text: the single capacitively - compensated microstrip directional coupler 10 shown in fig1 includes four ports 1 , 2 , 3 , and 4 and two symmetrical inner conductors 12 and 14 separated by a gap 16 on a dielectric substrate with relative dielectric constant , ε r . at the far edge of the coupled section , between ports 3 and 4 , there is one lumped capacitor 18 , implemented on microstrip as shown in fig1 . the method of analysis makes use of the physical symmetry of this directional coupler . by applying symmetric ( even mode ) and antisymmetric ( odd mode ) excitation to two colinear ports of the directional coupler , the four - port problem is reduced to that of solving two two - port problems . for example , for the capacitively - compensated case in fig1 the pair of two - ports to be analyzed are shown schematically in fig2 and 3 . the even mode is characterized by a transmission line 20 of electrical length θ e and characteristic impedance z oe . note that the compensating capacitance does not affect the even mode representation . fig3 shows the coupled odd mode representation which is characterized by a transmission line 24 of electrical length θ o , with odd mode characteristic impedance , z oo . the overall characteristic impedance is z o and is the square root of z oo * z oe . capacitor 28 has capacitance 2c . the standard practice is to describe the two circuits represented in fig2 and 3 using the abcd matrix approach , which leads directly to the development of the overall scattering parameters of the directional coupler . scattering parameters s 11 , s 12 , s 13 . and s 14 ; parameters a , b , c , and d ; transmission coefficients t e , t o , and t for even mode , odd mode and overall transmission , respectively ; reflection coefficients γ e , γ o , and γ for even mode , odd mode and overall reflection , respectively ; characteristic impedance z o ; and , characteristic admittance y o are related as follows : ## equ1 ## directivity is defined as the difference between isolation and coupling expressed in decibels ( db ). both isolation i and coupling p are deduced from the scattering matrix of the directional coupler , i . e . : ## equ2 ## for matched directional couplers and maximum isolation or directivity , the following results are necessary : the abcd matrix for the even mode of the single capacitive compensation of fig2 is as follows : ## equ3 ## for the odd mode , as in fig3 : ## equ4 ## where ω is the frequency of the input signal and y oe and y ooa are the even mode and actual characteristic admittances , respectively . it is not possible to satisfy both of the latter two equations in the same circuit architecture , since scrutiny of equation 11 reveals : it is possible , however , to provide an ideal match or directivity by satisfying either : for an ideally - matched microstrip directional coupler , it will be necessary to deal with the reflection coefficients between the actual realization and the ideal odd - mode representation . fig4 illustrates the correspondence , in which z ooa and z ooi are the actual odd - mode and the ideal odd - mode characteristic impedances , respectively , and θ o and θ e are the actual odd - mode and the even - mode electrical lengths of the coupled sections 30 and 36 , respectively . capacitor 34 of capacitance 2c is connected as shown in fig4 in the odd mode equivalent circuit representation . the ideal odd - mode electrical length is made equal to the even - mode electrical length . furthermore , the actual characteristic impedance of the odd mode z ooa is different from the ideal z ooi . the abcd circuit representation is used to find the actual odd - mode reflection or transmission coefficients by equating them to the ideal condition . use of equation ( 11 ) in conjunction with equation ( 1 ) determines the odd - mode reflection coefficient for the actual representation : ## equ5 ## the matrix description for the ideal representation is given by : ## equ6 ## and the reflection coefficient is given by : ## equ7 ## and , recognizing that at the center frequency : ## equ8 ## and the ideal odd - mode reflection coefficient becomes : equations ( 20 ), ( 21 ), and ( 14 ), after equating , separating the result into real and imaginary components , and solving for the compensating capacitance and the new odd - mode characteristic impedance yield : ## equ9 ## equation ( 22 ) demands that : which can be achieved by making the inner conductor narrower and increasing the separation to keep the even - mode characteristic impedance constant . a single inductively - compensated microstrip directional coupler is shown in fig5 . the single inductively - compensated microstrip directional coupler 40 shown in fig5 includes four ports 1 , 2 , 3 , and 4 and two symmetrical inner conductors 42 and 44 separated by a gap 46 in a dielectric substrate with relative dielectric constant , ε r . at the far edge of the coupled section , between ports 3 and 4 , there is one lumped capacitor 48 , implemented on microstrip as shown . following the same method of analysis as described for the single capacitively - coupled case , the four - port configuration is reduced to a two - port configuration with odd mode representation as shown in fig6 . the coupled region is characterized by a transmission line 50 of electrical length θ o , with actual odd mode characteristic impedance z ooa . inductor 54 has inductance l / 2 and is positioned as indicated in the fig6 odd mode representation . the abcd matrix which corresponds to the circuit is given by : ## equ10 ## combining equations ( 25 ), ( 1 ), and ( 20 ), and separating real and imaginary parts yields : ## equ11 ## note that equation ( 26 ) demands that : the inner conductor can be made wider and the separation decreased to keep the even - mode characteristic impedance constant . for the case of ideal isolation or directivity of a microstrip directional coupler , the transmission coefficients are equated between the actual realization and the ideal odd - mode representations . the first structure to be considered is that of a single capacitive compensation between ports 3 and 4 of fig1 . use of equation 11 in conjunction with equation ( 2 ) determines the odd - mode transmission coefficient for the actual representation . the ideal odd - mode transmission coefficient at the center frequency of operation is given by : ## equ12 ## equating real and imaginary