Microwave System - Hanyang Antennas & RF Devices Lab

1/33 Antennas & RF Devices Lab.

Microwave System

Sehwan Choi

Chapter 6. Microwave Resonators

Chapter 7. Power Dividers and Directional Couplers


Contents

6.7 Cavity Perturbations

7. Power Dividers and Directional Couplers

7.1 Basic Properties of Dividers and Couplers


6.7 Cavity Perturbations

Cavity resonators are often modified by making small changes in their shape, or by

introducing small pieces of dielectric or metallic materials

For example, the resonant frequency of a cavity resonator can be easily tuned with a small screw

(dielectric or metallic) that enters the cavity volume, or by changing the size of the cavity with a movable

wall. Another application involves the determination of dielectric constant by measuring the shift in

resonant frequency when a small dielectric sample is introduced into the cavity.

One useful technique for approximations of the perturbation effect is the

perturbational method.

This method assumes that the actual fields of a cavity with a small shape or material perturbation are not

greatly different from those of the unperturbed cavity.

In this section we derive expressions for the approximate change in resonant frequency

when a resonant cavity is perturbed by small changes in the material filling the cavity,

or by small changes in its shape.


Material Perturbations

If E0, H0 are the fields of the original cavity, and E, H are the fields of the

perturbed cavity, then Maxwell’s curl equations can be written for the two

cases as

where ω0 is the resonant frequency of the original cavity, and ω is the resonant

frequency of the perturbed cavity.

< Original cavity > < Perturbed cavity >

HE ,

0V

0Sω

ΔΔ , εε

n̂

00 HE ,

0V

0S0ω

, ε

n̂


Multiply the conjugate of (6.95a) by 𝐻, and multiply (6.96b) by 𝐸0∗, to get

Similarly, multiply the conjugate of (6.95b) by 𝐸, and multiply (6.96a) by 𝐻0

∗, to get

Subtracting these two equations and using the vector identity



Add (6.97a) and (6.97b), integrate over the volume V0 , and use the

divergence theorem to obtain

where the surface integral is zero because 𝑛× 𝐸 = 0 on S0. Rewriting gives

This is an exact equation for the change in resonant frequency due to material

perturbations, but is not in a very usable form since we generally do not know 𝐸 and 𝐻, the exact fields in the perturbed cavity.


Assume that Δε and Δµ are small,

This result shows that any increase in ε or µ at any point in the cavity will

decrease the resonant frequency.


EXAMPLE 6.7 MATERIAL PERTURBATION OF A RECTANGULAR CAVITY

A rectangular cavity operating in the TE101 mode is perturbed by the insertion

of a thin dielectric slab into the bottom of the cavity, as shown in Figure 6.24.

Use the perturbational result of (6.100) to derive an expression for the change

in resonant frequency.

From (6.42a)–(6.42c), the fields for the unperturbed TE101 cavity mode can be

written as

In the numerator of (6.100), Δε = (εr − 1) ε0 for 0 ≤ y ≤ t and zero elsewhere.


The integral can then be evaluated as

The denominator of (6.100) is proportional to the total energy in the

unperturbed cavity, which was evaluated in (6.43); thus,

Then (6.100) gives the fractional change (decrease) in resonant frequency as


Commercial Instruments for measuring Permittivity and Permeability


Shape Perturbations

Maxwell’s curl equations can be written for the two cases as

< Original cavity > < Perturbed cavity >

HE ,

VSω

n̂

SΔ

VΔ00 HE ,

0V

0S0ω

n̂


Multiply the conjugate of (6.101a) by 𝐻, and multiply (6.102b) by 𝐸0∗, to get

Similarly, multiply the conjugate of (6.101b) by 𝐸 and (6.102a) by 𝐻0∗ to get




Add (6.103a) and (6.103b), integrate over the volume V, and use the

divergence theorem to obtain

since 𝑛× 𝐸 = 0 on S.

Because the perturbed surface S = S0 − ΔS, we can write

because 𝑛× 𝐸 = 0 on S0 .


Using this result in (6.104) gives

which is an exact expression for the new resonant frequency, but not a very

usable one since we generally do not initially know 𝐸, 𝐻 or ω. If we assume

ΔS is small,

In terms of stored energies

These results show that the resonant frequency may either increase or

decrease, depending on where the perturbation is located and whether

it increases or decreases the cavity volume.


The fields for the unperturbed TE 101

cavity can be written as

If the screw is thin,

EXAMPLE 6.8 SHAPE PERTURBATION OF A RECTANGULAR CAVITY

A thin screw of radius r0 extends a

distance l through the center of the

top wall of a rectangular cavity

operating in the TE 101 mode.

Derive an expression for the change

in resonant frequency from the

unperturbed cavity.


Then the numerator of (6.107) can be

evaluated as

where ΔV = πl𝑟02is the volume of the

screw. The denominator of (6.107) is,

from (6.43),

where V0 = abd is the volume of the

unperturbed cavity. Then (6.107) gives

=> Lowering of the resonant

frequency is proportional to l and r.

<Cavity Filter>


7.1 BASIC PROPERTIES OF DIVIDERS AND COUPLERS

In this section we will use properties of the scattering matrix developed in

Section 4.3 to derive some of the basic characteristics of three- and four-port

networks.

