Impedance Matching & Smith Chart Microwave Engineering EE 172 Dr. Ray Kwok
Impedance Matching - Dr. Ray Kwok
Why match? Z − Zo Γ= Z + Zo Zo
Minimize reflection, maximize transmission
l
ZL
If ZL = Zo, line length is not important.
Zg Vg
Zin
If Zin = Zg, max power to the load.
Goal: to design matching networks so all Z’s are the same = 50Ω Zin = 50Ω
matching network
ZL
Impedance Matching - Dr. Ray Kwok
Technician tuning even for 50Ω designs input 50 Ω
50 Ω
50 Ω
50 Ω 50 Ω
Think of a block of cracked glass…
output 50 Ω
Impedance Matching - Dr. Ray Kwok
Stub tuning ?
Length = ?
Zin = 50Ω
Short piece of open stub → capacitor e.g. ZL = 50 − j 10 Ω
50Ω
C Y1
want Y1 = Yo - jB
t = 0.0574 = tan βl βl = 0.0573 λ l = 0.0573 = 0.009λ 2π − B = 0.04 / t − 0.2 = 0.497
ZL = 1 − j0.2 YL = Y1 =
1 = 0.9615 + j0.1923 ZL YL + j tan β l 1 + jYL tan β l
1 − jB =
ZL
0.9615 + j0.1923 + jt 0.9615 + j( t + 0.1923) = 1 + j(0.9615 + j0.1923) t (1 − 0.1923t ) + j0.9615t
(1 − 0.1923t ) + j0.9615t − j B(1 + 0.1923t ) + 0.9615t B = 0.9615 + j( t + 0.1923) (1 − 0.1923t ) + 0.9615t B = 0.9615 real 0.0385 − 0.1923t 0.04 − 0.2t −B= = 0.9615t t 0.9615t − B(1 + 0.1923t ) = t + 0.1923 imaginary t − 0.7692 0.04 − 0.2 t −B= = 1 + 0.1923t t 2 2 t − 0.7692t = −0.0385t − 0.0123t + 0.04 0 = t 2 − 0.7288t − 0.0385 t = 0.0574 or 0.671
B = BYo = −0.497 / 50 = −0.01 B (50 − R )
R (1 − R ) > X 2
conditions
R < 50
Z ≠ 50R ??
50X 2 Z = 50R − (50 − R ) Z (50 − R ) tan β l = c 50X
)
ok…. that means Zc is real, and transmission line is a λ/4 transformer.
R (50 − R ) > X 2 c
If ZL = 25 + j20 Ω, ZL = 0.5 + j0.4 normalized to 50 Ω R (0.4)2 conditions ok
X2 0.4 2 = 0.5 − Z =R− (1 − R ) (1 − 0.5) 2 c
Z c = 0.424 Z c (1 − R ) 0.424(1 − 0.5) = = 0.53 X 0.4 λ l = (0.488) = 0.0776λ 2π tan βl =
Impedance Matching - Dr. Ray Kwok
In Smith Chart ?
0.172λ 0.172 +0.078 = 0.250λ
Last example ZL = 25 + j20 Ω Zo = 21.2 Ω Length = 0.078λ Zin = 50 Ω
clockwise ZL (1.19, 0.94)
ZL = ZL/Zo = 1.19 + j0.94 Zin = 50/21.2 = 2.356 0.078λ λ Zin = 50Ω
21.2 Ω
ZL
Matching doesn’t necessary means center of the chart !! Depends on Z of the line & system.
Zin(2.35, 0)
0.250λ
Impedance Matching - Dr. Ray Kwok
Single Stub Tuning refers to sliding a stub (any kind) along a transmission line. In practice, usually shunt stubs, short stub for waveguides, open stub for microstrip.
0.305λ
0.064λ λ ZL (0.4, 1) YL (0.34, -0.87) Y (1, -1.9)
previous example
50 Ω
ZL =0.4+j1
short shunt stub jB = -j1.9 = -jcotβl
Y (1, 1.9) 0.434λ 50 Ω
ZL =0.4+j1 0.435λ λ
open shunt stub jB = j1.9 = jtanβl
0.130λ λ
Impedance Matching - Dr. Ray Kwok
Any Stub
0.130λ λ
tanβl can be “+” or “-”
0.064λ λ ZL (0.4, 1) YL (0.34, -0.87)
can use any type depends on realization. e.g. use shunt open stub only… Y (1, -1.9) 0.305λ
previous example
50 Ω
ZL =0.4+j1
open shunt stub jB = - j1.9 = jtanβl βl = -1.086 + π = 2.055 length = 0.327λ > λ/4 Y (1, 1.9)
0.434λ 50 Ω
open shunt stub jB = j1.9 = jtanβl βl = 1.086 length = 0.173λ
ZL =0.4+j1
0.435λ λ
Impedance Matching - Dr. Ray Kwok
Double-Stub Tuning • • • • • • • •
Preferable because it’s not sensitive to the initial line length. Useful in tuning (especially in waveguide) with variable load. 2 adjustable stubs connected to a fixed-length transmission line. In principle, can use any series or shunt stubs. In practice, mostly shunt short stubs for waveguide. Impedance of the stubs are arbitrary, and they don’t have to be the same. Impedance of the connecting line doesn’t have to be Zo of the system. 2 parameters to tune: d1 & d2 fixed L Zo
s ju ad
Z1
e bl ta
e bl ta
s ju ad
Z2
ZL
d1
d2
Impedance Matching - Dr. Ray Kwok
Double Stub Tuning e.g. 2 shunt short stubs (50Ω) separated by a 50Ω line of 0.2λ ZL(0.4,1) YL(0.34,-0.87)
0.2λ
ZL =0.4+j1
50 Ω 50
d2 -cotβd2 = −1.22 βd2 = 0.687 d2 = 0.109λ
• • • • • •
50
Ω
Y (0.34,-0.2)
Ω
d1 -cotβd1 = +0.67 βd1 = -0.98 +π = 2.16 d1 = 0.344λ
0.5λ λ
rotate the 1-circle by line length adjust d1 along constant-G circle stop at the rotated blue-circle xline will bring it to the green circle adjust d2 along the green circle to Zo not for all ZL !! Forbidden zone.
Y (1,1.22) – 0.2λ λ 0.3λ λ
Impedance Matching - Dr. Ray Kwok
Quarter-Wave Transformer • • • •
Zc2 = ZoZ1 Transformer real-to-real, complex-to-complex impedance. Usually needs one more element to match (before λ/4) (lumped elements, stubs or transmission line). normalized to Zo,
Zc = Z1 λ/4 Zo
Zc
Z1
matching element
ZL
Impedance Matching - Dr. Ray Kwok
e.g. Quarter-Wave e.g. shunt stub (100Ω) then λ/4 ZL = 20 + j 50 Ω ZL(0.4,1) YL(0.34, -0.87)
λ/4
50 Ω
ZL =0.4+j1
Zc
10 0Ω
d (1/100) tanβd= (0.87)(1/50) βd = 1.05 d = 0.167λ
• move Z to the real axis • normalized Zc = √Z = √3.1; Zc = 1.76 • Zc = 50(1.76) = 88 Ω
Z (3.1,0) Y(0.34,0)
Impedance Matching - Dr. Ray Kwok
Advanced Impedance Matching So far…. • single frequency • 1-port network Need: • wideband matching – multiple sections • multi-ports (simultaneously tuned) require knowledge of multi-port network analysis
Impedance Matching - Dr. Ray Kwok
Homework Smith Chart Exercise Matching Exercise Double Stub Exercise