components , and solving for the compensating capacitance and a new odd mode characteristic impedance yields : ## equ13 ## the single central capacitively - compensated microstrip directional coupler 60 shown in fig7 consists of four ports 1 , 2 , 3 , and 4 and two symmetrical inner conductors 62 and 64 separated by a gap 66 in a dielectric substrate with relative dielectric constant , ε r . at the center of the coupled section there is one lumped capacitor 68 , implemented as shown . the odd - mode equivalent circuit for the directional coupler in fig7 can be represented as in fig8 . the equivalent circuit coupled region is characterized by two transmission lines 70 and 74 , each of electrical length θ o / 2 and characteristic impedance z ooa . capacitor 78 has capcitance 2c . the corresponding abcd matrix representation is : ## equ14 ## solving yields : ## equ15 ## at the center frequency , the matrix representation for the symmetrical single capacitive representation reduce to : ## equ16 ## this leads to an ideal directional coupler with the following s - parameters : ## equ17 ## the use of the developed formulas are demonstrated via design examples of an edge - coupled microstrip directional coupler . the preferred embodiments use a substrate of gallium arsenide ( gaas ) with a metal thickness of approximately three micrometers ( 3 μm ), height of approximately one hundred micrometers ( 100 μm ) and ε r of approximately 12 . 9 . table 1 shows the pertinent data regarding the microstrip directional coupler requirements and coupled line realization for both uncompensated , capacitively compensated and inductively compensated structures for the ideally matched case . table 2 provides similar information for the asymmetric ( ideal isolation ) and symmetric ( ideal coupler ) capacitive compensation . table 1______________________________________microstrip directional coupler ideallymatched case asymmetric asymmetricparameters uncompensated capacitor inductor______________________________________center frequency 35 ghz 35 ghz 35 ghzcoupling - 7 . 25 db - 7 . 0 db - 7 . 0 dbz . sub . o 50 ohms 50 ohms 50 ohmsz . sub . oe 80 . 85 ohms 80 . 85 ohms 80 . 85 ohmsz . sub . oo 30 . 92 ohms 31 . 59 ohms 29 . 09 ohmsε . sub . effe 8 . 83 8 . 83 8 . 83ε . sub . effo 6 . 14 6 . 17 6 . 04coupled line : width 45 . 81 μm 45 . 32 μm 47 . 26 μmseparation 18 . 70 μm 19 . 93 μm 15 . 28 μmparallel length 785 . 82 μm 721 μm 721 μmdirectivity 13 . 25 db finite finitematch finite 0 0capacitive 0 . 012 pfcompensationinductive 0 . 614 nhcompensation______________________________________ table 2______________________________________microstrip directional coupler idealsolution and ideal coupler capacitivecompensation cases asymmetric symmetricparameters uncompensated capacitor capacitor______________________________________center frequency 35 ghz 35 ghz 35 ghzcoupling - 7 . 25 db - 7 . 0 db - 7 . 0 dbz . sub . o 50 ohms 50 ohms 50 ohmsz . sub . oe 80 . 85 ohms 80 . 85 ohms 80 . 85 ohmsz . sub . oo 30 . 92 ohms 35 . 72 ohms 40 . 26 ohmsε . sub . effe 8 . 83 8 . 83 8 . 81ε . sub . effo 6 . 14 6 . 17 6 . 43coupled line : width 45 . 81 μm 42 . 07 μm 39 . 47 μmseparation 18 . 70 μm 28 . 25 μm 35 . 72 μmparallel length 785 . 82 μm 721 μm 721 μmdirectivity 13 . 25 db infinite infinitematch finite finite 0capacitive 0 . 034 pf 0 . 027 pfcompensation______________________________________ scrutiny of results indicate ideal directivity , on frequency operation , and no change in coupling value for the symmetric case . also , the ideal match case has an improved isolation and the ideal isolation case has an improved match as compared to the non - compensated case . thus , a directional coupler with single capacitive or inductive compensation has been described which overcomes specific problems and accomplishes certain advantages relative to prior art methods and mechanisms . the improvements over known technology are significant . traditional methods of improving the directivity of such couplers , such as adding an additional layer of dielectric over the conductors for symmetry , serrating the gap between the conductors , adding lumped capacitors at each end of the coupler , or selecting two or more different materials of different thicknesses and permittivities for the multi - level substrate are associated with particular disadvantages . adding a slab of dielectric adds material and introduces adhesive between the metallization and the substrate . such a structure may require handcrafting , or at least additional fabrication steps . serrating the gap between the conductors does not produce a satisfactory compensation for all values of the coupling . for lumped capacitance added at each end of the coupler are nearly true for tight coupling , the center frequency predicted is lower than desired . this result necessitates foreshortening the coupled section . furthermore , for loosely coupled sections , the equations are no longer valid . the traditional methods lack a design method for determining appropriate compensation without resorting to empirical means . none of the traditional methods has associated with it a closed form solution for the compensating lumped capacitance and odd mode characteristic impedance necessary to realize an ideal microstrip directional coupler . the directional coupler described here overcomes these previous shortcomings and has associated with it a closed form solution for the compensating lumped capacitance and a new odd mode characteristic impedance necessary to realize an ideal microstrip directional coupler . the results are accurate for either tight or loosely - coupled sections . the method results in embodiments for both antisymmetric and symmetric microstrip directional couplers with single inductive or capacitive compensation . thus , there has been provided , in accordance with an embodiment of the invention , a directional coupler with single capacitive or inductive compensation that fully satisfies the aims and advantages set forth above . while the invention has been described in conjunction with a specific embodiment , many alternatives , modifications , and variations will be apparent to those of ordinary skill in the art in light of the foregoing description . accordingly , the invention is intended to embrace all such alternatives , modifications , and variations as fall within the spirit and broad scope of the appended claims .