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# Help with Phase locked loops (PLL) what does phase mean here?

#### george2525

Jan 30, 2015
170
Hi,

So im realy stuck trying to understand PLL's in terms of the calculations ive been looking at. This is a good example of the type of thing - taken from phase locked loops by Roland Best

OK so the U(t) signal seems to tell me that its a standard sine with a changing "phase"

The example and figure show a linear frequency increase from t=0

I have no idea what delta*w^dot means in the notation. I have never seen this before. it seems to be saying the change of gradient? anyway...

Then I keep hearing that "the derivative of phase is frequency" but in this context what does "phase" mean?

if we take the derivative of the "Argument" I would generally get w , IF the other term were constant right... but I dont understand how taking a derivative of that extra term added to wt would give the frequency!, in any context

Im very confused so if anyone has the time to talk me through some maths and concepts here id be very grateful. I understand the idea of PLL's to some extent but I have big problems deciphering notation and would realy benefit from a simple breakdown of what is happening here.

Thanks to anyone who has the time

#### Harald Kapp

##### Moderator
Moderator
Nov 17, 2011
12,735
Then I keep hearing that "the derivative of phase is frequency" but in this context what does "phase" mean?
Have a look at the phasor representation of AC signals, e.g. described here. The phasor is an arrow with a length (aka amplitude) and an angle to the x-axis. This angle is the phase of the phasor (it's no coincidence the arrow has been dubbed phasor).
To create a sinusoidal wave, this phasor rotates (typically counter-clockwise) around the origin. The length of the phasor stays the same all the time. The rotation can be describes as a change in angel (phase). The faster the phasor rotates, the more rottaions fit into a given timeslot (e.g. 1 second). The number of rotations per second is therefore the frequency of the sine wave. The number of rotations per second is equivalent to the change in angle (phase) over that time: each rotation adds 2*PI to the phase (you accumulate the phase, you do not re-start at 0 when the phasor corsses the rightward pointing x-axis). Therefore the derivative of the (accumulated) phase with respect to time is equivalent to the frequency.

The possible issue with the term 'phase' here is that it may seem to have different meanings depending on the context. But in reality there is no difference.

One meaning which may be better known to you is the phase between two sines. This is in reality the phase difference. If both sines have the exact same frequency but their zero crossings happen at different points in time, the phase difference is fixed and can be calculated from the frequency and the time difference of the zero crossings.
If both sines have different frequencies, the phase difference changes with time.
You can easily draw this in a simple diagram.

The other meaning of phase is as described above as the angle of a phasor with reference to the x-axis and a fixed point in time where phase is by definition 0 ° (imagine the phasor being at rest and starting to rotate at time T0, then this is the reference point). With each full rotation of the phasor the accumulated phase is increased by 2*PI.

The phase thus can be described as phi1(t) = 2*PI*f*t where T0 = 0, f = frequency, t = time
A second signal with an offset in phase to this first signal can be described as phi2(t) = 2*PI*f*t +delta_phi where delta_phi is the fixed offset in phase.
The phase difference of these two signals is phi2(t)-phi1(t) = (2*PI*f*t +delta_phi) - (2*PI*f*t) = delta_phi as expected.
The derivative of e.g. phi1(t) with respect to time is dphi1(t)/dt = d(2*PI*f*t)/dt = 2*PI*f. This is the angular frequency of the signal, the 'speed' at which the phasor rotates.

Hope that helps you get along.

By the way: congratulations to your choice of book. I don't know the one on PLLs, but I know 2 other books by Roland Best and I think he's a good author, explaining matters well.

#### george2525

Jan 30, 2015
170
Thanks. I actually know most of that but it was a very good explanation and nice to read.

My main issue with the example is that he takes the derivative of theta1 and states that it equals w1

if, as in your example he labelled the argument as phi1(t) and took the derivative of that, it would make sense
but is his example the theta1 seems to be referring to the phase "offset" alone. Perhaps this is just bad notation?

my second issue is the term delta*w^dot. its stated that this means "rate of change of angular frequency"
should this not simply be dw/dt ? as it is the gradient of the increasing ramp in the fig 2.4.

You seem to have confirmed to me that "phase" in this PLL context just means the argument of the sine (or angle in the phasor) which is helping. any thoughts on the other stuff?

#### LvW

Apr 12, 2014
604
My main issue with the example is that he takes the derivative of theta1 and states that it equals w1
For my opinion, it is best to forget this formula. It is just a general expression which should serve as an additional explanation for the sentence before. Unfortunately the author has chosen the subscript "1" which is also used in the first line (formula for u1(t). Another subscript (like "x") would be better.

my second issue is the term delta*w^dot. its stated that this means "rate of change of angular frequency"
should this not simply be dw/dt ? as it is the gradient of the increasing ramp in the fig 2.4.
Yes - I think so. This expression is nothing else than the slope of the w-function vs. time.

#### Syncopator

Jan 12, 2017
8
ω is the small (we would call it lower case) Greek letter omega. Ω is the large omega.

ω in a formula means 2πf, and is called pulsatance.

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