Filters to microwaves

Filters not commensurati

1) progettuali Options for filters pass to distributed constants:

to)       filters not commensurati which use line features that they do not have all the same length

b)       filters commensurati in which have be a matter of line having all the same length which frequency transformation is reached by means of one

c)       I use of the parameters image in order to characterize a sure cell base, this approach has fallen in disuse

 

2) PLR:

The Power Lost Ratio is the relationship between the P power_in in income to the net and the power P_T transmitted to the cargo, in particular has , for P _LR low-pass filter is worth 1 until to the frequency of cut in order then stretching to ¥ in more or less steep way to second that the filter is ideal or real.

 

3) Typology of approximation:

P_LR in the case of approximation of Butterworth is worth and has therefore the shape of one parabola centered in w = 0. In the case of approximation of Chebyshev instead is had where is the polinomi of Chebyshev.

 

4) Realization of filters lowpass :

The typology of filters lowpass that they realize the approximations of Butterworth and Chebyshev are different to second that it is equal N or N uneven but after all draft of scale nets constituted from ability parallel and inductances series always fed from an endowed generator of tension of inner stiffness and sluices on one unitary resistance. We must change this net to parameters concentrates valid to vlf to you in a net to distributed parameters valid to frequencies at least some GHz, in order to obtain that are two approaches:

to)       the inductances they come replaced with a short feature of line and to high Z_C (…therefore one line much grip) while the abilities parallel whose matrix of transmission is come replaced with a feature of having line q small and Z_C small, in such case is had in fact from which equaling the two matrices I obtain therefore has where n it is the propagation speed on the line. The true incognito is but the length l of the line, it is worth .

b)       remembering that the Z matrix of the line log is and being notes the relations between the elements of a net to T and the coefficients of its Z matrix it is obtained for the longitudinal stiffnesses of the same while for the cross-sectional stiffness it is had simply . Assuming q small and thinking to realize Z_to and Z_b with an inductance, I obtain for it the expression analogous imagining to realize Z_c with one ability I obtain for it . This model is led back to that one of the simple ability in the case makes the hypothesis of Z_c small. In analogous way it can be thought to a net to p that inductance is behaved like one

The turning out filter is given from the series of tight features of line that simulate the inductance and wide features of line that simulate the abilities, in truth then would be also to consider adorned you due to the abrupt transitions.

 

5) Realization of filters bandpass :

In order to pass from the lowpass of Chebyshev or Butterworth to the correspondents bandpass the transformation is used dove w' is the frequency of the lowpass and w it is the frequency of the bandpass moreover. The circuita impact them of this transformation pu² to obtain itself that is uguagliando the stiffness of every presentL inductance _k in lowpass with the stiffness of the series of a inductance and a ability , obtains and . In way reasoning on the admittances one obtains that the abilities to the lowpass transform in the parallel of one ability and one inductance.

 

6) Invester of stiffness and invester of admittance:

Draft of nets that concur us to realize not commensurate nets bandpass with distributed parameters in particular the stiffness invester is a net that sluice on a Z cargo_b introduces in income a stiffness while the admittance invester is a net that sluice on a Y cargo_b introduces in income one admittance . The elements of the Z matrix of the stiffness invester gain themselves from the imposing in fact that he is equal to then obtains that must beZ ₁₁ =₂₂ Z = 0 and , remembering the relation between the elements of the Z matrix and those of the transmission matrix it is had that for it To = D = 0.

The elements of the Y matrix of the admittance invester obtain themselves from the equality, must themselves be had in fact Y₁₁ = Y₂₂ = 0 e also in this case therefore for the correspondent transmission matrix have themselves To = D = 0.

 

7) practical Realization of an admittance invester:

The admittance invester is used in the realization of bandpass not commensurati in mstrip, one possible realization is given from the cascade of a feature of having line stiffness ₀Z and length f/2, one admittance series of value and according to feature of equal line to the previous one. The matrix of turning out transmission is the product of the three matrices, and to we it interests to place A=0 that it is one of the conditions in order to realize an admittance invester, obtains while placing is obtained and therefore .

 

8) Realization of passes band with stiffness invester:

In parallel to the cargo R_L of the bandpass we have a resonant cell parallel constituted from and continuation from a resonant cell series constituted from and , the stiffness of income of this sottorete is: , uguagliamo it to the stiffness of income of a net constituted from the R cargo_0L , an invester of stiffness K₁₀ , a resonant cell series L₀₁ C₀₁ , an invester of stiffness K₂₁ and a second resonant cell series ₀₂L C₀₂ , obtain the values of the investers e that is e .

It is possible to go back simply towards the generator replacing the cargo resistance R_L with a stiffness in the previous schematizzazione without investers and replacing the R resistance_0L with a stiffness in the schematizzazione with investers, obtain therefore the values of the stiffness investers, in particular e . We pass hour to analyze the resonant cell and nearer the generator and its inner resistance R_g , in the circuit with investers we will have the resonant cell and and the inner resistance of the generator, we can therefore by means of a proportion gain the inner stiffness of the generator , if it is various from the wished value enough simply to insert an ulterior invester who goes inserted also in the case of uneven N for which the resonant cell more neighbor to the generator is of type parallel. They remain hour to realize the risonanti series with of the long lines of transmission l/2 obtains from which pu² to determine the characteristic stiffness of these lines.

 

9) Realization of passes band with stiffness invester:

In complementary way to how much fact with the stiffness investers, the net bandpass can be realized using of the having investers of admittance values e spaces out you from risonanti circuits parallel that can be realize to you by means of of long lines l/2, in particular uguagliando the derivatives respect to w of elements C of the transmission matrices is reached the relation that concurs to determine the characteristic stiffness of long line l/2.

