[SOLVED] what's the relation between the frequency and the phase of a signal?

hello~everybody.This is my first post on this board.I have a simple question:
what's the relation between the frequency and the phase of a signal?I know frequency is derivate of phase.But Razavi's book says that the relation of this two terms is linear.therefore,I cannot understand what it means.Please help me. Thank you in advance.
ps:I am learning PLL,it is a hard work.:-(

Hi djallon,

your question is not as simple as it looks.
Yes, there is a fixed relationship between frequency and phase of a frequency dependent linear circuitry - provided it is a so called "minimum phase" system (MPS) - which means: without zeros in the right-hand part of the s-plane (RHP).
Most circuits (without allpass elements) are MPS circuits.
For these systems BODE has developed an equation showing the relation between phase shift PHI and magnitude H (in lg form) that can be approximated as follows:

PHI (w=wx)=(Pi/2)*dA/du

with A=lg|H(jw)| and u=lg(w/wx).

Thus, due to the differential quotient the phase function depends on the SLOPE of the magnitude.
Please note, this is an approximation which is sufficient for most applications.
For the exact formula (containing an integral) search for "BODE, gain-phase relation".
LvW

- - - Updated - - -

For BODE's formula see here (chapter 8.5)
http://www.ece.jhu.edu/~pi/Courses/454/Notes8.pdf

I have done systems where it was important to know or re-create the phase of signals that were different frequencies. If you just compare 2 of such frequencies, there is no obvious relationship, as seen on an osciloscope. But if there is some common timing event that can be used to set zero time, then the relative phase of one signal to the other is obvious (even though they might not be the exact same frequency). If you are creating the signals, you can set zero time with your system clock, perhaps triggering a DDS synthesizer. If it is not under your control, as in distant frequency ofsetting repeaters, you might define zero time as some modulation event, or a zero crossing of two tones beating with each other, etc.

Thanks for your reply.

Imagine you have two signal sources, both at exactly the same frequency, and you are displaying them together.

If the displayed waves go up and down together, they are in phase. The time difference is zero.
If one is delayed such that its displayed wave lags behind, reaching its peaks later, the waves are not in step.
The PHASE of the delayed signal is no longer zero with reference to the first signal.

If a signal is briefly slightly increased in frequency, then brought back, its little excursion will have involved the waveform departing from a strict sine-wave, and the peaks of the wave will have gained a lead on the reference signal. once settled back on frequency, you have a signal where the PHASE is some number of degrees ahead. In this context, a whole wavelength means 360 degrees, or 2*PI radians.

The sine-wave can be plotted by thinking of a rotating arrow vector (called a phasor). The length of the arrow is the amplitude, and the arrow is rotating about its base at the centre of a circle. The revolutions per second is the frequency. It turns through 2*PI radians each revolution, and the motion of the tip can plot out a regular sinewave. The angle it has turned through is its PHASE.

A sine-wave is plotted as y=A*sine(w*t+PHASE) where w (the angle) = 2*PI*frequency.
If a wave has PHASE =0, and another has PHASE=(some angle) , then the PHASE of the second will be ahead of the first.

You can see now that wiggling about with the PHASE will give PHASE MODULATION.
You can also see that FM (frequency modulation) involves a continuous messing with the phase.

Hope this helps.

Thank you.Now I know the differences between frequency and phase.
hah~