CN101539961A - Design method of two-band Wilkinson power divider based on genetic algorithm - Google Patents

Abstract

The invention discloses an optimized design method of a two-band Wilkinson power divider based on a genetic algorithm, and relates to the design of special devices used for mobile communication. For a two-band equant Wilkinson power divider working at any two frequencies, in the traditional numerical computation method, when a transmission-line theory is adopted for calculating structural parameters, a transcendental equation needs to be solved, and a plurality of structural parameters of the power divider can not be guaranteed to reach an optimal value simultaneously. The genetic algorithm is adopted in the invention, optimum retaining operation, self-adaptive probability crossing and mutation operation and the like are added, whether the fitness value F is converged is judged according to the fitness function F corresponding to each chromosome, and the optimal solution is output, therefore, the optimal electronic parameter is determined. The method only needs the objective function which affects the search direction and corresponding fitness, greatly reduces the calculation quantity, and can easily obtain the optimal physical parameter of the power divider.

Description

Design method of two-band Wilkinson power divider based on genetic algorithm

Technical field

The present invention relates to the mobile communication technology field, relate in particular to a kind of power divider.

Background technology

Along with the widespread use of numerous communication standards such as WCDMA, GPS, WLAN, the required frequency band that takies is more and more, therefore need use the communication facilities that works in two frequencies at the receiving and transmitting front end of these systems.In recent years, more and more come into one's own for the research of double frequency communication facilities, document [1] has proposed a kind of double frequency-band power divider based on two joint impedance transformers, and it operates mainly in fundamental frequency f ₁With frequency multiplication 2f ₁Two frequencies.On this basis, people such as Wu ^[2]Further proposed to work in the power splitter of two optional frequencies.Wherein, document (1, S.Srisathit, M.Chongcheawchamnan and A.Worapishet, Designand realization of a dual-band 3-dB power divider based on a two-sectiontransmission-line topology[J] .Electronics Letters, 2003,39 (9): 723-724; 2, Lei Wu, Zengguang Sun, Hayattin Yilmaz, ADual-Frequncy Wilkinson Power Divider[J] .IEEE Trans.Microw.TheoryTech., 2006,54 (1): 278-284) disclose a kind of method for designing of power divider, this method mainly is that power splitter is structurally improved, when using transmission line theory to carry out structural parameters calculating again, find the solution transcendental equation, obtain a design equation formula and obtain structural parameters, but this method there is strict constraint to the characteristic impedance of every section transmission line and electrical length, thereby can not guarantee that a plurality of structural parameters of power splitter reach optimum value simultaneously.Adopt the interport isolation of the designed power splitter of this method not high enough, ideal situation can only reach 18dB.In order to make power divider work in two frequencies arbitrarily, and improve its interport isolation, by device architecture being improved and adopting Optimization Design then can avoid above problem.Traditional double frequency power divider method for designing mainly is that power splitter is structurally improved, when using transmission line theory to carry out structural parameters calculating again, need find the solution transcendental equation, characteristic impedance and electrical length to every section transmission line have strict constraint, thereby can not guarantee that a plurality of structural parameters of power splitter reach optimum value simultaneously.

In numerous optimization methods, genetic algorithm (Genetic Algorithm, abbreviation GA) the random search algorithm of natural selection of this reference organic sphere and natural genetic mechanism, owing to have the optimisation technique of global search performance preferably and strong robustness, can good treatment multiparameter, non-linear, non-differentiability continuous problem not even, and genetic algorithm does not require analytical expression, do not need supplementary or supplementary knowledge, it only needs to influence the objective function and the corresponding fitness of the direction of search, but the precocity and the convergence problem that exist.

Summary of the invention

The present invention is directed to the branch Wilkinson power dividers such as double frequency that work in two optional frequencies, when the tradition numerical computation method carries out structural parameters calculating in the utilization transmission line theory, need find the solution transcendental equation, characteristic impedance and electrical length to every section transmission line have strict constraint, thereby can not guarantee that a plurality of structural parameters of power splitter reach optimum value simultaneously.To working in the branch Wilkinson Wilkinson power dividers such as double frequency of two optional frequencies, the Optimization Design based on genetic algorithm has been proposed.In order to solve precocity and the convergence problem that exists in the genetic algorithm, the intersection of optimum maintenance operation, adaptive probability and mutation operation etc. in standard genetic algorithm, have been added.Utilize the ability of searching optimum of genetic algorithm to obtain optimal physical.