10) Filters pass band gap capacitivo:

This realization uses investers of stiffness constituted from 2 inframmezzati features of linef /2 from an ability, the two investers are then separate to you from a risonatore parallel realized with a feature of line along l/2 and having characteristic stiffness Z_c that is placed equal to Z₀ , the values of the abilities can be gain to you with the formula can therefore determine the being the resonant described series from while the resonant parallel it is described from where g_{the K} is the coefficients of the lowpass of departure.

In the fattispecie for the first invester is had while for the next invester to the generator in the case of uneven N and in the case of equal N is had.

You notice the J is possible to determine the suscettanze of the investers by means of the relation and the lengths follow some that the suscettanze are separated from long features of line . The main problem of this filter is the excessive dimensions.

 

11) Filters pass band to coupled lines:

To tight band it is had that a brace of admittance investers separates to you from a feature of line along l/2 (…the matrix of transmission of the structure has e ) is equivalent to of the coupled lines sluices on an open to opposite sides (… characterized from e ).

Equaling between they the equations of To and the equations of B let alone of their derivatives it is reached the two equations of plan e .

 

12) practical Realization of microstrip filters bandpass :

Part from the two frequencies of cut to â?"3dB of the bandpass , f₁ and f₂ , gains the frequency centers them and through the transformation in frequency the w is determined of the lowpass , it with to the wished slope of the filter concurs to determine the N order and therefore to extract from the tables the values of the members it concentrates you of the filter of Butterworth. To this point we have N 1 investers separates to you from long lines l/2 and having stiffness Z₀ , the coefficient of the first invester finds with the formula while those of remaining with it we can implement the two following realizations:

to)       for the filter to gap it enables the suscettanze to you are immediately calculable and from they the abilities and the distance between two successive abilities.

b)       For the filter to lines coupled from the J_{K 1, K} gain Z _0eand Z immediately_0o that they determine the widths of the lines while the lengths are all pars to l/2.

Commensurati filters

13) Transformation of Richard:

It is a periodic transformation in how much uses the tangent, is had where w' it is the pulsation in low-pass filter while w it is the pulsation in passes band, of the rest in truth also in the case of commensurati filters were not obtained of the periodic answers even if we have analyzed to the single behavior pass-band.

The usefullness of this transformation is that it concurs to realize one ability C_K with one stub opened along l/4 and having characteristic stiffness C_K while an inductance of value L_K comes realized with one stub in short along l/4 and having characteristic stiffness L_K .

The problem that rises is that in the filters low-pass filter from which part in order to obtain itself passes to band the inductances is in series and therefore they cannot be replaced with stub in how much the stub is always in parallel, for the abilities instead not there are problems and to they it will be attempted of ricondursi by means of I use it of the unitary elements of Kuroda.

 

14) unitary Element of Kuroda:

It is a device two doors characterized from a characteristic stiffness Z₁ and from a length therefore to the job frequency it is substantially l/4, its matrix of transmission is gained to leave from the matrix of transmission of a feature of line of transmission along q that is , in it pu² to replace infatti follows some that therefore from obtains the value to us of cosq. Of the rest is had also therefore the transmission matrix becomes .

 

15) Equivalences that are involved the unitary elements of Kuroda:

Two following equivalences can be demonstrated:

to)       a unitary element with Z stiffness₁ having in income one ability parallel C is equivalent to one unitary element with stiffness Z₂ having in escape one inductance L series

The matrix of transmission of the structure with the ability is while that one of the structure with the inductance is the terms To is equal while from the B term we obtain the relation, from term C the relation and finally from the D term the relation , to this point is possible the two following ways:

·         We suppose you notice Z₁ and C and we place , replacing in the obtains while replacing in the it is obtained

·         We suppose you notice ₂Z and L and we place , replacing in the obtains while replacing in the it is obtained

b)       a unitary element with Z stiffness_{a 1} having in income inductance L series is equivalent to a unitary element having in escape one ability parallel C

The matrix of transmission of the structure with the inductance is while that one of the structure with the ability is the D terms is equal while from the term To we obtain the relation, from the B term the relation and finally from term C the relation , to this point is possible the two following ways:

·         We suppose you notice Z₁ and L and we place , replacing obtains e

·         We suppose you notice ₂Z and C and we place , replacing it is obtained and

 

16) Realization of filters passes band by means of the unitary elements of Kuroda:

We consider low-pass filter of 4° the order constituted from 2 inductances series and 2 abilities parallel, we insert to mount a unitary element of Kuroda with stiffness Z₀ , it not modification the amplitude of the escape, then transform the inductance series mail to its escape in an ability parallel mail to its income to mount of which both hour joins to an other U.E. has to their escape an ability in parallel and comes transforms in others 2 U.E. to you having in income an inductance series, to this point an other U.E. joins to mount and it uses for all and three the transformation that concurs to pass from a U.E. with one inductance series to its escape to a U.E. with one ability parallel to its income. The abilities to the obtained filter come realized with of the stub opened while the U.E. are come true with of the features of line all of length l/4 to the job frequency therefore the filter are of commensurato type.

 

16bis) I use of the lines coupled in the commensurati filters:

Leaving from it low-pass filter of Butterworth, reaches to a configuration only constituted from unitary elements of Kuroda having in escape an inductance series, such block can be replaced with of the coupled lines in particular marks them of income fuoriesce to the other head of the same line so as to to concur the passage of the continuous one while the coupled line is sluice in short on one side and on an open from the other side.

The condition so that the substitution are valid are that they have the same matrix of transmission, proceeding therefore in the customary way and are obtained from which with notes eguaglianze e moreover is had.

To this point equaling the B found for the lines coupled to the B previously found for U.E. more inductance obtains the equation while equaling terms C we obtain the equation therefore resolving the quadratic equation obtain e that concur the planning of the filter.