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When we used the microwave frequency divider, such like: RF Bay FBS-N-14 or analog dividers in wiki.

The input signal is $\cos(\omega t+\phi)$, and the output signal is $\cos(\frac{1}{2}\omega t+\theta)$ when it passes the divide-by-2 prescaler.

what is the phase $\theta$ equal to? $\frac{\phi}{2}$ or $\frac{\phi}{2}+\pi$?

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That can't generally be said. It depends on the actual design of the device.

In fact, it's not going to be constant for all frequencies, because that's physically pretty much impossible to achieve.

You'll find multiple descriptions in literature for three- and four-port devices like these, and you'll have to learn how to infer the relative phases for different frequencies from the measurements that the manufacturer of a divider will provide to you.

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  • $\begingroup$ Thank you for your answer. $\endgroup$ – Hewie Ding Sep 24 at 2:21
  • $\begingroup$ You're welcome! If you think this answers your question, please don't forget to "accept" this answer. $\endgroup$ – Marcus Müller Sep 24 at 20:44

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