The present invention uses the concrete steps of Genetic Algorithm optimized design two-band Wilkinson power divider to comprise:

Step 1:, produce and contain characteristic impedance Z according to its chromosome of structure construction of Wilkinson power divider ₁, Z ₂, the electrical length θ of two sections transmission lines ₁, θ ₂, isolation resistance R ₁And R ₂Initial population;

Step 2: power divider circuits is turned to strange moding circuit to employing even-odd mould analytical technology and even moding circuit is analyzed, and calculates its strange mould reflection coefficient Γ _OddWith even mould reflection coefficient Γ _Even

Step 3: calculate the pairing fitness function F of each chromosome, the reflection coefficient average absolute of choosing required Frequency point is fitness function F: $F=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\frac{\left|{\mathrm{\Γ}}_{\mathrm{even}}\left(i\right)\right|+\left|{\mathrm{\Γ}}_{\mathrm{odd}}\left(i\right)\right|}{n}.$ According to initial population, for the design of dual-frequency power divider, can choose two required Frequency points, n=2 is 2 Frequency points of required coupling;

Step 4: adopt the optimization conversation strategy, the highest individuality of fitness in the population is not matched intersect and directly copy among the next generation.

Step 5: the intersection of adaptive probability and mutation operation, call formula and calculate crossover probability P _cWith the variation probability P _m,

$P_c=\left\{\begin{array}{cc}{k}_1(f_{\max }-f^{\&prime;})/(f_{\max }-f_{\mathrm{avg}})& f\&GreaterEqual;f_{\mathrm{avg}}\\ k_2& f<f_{\mathrm{avg}}\end{array}\right.$

$P_m=\left\{\begin{array}{cc}{k}_3(f_{\max }-f^{\&prime;})/(f_{\max }-f_{\mathrm{avg}})& f\&GreaterEqual;f_{\mathrm{avg}}\\ k_4& f<f_{\mathrm{avg}}\end{array}\right.$

Wherein, f _MaxBe the maximum adaptation degree value in the per generation population, f _AvgBe the average fitness value in the per generation population, f ' is bigger fitness value in two individualities that will intersect, and f is the individual fitness value of variation.k _iValue between ∈ [0,1].Along with the evolution of individuality, too much intersection, variation are easy to generate infeasible individuality, and crossover probability and variation probability should descend along with the carrying out of evolving;

Step 6: judge whether to satisfy the condition of convergence according to fitness value, judge promptly whether fitness value F restrains, satisfied then export optimum solution; Otherwise forward step 3 to, the optimum individual target function value reaches optimal value or current algebraically reaches maximum genetic algebra up to obtaining, the optimum electronic parameter of output power splitter.

The present invention is applied to genetic algorithm in the optimal design of two-band Wilkinson power divider, compare with traditional method for designing, be not subjected to the restriction of initial value, do not need to understand the inwardness of power divider, only need the impedance of each section of power divider, electrical length etc. are encoded, and set fitness and constraint condition is controlled search at index request, and can handle the objective function and the constraint condition of arbitrary form, no matter be linear or nonlinear, discrete is still continuous, even the search volume of mixing, and can both carry out global search effectively, obtain ideal solution, thereby improved design efficiency.The power divider that adopts this method to design can not only be in frequency f ₁And 2f ₁The place reaches good effect, and in any two interested frequency f ₁And m*f ₁All be suitable for, wherein m＞1 no longer is confined to simple integer, by the genetic Algorithm Design power splitter, avoided complicated derivation of equation process, and the strictness of parametric relationship has been limited, can be optimized relevant parameter neatly, thereby make power divider reach good performance index.

