If v(t) is a time-varying voltage, then v(t)/A is NOT a constant.

θ is the phase of the sinusoids at t=0. Hence, sin(θ) = v(0)/A.

True, θ is not A CONSTANT, but it can be CONSTANT. If time t dos not vary.

So, in a circuit, if something is set to trigger say when t = 1.222222...ms, phase offset, or the phase will be constant.

I think you would write: ν (1.222222 ms) = A sin θ.

And about word usage, yes, just the word phase is probably better than phase offset, in this context, although phase offset would not be wrong. It seerms.

 

I don't know what train you are on and I don't want to derail it. Your train has not left the station.
Why do you continue to make guesses and assumptions?

There is a lot of information in the equation

v(t) = A sin ( 2πf t + θ )

There are many folks here on AAC that can get your train moving.

What is your question?

 

I don't know what train you are on and I don't want to derail it. Your train has not left the station.
Why do you continue to make guesses and assumptions?

There is a lot of information in the equation

v(t) = A sin ( 2πf t + θ )

There are many folks here on AAC that can get your train moving.

What is your question?

I don't have a single question. I'm unpicking v(t) = A sin ( 2πf t + θ ). What you call assumptions is just me telling you what my understanding is.

 

You know, I don't like being crtiticized on this thread for simply expressing what my understanding currently is. I think it's bad form to criticize on something like that. That's a bad train to be on IMHO.

 

Ok, after some studying etc, let me now just put down what I feel I know.

When I've been studying PLL, one of the inputs is marked A sin ( 2πf t + θ ). What bothered me about it, is that it's not an equation. That was somewhat offputting.

I now see that you are meant to mess with A sin ( 2πf t + θ ) and make it work for you. And you don't have to use all of it.

Consider an AC signal v (t) = 200V; f = 50 hz; t = 1.222222 ms.

Anyway, from the statement, A sin ( 2πf t + θ ) I see that we can write equations for the following:

Instantaneous Voltage:
ν (t) = A sin θ

Phase or Phase Offset:
Sin θ = ν (t) / A

Amplitude:
A = ν (t) / sin θ

Considering a frequency of 50Hz, my understanding is that t could be anything, but in practical terms, t would most often be a value of 0 s and < 20 ms. Because in practical electronics, once t goes over 20ms, you are repeating. Also, I imagine, practically, t would not tend in general to be 20 ms, 40 ms, 60 ms etc. Here I'm touching upon the practical side of the math, usage in a maner of speaking. And that is part of this thread's purpose as well. To draw out some practicals, not providing the answers to exam type questions and use of equations in that "setting".

With the above equations there is no frequency information present. It's all based on trigonometry. You could get frequency if you were given the period (20ms).

So far this thread has basically been a focus on messing with A sin ( 2πf t + θ ) in say an exam setting. But there is a practical setting which I want to examine as well.

(To be continued)....

 

And you don't have to use all of it.

You do if you want to know what V is at time t.

 

Before I post something about the 2πf t part.

I'm thinking that there is a an exam setting and a practical setting for use of A sin ( 2πf t + θ ).

Exam setting - (working with A sin θ):

A = 200v; θ = 22.02431°: calculate v(t)

(Actually, what's interesting is that you cannot know t. So, one wonders whether it's strictly correct to say v (t) = A sin θ). So, the v(t) would only apply to the 2πf t part of A sin ( 2πf t + θ )?)

So, maybe you have to write

θ = 22.02431°; v = +75V : calculate A

or

sin θ = - 0.375; v = +75: calculate A

Leaving out t, because t is not available. Perhaps someone could comment on this.

Now, I'm wondering, what would be a practical use of these equations, like, on the benchtop, or in circuit design, outside of an exam setting. Is there really an exam setting and a practical setting?

 

If you knew t the fraction of the cycle the phase has gone through, and we had the period, then I think you might be able to write:

v (t) = A sine θ

t would be the time of the fraction of the cycle.

Because you could work out θ by 360 x ( t (fraction) / t (period))

Otherwise I think it's:

v = A sine θ

Yet, there is an element of time t, in θ.

