We start with the definition . (B-1) From we write Figure B-1 . (B-2) Substituting Equation B-2 into Equation B-1 yields . (B-3) At this point we recognize that a practical planar array only radiates in its forward hemisphere. This means that will be zero for . If we use this, and cancel the terms, we get . (B-3) For the next step we need to relate and to and . To do so, we want to write the triple of Equation 1-81 in terms of the angles and . In the center of the patch Figure B-1 is the point . Thus, we can write . (B-4) By analogy to Equations 1- 86 through 1-90 we can find that (B-5) and . (B-6) and that . (B-7) Using this we can write Equation B-3 as . (B-8) We now make the substitution . This leads to and (B-9) we have made use of the fact that we can write since . We next make the change of variables . This leads to . We write the term multiplying as and use Equation B-4 to note that (B-10) which leads to . (B-11) and . (B-12)
We start with the definition . (B-1) From Figure B-1 we write . (B-2) Substituting Equation B-2 into Equation B-1 yields . (B-3) At this point we recognize that a practical planar array only radiates in its forward hemisphere. This means that will be zero for . If we use this, and cancel the terms, we get . (B-3) For the next step we need to relate and to and . To do so, we want to write the triple of Equation 1-81 in terms of the angles and . In Figure B-1 the center of the patch is the point . Thus, we can write . (B-4) By analogy to Equations 1- 86 through 1-90 we can find that (B-5) and . (B-6) and that . (B-7) Using this we can write Equation B-3 as . (B-8) We now make the substitution . This leads to and (B-9) we have made use of the fact that we can write since . We next make the change of variables . This leads to . We write the term multiplying as and use Equation B-4 to note that (B-10) which leads to . (B-11) and . (B-12)