Description of drawings

Fig. 1 the present invention adopts the genetic algorithm operational flowchart

Fig. 2 two-band Wilkinson power divider structural representation

Strange mould of Fig. 3 and even mould schematic equivalent circuit

(1) the strange mould equivalent electrical circuit of even mould equivalent electrical circuit (2)

Embodiment

Be illustrated in figure 1 as the present invention and adopt genetic Algorithm Design power splitter schematic flow sheet, comprise,, produce and contain characteristic impedance Z according to its chromosome of Wilkinson power divider circuits structure construction and coding ₁, Z ₂, the electrical length θ of two sections transmission lines ₁, θ ₂, isolation resistance R ₁And R ₂Initial population; Power divider circuits is turned to strange moding circuit and even moding circuit is analyzed, calculate its strange mould reflection coefficient Γ _OddWith even mould reflection coefficient Γ _EvenAccording to strange mould reflection coefficient Γ _OddWith even mould reflection coefficient Γ _EvenCalculate the pairing fitness function F of each chromosome; Select the highest individuality of fitness in the population do not matched and intersect and directly copy among the next generation, according to the average fitness value f in the fitness value f of variation individuality and the per generation population _AvgSize, determine fitness function F crossover probability and the variation probability; Judge whether fitness value F restrains, as satisfying convergence, the output optimum solution is as the optimum electronic parameter of power splitter, otherwise fitness function is calculated in continuation.

Be illustrated in figure 2 as the two-band Wilkinson power divider structural representation, power splitter is made up of two joint impedance transformers, and its characteristic impedance is Z ₁And Z ₂, isolation resistance is R ₁And R ₂, be used for isolating two output ports.According to this power divider structure construction chromosome, with characteristic impedance Z ₁And Z ₂, transmission line electrical length θ ₁And θ ₂, isolation resistance R ₁And R ₂Carry out binary coding, be about to 6 Parameters Transformation and become 6 binary-coded symbol strings, then 6 string structures are constituted an initial population.In genetic algorithm, electrical length θ scope is 0 °～90 °.According to test, for satisfying the practical application needs, population scale is 100 in the definition basic parameter, and maximum genetic algebra is 200, and the number of bits of variable is 20, is the precision of decision variable.

Because the Wilkinson power splitter is a symmetrical structure, power divider circuits is turned to strange moding circuit to employing even-odd mould analytical technology and even moding circuit is analyzed, and is illustrated in figure 3 as strange moding circuit and even mould schematic equivalent circuit:

1) according to even-odd mould analytical technology, for even mould excitation, at two ports 2., 3. add the voltage of constant amplitude, homophase on, this moment, isolation resistance two terminal potentials were equal as can be known by symmetry, therefore do not have electric current to flow through on the isolation resistance, be equivalent to short circuit between two transmission lines of input port, can not consider the effect of isolation resistance.So just obtained by characteristic impedance Z ₁And Z ₂, transmission line electrical length θ ₁And θ ₂, source and loaded impedance Z _SAnd Z _L, isolation resistance R ₁And R ₂Through the even mould equivalent electrical circuit of connection in series-parallel formation shown in Fig. 3 (1), thus, obtain the output impedance Z at this place _aAnd Z _Even, and should the place even mould reflection coefficient Γ _Even

2) for the excitation of strange mould, two constant amplitudes, anti-phase voltage from two-port 2., 3. input, at this moment, isolation resistance two terminal potentials do not wait, and therefore have electric current to flow through, so can obtain, by characteristic impedance Z ₁And Z ₂, transmission line electrical length θ ₁And θ ₂, loaded impedance Z _L, isolation resistance R ₁And R ₂Connection in series-parallel constitutes the strange mould equivalent electrical circuit shown in Fig. 3 (2).Thus, obtain the output impedance Z at this place _A1, Z _B1, Z _C1And Z _Odd, and the strange mould reflection coefficient Γ that should locate _Odd

The following specifically describes the mould reflection coefficient Γ that seeks a spouse _EvenMethod:

