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ELEC 401 – Microwave Electronics ELEC 401 MICROWAVE ELECTRONICS Lecture on Smith Chart Instructor: M. İrşadi Aksun Acknowledgements: 1. Some Smith Chart figures were taken from the textbook “Fundamentals of Applied Electromagnetics” by F. T. Ulaby. M. I. Aksun Koç University 1/18 ELEC 401 – Microwave Electronics Outline Chapter 1: Motivation & Introduction Chapter 2: Review of EM Wave Theory Chapter 3: Plane Electromagnetic Waves Chapter 4: Transmission Lines (TL) Chapter 5: Smith Chart & Impedance Matching Chapter 6: Microwave Network Characterization Chapter 7: Passive Microwave Components M. I. Aksun Koç University 2/18 ELEC 401 – Microwave Electronics Smith Chart - Introduction Smith chart is a graphical tool that represents a mapping between impedance and reflection coefficient; It was introduced by Phil Smith of RCA in 1936; It is nothing but the polar plot of reflection coefficient with the corresponding impedances written on it; It is a very convenient tool for presentation purpose. M. I. Aksun Koç University 3/18 ELEC 401 – Microwave Electronics Smith Chart - Introduction Polar plot of e j r Point A: A 0.5e j 53 0 0.3 j 0.4 Point B: B 0.54e M. I. Aksun Koç University j 202 0 0.5 j 0.2 4/18 ELEC 401 – Microwave Electronics Smith Chart - Introduction To show how the chart is constructed, we first write the relationship between input impedance and the reflection coefficient: 1 in 1 in Z in Z 0 Z in 1 in 1 in R jX 1 e j 1 e j 1 u jv 1 u jv By equating the real and imaginary parts 2 u 12 v 1 12 X X Reactance circles 2 R 1 u v 2 R 1 R 12 M. I. Aksun Koç University Resistance circles 5/18 ELEC 401 – Microwave Electronics Smith Chart - Introduction 2 R 1 u v 2 R 1 R 12 (v ) Resistance circles Equation of a circle with radius R located at (x0,y0) ( x x 0 ) 2 ( y y0 ) 2 R 2 (u) 2 u 1 v 1 12 X X 2 Reactance circles M. I. Aksun Koç University 6/18 ELEC 401 – Microwave Electronics Smith Chart - Introduction Unit circle Point P represents a normalized impedance Z=2-j1. The reflection coefficient has a magnitude OP/OR, and an angle -26.6 degrees. R O -26.60 P Point R is an arbitrary point on the R=0 circle, which is also ||=1.0 circle. M. I. Aksun Koç University 7/18 ELEC 401 – Microwave Electronics Smith Chart - Application Show the locations of the following impedances: Z A (1.0 j 0.0) Z B (1.0 j1.0) YE Z C ( j 0.0) o.c Z D (0.0 j 0.0) s.c Z E (1.0 j1.0) Show the locations of the following admittance: Y A (1.0 j 0.0) ZD ZB ZA ZC YA YD YC ZE YB YB (0.5 j 0.5) YC (0.0 j 0.0) o.c YD ( j 0.0) s.c YE (0.5 j 0.5) M. I. Aksun Koç University 8/18 ELEC 401 – Microwave Electronics Smith Chart - Applications Find the admittance from a given impedance, or vice versa: Z Constant || circle 1 1 1 1 e j 1 e j ( ) Y 1 1 e j 1 e j ( ) Z 1.0 j1.0 Y 0.5 j 0.5 M. I. Aksun Koç University 9/18 ELEC 401 – Microwave Electronics Smith Chart - Applications Find the input impedance of a TL terminated in a load impedance ZL. Z L 1.0 j1.0 Z0 , z=-l Constant || circle ZL z=0 Z L 1.0 j1.0 Angle 2 l l 0.148 1 