Portland State University
Microwave Circuit Design – ECE 531
Chapter 5 – Sections 5.5 – 5.9
By David M.Pozar
H.Imesh Neeran Gunaratna
PSU ID: 901129894
Index:
• Introduction: The Quarter-Wave Transformer………………………… slide 3
• Section 5.5 – The Theory of Small Reflections…………………………. slide 7
• Section 5.6 – Binomial Multisection Matching Transformers…… slide 17
• Section 5.7 – Chebyshev Multisection Matching Transformers… slide 24
• Section 5.8 – Tapered Lines………………………………………………….. slide 28
• Section 5.9 – The Bode- Fano Criterion………………………………….. slide 31
Introduction: Quarter-Wave Transformer
• Definition: The quarter-wave transformer is a useful and
practical circuit for impedance matching and also provides a simple
transmission line circuit that further illustrates the properties of
standing waves on a mismatched line.
Let me explain with an example.
• Say the end of a transmission line with characteristic impedance Z0
is terminated with a resistive (i.e., real) load.
• We typically would like all power traveling down the line to be
absorbed by the load RL.
• But if RL ≠Z0 , the line is unmatched and some of the incident
power will be reflected.
Can all incident power be delivered to
a resistive load if RL ≠Z0 ??
• The answer to this question is “YES”.
• We can insert a matching network between the transmission line
and the load.
• A matching network is a lossless, 2-port device. Its job is to
transform the load RL ( or even ZL ) to a value Z0.
• In other words, we want the input impedance of the matching
network to be Zin =Z0, so that in Γin = 0 --no reflection!
• Since none of the incident power is reflected, and none is
absorbed by the lossless matching network, it all must be
absorbed by the load RL.
How can we build a matching network?
• The easiest is the quarter-wave transformer
• First, insert a transmission line with characteristic impedance
• Z1 and length l = λ/4 (i.e., a quarter-wave line) between the load
and the Z0 transmission line.
The λ 4
line is the matching network
•
The quarter wavelength case is one of the special cases that we studied. We know that the input
impedance of the quarter wavelength line is:
•
Thus, if we wish for Zin to be numerically equal to Z0, we find:
•
Solving for Z1, we find its required value to be:
• Thus, all power is delivered to load RL
• Important Note: We find that in Z0 =Zin only if the matching quarterwave transmission line is exactly one-quarter wavelength in length
l = λ/4.
• The problem with this, of course, is that a physical length l of
transmission line is exactly one-quarter wavelength at only one frequency f.
• Remember, wavelength is related to frequency as:
• One drawback of the quarter-wave transformer is that it can only match a real
load impedance. A complex load impedance can always be transformed to a real
impedance by using an appropriate length of transmission line between the
load and the transformer, or an appropriate series or shunt reactive stub.
• Another drawback of the quarter-wave transformer is that a quarter-wave
transformer provides a perfect match ( Γ = 0 ) at only one signal
frequency
Section 5.5-The Theory of Small Reflections
• The quarter-wave transformer provides simple means of matching
any real load impedance to any line impedance. For applications
requiring more bandwidth than a single quarter-wave section can
provide, multisection transformers can be used.
• When designing these transformers, total reflection coefficient
caused by the partial reflections from small discontinuities in the
transmission line have to be considered.
What are discontinuities?
By either necessity or design, microwave networks
often consist of transmission lines with various types of
transmission line discontinuities. Discontinuities may be
unavoidable result of mechanical or electrical transitions from one
medium to another. For instance like a junction between a two
waveguides.
Some Common Microstrip Discontinuities
Single Section Transformer
• Consider the single-section transformer shown below.
• We will derive an approximate expression for the overall reflection
coefficient Γ.
Γ1
-T12T21 Γ3
T12T21 Γ3 Γ3 Γ2
This expression can be further simplified
using the geometric series
Multisection Transformer
• Consider a sequence of N transmission line sections; each section has
equal length “l”, but dissimilar characteristic impedances:
• The partial reflection coefficients can be defined at each junction, as
follows,
• Note that since RL is real, and since we assume lossless
transmission lines, all Γn will be real.
• If the load resistance RL is less than Z0 , then we should
design the transformer such that:
Z 0>Z1 >Z2 >Z3……> ZN >RL
• Conversely, if RL is greater than Z0 , then we will design
the transformer such that:
Z 0 <Z1 <Z2 <Z3…..<ZN <RL
In other words, we gradually transition from Z0 to RL.
• Hence, this proves the fact that Zn increase or decrease
monotonically across the transformer, and that ZL is real.
• Here, we can apply the “theory of small reflections”
to analyze this multi-section transformer.
• The theory of small reflections allows us to approximate the input
reflection coefficient of the transformer as follows,
•
•
Signal Flow Diagram (read section 4.5)
What do we want from multisection?
• We want to make Γin (ω) = (0) (i.e., the reflection coefficient
is zero for all frequencies).
• But to achieve Γin (ω) = (0) would require an infinite
number of sections (i.e., N = ∞ )
• Instead, we seek to find an “optimal” function for Γin (ω ),
given a finite number of N elements.
• Therefore, assuming that we can make a symmetrical transformer.
We can write for “N” even as,
• We can write for “N” odd as,
• Hence, given an optimal and realizable function Γin(ω) , we can
determine the necessary number of sections N.
Section 5.6 – Binomial Multisection
Matching Transformers
• The pass band response of a binomial matching transformer is
optimum in the sense that for a given number of sections. It can be
known as maximally flat. This type of response is designed for an Nsection transformer, by setting the first N-1 derivatives of l Γ(θ) to
zero, at the center frequency f0.