Utilize two port theories of transmission line, according to transmission line in frequency f ₁The time electrical length θ ₁, θ ₂, characteristic impedance Z ₁And Z ₂, source and loaded impedance Z _SAnd Z _L, and arbitrfary point frequency f, two-port 2., the output impedance Z that 3. locates _aAnd Z _EvenDetermine by following formula,

$Z_a={\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_1\frac{2Z_S+\mathrm{<figure-callout\; id="1"\; label="j\; Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">j</figure-callout>}{Z}_{}{}_1\tan ({\mathrm{\&theta;}}_{1}f/f_1)}{{\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_1+2j{Z}_S\tan ({\mathrm{\&theta;}}_{1}f/f_1)}$

$Z_{\mathrm{even}}=Z_2\frac{Z_a+\mathrm{<figure-callout\; id="2"\; label="j\; Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">j</figure-callout>}{Z}_{}{}_2\tan ({\mathrm{\&theta;}}_{2}f/f_1)}{{\mathrm{<figure-callout\; id="2"\; label="Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_2+j{Z}_a\tan ({\mathrm{\&theta;}}_{2}f/f_1)}$

Idol mould reflection coefficient is: ${\mathrm{\&Gamma;}}_{\mathrm{even}}=\frac{Z_{\mathrm{even}}-Z_L}{Z_{\mathrm{even}}+Z_L}$

Ask strange mould reflection coefficient Γ _OddMethod be:

Two port theories according to transmission line can obtain from figure:

According to transmission line in frequency f ₁The time electrical length θ ₁, θ ₂, characteristic impedance Z ₁And Z ₂,, isolation resistance R ₁And R ₂, and arbitrfary point frequency f, two-port 2., the output impedance of 3. locating is:

$Z_{a1}={\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_1\frac{0+\mathrm{<figure-callout\; id="1"\; label="j\; Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">j</figure-callout>}{Z}_{}{}_1\tan ({\mathrm{\&theta;}}_{1}f/f_1)}{{\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_1+0\&times;j\tan ({\mathrm{\&theta;}}_{1}f/f_1)}=\mathrm{<figure-callout\; id="1"\; label="j\; Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">j</figure-callout>}{Z}_{}{}_1\tan ({\mathrm{\&theta;}}_{1}f/f_1)$

Because Z _A1With R ₁/ 2 parallel connections, so have:

Z _b1＝Z _a1×R ₁/(R ₁+2Z _a1)

${\mathrm{<figure-callout\; id="1"\; label="Z\; c"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_c=Z_2\frac{{\mathrm{<figure-callout\; id="1"\; label="Z\; b"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_b+\mathrm{<figure-callout\; id="2"\; label="j\; Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">j</figure-callout>}{Z}_{}{}_2\tan ({\mathrm{\&theta;}}_{2}f/f_1)}{{\mathrm{<figure-callout\; id="2"\; label="Z"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">Z</figure-callout>}}_2+\mathrm{<figure-callout\; id="1"\; label="j\; Z\; b"\; filenames="A20091010370500122.png,A20091010370500131.png"\; state="\{\{state\}\}">j</figure-callout>}{Z}_b{1}_{}\tan ({\mathrm{\&theta;}}_{2}f/f_1)}$

Thereby strange mould output impedance is: Z _Odd=Z _C1* R ₂/ (R ₂+ 2Z _C1)

So strange mould reflection coefficient is: ${\mathrm{\&Gamma;}}_{\mathrm{odd}}=\frac{Z_{\mathrm{odd}}-Z_L}{Z_{\mathrm{odd}}+Z_L}$

Even mould reflection coefficient and the Qi Mo reflection coefficient average absolute of choosing required two Frequency points are fitness function F:

$F=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{\left|{\mathrm{\&Gamma;}}_{\mathrm{even}}\left(i\right)\right|+\left|{\mathrm{\&Gamma;}}_{\mathrm{odd}}\left(i\right)\right|}{n}$

