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Microwave System
Sehwan Choi
Chapter 6. Microwave Resonators
Chapter 7. Power Dividers and Directional Couplers
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Contents
6.7 Cavity Perturbations
7. Power Dividers and Directional Couplers
7.1 Basic Properties of Dividers and Couplers
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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.
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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
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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
Subtracting these two equations and using the vector identity
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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.
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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.
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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.
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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
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Commercial Instruments for measuring Permittivity and Permeability
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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
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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
Subtracting these two equations and using the vector identity
Subtracting these two equations and using the vector identity
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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 .
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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.
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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.
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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>
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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.
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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
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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)
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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:
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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)
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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.
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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)
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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.
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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>
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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.
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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
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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|
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1. A Symmetric Coupler: θ = φ = π/2.
2. An Antisymmetric Coupler: θ = 0, φ = π.
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The following quantities are commonly used to characterize a directional
coupler:
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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
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POINT OF INTEREST: Measuring Coupler Directivity
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Thank you