5. Impedance Matching and Tuning

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5. Impedance Matching and Tuning

• Apply the theory and techniques of the previous
  chapters to practical problems in microwave
  engineering.
• Impedance matching is the 1st topic.

Figure 5.1 (p. 223)A lossless network matching
an arbitrary load impedance to a transmission
line.
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• Impedance matching or tuning is important since
• Maximum power is delivered when the load is
  matched to the line, and power loss in the feed
  line is minimized.
• Impedance matching sensitive receiver components
  improves the signal-to-noise ratio of the system.
• Impedance matching in a power distribution
  network will reduce the amplitude and phase
  errors.

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• Important factors in the selection of matching
  network.
• Complexity
• Bandwidth
• Implementation
• Ajdustability

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5.1 Matching with Lumped Elements

• L-section is the simplest type of matching
  network.
• 2 possible configurations

Figure 5.2 (p. 223)L-section matching networks.
(a) Network for zL inside the 1 jx circle.
(b) Network for zL outside the 1 jx circle.
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Analytic Solution

• For Fig. 5. 2a, let ZLRLjXL. For zL to be
  inside the 1jx circle, RLgtZ0. For a match,
• Removing X ?

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• For Fig.5.2b, RLltZ0.

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Smith Chart Solutions

• Ex 5.1

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Figure 5.3b (p. 227)(b) The two possible
L-section matching circuits. (c) Reflection
coefficient magnitudes versus frequency for the
matching circuits of (b).
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Figure on page 228.
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5.2 Single Stub Tuning
Figure 5.4 (p. 229)Single-stub tuning circuits.
(a) Shunt stub. (b) Series stub.
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• 2 adjustable parameters
• d from the load to the stub position.
• B or X provided by the shunt or series stub.
• For the shunt-stub case,
• Select d so that Y seen looking into the line at
  d from the load is Y0jB
• Then the stub susceptance is chosen as jB.
• For the series-stub case,
• Select d so that Z seen looking into the line at
  d from the load is Z0jX
• Then the stub reactance is chosen as jX.

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Shunt Stubs

• Ex 5.2 Single-Stub Shunt Tuning
• ZL60-j80

Figure 5.5a (p. 230)Solution to Example 5.2.
(a) Smith chart for the shunt-stub tuners.
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Figure 5.5b (p. 231)(b) The two shunt-stub
tuning solutions. (c) Reflection coefficient
magnitudes versus frequency for the tuning
circuits of (b).
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• To derive formulas for d and l, let ZL 1/YL RL
  jXL.
• Now d is chosen so that G Y01/Z0,

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• If RL Z0, then tanßd -XL/2Z0. 2 principal
  solutions are
• To find the required stub length, BS -B.
• for open stub
• for short stub

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Series Stubs

• Ex 5.3 Single Stub Series Tuning
• ZL 100j80

Figure 5.6a (p. 233)Solution to Example 5.3.
(a) Smith chart for the series-stub tuners.
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Figure 5.6b (p. 232)(b) The two series-stub
tuning solutions. (c) Reflection coefficient
magnitudes versus frequency for the tuning
circuits of (b).
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• To derive formulas for d and l, let YL 1/ZL GL
  jBL.
• Now d is chosen so that R Z01/Y0,

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• If GL Y0, then tanßd -BL/2Y0. 2 principal
  solutions are
• To find the required stub length, XS -X.
• for short stub
• for open stub

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5.3 Double-Stub Tuning

• If an adjustable tuner was desired, single-tuner
  would probably pose some difficulty.
• Smith Chart Solution
• yL ? add jb1 (on the rotated 1jb circle) ?
  rotate by d thru SWR circle (WTG) ? y1 ? add jb2
  ? Matched
• Avoid the forbidden region.

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Figure 5.7 (p. 236)Double-stub tuning. (a)
Original circuit with the load an arbitrary
distance from the first stub. (b)
Equivalent-circuit with load at the first stub.
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Figure 5.8 (p. 236)Smith chart diagram for the
operation of a double-stub tuner.
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Figure 5.9a (p. 238)Solution to Example 5.4.
(a) Smith chart for the double-stub tuners.
Ex. 5.4 ZL 60-j80 Open stubs, d ?/8
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Figure 5.9b (p. 239)(b) The two double-stub
tuning solutions. (c) Reflection coefficient
magnitudes versus frequency for the tuning
circuits of (b).
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Analytic Solution

• To the left of the first stub in Fig. 5.7b,
• Y1 GL j(BLB1) where YL GL jBL
• To the right of the 2nd stub,
• At this point, ReY2 Y0

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• Since GL is real,
• After d has been fixed, the 1st stub susceptance
  can be determined as
• The 2nd stub susceptance can be found from the
  negative of the imaginary part of (5.18)

