Look at a distance z = - L toward the generator -z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0

-z ZLZL Z c1 z = 0 Z c2

-z ZLZL ZcZc z = 0

X = Z c tan kL L

Should I add something in series or in parallel?

Eureka! Use a stub. -z ZLZL ZcZc z = 0 -z ZLZL ZcZc z = 0 -z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0 

-z ZLZL ZcZc z = 0 Not matched! -z ZLZL ZcZc z = 0 Needs some -jB somewhere, but where? -z ZLZL ZcZc z = 0 Eureka! Use a stub, somewhere. How long should it be?

An article appeared in the January, 1939 issue of Electronics that changed forever the way radio engineers think about transmission lines. Phil Smith [ ] devised an extraordinarily clever circular chart that revealed graphically the complex impedance anywhere along a line. No math and minimum fuss. There's a marvelous symmetry in its design - everything fits together neatly. So ingenious was his invention that it has been the standard of the industry - for over sixty years.

-z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0

-z ZLZL ZcZc z = 0

a x b y r Equation of circle (x - a) 2 + (y - b) 2 = r 2

clf; clear; plot([-1 1], [0 0], ’y’) axis equal axis off hold on

for r = [ ] rr = 1 / (r + 1); cr = 1 - rr; tr = 2 * pi * (0 :.01: 1); plot(cr + rr * cos (tr), rr * sin (tr), ‘y’); end

RrRr RiRi R r = 1

for x = [ ] rx = 1 / x; cx = rx; tx = 2 * pi * (0 :.01: 1); plot(1 - rx * sin (tx), cx - rx * cos (tx), ‘y’); plot(1 - rx * sin (tx),- cx + rx * cos (tx), ‘y’); end

RrRr RiRi R r = 1

for x = [ ] rx = 1 / x; cx = rx; tx = 2 * atan(x) * (0 :.01: 1); plot(1 - rx * sin (tx), cx - rx * cos (tx), ‘y’); plot(1 - rx * sin (tx),- cx + rx * cos (tx), ‘y’); end

RrRr RiRi R r = 1

RrRr RiRi

To load To generator

|R| 1 0

Movie to illustrate the frequency dependence of the impedance of a series resonant circuit

Movie to illustrate the frequency dependence of the impedance of a parallel resonant circuit

Movie to illustrate the transformation of a load impedance at various locations on the transmission line  

Movie to illustrate the transformation of an impedance to an admittance

y =  z = 0

Z c = 50  Z in = ? 

z = j1 z = 0

Z c = 50  Z in = ? 

0.434 Z c = 100  Z in = ?

z L = j1.8 z = j1.2

0.434 Z c = 100  Z in = ?

Single stub matching

y L = 2 + j1 y = 1 - j1 y = 1 + j1

y L =  y = - j1 y = + j1