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topics

Basic Transmission Line EquationsNine Power Gains of AmplifiersLinear and Nonlinear

Synthesis/AnalysisFull-Wave Analysis for Microstrip

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basic transmission line equations

04/22/23 3

basic transmission line equations

L

Oin Z

ZZ2

L

LOL ZZ

11

djZZdjZZZZ

LO

OLOin

tantan

OL

OLL ZZ

ZZ

Important Conclusions from the above equations : When ZL =ZO, ΓL = 0 and Zin = ZO For an open transmission line ΓL = 1, Zin = -jZocotθ. Under the condition θ

< π/2, the behavior of the input impedance is like that of a capacitance. Hence a short open-circuit transmission line can be used as a capacitance element.

For a shorted transmission line ΓL = -1, Zin = -jZotanθ. Under the condition θ < π/2, the behavior of the input impedance is like that of an inductance. Hence a short short-circuit transmission line can be used as an inductive element.

When an electrical length θ = π/2 (of physical length l = λ/4), the transmission line is called a quarter-wave transformer. The quarter-wave transformer has the following important property:

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basic transmission line equations

2

1

2

1

00

aa

ee

bb

j

j

2

2

1

1

cos/sinsincos

iv

ZjjZ

iv

O

O

For a Matched Lossless two-port Transmission Line with electrical length θ :S matrix

ABCD matrix

VSWR is the ratio of Vmax to Vmin. The relationship between VSWR and reflection coefficient is as follows:

L

L

xVxVVSWR

11

)()(

min

max

11

VSWRVSWR

L

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basic transmission line equations

Shift in Reference Planes

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basic transmission line equations

00

1

1

1

j

j

ee

S

00

2

2

2

j

j

ee

S

212

211

222

)(221

)(211

211

2221

1211

jj

jj

eSeSeSeS

SSSS

Shift in Reference PlanesThe S-matrices of the 50Ω transmission lines are represented by S1 and S2 :

The S-matrix of the device can be represented by S1 and S2 and the measured matrix S’ as follows :

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nine power gains of amplifiers

Power Gains of different amplifiers are determined using S- parameters to get the following results :Transducer power gain in 50-

Ω system2

21SGT

221122211

2221

2

)1)(1(

)1()1(

LGLG

LGT

SSSS

SG

222

211

2221

2

11

)1()1(

LG

LGTU

SS

SG

211

221

222

211

2221

11)1(

)1(

S

S

SS

SG

L

L

Transducer power gain for arbitrary ΓG and ΓL

Power gain with input conjugate matched

Unilateral transducer power gain

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nine power gains of amplifiers

)1( 2

12

21 kkSSGma

)1)(1( 222

211

221

maxSS

SGTU

12

21

SSGms

)Re(121

12211221

21221

SSSSkSS

U

222

221

211

222

2221

11)1(

)1(

S

S

SS

SG

G

GA

Maximum available power gain

Maximum unilateral transducer power gain

Maximum stable power gain

Unilateral power gain

Available power gain with output conjugate matched

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full wave analysis for microstrip

METHOD FEATURESDISADVANTAGESSpectral Domain Method

One of the most popular methods for infinitesimally thin conductors on multilayer structures Closed-form expressions for Fourier-transformed Green’s functions Numerical efficiency

Cannot handle thick conductor structures For tight coupling the number of basic functions becomes large; would involve convergent problems

Finite Difference

Method

Mathematical preprocessing is minimal Can be applied to a wide range of structures

Numerically inefficient Precautions must be taken when the method is applied to an open-region problem Need layer computer storage for accurate solution

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full wave analysis for microstrip

METHOD FEATURESDISADVANTAGESFinite Element Method

Similar to the finite difference method Has variational features in the algorithm and is more flexible in the application

Developed to solve very large matrix equations Numerically inefficient Existence of so-called spurious (unphysical) zeros

Mode- Matching Method

Typically applied to the problem of scattering at the waveguide discontinuity Often used to solve enclosed planar structures, including metal thickness effects

Several different formulations possible, all theoretically equivalent; however, they may be different numerically Precautions must be taken on relative convergence for some problems

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full wave analysis for microstrip

METHOD FEATURESDISADVANTAGESEquivalent

Waveguide Model

Very useful method for analysis of microstrip discontinuity problem

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linear and nonlinear synthesis

1. Matching networks for single-frequency and wide-frequency bands (e.g., a 4:1 for complex loads and termination).

2. Narrowband/wideband lumped and distributed filter synthesis.

3. Oscillator synthesis from small- and large-signal S parameters. Parallel-series design, determination of all components, determination of efficiency, output power, phase noise, and other relevant data.

4. Open and closed loop, PLL design, phase nosie determination, nonlinear switching, frequency lock phase lock.

5. System analysis and optimization for noise figure IMD performance.

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linear and nonlinear analysis

1. When load impedance ZL equals ZO, the characteristic impedance of the transmission line, the load reflection coefficient ΓL = 0, and the input impedance equals the characteristic impedance of the transmission line, namely Zin = ZO.

2. For an open transmission line ΓL = 1, Zin = -jZOcotθ. Under the condition θ < π/2, the behavior of the input impedance is like that of a capacitance. Hence a short open-circuit transmission line can be used as a capacitance element.

3. For a shorted transmission line ΓL = -1, Zin = -jZOtanθ. Under the condition θ < π/2, the behavior of the input impedance is like that of an inductance. Hence a short short-circuit transmission line can be used as an inductive element.

4. When an electrical length θ = π/2 (of physical length l = λ/4), the transmission line is called a quarter-wave transformer. The quarter-wave transformer has the following important property: Zin = ZO

2 /ZL.