ECE 451 Advanced Microwave Measurements

The Smith Chart

Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois [email protected]

ECE 451 – Jose Schutt‐Aine 1 Derivation of the Smith Chart The relationship between impedance and is given by: 1(z)  Z( z ) Zo  1(z)  where Zo is the of the system. The normalized impedance is 1(z)1 Z(z)n  1(z)1 The reflection coefficient and the normalized impedance have the form:

ri j and Zn rjx

ECE 451 – Jose Schutt‐Aine 2 Derivation of the Smith Chart Therefore

1jri((1)j1)j   rjx = ri ri 1j 22 ri (1)ri  Separating real and imaginary components,

22 1j1)j1)ri((  r   i  ri  rjx = 22 (1)ri

Isolating the real part from both sides

22 1ri r  22 (1)ri

ECE 451 – Jose Schutt‐Aine 3 Derivation of the Smith Chart

Multiplying through by the denominator, 222 r1rri 2   1  ri

22 ri(r 1) (r  1)  2r r  1 r  2r r1rr22 22r    ri1r (1r)1r22  (1r)

Completing the square

 2 r12 222r r 1r    or ri2 ri1r 1r 1r (1r)

ECE 451 – Jose Schutt‐Aine 4 Derivation of the Smith Chart

2 r12 ri2 1r (1r) This is the equation of a circle centered at r 1 ,0 and of radius 1r 1r

Equating the imaginary parts gives 2 x  i 22 (1ri )

x122 2   2  22 rriior rriix2x x 2 x

ECE 451 – Jose Schutt‐Aine 5 Derivation  of the Smith Chart

2 11 2221i 11 rr ixxx22

2 2 11 ri1   2 xx This is the equation of a circle centered at

1 1 1, of radius x x

ECE 451 – Jose Schutt‐Aine 6 The Smith Chart

The reflection coefficient is given by Z 1 r1 jx  n Zn 1r1jx

We also have 1/ 2 (r 1)22 x  22 1 (r 1) x 1(z)  Zn  1(z)  Thus, going from normalized 11  (z) impedance to normalized y  corresponds to a 180 Zn 1(z)  shift.

ECE 451 – Jose Schutt‐Aine 7 The Smith Chart

3 ways to move on the Smith chart Constant SWR circle  displacement along TL Constant resistance (conductance) circle addition of reactance () Constant reactance (susceptance) arc  addition of resistance (conductance)

ECE 451 – Jose Schutt‐Aine 8 The Smith Chart

ECE 451 – Jose Schutt‐Aine 9 Smith Chart Example

FIND: 1. Reflection coefficient at load

j 227 zRR03.j. 04  06 .e

2. SWR on the line SWR=4.0

3. dmin

d..min 05 0435  0065 .

ECE 451 – Jose Schutt‐Aine 10 Smith Chart Example

4. Line impedance at 0.05 to the left 50 0. 26j 0. 09 13j 4. 5

5. Line admittance at 0.05l 35.j./ 12 50 0068 . j. 0025

6. Location nearest to load where Real[y]=1 0.. 14 0 325 j 0.. 185  0 14 

ECE 451 – Jose Schutt‐Aine 11 Smith Chart Example

ECE 451 – Jose Schutt‐Aine 12