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[Design of Analog CMOS inteagrtaed circuits by Razavi]

Can someone explain what Lapalce transform rules states that this is possible? They have a function of excess output phase / excess input phase in s-domain and they say that this can be extended to w_out/w_in since the derivative is a linear operator? How?

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  • \$\begingroup\$ But you should not forget that the given transfer function results from a linearization process and applies under locked conditions only. That means: Both frequencies must not differ too much (frequency difference smaller than the natural PLL frequency) \$\endgroup\$
    – LvW
    Commented Dec 12, 2023 at 16:11

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This has nothing to do with PLL theory, aside from the fact that they're using a linear system approximation.

You have the relationship

$$\frac{\phi_{out}}{\phi_{in}} = H_{cl}(s) \tag 1$$

By definition, \$\omega_{out} = \phi_{out} s \$ and \$\omega_{in} = \phi_{in} s\$.

So

$$\frac{\omega_{out}}{\omega_{in}} = \frac{\phi_{out} s}{\phi_{in} s} = \frac{\phi_{out}}{\phi_{in}} = H_{cl}(s) \tag 2$$

QED

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  • \$\begingroup\$ Can't believe I missed that. Thank you! \$\endgroup\$
    – muosac
    Commented Dec 12, 2023 at 16:35

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