λ/4 coupling sections are used. The design of such couplers proceeds in much the same way as the design of distributed-element filters. The sections of the coupler are treated as being sections of a filter, and by adjusting the coupling factor of each section the coupled port can be made to have any of the classic filter responses such as maximally flat (Butterworth filter), equal-ripple (Cauer filter), or a specified-ripple (Chebychev filter) response. Ripple is the maximum variation in output of the coupled port in its passband, usually quoted as plus or minus a value in dB from the nominal coupling factor. It can be shown that coupled-line directional couplers have purely real and purely imaginary at all frequencies. This leads to a simplification of the S-matrix and the result that the coupled port is always in quadrature phase (90°) with the output port. Some applications make use of this phase difference. Letting , the ideal case of lossless operation simplifies to, Branch-line coupler The branch-line coupler consists of two parallel transmission lines physically coupled together with two or more branch lines between them. The branch lines are spaced λ/4 apart and represent sections of a multi-section filter design in the same way as the multiple sections of a coupled-line coupler except that here the coupling of each section is controlled with the impedance of the branch lines. The main and coupled line are of the system impedance. The more sections there are in the coupler, the higher is the ratio of impedances of the branch lines. High impedance lines have narrow tracks and this usually limits the design to three sections in planar formats due to manufacturing limitations. A similar limitation applies for coupling factors looser than ; low coupling also requires narrow tracks. Coupled lines are a better choice when loose coupling is required, but branch-line couplers are good for tight coupling and can be used for hybrids. Branch-line couplers usually do not have such a wide bandwidth as coupled lines. This style of coupler is good for implementing in high-power, air dielectric, solid bar formats as the rigid structure is easy to mechanically support. Branch line couplers can be used as crossovers as an alternative to air bridges, which in some applications cause an unacceptable amount of coupling between the lines being crossed. An ideal branch-line crossover theoretically has no coupling between the two paths through it.