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1、2-Smith Chart Contents Introduction to the Smith Chart Construction and axes Plotting on the Smith Chart Analysis using the Smith Chart Bilinear Transform Mapped Circles in LNA Design Introduction First introduced by P.H. Smith in 1944 (Remains the most versatile matching chart in use today!) Used f
2、or analysis and design CAD tools and VNAs equipped with this display format Powerful: displays vector quantities Reflects periodic nature of transmission line phenomena Looks complicated at first but really straightforward to use Z and Complex Plane Complex Plane Re +1 -1 Im +1 -1 1 1 0 0 z z ZZ ZZ
3、Z Complex Plane Im Re z=r+jx 1 1 0 Z Z z Example Complex Plane Re +1 -1 Im +1 -1 8262 0.6 Z Complex Plane Re z=0.5+j1 0.512 8362. 0 1 1 5 . 0 z z jz z Im jbg z y jxrz 1 Impedance Reactance Resistance susceptance conductance Admittance Constant Resistance convert to Complex Plane Complex Plane Z Comp
4、lex Plane +1 -1 Re +1 120.50 -1 00.40.20.60.81.0-0.4 -0.2-0.6-0.8-1.0 Constant Resistance Lines Mapped into Constant Resistance Circle in Complex Plane Z Complex Plane Re 0.5120 Complex Plane 00.40.20.60.81.0-0.4 -0.2-0.6-0.8-1.0 Constant Reactance Lines mapped into to Constant Reactance circle in C
5、omplex Plane Z Complex Plane Complex Plane Re 0.5 1.0 2 0 -0.5 -1.0 x=1 x=0.5 x=0 x=-1 x=-0.5 x=1 x=0.5 x=0 x=-0.5 x=-1 00.40.20.60.81.0-0.4 -0.2-0.6-0.8-1.0 Z mapping into Complex Plane Re -1 Im +1 -1 Re +1 Im Z mapping into Complex Plane Im Re The Impedance Smith Chart simplified. This is an imped
6、ance chart transformed from rectangular Z. Normalized to 50 ohms, the center = R50+J0 or Zo (perfect match). For S11 or S22 (two-port), you get the complex impedance. Bottom Half:Bottom Half: CapacitiveCapacitive Reactance (Reactance (- -jx)jx) Top Half:Top Half: InductiveInductive Reactance (+jx)Re
7、actance (+jx) OPENSHORT Circles of constant Resistance Lines of constant Reactance (+jx above and -jx below) Zo (characteristic impedance) = 50 + j0 16.750150 25 50100 More Smith chart. The Smith chart in ADS Data Display Z=0+j1 Z=0-j1 Z=infinity + j infinity Gamma or S11= 1 / 0 m1m1 ADS marker defa
8、ults to: S(1,1) = 0.8/ -65 Z0 * (0.35 - j1.5) but can be changed to give Z in ohms. Z=0-j0.5 Z=0-j2 S(1,1): mag / phase 0 to 1 / 0 to +/- 180 Reflection Coefficient: gamma Z = real / imaginary 0 to +infinity / -infinity to + infinity Z= 0 + j0 Gamma or S11=1 / 180 Impedance: Z Z = 1 / 0 - -0.50.5+0.
9、5+0.5 S-parameter controller. Circles of Constant Resistance Always normalize Z to ZO As RNdecreases: circle radius increases center increases away from open circuit point outer circle RN=0 Circles of Constant Reactance Upper half: +jX - inductive: +j L Lower half: -jX - capacitive: 1/ j C Remember:
10、 Normalized reactance Reflection Coefficient Circles: Center ZO Radius Also known as: constant VSWR circle Find phase of as from horizontal Plotting on the Smith Chart Normalized load: ZL=0.3+j0.5 Plot to find: S11=0.62 123 Const. VSWR circle: 4.23 ZL* is refln. in x-axis YLis 180Orotation from ZL Z
11、L=0.3+j0.5 S11=0.62 123 ZL*=0.3-j0.5 YL Unit Z and Y Circle Open circuit point ZL= Short circuit point ZL=0 Match point ZL=1=ZO Unit Z circle: Z=1 jX Unit Y circle: Y=1 jB Input Impedance Along Tx Line Rmax Rmin Rmax C L RLZ O ZO g/2 /4 Transformer - - 90 90 90 90 L O Sin l LO OL Oin Z Z ZZ ljZZ ljZ
12、Z ZlZ 2 4/ tan tan Recall: Frequency Response of R-L-C Networks f f f f Cj RZin 1 LjRZin LjR Yin 11 Cj R Yin 1 Frequency Response of Parallel Resonator Parallel Resonator As F0: Zin=0 inductor short circuit As F: Zin=0 capacitor short circuit At F=fO: Im (Zin)=0 Zin=R F=fo o F=0 F= F= F=0 Frequency
13、Response of Series Resonator Series Resonator As F0: Zin= Capacitor Open circuit As F: Zin= Inductor Open circuit At F=fO: Im (Zin)=0 Zin=R F=fo o F=0 F= F= F=0 Bilinear Transformation )0(bcad dcz baz w z and w are complex plane. a,b,c,d are constant The bilinear transformation maps line in z plane
