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As shown in the example 6 of here, the Nyquist plot of a open-loop transfer function that has a delay term in the numerator will encircle the origin several times (in fact, infinitely many times). However, is that a contradiction to the argument principle?

To be specific, that example is repeated below:

\$L(s)=90e^{-0.05s}/(s+3)(s+6)\$

Since \$L(s)\$ has no poles neither zeros in the RHP, and since \$e^{-0.05s}\$ is an entire function, according to the argument principle, the number of encirclement of the origin equals \$N-P=0\$, where N and P are the numbers of zeros and poles of \$L(s)\$ in the RHP.

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