ENE 428 Microwave Engineering

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ENE 428Microwave Engineering

• Lecture 12 Power Dividers and Directional Couplers

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Power dividers and directional couplers

• Passive components that are used for power
  division or combining.
• The coupler may be a three-port or a four-port
  component
• Three-port networks take the form of T-junctions
• Four-port networks take the form of directional
  couplers and hybrids.
• Hybrid junctions have equal power division and
  either 90? or a 180? phase shift between the
  outport ports.?

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Types of power dividers and directional couplers

• T-junction power divider
• Resistive divider
• Wilkinson power divider
• Bethe Hole Coupler
• Quadrature (90?) hybrid and magic-T (180?)
  hybrid
• Coupled line directional coupler

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Basic properties of dividers and couplers

• The simplest type is a T-junction or a
  three-port network with two inputs and one
  output.
• The scattering matrix of an arbitrary three-port
  network has nine independent elements

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The scattering parameters lossless property

• The unitary matrix
• This can be written in summation form as
• where ?ij 1 if i j and ?ij 0 if i ? j thus
• if i j,
• while if i ? j ,

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It is impossible to construct a three-port
lossless reciprocal network. (1)

• If all ports are matched, then Sii 0, and if
  the network is reciprocal the scattering matrix
  reduces to
• If the network is lossless, the scattering
  matrix must be unitary that leads to

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It is impossible to construct a three-port
lossless reciprocal network. (2)

• Two of the three parameters (S12, S13, S23) must
  be zeros but this will be inconsistent with one
  of eq. (1a-c), implying that a three-port network
  cannot be lossless, reciprocal, and matched at
  all ports.

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Any matched lossless three-port network must be
nonreciprocal. (1)

• The S matrix of a matched three-port network
  has the following form
• If the network is lossless, S must be unitary,
  which implies the following

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Any matched lossless three-port network must be
nonreciprocal. (2)

• Either of these followings can satisfy above
  equations,
• or

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Any matched lossless three-port network must be
nonreciprocal. (3)

• This results show that Sij ? Sji for i ? j,
  therefore the device must be nonreciprocal.
• These S matrices represent two possible types of
  circulators, forward and backward.

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A lossless and reciprocal three-port network can
be physically realized if only two of its ports
are matched. (1)

• If ports 1 and 2 are matched ports, then
• To be lossless, the following unitary conditions
  must be satisfied

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A lossless and reciprocal three-port network can
be physically realized if only two of its ports
are matched. (2)

• From (3a-b), , so (3d) shows that
  S13 S23 0. Then S12S331.

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A lossless and reciprocal three-port network can
be physically realized if only two of its ports
are matched. (3)

• The scattering matrix and signal flow graph are
  shown below.
• If a three-port network is lossy, it can be
  reciprocal and matched at all ports.

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Four-port networks (Directional Couplers)

• Power supplied to port 1 is coupled to port 3
  (the coupled port), while the remainder of the
  input power is delivered to port 2 (the through
  port)
• In an ideal directional coupler, no power is
  delivered to port 4 (the isolated port).

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Basic properties of directional couplers are
described by four-port networks.(1)

• The S matrix of a reciprocal four-port
  network matched at all ports has the above form.
• If the network is lossless, there will be 10
  equations result from the unitary condition.

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Conditions needed for a lossless reciprocal
four-port network (1)

• The multiplication of row 1 and row 2, and the
  multiplication of row 4 and row 3 can be arranged
  so that
• (4)
• The multiplication of row 1 and row 3, and the
  multiplication of row 2 and row 4 can be arranged
  so that
• (5)
• If S14 S23 0, a directional coupler can be
  obtained.

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Conditions needed for a lossless reciprocal
four-port network (2)

• Then the self-products of the rows of the
  unitary S matrix yield the following equations
• which imply that S13S24and that S12S24.

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Symmetrical and Antisymmetrical coupler (1)

• The phase references of three of the four ports
  are chosen as S12 S34 ?, S13 ?ej?, and S24
  ?ej?, where ? and ? are real, and ? and ? are
  phase constants to be determined.
• The dot products or rows 2 and 3 gives
• which yields a relation between the remaining
  phase constant as
• ? ? ? ? 2n?.

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Symmetrical and Antisymmetrical coupler (2)

• If 2? is ignored, we yield
• 1. The symmetrical coupler ? ? ?/2.
• 2. The antisymmetrical coupler ? 0, ? ?.

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Symmetrical and Antisymmetrical coupler (3)

• The two couplers differ only in the choice of
  the reference planes. The amplitudes ? and ? are
  not independent, eq (6a) requires that
• ?2 ?2 1.
• Another way for eq. (4) and (5) to be satisfied
  is if S13S24 and S12S34.
• If phase references are chosen such that
  S13S24? and S12S34j?, two possible solutions
  are given. First S14S230, same as above.
• The other solution is for ? ? 0, which
  implies S12S13S24S340, the case of two
  decoupled two-port network.

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Directional couplers characterization (1)

• Power supplied to port 1 is coupled to port 3
  (the coupled port) with the coupling factor
• The remainder of the input power is delivered to
  port 2 (the through port) with the coefficient
• In an ideal coupler, no power is delivered to
  port 4 (the isolated port).
• Hybrid couplers have the coupling factor of 3 dB
  or ? ? The quadrature hybrid
  coupler has a 90? phase shift between ports 2 and
  3 (? ? ?/2) when fed at port 1.

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Directional couplers characterization (2)

• Coupling C -20log? dB,
• Directivity D 20log
  dB,
• Isolation I -20logS14
  dB.
• The coupling factor indicates the fraction of
  the input power coupled to the output port.
• The directivity is a measure of the couplers
  ability to isolate forward and backward waves, as
  is the isolation. These quantities can be related
  as
• I D C dB.

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Ideal coupler

• The ideal coupler would have infinite
  directivity and isolation (S14 0).

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The T-junction power divider

• The T-junction power divider can be implemented
  in any type of transmission line medium.

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Lossless divider (1)

• A lumped susceptance, B, accounts for the stored
  energy resulted from fringing fields and higher
  order modes associated with the discontinuity at
  the junction.
• In order for the divider to be matched to the
  input line impedance Z0, and assume a TL to be
  lossless, we will have

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Lossless divider (2)

• The output line impedances Z1 and Z2 can then be
  selected to provide various power division
  ratios.
• In order for the divider to be matched to the
  input line impedance Z0, and assume a TL to be
  lossless, we will have

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Ex1 A lossless T-junction power divider has a
source impedance of 50 ?. Find the output
characteristic impedances so that the input power
is divided in a 31 ratio. Compute the reflection
coefficients seen looking into the output ports.
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Resistive divider

• A lossy three-port divider can be made to matched
  at all ports, although the two output ports may
  not be isolated.

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The Wilkinson power divider

• The lossless T-junction divider cannot be matched
  at all ports and does not have any isolation
  between output ports.
• The resistive divider can be matched at all ports
  but the isolation is still not achieved.
• The Wilkinson power divider can be matched at all
  ports and isolation can be achieved between the
  output ports.