Formulas useful for the synthesis and optimization of general, uniform contradirectional couplers

Explicit formulas allowing the synthesis and optimization of general, uniform, contradirectional couplers are derived. The well-known method of compensating for different phase velocities of the coupler modes with shunting capacitors is also mathematically formulated in a stringent form so that a lossless coupler of the general class mentioned above can be made ideal at any chosen frequency. Previously, only specialized cases have been treated in closed mathematical form. Compensated and uncompensated microstrip coupler designs are compared under the influence of realistic losses. It is shown that a considerably better directivity is attainable with capacitive compensation, even over a frequency range as large as one octave. The value of the formulas is confirmed by application to previously published experimental results