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

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 with length l and Reflection coefficient is 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 → 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 lines?

Difficulty in measurement of voltage at high .

probe

oscilloscope 1. Cannot ignore distributed 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 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