For the three-port network of symmetry, call following formula and can obtain following S parameter: input port return loss S ₁₁, output port return loss S ₂₂And S ₃₃, output port isolation S ₂₃

|S ₁₁|＝|Γ _even|； $S_{22}=S_{33}=\frac{1}{2}({\mathrm{\&Gamma;}}_{\mathrm{even}}+{\mathrm{\&Gamma;}}_{\mathrm{odd}});$ $S_{23}=\frac{1}{2}({\mathrm{\&Gamma;}}_{\mathrm{even}}-{\mathrm{\&Gamma;}}_{\mathrm{odd}}).$ If the Γ in the frequency band _EvenAnd Γ _OddObtain, then can obtain each parameter S, and from the Γ as can be seen of the relation between them _EvenAnd Γ _OddThe smaller the better, promptly work as Γ _EvenAnd Γ _OddS during optimal response ₁₁, S ₂₂And S ₂₃Also approaching optimal response, is fitness function F so can obtain choosing the reflection coefficient average absolute of required two Frequency points:

$F=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{\left|{\mathrm{\&Gamma;}}_{\mathrm{even}}\left(i\right)\right|+\left|{\mathrm{\&Gamma;}}_{\mathrm{odd}}\left(i\right)\right|}{n},$ (wherein n is required Frequency point, as n=2, and 2 Frequency points of required coupling)

According to initial population, calculate the pairing fitness value F of each chromosome that contains 6 variablees.

According to the genetic algorithm committed step in algorithm flow chart shown in Figure 1 and the fitness function F design dual-frequency power divider be:

1) selection operation method: for optimizing designed dual-frequency power divider, can adopt the optimization conversation strategy, the individuality that fitness F is the highest in the colony not carried out interlace operation and mutation operation, and directly copy among the next generation.Adopt the advantage of this system of selection to be, the optimum solution of certain generation can not destroyed by intersection and mutation operation in the evolutionary process.

2) intersection of adaptive probability and mutation operation method: the interlace operation method is right in this generation colony the fitness value F of two individualities being mixed at random, with certain or some position of crossover probability exchange between them, thereby produces two new individualities.Then the colony of new generation that produces is carried out mutation operation, the method of mutation operation for selecting the fitness value F of body one by one in generation at random in this colony, and change in string structure represent fitness value F certain position or some changes, thereby produce new individuality with the variation probability.

Determine crossover probability and the variation probability of fitness F.Average fitness value f in fitness value f that variation is individual and the per generation population _AvgCompare, when f more than or equal to f _AvgThe time, according to the maximum adaptation degree value f in the per generation population _Max, bigger fitness value f ' and average fitness value f in two individualities that will intersect _AvgDetermine crossover probability P _cWith the variation probability P _m, when f less than f _AvgThe time, crossover probability P _cWith the variation probability P _mIt is a constant constant.

Calculate crossover probability P _cWith the variation probability P _m, formula as follows:

$P_c=\left\{\begin{array}{cc}{k}_1(f_{\max }-f^{\&prime;})/(f_{\max }-f_{\mathrm{avg}})& f\&GreaterEqual;f_{\mathrm{avg}}\\ k_2& f<f_{\mathrm{avg}}\end{array}\right.$

$P_m=\left\{\begin{array}{cc}{k}_3(f_{\max }-f^{\&prime;})/(f_{\max }-f_{\mathrm{avg}})& f\&GreaterEqual;f_{\mathrm{avg}}\\ k_4& f<f_{\mathrm{avg}}\end{array}\right.$ Wherein, k _iValue between ∈ [0,1].

3) judge whether to satisfy the condition of convergence, promptly calculate fitness value F and whether restrain and reach minimal value, satisfied then export optimum solution, i.e. the structural parameters of power splitter (characteristic impedance Z ₁And Z ₂, transmission line electrical length θ ₁And θ ₂, isolation resistance R ₁And R ₂).Otherwise continue to calculate the pairing fitness function F of each chromosome, repeat said process.The optimum individual target function value reaches optimal value or current algebraically reaches maximum genetic algebra up to obtaining, the optimum electronic parameter Z of output power splitter ₁, Z ₂, θ ₁, θ ₂, R ₁And R ₂

The present invention adopts genetic algorithm, the intersection of optimum maintenance operation, adaptive probability and mutation operation etc. have been added, according to the pairing fitness function F of each chromosome, judge whether fitness value F restrains, F reaches optimal value or reaches maximum genetic algebra up to optimum individual fitness function value, output optimum solution, the optimum electronic parameter of definite power splitter thus.