• We need to define N independent design equations, which we
can then use to solve for the N values of characteristic
impedance Zn .
• First, we start with a single design frequency ω0 , where we
wish to achieve a perfect match:
Γin (ω = ω0 ) = 0
• That’s just one design equation: we need N-1 more.
• One such criterion is to make the function Γin(ω) maximally flat
at the point ω0 = ω .
• We found that this function has a matching network of Γ(θ ) = 0 at
θ = π/2--a perfect match
• Additionally, the function is maximally flat at θ = π/2, therefore
Γ(θ ) ≈ 0 over a wide range around θ = π/2--a wide bandwidth.
• But how does θ = π/2 relate to frequency ω????
•
θ = βl
• Where β = 2π/λ and λ = V/ω
• This (ω0 ) is our design frequency—the frequency where we
have a perfect match.
• Note that the length “l” has an interesting relationship with this
frequency:
•
When l = λ/4
• Binomial Multi-section matching network will have a
perfect match at the frequency where the section lengths “l”
are a quarter wavelength.
• First design rule: Set section lengths “l” so that they are a
quarter wave length ( λ0/4) at the design frequency ω0 .
•
•
•
•
What is the value of A ??
We can determine this value by evaluating a boundary condition.
We can easily determine the value of Γ(ω ) at ω = 0.
Note as ω approaches zero, the electrical length βl of each
section will likewise approach zero. Thus, the input
impedance Zin will simply be equal to RL as ω → 0.
• As a result, the input reflection coefficient Γ(ω = 0) must be:
• And we found out that,
• By equating the two expressions we get,
•
Hence
• Now have a form for the partial reflection coefficients Γn :
• But we also know that the partial reflection coefficients Γn is also,
• We know that the values of n 1 Z + and n Z are typically very close,
such that Zn+1 - Zn is small.
• Note that as we increase the number of sections, the matching
bandwidth increases.
• As we move from the design (perfect match) frequency f0 the value Γ(f ) will
increase. At some frequency (fm, say) the magnitude of the reflection
coefficient will increase to some unacceptably high value ( Γm , say).
At that point, we no longer consider the device to be matched.
• Note there are two values of frequency fm —one value less than
design frequency f0, and one value greater than design frequency f0.
These two values define the bandwidth of the matching network.
let Γm be the maximum value
of reflection coefficient that can
be tolerated over the passband
Fractional bandwidth
Section 5.7 – Chebyshev Multisection
Matching Transformers
• We can also build a multisection matching network such that the
function Γ(f ) is a Chebyshev function.
• Chebyshev functions maximize bandwidth, although at the
cost of pass-band ripple.
• Chebyshev solutions can provide functions Γ(ω ) with wider
• bandwidth than the Binomial case—although at the
“expense” of passband ripple.
• Chebyshev transformers are symmetric.
• The reflection coefficient of a Chebyshev matching network has
the form:
where θm =ωmT
• The function TN(cos θ sec θ) is a Chebyshev polynomial of order N.
• The first four Chebyshev polynomials are,
• Inserting the substitution: x = cosθsecθm into the Chebyshev
polynomials above
• We can now synthesize a Chebyshev equal-ripple passband by
making Γ(θ) proportional to TN(secθmcosθ) where N is the number
of sections in the transformer.
• As in the binomial transformer case we can find the constant “A” by
letting θ = 0 at zero frequency.
• The maximum allowable reflection coefficient magnitude in the
passband is Γm.
• But Γm = lAl
• The maximum value of TN(secθmcosθ) is in the passband is unity.
• We can find θm by using,
• The fractional bandwidth can be calculated using the following
equation once θm is known.
• Summarizing the he Chebyshev matching network
design procedure
• 1. Determine the value N required to meet the bandwidth and
ripple m Γ requirements.
• 2. Determine the Chebychev function.
• 3. Determine all Γn by equating terms with the symmetric
multisection transformer expression:
• 4. Calculate all Zn using the approximation:
• 5. Determine section length l = λ0/4.
Section 5.8 – Tapered Lines
• What is taper? It’s the gradual diminution of thickness, diameter, or
width in an elongated object.
• In this section the matching is done by discrete sections, while line
is continuously tapered.
The total reflection coefficient at z = 0
can be found by summing all the
partial reflections with their
appropriate phase shifts.
Exponential, Triangular &Klopfenstein Taper
Consider first an exponential taper, where
Z(z) = Z0eqz
for 0 < z < L ,
At z = 0, Z (0) = Z0
At z = L, we want to have Z (L) = ZL = Z0eaL
Constant “a” is derived as,
•We can find Γ(θ) by using the previous equation.
•The magnitude of the reflection coefficient
decreases as length increases as shown above.
Reflection coefficient magnitude vs frequency for tapers
Section 5.9 – The Bode-Fano Criterion
• When a lossless network is used to match an arbitrary complex load,
generally over a nonzero bandwidth. Some questions may arise.
• Can we achieve a perfect match (zero reflection) over a specified
bandwidth?
• If not, how well can we do? What is the trade-off between Γm, the
maximum allowable reflection in the passband, and the bandwidth?
• How complex must the matching network be for a given specification?
• These questions can be answered by the Bode-Fano criterion.
• Bode-Fano Criterion gives certain canonical types of load impedances, a
theoretical limit on the minimum reflection coefficient magnitude that
can be obtained with an arbitrary matching network. Thus represents
the optimum result that can be ideally achieved, even though such a
result may only be approximated in practice.
Bode-Fano Limits for RC and RL loads
matched with networks