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1. Transmission Lines1.1 Ideal Transmission Line Theory

:series resistance per unit length in .

:series inductance per unit length in .:shunt conductance per unit length in .:shunt capacitance per unit length in .

By Kirchhoff’s voltage law:

By Kirchhoff’s current law:

As ,

For time-harmonic( ) circuits

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Thus

where:complex propagation constant.

We have the solutions

: positive z-direction propagation wave.: negative z-direction propagation wave.

: constants.Also

Define characteristic impedance

Then

and

For lossless line

Terminated Lossless Transmission Line

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Assume incident wave , reflected wave then

At ,

Define return loss:

Special case:1. (short): .2. (open): .3. Half wavelength line:

4. Quarter wavelength line:

Two-transmission Line Junction

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At ,

: transmission coefficient.

Define Insertion loss: Conservation of energy

Incident power:

Reflected power:

Transmitted power:

Voltage Standing Wave Ratio (VSWR)

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Define Standing Wave Ratio

1.2 Coaxial LinesTEM modeLet be the inner radius of the coaxial line and be the outerradius of the coaxial line.Let be the potential function of the TEM mode, then satisfies Laplace’s equation . In polar coordinate

and the boundary condition

Due to symmetry, , we have

Use the boundary condition to solve and , we have

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1.3 Microstrip Line

Formulas,

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Or

where

Loss, where

where

Operating frequency limits

The lower-order strong coupled TM mode:

The lowest-order transverse microstrip resonance:

Frequency Dependence

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where

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1.3 Strip Line

Formulas

where

.

Or

where.

Loss

where

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1.4 Coplanar Waveguide (CPW)

Benefit:1. Lower dispersion.2. Convenient connecting lump circuit elements.

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1.5 Generator and Load Mismatches

1. Load Matched to Line

2. Generator Matched to Loaded Line

3. Conjugate Matched

Note this result means maximum power delivered to theload under fixed . In reality, our concern is efficiency or howmuch portion of total power is delivered to the load which is

related to .

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1.6 Calculation of Transmission Line Parameters

Note:

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1.7 Criteria of Ideal Transmission LinesTEM WaveAssume and dependence of the form .Substitute to Maxwell’s equations, we have

where and . These lead to

1. The propagation constant of any TEM wave is the intrinsicpropagation constant of the media.

Also,

2. . The z-directed wave impedance of any TEM wave isthe intrinsic wave impedance of the medium.

Let , then from wave equation we have

.Similarly,

The boundary conditions at perfect conductors are

3. The boundary-value problem for and is the same asthe 2-dimensional electrostatic and magnetostatic

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problem. Thus, static capacitances and inductances canbe used for transmission lines even though the field istime-harmonic.

4. The conductor must be perfect, otherwise will exist.5. Voltage is uniquely defined on the cross-section of the

waveguide.

To sum up, two conditions must be satisfied to support idealtransmission lines:1. Homogeneous, i.e., or are independent of location.

(Why?)2. Two conductors. (Why?)

Multiplying both , we have

.

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Equating both , we have

1.8 Not Ideal Transmission Lines

Introduce mode functions , , mode voltages and mode currents according toTM:

TE:

We can choose forTM:

TE:

Also all modes are normalized according to

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Then, the characteristic impedance is

These is also the wave impedance. Also and willsatisfy transmission-line equations

The power transmitted is

Since

Then for

TE: TM:

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1.9 Composite Right-Left Hand (CRLH) Transmission Lines

Let

Then,

where

and

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Applications1. Dual-band Components.2. Bandwidth Enhancement.3. Zeroth-order Resonator2. Microwave Network Analysis

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2.1 Concept of Impedance

1. Intrinsic impedance:

2. Wave impedance (of a propagation mode):

3. Characteristic impedance (of a transmission line):

2.2 Properties of One Port

Complex power (Poynting vector)

where : real positive. The average power dissipated.: real positive. The stored magnetic energy.: real positive. The stored electric energy.

Define real transverse model fields and such that

and

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then,

Thus, the input impedance

Properties:1. is related to . equals zero if lossless.2. is related to . , inductive load.

, capacitive load.

Even and Odd Properties of and

since . Similarly,

.

Summary1. Even functions: .2. Odd functions: 3. Even functions: .

2.3 Properties of N-Port

2.3.1 Impedance and Admittance MatrixLet the total voltage and current at each port be

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where + and - sign mean the voltage or current entering theport and leaving the port, respectively.Define impedance matrix , such that , i.e.,

where

and admittance matrix , such that

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where

Obviously,

Reciprocal Networks:1. No source in the network.2. No ferrite or plasma.

Lossless networks:

Example

2.3.2 The Scattering Matrix

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Define scattering matrix

where

Relationship with

Let be the matrix formed by the

characteristic impedance of each port.

Thus

If lossless

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Therefore,

. When all ports are equal, ,

unitary.

Since

Also

Therefore, , or

If reciprocal

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Example

1. Reciprocal?

2. Lossless?

3. Return loss at port 1 when port 2 is matched.

4. Return loss at port 1 when port 2 is shorted.

Shift in Reference Planes

If at port n, the reference plane is shifted out by a length of , the

voltage at the reference plane will be

where . Let

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We have

2.3.3 Generalized Scattering Parameters

Define the scattering parameters based on the amplitude of the incidentand reflected wave normalized to power.

