Use negative feedback opamp to understand the feedback divider in Phase-locked loop

Phase-locked loop is a feedback system. In the feedback loop, a divider is usually used to get a frequency multiplication.

At the first sight, it is hard to understand why the output frequency feedback with a divider can get a increased frequency clock signal.

I found a article in ALL ABOUT CIRCUITS website. It is very useful to explain it by using the negative feedback opamp knowledge.

In the opamp non-inverting amplifier, if the output signal is connected directly back to the inverting input termal, it is used as a voltage follower, or unity-gain buffer. As a result, the opamp does whatever it needs to do to make the output voltage equal to the input voltage.

What if we need a gain? We just use some resistors to turn the feedback loop into a voltage divider. The feedback voltage, Vfb, will be a division of output voltage, Vout.

As we know, the negative-feedback arrangement causes the op-amp to modify its output with one goal: make the voltage at two input nodes equal. When it is configured as unity-gain buffer, this means the Vout must be equal Vin.

But the voltage divider in the feedback loop changes everything. Now, the voltage at the inverting input is DIV times smaller than the voltage at the output. Thus, in order to make the inverting-input voltage equal to the noninverting-input voltage, the output voltage must be DIV times larger than the input voltage.

With an op-amp, then, we create voltage gain by reducing the amplitude of the feedback voltage; with a PLL, we create frequency gain by reducing the frequency of the feedback waveform. To continue the analogy, the gain of a noninverting opamp circuit is equal to the factor by which the feedback voltage is divided, and the amount of frequency multiplication perfomed by the PLL is equal to the factor by which the feedback signal's frequency is divided.