We will also define isolation, coupling, and directivity, which are important

quantities for the characterization of couplers and hybrids.


Three-Port Networks (T-Junctions)

Three-port network : Two inputs and one output.

The scattering matrix of an arbitrary three-port network has nine independent elements:

The device is passive and isotropic => Reciprocal and symmetric (Sij = Sji ).

To avoid power loss = > matched at all ports.

If all ports are matched, then Sii = 0, and if the network is reciprocal, the

scattering matrix of (7.1) reduces to


Lossless

If the network is also lossless, then energy conservation requires that the

scattering matrix satisfy the unitary properties of (4.53), which leads to the

following conditions :

Equations (7.3d)–(7.3f) show that at least two of the three parameters (S12,

S13, S23 ) must be zero. However, this condition will always be inconsistent

with one of equations (7.3a)–(7.3c).

It implies that a three-port network cannot be simultaneously lossless,

reciprocal, and matched at all ports.

If any one of these three conditions is relaxed, then a physically realizable

device is possible.

jiSS ij

N

k

kjki ,1

*

If ,

If ,

1ijji

ji 0ij

(4.52,53)


Nonreciprocal

If the three-port network is nonreciprocal, then Sij ≠ Sji , and the conditions of

input matching at all ports and energy conservation can be satisfied.

>>> Such a device is known as a circulator.

The scattering matrix of a matched three-port network has the following form:


If the network is lossless, [S] must be unitary, which implies the following

conditions:

These equations can be satisfied in one of two ways. Either

Or

*

31 32

*

21 23

*

12 13

0

0

0

S S

S S

S S

S S

S S

S S

2 2

21 31

2 2

12 32

2 2

13 23

1

1

1

(7.5a)

(7.5b)

(7.5c)

(7.5d)

(7.5e)

(7.5f)


These results shows that Sij ≠ Sji for i ≠ j, which implies that the device must

be nonreciprocal.

The scattering matrices for the two solutions of (7.6) are shown in Figure 7.2,

together with the symbols for the two possible types of circulators.

The only difference between the two cases is in the direction of power flow

between the ports.


Mismatched

A lossless and reciprocal three-port network can be physically realized if only

two of its ports are matched. If ports 1 and 2 are the matched ports, then the

scattering matrix can be written as

To be lossless, the following unitarity conditions must be satisfied:

S S

S S S S

S S S S

*

13 23

* *

12 13 23 33

* *

23 12 33 13

0

0

0

2 2

12 13

2 2

12 23

2 2 2

13 23 33

1

1

1

S S

S S

S S S

(7.8a)

(7.8b)

(7.8c)

(7.8d)

(7.8e)

(7.8f)


Equations (7.8d) and (7.8e) show that |S13| = |S23|, so (7.8a) leads to the result

that S13 = S23 = 0. Then, |S12| = |S33| = 1.

The scattering matrix and corresponding signal flow graph for this network

are shown in Figure 7.3, where it is seen that the network actually degenerates

into two separate components — one a matched two-port line and the other a

totally mismatched one-port.


Lossy

If the three-port network is allowed to be lossy, it can be reciprocal and

matched at all ports

=> Resistive divider (Section 7.2)

In addition, a lossy three-port network can be made to have isolation between

its output ports (e.g., S23 = S32 = 0).

=> Wilkinson Power Divider (Section 7.3)

<Resistive divider>


Four-Port Networks (Directional Couplers)

The scattering matrix of a reciprocal four-port network matched at all ports

has the following form:

If the network is lossless, 10 equations result from the unitarity, or energy

conservation.


Consider the multiplication of row 1 and row 2, and the multiplication of row

4 and row 3:

Similarly, the multiplication of row 1 and row 3, and the multiplication of row

4 and row 2, gives

Multiply (7.12a) by S12 , and (7.12b) by S34, and subtract to obtain

Multiply (7.10a) by 𝑆24∗ , and (7.10b) by 𝑆13

∗ , and subtract to obtain


One way for (7.11) and (7.13) to be satisfied is if S14 = S23 = 0, which results

in a directional coupler. Then the self-products of the rows of the unitary

scattering matrix of (7.9) yield the following equations:

We choose S12 = S34 = α, S13 = βejθ , and S24 = βejφ.

The dot product of rows 2 and 3 gives

which yields a relation between the remaining phase constants as

|S13| = |S24|

|S12| = |S34|


1. A Symmetric Coupler: θ = φ = π/2.

2. An Antisymmetric Coupler: θ = 0, φ = π.


The following quantities are commonly used to characterize a directional

coupler:


Hybrid couplers are special cases of directional couplers, where the coupling

factor is 3 dB, which implies that α = β = 1/ √ 2.

There are two types of hybrids.

The quadrature hybrid has a 90 ◦ phase shift between ports 2 and 3 (θ = φ =

π/2) when fed at port 1.

=> Symmetric coupler

The magic-T hybrid and the rat-race hybrid have a 180 ◦ phase difference

between ports 2 and 3 when fed at port 4.

=> Antisymmetric coupler


POINT OF INTEREST: Measuring Coupler Directivity


Thank you

Microwave System - Hanyang Antennas & RF Devices Lab

Documents

Transcript of Microwave System - Hanyang Antennas & RF Devices Lab