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• B2
• The open-circuited stub length is
• The short-circuited stub length is

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5.4 The Quarter-Wave Transformer

• Single-section transformer for narrow band
  impedance match.
• Multisection quarter-wave transformer designs for
  a desired frequency band.
• One drawback is that this can only match a real
  load impedance.
• For single-section,

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Figure 5.10 (p. 241)A single-section
quarter-wave matching transformer.
at the design frequency f0.
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• The input impedance seen looking into the
  matching section is
• where t tanßl tan?, ? p/2 at f0.
• The reflection coefficient
• Since Z12 Z0ZL,

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• The reflection coefficient magnitude is

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• Now assume f f0, then l ?0/4 and ? p/2.
  Then sec2 ? gtgt 1. ?

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• We can define the bandwidth of the matching
  transformer as
• For TEM line,
• At ? ?m,

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• The fractional bandwidth is
• Ex. 5.5 Quarter-Wave Transformer Bandwidth
• ZL 10, Z0 50, f0 3 GHz, SWR 1.5

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Figure 5.12 (p. 243)Reflection coefficient
magnitude versus frequency for a single-section
quarter-wave matching transformer with various
load mismatches.
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5.5 The Theory of Small Reflection

• Single-Section Transformer

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Figure 5.13 (p. 244)Partial reflections and
transmissions on a single-section matching
transformer.
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Multisection Transformer

• Assume the transformer is symmetrical,

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• If N is odd, the last term is
• while N is even,

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5.6 Binomial Multisection Matching Transformer

• The response is as flat as possible near the
  design frequency. ? maximally flat
• This type of response is designed, for an
  N-section transformer, by setting the first N-1
  derivatives of G(?) to 0 at f0.
• Such a response can be obtained if we let

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• Note that G(?) 0 for ?p/2, (dn G(?)/d?n )
  0 at ?p/2 for n 1, 2, , N-1.
• By letting f ? 0,

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• Gn must be chosen as
• Since we assumed that Gn are small, ln x
  2(x-1)/(x1),
• Numerically solve for the characteristic
  impedance ? Table 5.1

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• The bandwidth of the binomial transformer
• Ex. 5.6 Binomial Transformer Design

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Figure 5.15 (p. 250)Reflection coefficient
magnitude versus frequency for multisection
binomial matching transformers of Example 5.6 ZL
50O and Z0 100O.
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5.7 Chebyshev Multisection Matching Transformer

• Chebyshev Polynomial
• The first 4 polynomials are
• Higher-order polynomials can be found using

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Figure 5.16 (p. 251)The first four Chebyshev
polynomials Tn(x).
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• Properties
• For -1x 1, Tn(x)1 ? Oscillate between 1 ?
  Equal ripple property.
• For x gt 1, Tn(x)gt1 ? Outside the passband
• For x gt 1, Tn(x) increases faster with x as n
  increases.
• Now let x cos? for x lt 1. The Chebyshev
  polynomials can be expressed as
• More generally,

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• We need to map ?m to x1 and p- ?m to x -1.
  For this,
• Therefore,

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Design of Chebyshev Transformers

• Using (5.46)
• Letting ? 0,

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• If the maximum allowable reflection coefficient
  magnitude in the passband is Gm,

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• Once ?m is known,
• Ex 5.7 Chebyshev Transformer Design
• Gm 0.05, Z0 50, ZL 100
• Use Table 5.2

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Figure 5.17 (p. 255)Reflection coefficient
magnitude versus frequency for the multisection
matching transformers of Example 5.7.
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Figure 5.18 (p. 256)A tapered transmission line
matching section and the model for an incremental
length of tapered line. (a) The tapered
transmission line matching section. (b) Model
for an incremental step change in impedance of
the tapered line.
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Figure 5.19 (p. 257)A matching section with an
exponential impedance taper. (a) Variation of
impedance. (b) Resulting reflection coefficient
magnitude response.
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Figure 5.20 (p. 258)A matching section with a
triangular taper for d(In Z/Z0/dz. (a) Variation
of impedance. (b) Resulting reflection
coefficient magnitude response.
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Figure 5.21 (p. 260)Solution to Example 5.8.
(a) Impedance variations for the triangular,
exponential, and Klopfenstein tapers. (b)
Resulting reflection coefficient magnitude versus
frequency for the tapers of (a).
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Figure 5.22 (p. 262)The Bode-Fano limits for RC
and RL loads matched with passive and lossless
networks (?0 is the center frequency of the
matching bandwidth). (a) Parallel RC. (b) Series
RC. (c) Parallel RL. (d) Series RL.
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Figure 5.23 (p. 263)Illustrating the Bode-Fano
criterion. (a) A possible reflection coefficient
response. (b) Nonrealizable and realizable
reflection coefficient responses.