電子回路論第8回 Electric Circuits for Physicists #8 東京大学理学部・理学系研究科 物性研究所 勝本信吾 Shingo Katsumoto Outline Various transmission lines Termination and connection Smith chart S-matrix (S-parameters) Transmission lines with TEM mode Transmission lines with two conductors are “families”. Electromagnetic field confinement with parallel-plate capacitor Roll up to coaxial cable Open to micro-strip line Commonly TEM is the primary mode. Shrink to dipole (Lecher line) Lecher line Micro strip line Wide (W/h>3.3) strip Narrow (W/h<3.3) strip Waveguide Electromagnetic field is confined into a simply- cross section connected space. metal TEM mode cannot exist. hollow Maxwell equations give Helmholtz equation 퐸푧 = 0: TE mode, 퐻푧 = 0: TM mode Optical fiber Difference in dielectric constant step-type optical fiber no dispersion Connection and termination 퐽 = 퐽 + 퐽 (definition right positive) 푍0 + − At 푥 = 0: progressive retrograde 푍1 푉 = 푉+ + 푉− = 푍0 퐽+ − 퐽− x -l 0 Termination of a transmission line with length l and Reflection coefficient is characteristic impedance 푍0 at 푥 = 0 with a resistor 푍1. 푍1 = 푍0 : no reflection, i.e., impedance matching Comment: Sign of Ohm’s law in transmission lines 푍1 = +∞ (open circuit end) : 푟 = 1, i.e., free end reflects direction of waves (depends on the definitions). 푍1 = 0 (short circuit end) : 푟 = −1, i.e., fixed end Connection and termination Finite reflection → Standing wave Voltage-Standing Wave Ratio (VSWR): 1 − |푟| 1 + |푟| At 푥 = −푙 Then at 푥 = −푙 (at power source), the right hand side can be represented by Reflection coefficient: Connection and termination Transmission line connection. 푍0 푍′0 Characteristic impedance 푍0, 푍0′ At the connection point, only the local relation 푍0 푍′0 between V and J affects the reflection coefficient. The local impedance from the left hand side is 푍0′. 푍′0 antenna 푍0 푍′0: input impedance of antenna SWR measurement SWR Meters: desktop types cross-meter handy type directional coupler 5.2.3 Smith chart, immittance chart End impedance 푍1: Normalized end impedance 푍푛 ≡ 푍1/푍0 real: imaginary: y: erase constant reactance circle x: constant x: erase constant reactance circle y: constant 5.2.3 Smith chart, immittance chart r : reflection coefficient Z0 푉 푍 − 푍 Im r 푟 = − = 0 Z 푉+ 푍 + 푍0 y = 2 y = 4 Im Z x-constant Im r Re r y = 0 y-constant Re r x = 4 Re Z y = - 4 y = - 2 Circles with constant x w Common point: u Circles with constant y w y =1 y =2 y =0.5 y =4 y =0.2 Common point: y =0 u y = -0.2 y = -0.5 y =-4 y =-1 y =-2 Smith chart y =1 y =2 y =0.5 y =4 y =0.2 y =0 y = -0.2 x=2 x=4 y =-4 y = -0.5 y =-1 y =-2 Smith chart, immittance chart Admittance chart Immittance chart 5.3 Scattering (S) matrix (S parameters) How to treat multipoint (crossing point) in transmission lines systematically? Transmission lines: wave propagating modes → Channels 2 2 Take |푎푖| , |푏푖| to be powers (energy flow). output 푎1 S-matrix input 푏1 푎2 푏2 S 푎3 푏3 푎4 푏4 S-matrix (S parameters) S-matrix symmetries Reciprocity (In case no dissipation, no amplification) Unitarity Incident power wave Propagation with no dissipation Reflected (transmitted) power wave S matrix (S parameters) 퐽1 퐽2 푍푆 Simplest example: series impedance 푍푆 푉 푉 Take voltage as the “flow” quantity. 1 2 (assume common characteristic impedance) input direction input direction Measurement (calculation) of the S-matrix in the case the above is placed in transmission line with characteristic impedance 푍0. Consider the case that the ends of 푍푆 푍 푍 the line is terminated with 푍 and the 푍0 0 0 0 푉 푉 푍0 source is connected at the left end. 1 2 Simple example of S-matrix Terminate 2 with 푍0 → 푎2 = 0 (no reflection, no wave source at the right end) From symmetry of S-matrix: When the impedance matrix 푍 = 푍푖푗 is given, the S-matrix is obtained as Cascade connection of S-matrices 푟퐿,푅, 푡퐿,푅: complex reflection, transmission coefficients satisfying Why we need S-matrix (S-parameters) for high frequency lines? Difficulty in measurement of voltage at high frequencies. probe oscilloscope 1. Cannot ignore distributed capacitance 2. Probe line should also be treated as a transmission line Vector network analyzer On the other hand, S-parameters can be obtained from power measurements. S-parameters give enough information for designing high frequency circuits. Application of S-matrix to quantum transport Electron (quantum mechanical) waves also have propagating modes in solids. → Conduction channel Landauer equation: the conductance of a single perfect quantum channel is Rolf Landauer AB ring S-matrix model conductance magnetic flux Practical impedance matching with Smith chart Impedance matching designer http://home.sandiego.edu/~ekim/e194rfs01/jwmatcher/matcher2.html http://leleivre.com/rf_lcmatch.html An example FET power matching 푆11 푆12 C band: 4 – 8 GHz X band: 8 – 12 GHz Ku band: 12 – 18 GHz 푆21 푆22 The datasheet tells that we need impedance matching circuits with transmission lines. S parameter representation and matching circuits 푍0 input output 푍0 푍0 matching FET matching 푍0 circuit circuit Γs Γin Γout Γl FET The situation can be seen as: Source with reflection coefficient Γ푠, load FET S-matrix with reflection coefficient Γ푙 are connected. Power matching with S-parameters 푎1 푎2 That is 훤푠 S 훤푙 푏1 푏2 Then the total coefficients are The power matching conditions are The quadratic equation gives Then the problem is reduced to tune the passive circuits to .