The present invention can not only be in frequency f ₁And 2f ₁The place reaches good effect, and in any two interested frequency f ₁And m*f ₁All be suitable for, m＞1 no longer is confined to simple integer herein, as 2 or 3 etc.This method only need influence the objective function and the corresponding fitness of the direction of search, has simplified operand greatly, can obtain optimal physical easily, can be widely used in mobile communication dedicated devices design aspect.

Claims (5)

1, based on the two-band Wilkinson power divider design optimization method of genetic algorithm, it is characterized in that, comprise,

Step 1:, produce and contain characteristic impedance Z according to its chromosome of Wilkinson power divider circuits structure construction ₁, Z ₂, the electrical length θ of two sections transmission lines ₁, θ ₂, isolation resistance R ₁, R ₂Initial population;

Step 2: power divider circuits is turned to strange moding circuit and even moding circuit, calculate its even mould reflection coefficient Γ _EvenWith strange mould reflection coefficient Γ _Odd

Step 3: the reflection coefficient average absolute of choosing required Frequency point is the pairing fitness function F of each chromosome, that is: $F=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{\left|{\mathrm{\&Gamma;}}_{\mathrm{even}}\left(i\right)\right|+\left|{\mathrm{\&Gamma;}}_{\mathrm{odd}}\left(i\right)\right|}{n},$ Wherein, n is the Frequency point number of required coupling;

Step 4: the average fitness value f in individual fitness value f of variation and the per generation population relatively _AvgSize, determine fitness function F crossover probability and the variation probability;

Step 5: judge whether fitness value F restrains,, then export the optimum electronic parameter of optimum solution, otherwise forward step 3 to as power splitter as satisfying convergence.

2, power divider design optimization method according to claim 1 is characterized in that, adopts the optimization conversation strategy further to optimize dual-frequency power divider, and the highest individuality of fitness in the per generation population is not matched intersection and directly copying among the next generation.

3, power divider design optimization method according to claim 1 is characterized in that, for dual-frequency power divider, chooses two required Frequency points, i.e. n=2.

4, power divider design optimization method according to claim 1 is characterized in that, step 4 specifically comprises, when f more than or equal to f _AvgThe time, according to the maximum adaptation degree value f in the per generation population _Max, bigger fitness value f ' and average fitness value f in two individualities that will intersect _AvgDetermine crossover probability P _cWith the variation probability P _m, when f less than f _AvgThe time, crossover probability P _cWith the variation probability P _mBe a constant constant, that is,

$P_c=\left\{\begin{array}{cc}{k}_1(f_{\max }-f^{\&prime;})/(f_{\max }-f_{\mathrm{avg}})& f\&GreaterEqual;f_{\mathrm{avg}}\\ k_2& f<f_{\mathrm{avg}}\end{array}\right.$

$P_m=\left\{\begin{array}{cc}{k}_3(f_{\max }-f^{\&prime;})/(f_{\max }-f_{\mathrm{avg}})& f\&GreaterEqual;f_{\mathrm{avg}}\\ k_4& f<f_{\mathrm{avg}}\end{array}\right..$

5, power divider design optimization method according to claim 1 is characterized in that, step 2 specifically comprises, calls formula ${\mathrm{\&Gamma;}}_{\mathrm{even}}=\frac{Z_{\mathrm{even}}-Z_L}{Z_{\mathrm{even}}+Z_L}$ With ${\mathrm{\&Gamma;}}_{\mathrm{odd}}=\frac{Z_{\mathrm{odd}}-Z_L}{Z_{\mathrm{odd}}+Z_L}$ Mould reflection coefficient Γ seeks a spouse _EvenWith strange mould reflection coefficient Γ _Odd, Z wherein _EvenBe even mould output impedance, Z _OddBe strange mould output impedance, Z _LBe loaded impedance.

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