# Sinusoidal Signal Equation

#### Sparky49

Joined Jul 16, 2011
833
Hi all,

I've just got into another book on electronics, but I've come into a wee problem with the maths, in one part.

Re. sinusoidal signals, I believe the equation to find the voltage is:

V=Asin2∏ft

That's fine with me. However, what I don;t quite understand is when you need to knwo the voltage at some arbitrary point in time, ie when t=0. The book says that the equation requires a phase in this equation:

V=Asin(2∏ft+ϕ)

What I don't understand is the need for the 2∏ft, as if t=0 then the whole bracket would come to θ!

I'm clearly missing something here, could someone explain this to me please?

The book is The Art of Electronics, by Horowitz and Hill.

Sparky

#### steveb

Joined Jul 3, 2008
2,436
Hi all,

I've just got into another book on electronics, but I've come into a wee problem with the maths, in one part.

Re. sinusoidal signals, I believe the equation to find the voltage is:

V=Asin2∏ft

That's fine with me. However, what I don;t quite understand is when you need to knwo the voltage at some arbitrary point in time, ie when t=0. The book says that the equation requires a phase in this equation:

V=Asin(2∏ft+ϕ)

What I don't understand is the need for the 2∏ft, as if t=0 then the whole bracket would come to θ!

I'm clearly missing something here, could someone explain this to me please?

The book is The Art of Electronics, by Horowitz and Hill.

Sparky
Well, you kind of have a good point here. Most often we really don't care about the (2 pi f t) part. It is quite common to use the "Phasor" representation of signals which considers the amplitude and the phase (phi or theta part). Usually, it's the relative phase difference between signals that matter and the angle (phi or theta) contains that information. The time t=0 is somewhat arbitrary anyway. You can define t=0 to be at any point.

The use of the (2 pi f t) just provides a way to exactly specify the signal in time. There may be cases where you care about that information.

#### BillO

Joined Nov 24, 2008
990
...
What I don't understand is the need for the 2∏ft, as if t=0 then the whole bracket would come to θ! ...
Your math is right, but your understanding may need help. You need to know the initial conditions (phase, θ) and the progression (2piFt) for the complete description of the system at any particular point in time.

#### Sparky49

Joined Jul 16, 2011
833
Thanks guys.

#### samin

Joined Oct 14, 2011
32
The sine wave can be understood as the projection onto the real axis of a rotating vector on the complex plane, so its argument is the total phase 2∏ft + θ, The phase constant θ represents the angle that the complex vector forms with the real axis at t = 0.
Then based on the equation, you complete the description of the system:
when t=0 ; t=T/4 ; t = T / 2 and so on.