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ECE 5317-6351 Microwave Engineering. Fall 2011. Prof. David R. Jackson Dept. of ECE. Notes 2. Smith Charts. Generalized Reflection Coefficient . Recall,. Generalized reflection Coefficient:. Generalized Reflection Coefficient (cont.) . For. Proof:.

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slide1

ECE 5317-6351

Microwave Engineering

Fall 2011

Prof. David R. Jackson

Dept. of ECE

Notes 2

Smith Charts

slide2

Generalized Reflection Coefficient

Recall,

Generalized reflection Coefficient:

slide3

Generalized Reflection Coefficient (cont.)

For

Proof:

Lossless transmission line ( = 0)

slide4

Decreasing l (toward load)

l

Increasing l (toward generator)

Complex  Plane

Lossless line

slide5

Impedance (Z) Chart

Define

Substitute into above expression for Zn(-l ):

Next, multiply both sides by the RHS denominator term and equate real and imaginary parts. Then solve the resulting equations for R and Iin terms of Rn and Xn. This gives two equations.

slide6

Impedance (Z) Chart (cont.)

1) Equation #1:

I

Equation for a circle in the  plane

R

slide7

Impedance (Z) Chart (cont.)

2) Equation #2:

I

Equation for a circle in the  plane:

R

slide8

Impedance (Z) Chart (cont.)

Short-hand version

Xn= 1

Rn= 1

Xn= -1

slide9

Impedance (Z) Chart (cont.)

Xn= 1

 plane

Rn= 1

Xn= -1

slide10

Admittance (Y) Calculations

Note:

Define:

Conclusion: The same Smith chart can be used as an admittance calculator.

Same mathematical form as for Zn:

slide11

Admittance (Y) Calculations (cont.)

Bn= 1

plane

Gn= 1

Bn= -1

slide12

Impedance or Admittance (Z or Y) Calculations

The Smith chart can be used for either impedance or admittance calculations, as long as we are consistent.

slide13

Admittance (Y) Chart

As an alternative, we can continue to use the original  plane, and add admittance curves to the chart.

Compare with previous Smith chart derivation, which started with this equation:

If (RnXn) = (a, b) is some point on the Smith chart corresponding to  = 0,

Then (GnBn) = (a, b) corresponds to a point located at  = - 0(180o rotation).

  • Rn= acircle, rotated 180o, becomes Gn= acircle.
  • and Xn= bcircle, rotated 180o, becomes Bn= bcircle.

Side note: A 180o rotation on a Smith chart makes a normalized impedance become its reciprocal.

slide15

Admittance (Y) Chart (cont.)

Short-hand version

Bn= -1

Gn= 1

Bn= 1

 plane

slide16

Impedance and Admittance(ZY) Chart

Short-hand version

Bn= -1

Xn= 1

Gn= 1

Rn= 1

Xn= -1

Bn= 1

 plane

slide17

Standing Wave Ratio

The SWR is given by the value of Rn on the positive real axis of the Smith chart.

Proof:

slide18

Electronic Smith Chart

At this link

http://www.sss-mag.com/topten5.html

Download the following .zip file

smith_v191.zip

Extract the following files

smith.exe smith.hlp smith.pdf

This is the application file