Let

thus

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The generalized scattering matrix is defined as

where

or

If lossless, or

If reciprocal,

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2.3.4 The Transmission (ABCD) Matrix

Define a transmission matrix of a two port network as

or in matrix form

Relationship to impedance matrix

If reciprocal,

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Cascading of ABCD matrix:

2.3.5 Two-Port Circuits

Example: prove the first entry of Table 4.1.

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Homework #2: 2.14, 2.17., 2.20.

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3. Impedance Matching

3.1 Quarter-Wave Transformer

Match a real load to by a section of transmission linewith characteristic impedance and length .

The reflection coefficient becomes

for a given , solve for , we have

Assume TEM mode,

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The bandwidth becomes

`

Example: Match a 10 load to a 50 line at =3 GHz.Determine the percent bandwidth for SWR 1.5.

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jX

ZLjBZ0

(a)

Z0 jB

jX

ZL

(b)

3.2 Matching Using L-Sections

Analytic Solutions(a)

(b)

Smith Chart Solutions

1. . Use (a)a. Convert to admittance plot.b. Move along constant conductance curve until

intercept with the constant resistance curve equal to1.

c. Convert back to impedance plot.d. Find the required reactance.

2. . Use (b)a. Move along constant resistance curve until intercept

with the constant admittance curve equal to 1.b. Convert to admittance plot.c. Find the required susceptance.

Example 2.5

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Smith chartLumped Elements

3.3 Single-Stub Tuning

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Analytic Solutions1. Shunt Stubs

Open stub:

Short stub:

Where

2. Series Stubs

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Open stub:

Short stub:

where

Smith Chart Solutions

Shunt (Series) Stubs1. Use admittance (impedance) plot.2. Rotate clockwise along constant curve until intercept

with the constant conductance (resistance) curve of value1.

3. Compensate the remaining susceptance (reactance) by asuitable length of open or short stub.

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Example

Smith Chart

3.4 Double-Stub Tuning

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Analytical Solution

where

Open stub:

Short stub:

where or Smith Chart SolutionsSmith Chart1. Use admittance plot.2. Rotate the constant conductance circle of value 1

counterclockwise by a distance d.3. Move along the constant conductance curve until

intercepting the rotated circle in 2. The difference of thesusceptance determines the length of the stub 2.

4. Rotate the intercepting point back to constantconductance circle of value 1. The susceptance valuedetermine the length of stub 1.

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ExampleSmith Chart

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4. Power Dividers and Directional Couplers

4.1 The T-Junction Power Divider

Lossless Divider

1. Lossless2. Match at the input port.3. Mismatch at the output ports.4. No isolation at the output ports.

4.2 Resistive Divider

1. Lossy.2. Match at all ports.3. No isolation.

From the figure

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4.3 The Wilkinson Power Divider

1. Matched at all ports.

2. Isolation between output ports.

3. No power loss from input to output ports.

4. Half power loss from output to input ports.

Analysis

1. Excite port 1.

Symmetry equal voltages at port 2 and 3 no current flows

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through the resistor open. The circuit becomes

From the figure

To compute , let and denote the voltages of

the forward and backward propagating modes in one of the two

lines. Assume the reference plane is located at port 1. Let

the voltage of the incident wave at port 1 be and port 3 .We have at port 1

At port 3,

2. Even and odd mode excitation at port 2 and 3Rearranging the circuit as follow

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a. Even mode: Symmetry equal voltages at port 2and 3 no current flows through the resistor open.The circuit becomes

b. Odd mode: Anti-symmetry opposite voltages atport 2 and 3 short at the middle of the resistor. Thecircuit becomes

Since

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From , we have

Unequal Power Division

If power ration between ports 2 and 3 is ,

N-way, equal-split, Wilkinson power divider

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4.4 Basic Properties of a Three Port Device

Impossible scenario: reciprocal, matching at all ports, lossless.Reciprocal and matching at all ports give the following S matrix

If lossless, the matrix is unitary, that is,

Two of must be zero to satisfy the last 3 equations.However, then, the first 3 equations will not be satisfied.

Possible scenario:1. Nonreciprocal, matching at all ports, lossless.

Lossless

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Two possible solutions

and

Example: Circulators

2. Reciprocal, lossless, matching only two ports.

Lossless

Possible solution

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3. Lossy, matching at all ports, reciprocal.

4.5 Basic Properties of a Four Port Device

Reciprocal, matched at all ports.

If lossless, the following conditions are required.

if and , the following conditions can be derived:

If , then

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real.

1. Symmetrical:

2. Anti-symmetrical: ,

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4.6 The Quadrature (90) Hybrid

Even-Odd Mode Analysis

Even Mode

Using ABCD matrix, we have

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Odd mode

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4.7 Coupled Line Directional Coupler

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Assume and , we have

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where .

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If , and . Also

For , choose the mid-band frequency such that

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Example Design a 20 dB single-section coupler.