to come up with component values- Inductance Lm shifts impedance up along reactance
curve- Capacitance Cm shifts impedance down along susceptance curve
Zin Rp=200Cp=1pFZL
Lm
Cm
Matching Network
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Add Reactance of Inductor Lm
0.2 0.5 1 2 5
j0.2
-j0.2
0
j0.5
-j0.5
0
j1
-j1
0
j2
-j2
0
j5
-j5
0
ZLn=0.37-j1.16
Z2n=0.37+j0.48
normalizedinductor
reactance= j1.64
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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!)
0.2 0.5 1 2 5
j0.2
-j0.2
0
j0.5
-j0.5
0
j1
-j1
0
j2
-j2
0
j5
-j5
0
ZLn=0.37-j1.16
Z2n=0.37+j0.48
1.0+j0.0
normalizedcapacitor
susceptance= j1.31
(note: Y2n=1.00-j1.31)
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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
H.-S. Lee & M.H. Perrott MIT OCW
Useful Web Resource
Play the “matching game” at
- 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
http://contact.tm.agilent.com/Agilent/tmo/an-95-1/classes/imatch.html