14、into circle in the w plane The bilinear transformation maps circle in z plane into circle in the w plane Circle Form in Complex Plane z0 r Z plane 2 2 0 * 0 2 2 2 00 * 0 2 2* 00 0 )Re(2 )( rzzzz rzzzzzz rzzzz rzz Circle in the w plane map into z plane(1) dcz baz w z0 r * )()(dczdczbazbaz 0 2 2 2 2 2
15、 2 2 2 2 * * 2 2 2 * 2 ca db ca badc z ca abcd zz 2 2 2 2 2 2 2 2 2 2 0 2 ca bcad r ca db zr 2 2 2 * 0 ca badc z Circle in the w plane map into z plane(2) z0 rw0 00 w dcz baz ww dcz bza dcz dwbzcwa 00 )()( 2 2 2 ca cbda r 2 2 2 * 0 )( ca badc z Constant Resistance Line in the z plane map into plane(
16、1) Complex Plane Z Complex Plane Re 0.5120 r rzz jxrz 2) 1 1 () 1 1 ( 2 * * 22 2 * 2 )1 ( 1 )1 (11rr r r r r r r r C 1r R 1 1 Constant Reactance Lines in the z plane map into plane(1) Z Complex Plane Re 0.5 1.0 2 0 -0.5 -1.0 r jxzz jxrz 2) 1 1 () 1 1 ( 2 * * 22 2 * 211 xx x x jx x jx x jx C x R 1 Co
17、mplex Plane Constant Q Lines in the z plane map into plane jQ jQ jQ jQ jxr jxr z z jxrz 1 1 ) 1 1 /() 1 1 ( 1 1 * * Re 2 r x Q 2 2 2 * 211 Q Q QQ j Q j Q j C Q Q R 2 1 Constant Conductance Line in the y plane map into plane(1) y Complex Plane Re 0.5120 g gyy jbgy 2) 1 1 () 1 1 ( 2 * * 22 2 * 2 )1 (
18、1 )1 (11gg g g g g g g g C 1g R 1 1 Complex Plane Constant Susceptance Lines in the z plane map into plane Complex Plane y Complex Plane Re 0.5 1.0 2 0 -0.5 -1.0 b jbyy jbgy 2) 1 1 () 1 1 ( 2 * * 22 2 * 211 bb b b jb b jb b jb C b R 1 Constant Magnitude Impedance in the z plane map into plane Re 0 z
19、 Complex Plane 1 2 1 1 1 1 1 1 2 2 2 2 2 2 22 K K R K K K Kzzz Kxrzjxrz c 2 2 2 2 2 2 2 2 2 ) 1 2 () 1 1 ( 1 1 1 1 K K K K K K K K ) 0 , 4 5 ( 4 3 R )0 , 4 5 ( 4 3 R 3 1 K 1K 3K Complex Plane Mapping Circles in LNA Design Source Stability Circle Load Stability Circle Noise Figure Circle In/Output VS
20、WR Circle Operation Power Gain Circle Available Gain Circle L L in S SS S 22 2112 11 1 SIN SIN a1 * Lout Lout b1 * 2 2 2 1)1( 4 min optS optSn r FF 2 22 2 2 1 1 2 21 1 1 L L IN S p SP 2 11 2 2 22 1 12 21 1 1 s s OUT SS A SP S S out S SS S 11 2112 22 1 Load Stability Condition and Circle L-Plane RL C
21、L S111 CL RL S111 Center and Radius of Available Gain Circle Radius of circleCenter of circle A AA GA gD SSgSSKg R 1 2 2112 2 2112 1 |21 A A GA gD Cg C 1 * 1 1 Where: | |2 |1 1221 2 22 2 11 2 SS SS K 22 111 SD * 22 2 111 SSC 21122211 SSSS dBnotvalue,absoluteGaindesierisGA 2 21 S G g A A Center and R
22、adius of Power Gain Circle Radius of circleCenter of circle p p Gp gD Cg C 2 * 2 1 p pp Gp gD SSgSSKg R 2 2 2112 2 2112 1 |21 | |2 |1 1221 2 22 2 11 2 SS SS K Where: 22 222 SD * 22 2 222 SSC 21122211 SSSS dBnotvalue,absoluteinGainDesiredisGp 2 21 S Gp g p Center and Radius of Noise Circle 2 2 2 1)1(
23、 4 min optS optSn r FF 2 1 2 2 |1 1 1 optii i N NN N R i opt N N C 1 2 min 1 4 opt n factor i r FF N Constant VSWR (Return Loss) Circle Output Constant VSWR CircleInput Constant VSWR Circle 2 2 2 2 2 2 * 2 2 * * 1 )1 ( )( 1 )1 ( )( 1 )1 ( )( 1 )1 ( )( 11 outb outb vo ina ina vi outb bout vo ina ain
24、vi Lout Lout b Sin Sin a RadiusRRadiusR centerCcenterC Lab2-1: Plot Mapping Circle on Data Display Lab2-2: Plot Mapping Circle Using Built-in MeasEq Plot source/load stability circle, available gain circle, noise circle, in/output constant VSWR circle using BFP640 SiGe BJT whose S parameters and noi
25、se parameters at 1.8 GHz, Vce=3V, Id=5.0mA 46.1-0.6103 42.80.0666 102.68.565 102.1-0.5521 22 12 21 11 S S S S 11. 0 300.15 0.65 min n opt r F Lab2-1:Source Stability Circle - - - - - - - Lab2-1:Load Stability Circle - - - - Lab2-1: 20,18 and 16 dB Available Gain Circle - Lab2-1 : 20,18 and 16 dB Pow
26、er Gain Circle - Lab2-1:Noise Circles - - - - - - - - Lab2-1:Constant -10dB Return Loss Circles - - - - - - - - - Lab2-2:Using the Build-in Mapping Circle MeasEq - - - - - - Result: Mapping Circle on S / L Smith Chart - - - Summary Smith Chart axis: impedance, Impedance & admittance charts Plotting, Z, Y and converting Input impedance along a line Interpreting network frequency response Bilinear Transformation