in 1 Le j 2 l Z in 1 in 1 Le j 2 l Z in Z in 1.4 j1.2 1 e j ( 2 l ) 1 e j ( 2 l ) M. I. Aksun Koç University 10/18 ELEC 401 – Microwave Electronics Smith Chart - Applications Find the SWR, voltage maxima and minima. Constant || circle Z L 1.0 j1.0 Z0 , ZL z=-l Z L 1.0 j1.0 1 1 Voltage Max. Current Min. M. I. Aksun Koç University s 2.8 Z in 2.8 l 0.148 VSWR s Angle 2 l z=0 Voltage Min. Current Max. Voltage Max. Current Min. 11/18 ELEC 401 – Microwave Electronics Smith Chart - Applications 0.13 0 Example: A 5.2 cm long, lossless 100 W line is terminated in a load impedance ZL=30+j50 W. a) Calculate |L|, fL, and VSWR: 30 j 50 ZL 0.3 j 0.5 100 Z 1 L L 0.62123.50 Z L 1 VSWR 93.60 Z L 0.3 j 0.5 1 0.62 4.2 1 0.62 b) Determine the impedance at the input for the frequency of 750 MHz and 0: c 3.0 1010 cm / s 0 40cm 6 f 750 10 / s 5 .2 2 l 5.2cm 0 0.13 0 2 l 2 l 93.60 40 0 M. I. Aksun Koç University VSWR=4.2 Constant || circle 12/18 ELEC 401 – Microwave Electronics Smith Chart - Applications Example: A 50W coaxial cable, filled with a dielectric material of e2.25, is connected to a generator with 50W internal impedance and 200 MHz signal frequency. The TL is terminated in an unknown impedance at a distance of 10 cm from the generator. a) Knowing that the input impedance at the source terminal was measured and noted as 25+j5 W, what is the load impedance? b) Find the VSWR; c) What should the length of the line be in order to have a real input impedance at the source terminal? 0 c 3.0 1010 cm / s 0 150 cm ; 100cm f 200 106 / s er Z in 25 j5 Z 0 50W z= -10cm M. I. Aksun Koç University ZL ? z=0 l 10cm Z in 10 2 0.1 2l 2 l 720 100 25 j 5 0.5 j 0.1 50 13/18 ELEC 401 – Microwave Electronics Smith Chart - Applications 1. Locate Zin on the S.C.; 2. Draw the constant || circle; 3. Starting from Zin move toward load by 0.1 on constant || circle; Constant || circle Z in 0.5 j 0.1 Z L 0.6 j 0.4 s 2.0 Z L Z L 50W (30 j 20)W Z in 0.5 j 0.1 Z L 0.6 j 0.4 Z 0 50W z= -0.1 M. I. Aksun Koç University ZL ? z=0 14/18 ELEC 401 – Microwave Electronics Smith Chart - Applications Example: A length of TL with 50W characteristic impedance is connected to a generator with 50W internal impedance and to an unknown load impedance ZL. The VSWR and the locations of the maximum and minimum of standing wave are measured and the following information is obtained: i) First voltage minimum occurs at a distance of /5 from the load terminals; ii) VSWR=Vmax/Vmin = 2.0 Find the load impedance by using the above information. Standing wave Vmax /5 Vmin Z 0 50W z= -l M. I. Aksun Koç University ZL ? z=0 15/18 ELEC 401 – Microwave Electronics Draw the SWR=2.0 circle 1. Find the s=2.0 point on the Smith Chart. 2. Draw a circle that is centered at s=1.0 and passes through s=2.0 point M. I. Aksun Koç University SWR=2 circle s=2.0 16/18 ELEC 401 – Microwave Electronics On the SWR=2.0 circle, start from Vmin point and go 0.2λ toward load. Normalized load impedance is: 1.55 - j0.65 Vmin , Imax Vmax , Imin Z L 1.55 j 0.65 Load impedance is: 77.5 – j32.5 0.2λ M. I. Aksun Koç University 17/18