So, can someone please give me a reading on whether it is correct to write v (t) = A sine θ. And if so, why. Thanks. If it is correct, it makes sense to think v must always be at some time t. Which it is. That would mean, you don't have to know t, but never-the-less the voltage is at some time t. You just don't know the value of t.

 

Geez, I don't know if I can help you any further. You keep making the same assumptions and the same mistakes and yet you refuse to pay attention. This is not a critique on your intelligence. As I pointed out from the start, there are holes in your knowledge base and you will not be able to fill those holes with your style of approach.

This is not an exam question. No one is asking to calculate anything at any time. You are obsessed with the math and refuse to get a good grasp on fundamentals.

The equation

v(t) = A sin ( 2πf t + θ )

has a lot of information and says it all. It has essential purposes in all sorts of electronic applications.

I am willing to go though all the fundamentals in complete detail in a step by step fashion in trying to get you to understand the math. There are no guesses or assumptions to be made. You are the one that is making guesses and assumptions. The math is perfectly clear.

I am quite happy to bow out of this thread if you no longer want any advice from me. I can leave amicably with no grief.

 

Folks here have to really listen to what I'm saying. I'm not looking at v(t) = A sin ( 2πf t + θ ) just yet.

We have started out by saying A sin ( 2πf t + θ ) is something you can use, but it's not an equation as it is written. And you can use what is appropriate or useful to you.

Such as v (t) = A sin θ

So, I say, oh, but that what I've written looks odd, because actually there is no frequency information to be used in that equation, culled from A sin ( 2πf t + θ ). It's trigonometry. So, it looks like one must say v = A sin θ . Now, why should saying that cause a problem for anyone?

 

Such as v (t) = A sin θ

I can't see what you are trying to achieve by removing parts of a valid equation.
If t is time, A is the amplitude of a sine wave and θ is a constant phase angle, that equation is saying that v is a function of time, but v(t) is the product of two constants; in other words v is independent of time.

 

Folks here have to really listen to what I'm saying. I'm not looking at v(t) = A sin ( 2πf t + θ ) just yet.

We have started out by saying A sin ( 2πf t + θ ) is something you can use, but it's not an equation as it is written. And you can use what is appropriate or useful to you.

Such as v (t) = A sin θ

So, I say, oh, but that what I've written looks odd, because actually there is no frequency information to be used in that equation, culled from A sin ( 2πf t + θ ). It's trigonometry. So, it looks like one must say v = A sin θ . Now, why should saying that cause a problem for anyone?

Because it is nonsense!

A sin ( 2πf t + θ )

is an EXPRESSION. It is IMPLIED that it is describing something, current, voltage, physical displacement, temperature, whatever the units on 'A' indicate, as a function of time. If A has units of voltage, then the EQUATION you so desperately seek is

v(t) = A sin ( 2πf t + θ )

You can't just then arbitrarily dismiss part of it on a whim.

 

Perhaps an explanation of standard notation would help.

In this context, the symbol θ usually stands for a numeric constant that represents the value of an angle, e.g., θ = 2π. The value of any trigonometric function taken at a constant is itself a constant. So, with θ = 2π, we know that

sin(θ) = 0

The expression above has no notion of time. When we want to describe a function that changes with time, we make that explicit by using t as the independent variable, e.g.:

f(t) = 2t

When we want to describe a function of a time-varying angle, we're dealing with angular frequency, which is typically notated with the symbol ω:

f(t) = sin(ωt)

If we define ω = 2π/T, with T the period, then ω is the angular frequency in radians per second. Notice that the product ωt has units of radians (angle). In other words, for each instant of time t, the product ωt spits out a numeric constant that represents an angle, like θ above. But now instead of just one angle and one value, as in the θ example, we get all values for all distinct angles. Also note that, because frequency in Hz is the reciprocal of time, ω = 2πf is an equivalent formulation, and so we can write

f(t) = sin(2πft)

In other words, for every tick of the independent variable t, f(t) tells us the value that a sine of frequency f would have at the corresponding angle.

But not all sinusoids have the same phase relationship, so we include an additive constant called the phase-offset term φ:

f(t) = sin(2πft + φ)

Here, the constant φ represents the phase relationship between f(t) and a hypothetical sine of the same frequency that has zero phase (i.e., angle = 0 at time = 0). In other words, φ tells us how much our particular waveform is shifted in time.

