Stability ProofA transfer function is stable if there are no poles in the right-half plane. That is, for each zero of , we must have re. If this can be shown, along with , then the reflectance is shown to be passive. We must also study the system zeros (roots of ) in order to determine if there are any pole-zero cancellations (common factors in and ).
Since re if and only if re, for , we may set without loss of generality. Thus, we need only study the roots of
At any solution of , we must have
To obtain separate equations for the real and imaginary parts, set , where and are real, and take the real and imaginary parts of Eq.(C.161) to get
For any poles of on the axis, we have , and Eq.(C.163) reduces to
It is well known that the ``sinc function'' is less than in magnitude at all except . Therefore, Eq.(C.164) can hold only at . We have so far proved that any poles on the axis must be at . The same argument can be extended to the entire right-half plane as follows. Going back to the more general case of Eq.(C.163), we have
Since for all real , and since for , this equation clearly has no solutions in the right-half plane. Therefore, the only possible poles in the right-half plane, including the axis, are at . In the left-half plane, there are many potential poles. Equation (C.162) has infinitely many solutions for each since the elementary inequality implies . Also, Eq.(C.163) has an increasing number of solutions as grows more and more negative; in the limit of , the number of solutions is infinite and given by the roots of ( for any integer ). However, note that at , the solutions of Eq.(C.162) converge to the roots of ( for any integer ). The only issue is that the solutions of Eq.(C.162) and Eq.(C.163) must occur together.
- Rotate the loci in Fig.C.49 counterclockwise by 90 degrees.
- Prove that the two root loci are continuous, single-valued functions of (as the figure suggests).
- Prove that for , there are infinitely many extrema of the log-sinc function (imaginary-part root-locus) which have negative curvature and which lie below (as the figure suggests). The and lines are shown in the figure as dotted lines.
- Prove that the other root locus (for the real part) has infinitely many similar extrema which occur for (again as the figure suggests).
- Prove that the two root-loci interlace at (already done above).
- Then topologically, the continuous functions must cross at infinitely many points in order to achieve interlacing at .
Reflectance of the Conical Cap