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 reflection coefficient is given by: 1(z) Z( z ) Zo 1(z) where Zo is the characteristic impedance 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: rij and Zn rjx ECE 451 – Jose Schutt‐Aine 2 Derivation of the Smith Chart Therefore 1jri((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 1ri r 22 (1)ri ECE 451 – Jose Schutt‐Aine 3 Derivation of the Smith Chart Multiplying through by the denominator, 222 r1rri 2 1 ri 22 ri(r 1) (r 1) 2r r 1 r 2r r1rr22 22r 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 ri2 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 i x 22 (1ri ) x122 2 2 22 rriior rriix2x x 2 x ECE 451 – Jose Schutt‐Aine 5 Derivation of the Smith Chart 2 11 2221 i 11 rr ixxx22 2 2 11 ri1 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 admittance corresponds to a 180 Z 1(z) n degree 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 (susceptance) 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 zRR03.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. 26j 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.