Ứng dụng đồ thị Smith

Ứng dụng đồ thị Smith để giải bài tập trường điện từ, siêu cao tần Điện tử viễn thông

The Smith Chart: I plot in the Complex Plane

¢ Smith’s chart is a graphical representation in the complex I’ plane of the input impedance, the load impedance, and the reflection

coefficient I’ of a /oss-free TL ImL'

¢ it contains two families

of curves (circles) in

the complex I’ plane

¢ each circle corresponds

to a fixed normalized

resistance or reactance |

Rel

The Smith Chart: Normalized Impedance and I

relation #1: normalized load impedance z, and reflection I’

The Smith Chart: Resistance and Reactance Circles

resistance circles reactance circles

let the abscissa be I’ and the ordinate be I’, (the I’ complex plane)

¢ resistance and reactance equations are circles in the I’

complex plane

¢ resistance circles have centers lying on the I axis (I; = 0 or ordinate = 0)

e reactance circles have centers with abscissa coordinate = 1

¢a complex normalized impedance z, =r, + jx, 1S a point on the Smith chart where the circle r, intersects the circle x,

Lecture 08: The Smith Chart 4

The Smith Chart: Resistance Circles

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The Smith Chart: Reactance Circles

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The Smith Chart: Nomographs

at the bottom of Smith’s chart, a nomograph is added to determine

¢ SWR and SWR in dB, 20 log, SWR

¢ return loss in dB, —20log,o | T |

¢ power reflection |I|* (P)

¢ reflection coefficient |I| (E or J), etc

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The Smith Chart: SWR Circles

> acircle of radius I’, centered

at I’ = 0 is the geometrical

place for load impedances

producing reflection of the

same magnitude, | I] =I,

The Smith Chart: Plotting Impedance and Reading Out I

relation #2: input impedance versus the TL length L

>» on the Smith chart, the point corresponding to z,, is rotated by

—2L (decreasing angle, clockwise rotation) with respect to the point corresponding to z; along an SWR circle

> one full circle on the Smith chart is 2Z,,, = 27, 1.€., Lina = 2/23 this reflects the periodicity of z,,

Lecture 08: The Smith Chart 10

The Smith Chart: Tracking Impedance Changes with L — 2

`

R for Z2 = 50 Q, the

transforms a load of Z¡=25- 725

The Smith Chart: Read Out Distance to Load

The Smith Chart: Reading Out SWR

*® normalized load admittance

* the relation between y,„ and y; 1s the same as that between z,„ and z

— one can get from load to input terminals (and vice versa) by

following a circle clockwise (counter-clockwise)

¢ standard Smith chart gives resistance and reactance values

¢ admittance Smith chart is exactly the same as the “impedance” (or standard) Smith chart but rotated by 180° [see eq (*) and sl 17]

Lecture 08: The Smith Chart 14

The Smith Chart: Admittance Interpretation — 2

15

The Smith Chart: Admittance Interpretation — 3

combined impedance and conductance Smith Chart

zy, (on impedance scales) 0}

: Ale

y,, (on admittance scales) 2}?

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16

¢ impedance values from a standard Smith chart can be easily

converted to admittance (conductance + susceptance) values by

rotation along a circle of exactly 180°

¢ rotation by 180° on the impedance Smith chart corresponds to

impedance transformation by a quarter-wavelength "2 8: 4 - z)

¢ the value diametrically opposite on the Smith chart from

an impedance value is the respective “admittance” value

Lecture 08: The Smith Chart 17

The Smith Chart: Admittance Interpretation — 5

Check whether in this

example the y, found

from the Smith chart

>» in this case the TL must be designed to have this specific Z,

Lecture 08: The Smith Chart 19

> the impedance match with the 4/4 transformer holds perfectly at one frequency only, fo, where L = 1,/4

> this impedance-match device is narrow-band

Optimal Power Delivery: Review (Homework)

> at the generator’s terminals, a loaded TL is equivalently represented

by its input impedance Z.,

> active (or average) power delivered to the loaded TL (this is also

the powel delivered to the load Z, if the line is loss-free) ,

> assume generator’s impedance Z,, = R, + jX, is known and fixed

> optimal matching is achieved when maximum active power 1s

delivered to the load Z,, — what is this optimal value of Z,,,?

Zi — Max Fin (Zin )

Lecture 08: The Smith Chart 22