the load reflection coefficient behavior is given

1
Notes on the reflection coefficient on lossless
lines

• the load reflection coefficient behavior is given
  
• as r and x vary, what exactly does GL do?
• lets plot G in the complex plane
• for any non-negative value of Re(Z) (i.e., r 0)
  and any value of Im(Z), G falls on or within the
  unit circle in the complex G plane

• pick a value of x, then vary r

• pick a value of r, then vary x

2
Relationships between G and Z

• note relation between G and the normalized
  impedance

• consider G in complex plane

• so the relationship between the normalized
  impedance and the real and imaginary parts of G is

3
Relationships between G and Z

• from the real part

4
Plot of reflection coefficient in complex plane

• plot G as a function of r
• these are circles!

• center

• radius

• r 0

• 0 lt r lt 1

• r 1

• r gt 1

curves of constant r Re(Z)
5
Plot of reflection coefficient

• from the Im part

• these are also circles!

• center

• radius

• plot G as a function of x
• x

• x gt 0

• x lt 0

• x 0

curves of constant x Im(Z)
6
Plot of reflection coefficient in complex plane

• the Smith Chart is a plot of the reflection
  coefficient in the complex plane, with contours
  of constant load resistance and load reactance
  superimposed

7
Plot of reflection coefficient in complex plane
Im(G)

• lets plot in the complex plane
• example
• ZLn 0.5 j0.5
• you go all the way around the circle when l is
  such that 2bl 2p
• recall b 2p/l, so you go all the way around
  when l l/2 !

-1
1
0
Re(G)
8
Plot of reflection coefficient in complex plane
Im(G)

• recall the input impedance is
• tangent is periodic with period p, so you repeat
  every time bl mp
• recall b 2p/l, so you get back to the same
  impedance when l ml/2 !
• you can find Z(l) directly from the Smith chart!
• applet

-1
1
0
Re(G)
9
The Smith Chart

• the Smith chart also has a phase angle scale
  measured using distance in units of wavelengths
• link to high res smith chart
• why? because the angle of G is

• direction of increasing l, or equivalently
• away from load
• towards generator

10
Example 1
Im(G)

• recall the input impedance is
• example
• ZLn 0.5 j0.5
• mark on Smith chart
• angle of GL is 0.088l
• what is the input impedance if l l/4?
• angle of G at this distance is (0.088 0.25)l
  0.338l
• read the Smith chart
• Z( l l/4) 1 j 0.9
• applet

-1
1
0
Re(G)
11
Example 2

• recall the input impedance is
• example
• ZLn 2 j(-1.2)
• mark on Smith chart
• angle of GL is 0.288l
• what is the input impedance if l 0.232l?
• angle of G at this distance is (0.288 0.232)l
  (0.52)l (0.50.02)l
• read the Smith chart
• Z( l 0.232l) 0.4 j 0.1
• applet

0.02l
0.5l
G
x -1.2
0.288l
12
Transmission line applets

• various
• http//www.eecg.toronto.edu/bradel/projects/trans
  missionline/
• http//fermi.la.asu.edu/w9cf/tran/
• matching (single stub) http//home3.netcarrier.com
  /chan/EM/PROGRAMS/STUBMATCH/
• Smith chart
• pointer on chart gives gamma and impedance
• http//www.ntu.edu.sg/eee/eee5/el/smithchart/
• http//www.ee.ust.hk/eee/course/elec241/java/Smith
  Chart.html
• local link
• how tos
• http//www.web-ee.com/primers/files/SmithCharts/sm
  ith_charts.htm
• http//cnx.rice.edu/content/m1060/latest/

13
L-C transmission line summary

• lossless L-C transmission line