6.776 high speed communication circuits and systems ... · h.-s. lee & m.h. perrott mit ocw...

6.776High Speed Communication Circuits and Systems

Lecture 5Generalized Reflection Coefficient, Smith Chart

Massachusetts Institute of TechnologyFebruary 15, 2005

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

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:

Determine Voltage and Current At Different Positions

Incident Wave

Reflected Wave

V+ejwtejkz

I+ejwtejkz

V-ejwte-jkz

I-ejwte-jkz

Incident and reflected waves must be added together

Determine Voltage and Current At Different Positions

Incident Wave

Reflected Wave

V+ejwtejkz

I+ejwtejkz

V-ejwte-jkz

I-ejwte-jkz

Define Generalized Reflection Coefficient

Recall:

Define Generalized Reflection Coefficient:

Generalized Reflection Coefficient Cont’d

Similarly:

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)}

∆ = 2kzΓL

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!

A Geometric View of |1 + Γ(z)|

Re{1+Γ(z)}

Im{1+Γ(z)}

|1+Γ(z)|

Reflections Cause Amplitude to Vary Across Line

Equation:Graphical representation:

directionof travel

tV+ejwtejkz

|1 + Γ(z)|

max|1+Γ(z)|

min|1+Γ(z)|

|1+Γ(0)|

Voltage Standing Wave Ratio (VSWR)

Definition

For passive load (and line)

We can infer the magnitude of the reflection coefficient based on VSWR

Reflections Influence Impedance Across The Line

From Slide 7

- Note: not a function of time! (only of distance from load)

Alternatively

- From Lecture 3:

Impedance as a Function of Location

We can now express Z(z) as

Note: Z(z) and Γ(z) are periodic in z with a period of λ/2

Example: Z(λ/4) with Shorted Load

Calculate reflection coefficient

Calculate generalized reflection coefficient

Calculate impedance

Z(λ/4)

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

Now Look At Z(∆) (Impedance Close to Load)

Impedance formula (∆ very small)

- A useful approximation:

- Recall from Lecture 2:

Overall approximation:

Example: Look At Z(∆) With Load Shorted

Z(∆)

Reflection coefficient:

Resulting impedance looks inductive!

Example: Look At Z(∆) With Load Open

Reflection coefficient:

Resulting impedance looks capacitive!

Z(∆)

Compare to an Ideal LC Tank Circuit

Calculate input impedance about resonance

L CZin

= 0 negligible

Transmission Line Version: Z(λ0 /4) with Shorted Load

As previously calculated

Impedance calculation

Relate λ to frequency

Z(λ0/4)

Calculate Z(∆ f) – Step 1

Wavelength as a function of ∆ f

Generalized reflection coefficient

Z(λ0/4)

Calculate Z(∆ f) – Step 2

ZLλ0/4

Impedance calculation

Recall

Z(λ0/4)

- Looks like LC tank circuit (but more than one mode)!

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

Real Impedance in Γ Coordinates (Equate Real Parts)

0.2 0.5 1 2 5

Im{ΓL}

Re{ΓL}

ΓL=1ΓL=-1

ΓL=-j

Zn=0.5

Imag. Impedance in Γ Coordinates (Equate Imag. Parts)

Im{ΓL}

Re{ΓL}

ΓL=0 ΓL=1ΓL=-1

ΓL=-j

Zn=j0.5

Zn=-j0.5

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?

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

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

Yn=0.5 Zn=0.5

Reactance and Susceptance on the Smith Chart

Im{ΓL}

Re{ΓL}

ΓL=0 ΓL=1ΓL=-1

ΓL=-j

Yn=-j2

Zn=-j2

Overall Smith Chart

0.2 0.5 1 2 5

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

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

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

Plot Starting Impedance (Admittance) on Smith Chart

0.2 0.5 1 2 5

YLn=0.25+j0.7854

(Note: ZLn=0.37-j1.16)

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

Matching Network

Add Reactance of Inductor Lm

0.2 0.5 1 2 5

ZLn=0.37-j1.16

Z2n=0.37+j0.48

normalizedinductor

reactance= j1.64

Inductor Value Calculation Using Smith Chart

From Smith chart, we found that the desired normalized inductor reactance is

Required inductor value is therefore

Add Susceptance of Capacitor Cm (Achieves Match!)

0.2 0.5 1 2 5

ZLn=0.37-j1.16

Z2n=0.37+j0.48

1.0+j0.0

normalizedcapacitor

susceptance= j1.31

(note: Y2n=1.00-j1.31)

Capacitor Value Calculation Using Smith Chart

From Smith chart, we found that the desired normalized capacitor susceptance is

Required capacitor value is therefore

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

6.776 high speed communication circuits and systems ... · h.-s. lee & m.h. perrott mit ocw...

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