서강대학교 전자공학과 윤상원 교수 2. Smith Chart

Microwave & Millimeter-wave Lab. 2 차 례차 례 1. Smith chart ; introduction Reflection coeff. and impedance Normalized impedance equation Impedance Transformation Some transmission line example Admittance transformation Parallel and series connections Single stub matching

Microwave & Millimeter-wave Lab Smith chart ; introduction Distributed nature of the transmission line  Impedance is repeated periodically One-to-one correspondence between the impedance and the reflection coeff.  Impedances of the passive circuits are mapped into unit circle of the reflection coefficient plane.  Impedance is treated as a reflection coefficient. Smith chart ; a unit circle which contains all the impedance of the passive circuits

Microwave & Millimeter-wave Lab Reflection coeff. and impedance Reflection coefficient in phasor form example 

Microwave & Millimeter-wave Lab. 5 Reflection coefficient and impedance(2)

Microwave & Millimeter-wave Lab Normalized impedance equation Input impedance ; reflection coefficient Normalize the impedance as A mapping from z in -plane to -plane

Microwave & Millimeter-wave Lab. Reflection Coeff. Plane 7

Microwave & Millimeter-wave Lab. 8 Normalized impedance equation(2) Normalized impedance vs. reflection coeff. Real and imaginary terms ; Normalized impedance is represented in terms of the reflection coefficient. impedance can be plotted in the reflection coefficient plane.

Microwave & Millimeter-wave Lab. Smith Chart 9  Normalized impedance is transformed to ;  Real Part(resistance)  Imaginary Part(reactance)

Microwave & Millimeter-wave Lab. Real Part(resistance) 10

Microwave & Millimeter-wave Lab. Imaginary Part(reactance) 11

Microwave & Millimeter-wave Lab. Smith Chart(1) 12

Microwave & Millimeter-wave Lab. 13 Smith Chart(2) Normalized impedance is transformed to ;

Microwave & Millimeter-wave Lab. 14 The Smith Chart(3)

Microwave & Millimeter-wave Lab Impedance Transformation For general load ; reflection coefficient Only phase changes as d increases or decreases for lossless transmission line. But, impedance changes its magnitude and phase Calculation of input impedance using reflection coefficient on the Smith Chart ; “ 6 steps in the textbook “

Microwave & Millimeter-wave Lab. 16 Impedance Transformation(2) 1. Normalize the load impedance Z L with Z 0 ; 2. Locate z L in the Smith Chart ; 3. Identify the load reflection coefficient in the Smith Chart in terms of its magnitude and phase. 4. Rotate  0 by its electrical length 2d to obtain  in (d). 5. Record the normalized input impedance at this spatial location d ; 6. Convert into the actual impedance ;

Microwave & Millimeter-wave Lab. 17 Impedance Transformation(3) Example 3-3 ; 1. Normalize the load impedance Z L with Z 0 ; 2. Locate z L in the Smith Chart ;

Microwave & Millimeter-wave Lab. 18 Impedance Transformation(4) 3. Identify the load reflection coefficient in the Smith Chart in terms of its magnitude and phase. 4. Rotate  0 by its electrical length 2d to obtain  in (d) ; 5. Record the normalized input impedance at this spatial location d ; 6. Convert into the actual impedance ; * exact calculation leads to

Microwave & Millimeter-wave Lab. 19 Impedance Transformation(5)

Microwave & Millimeter-wave Lab. Example The use of the transmission line chart is best shown by example. Let us again consider a load impedance, ZL=25+j50, terminating a 50- line. The line length is 60cm and the operating frequency is such that the wavelength on the line is 2m. We desire the input impedance. Solution. We have, and we read By drawing a straight line from the origin through A to the circumference, we note a reading of On the wtg scale. We have And it is, therefore, 0.3 from the load to the input We therefore find on the circle opposite Wtg reading of = And the point locating the input impedance is marked B. The Normalized input impedance is read as 0.28-j0.40, and thus. A more accurate analytical Calculation gives. 20

Microwave & Millimeter-wave Lab. D10.6 A load is located at z=0 on a lossless 50- line. The operating frequency is 200MHz and the wavelength on the line is 2m. (a)If the line is 0.8m in length, use the Smith chart to find the input Impedance. Therefore 0.4 from the load to the input. Using smith chart tool, the point locating the input impedance is marked B. The input impedance is read 79+j99 21

Microwave & Millimeter-wave Lab. 22 D10.6 (b)What is s?

Microwave & Millimeter-wave Lab Admittance transformation Parametric admittance equation  Normalized admittance equation  The conventional Smith Chart(Z-Smith Chart) gives  on the Smith Chart admittance located at 180 o away from the impedance point.

Microwave & Millimeter-wave Lab. 24 Admittance transformation(2)  Example 3-6 ; convert the normalized input impedance given as to normalized admittance. by direct inversion

Microwave & Millimeter-wave Lab. 25 Admittance transformation(3)  Admittance Smith Chart(Y-Smith Chart)

Microwave & Millimeter-wave Lab. 26 Admittance transformation(4)  ZY-Smith Chart

Microwave & Millimeter-wave Lab Parallel and series connections  Parallel connection of R and L elements

Microwave & Millimeter-wave Lab. 28 Parallel and series connections(2)  Parallel connection of R and C elements

Microwave & Millimeter-wave Lab. 29 Parallel and series connections(3)  Series connection of R and L elements

Microwave & Millimeter-wave Lab. 30 Parallel and series connections(4)  Series connection of R and C elements

Microwave & Millimeter-wave Lab. 31 Parallel and series connections(5)  Example of a T-network

Microwave & Millimeter-wave Lab. 32 Parallel and series connections(6)

Microwave & Millimeter-wave Lab Single stub matching  Impedance matching using distributed element Input admittance at d 1 ;

Microwave & Millimeter-wave Lab. 34 Single stub matching(2)  Stub length Stub length l 1 ;