Frequency of a sequential circuit

Question

I'm reading the book Digital Design and Computer Architecture. I came across this exercise:

From what I've learned so far, the frequency of a circuit is the reciprocal of the clock period. Since we have 3 inverters in a loop, I suppose the total time is \$N\times t_{pd}\$, therefore the oscillator should operate at a frequency between \$\frac{1}{N\times t_{pd}}\$ and \$\frac{1}{N\times t_{cd}}\$.

But instead the solution states that it operates at a frequency between \$\frac{1}{2\times N\times t_{pd}}\$ and \$\frac{1}{2\times N\times t_{cd}}\$. Where does the 2 come from?

Answer 1

The times \$N\cdot t_{pd}\$ and \$N\cdot t_{cd}\$ represent the max and min times needed for a single transition to make its way around the ring of N oscillators. However, one cycle requires two transitions to make it around the ring - one that brings the output high, and the second which brings the output low.