the set u03c6 nm n 5 6 m 1 2 shares common nulls with the nulls occurring at the second hybrid column for z there is only a single element with the null present at the final hybrid column the sets of elements highlighted all share the same set of nulls as the set u03c6 11 to u03c6 81 analyzed above for which matrix expansions and flow diagrams are presented a notable feature of the mpa observed from this table is if nulls are achieved for any one element in a given area x y or z then nulls will also be achieved for all other elements within that area for an ideal mpa in which all components hybrids and amplifiers are perfectly matched then nulls would also be produced for all the other sets x y amp z albeit at different points in the onet however perfect matching will not be achieved in practice and if nulls are achieved for a given x y amp z then for all the other sets x y amp z there will be a residual voltage dependent on the degree of mismatches in all cases the generation of a null can be considered as the result of a configuration as shown in fig8 of input hybrid 2 output hybrid 4 and pair of amplifier units 6 8 here u03b1 1 e j u03c81 2 and u03b1 2 e j u03c82 2 represent the total complex gains between the up and o p the factor of u00bd arises from the two hybrids one either side of the gain elements the relevance of this configuration to element u03c6 11 and all other u03c6 nm 1 lt n m u2267 4 may be.