Finally, the sine function ranges over the real interval [-1, 1]. If we want to shift the amplitude of a sine, we simply multiply it by some constant A:

f(t) = A sin(2πft + φ)

Now, depending on our choices of the constants A, f, and φ, we can express any sine we like. Thus, f(t) is the most general form of the sine function. When voltages and currents are sinusoidal, it's convenient to express them in the most general way possible, i.e., v(t) = f(t), or i(t) = f(t).

 

I think it ought to be pointed out, that right at the beginning it was said that the expression A sin ( 2πf t + θ ) was something that could be "messed" with. Meaning parts of the expression can be picked out and applied. So, I've been on a journey doing exactly that. That's why v = A sin θ came out, it was picked out. And where phase = 2πft would have also been posted, again picked out. Leading eventually to the whole thing v (t) = A sin (2πf t + θ) coming into focus.

But, perhaps it's a bit of a wrong steer to have said that that expression can be messed about like that, that one should not do that but use all the terms in the expression. Which is not messing with the expression.

Me messing with the expression seems to have been somewhat of a spanner in the works.

 

Me messing with the expression seems to have been somewhat of a spanner in the works.

Indeed. It's a bit like removing 3 spark plugs and 2 wheels from your car and still expecting it to work properly: or omitting flour from a cake recipe then wondering why the cake didn't rise when cooked.

 

Folks here have to really listen to what I'm saying. I'm not looking at v(t) = A sin ( 2πf t + θ ) just yet.

We have started out by saying A sin ( 2πf t + θ ) is something you can use, but it's not an equation as it is written. And you can use what is appropriate or useful to you.

Such as v (t) = A sin θ

So, I say, oh, but that what I've written looks odd, because actually there is no frequency information to be used in that equation, culled from A sin ( 2πf t + θ ). It's trigonometry. So, it looks like one must say v = A sin θ . Now, why should saying that cause a problem for anyone?

Hi,

Are you talking about constant frequency? That's used all the time in AC analysis.

We can even use a complex number:
v=a+b*j

where the amplitude is:
sqrt(a^2+b^2)

and the phase shift is:
atan2(b,a)

 

I think it ought to be pointed out, that right at the beginning it was said that the expression A sin ( 2πf t + θ ) was something that could be "messed" with. Meaning parts of the expression can be picked out and applied. So, I've been on a journey doing exactly that. That's why v = A sin θ came out, it was picked out. And where phase = 2πft would have also been posted, again picked out. Leading eventually to the whole thing v (t) = A sin (2πf t + θ) coming into focus.

But, perhaps it's a bit of a wrong steer to have said that that expression can be messed about like that, that one should not do that but use all the terms in the expression. Which is not messing with the expression.

Me messing with the expression seems to have been somewhat of a spanner in the works.

I think that that is a big part of the problem, but probably more in the way you did it than in what you did.

If I have

f(t) = At² + Bt + C

It's fine to say, "let's pull this apart and see what each term tells us." But to say f(t) = C muddies the water because f(t) does NOT equal C.

There are a couple ways to go about this. One is to use a different variable since f(t) is already used. So say

f(t) = g(t) + h(t) + k(t)

g(t) = At²
h(t) = Bt
k(t) = C

and then talk about the individual components.

You can also reuse f(t) if you make it clear that you are not talking about the same function. So perhaps something like.

Let's build up v(t) = A sin (2πf t + θ) one piece at a time.

To start with, what if we have

v(t) = A

That's just a constant, DC, voltage.

Then what about

v(t) = A·sin(θ)

This would also be a constant DC voltage that is somewhere between -A and +A.

Finally we have

v (t) = A·sin(2πf t + θ)

Now we have a time varying voltage. At t=0, it is A·sin(θ), it then oscillates between -A and +A, going through once complete cycle each time t increases by an amount of 1/f.

 

Indeed. It's a bit like removing 3 spark plugs and 2 wheels from your car and still expecting it to work properly: or omitting flour from a cake recipe then wondering why the cake didn't rise when cooked.

But it's reasonable to ask what role each ingredient plays in baking a cake, but the question gets lost if it is presented as picking apart the recipe and making a cake using only flour.