Prof. David R. Jackson Dept. of ECE

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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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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:

Lossless transmission line ( = 0)

l

Increasing l (toward generator)

Complex  Plane

Lossless line

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.

Impedance (Z) Chart (cont.)

1) Equation #1:

I

Equation for a circle in the  plane

R

Impedance (Z) Chart (cont.)

2) Equation #2:

I

Equation for a circle in the  plane:

R

Impedance (Z) Chart (cont.)

Short-hand version

Xn= 1

Rn= 1

Xn= -1

Impedance (Z) Chart (cont.)

Xn= 1

 plane

Rn= 1

Xn= -1

Note:

Define:

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

Same mathematical form as for Zn:

Bn= 1

plane

Gn= 1

Bn= -1

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.

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.

Short-hand version

Bn= -1

Gn= 1

Bn= 1

 plane

Short-hand version

Bn= -1

Xn= 1

Gn= 1

Rn= 1

Xn= -1

Bn= 1

 plane

Standing Wave Ratio

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

Proof:

Electronic Smith Chart