A phasor is a complex number representing an oscillating quantity. Phasors are really useful because oscillating quantities (often expressed with trigonometric functions such as [latex]\sin{x}[/latex] or [latex]\cos{x}[/latex]) are everywhere in science, and phasors allow them to be described with complex exponentials (like [latex]e^{ix}[/latex]), rather than trigonometric functions. That makes the maths a lot easier.
On this page we are going to see how complex exponentials can represent oscillating quantities. (If the first paragraph didn’t make sense to you, you should probably stop off first at Section 06: Complex numbers, and Section 12: Euler’s formula). We are going to lead up to these ideas through a discussion of circular motion. The content on this page will then be useful in Section 14, when we describe wave motion.
Uniform circular motion
In the diagram is a blue dot travelling in a circle. Imagine that its speed is constant (on this page, we will restrict ourselves to uniform circular motion – that is, where the speed is constant).
It could represent any object going in a circle – an orbiting planet, a hammer being thrown by an Olympic athlete, a point on a car tyre as it moves…
We have put a co-ordinate frame on the diagram, so that we can describe the position of the dot at any time in terms of the position vector, [latex]\mathbf{r}[/latex], from the origin (where the two axes intersect) to the dot, or the dot’s co-ordinates [latex](x,y)[/latex].
The dot is rotating anticlockwise, with an angular velocity [latex]\omega[/latex]. We’ll describe this quantity soon, but first we’ll introduce the units of angle we will be using.
Radians
When we talk of angles in everyday life, we almost always use the degree as the unit. There are 360 of those in a circle, or if you like, in a complete revolution. So when the blue dot in the picture above travels once around the circle, the radius vector [latex]\mathbf{r}[/latex] rotates through 360 degrees. Twice round is 720 degrees, and so on…
But in science, the degree is hardly ever the most convenient unit of angle. There is a really good reason why we would use a unit called the radian. At first sight the radian sounds a bit odd. For a start, there isn’t a whole number of them in a circle!? So what’s the deal?
Well, a radian is the angle that makes an arc on a circle the same length as the radius.
In the diagram, the blue arc length, [latex]s[/latex], is the same length as the radius, [latex]r[/latex]. 1 radian is the angle at the centre of the circle that makes that true. So if you wanted an arc length twice as long as the radius, you would need an angle of 2 radians… If you wanted an arc length [latex]\theta[/latex] times as long as the radius, you would need an angle of [latex]\theta[/latex] radians. So this means that [latex]s=r\theta[/latex].
This direct link between arc length and angle is the point to radians… If we measured angle in degrees, arc length would still be proportional to angle, but a conversion factor would be needed to make the numbers work. When measuring angles in radians, [latex]s[/latex] is not just proportional to [latex]r\theta[/latex], but equal to it. So from this point of view, radians simplify the geometry of circles.
We mentioned before, though, that there aren’t a whole number of radians in a circle. So how many radians are there in a circle? Well, the circumference of a circle is [latex]2\pi r[/latex], that is, [latex]2\pi[/latex] times the radius, so the corresponding angle is [latex]2\pi[/latex] radians. There are [latex]2\pi[/latex] radians in a circle.
Now we’ll use our knowledge of radians to help us deal with circular motion.
Angular velocity
Let’s return to the blue dot that we said was rotating with angular velocity, [latex]\omega[/latex].
In the same way that linear velocity is change in position per unit time, angular velocity is change in angle per unit time. So:
[latex]\displaystyle{\omega=\frac{\theta}{t}}[/latex]
We should probably say [latex]\omega=\frac{d\theta}{dt}[/latex], but we are only considering the case of constant angular velocity, so it amounts to the same thing on this page.
The unit of angular velocity is the radian per second.
It might be obvious, but we’ll re-state this relation in a rearranged form, one that we will use a lot on this page:
[latex]\theta=\omega t[/latex]
The angle covered, [latex]\theta[/latex], is a function of time, and depends on the angular velocity. Just like for linear motion, where [latex]x=vt[/latex] gives the distance covered at any time for a velocity [latex]v[/latex].
Frequency and phase
A related quantity is the frequency of circular motion. As applied to our diagram, the frequency, [latex]f[/latex], of the motion is the number of revolutions performed by the blue dot per unit time. When the unit of time used is the second, the frequency is the number of revolutions per second (s-1, or hertz, Hz).
Now, each of those revolutions amounts to an angle of [latex]2\pi[/latex] radians. So if the blue dot is covering [latex]f[/latex] revolutions per second, it is covering [latex]f\times2\pi[/latex] radians per second. And that’s the angular velocity, [latex]\omega[/latex]. In other words, [latex]\omega=f\times2\pi[/latex].
Or,
[latex]\omega=2\pi f[/latex]
Which, in turn means that
[latex]\theta=\omega t=2\pi ft[/latex]
You’ll see [latex]\omega t[/latex] or [latex]2\pi ft[/latex] a lot in the context of rotations and oscillations. It tells you ‘the angle covered’, which is known as the phase of the motion.
Where is the blue dot at time [latex]t[/latex]?
We have already said that we can describe the position of the blue dot in terms of the co-ordinates [latex](x,y)[/latex]. This position obviously changes with time [latex]t[/latex], so if we find equations for [latex]x[/latex] and [latex]y[/latex], we should expect those equations to have [latex]t[/latex] in them. We’ll take the coordinates one at a time, for simplicity, and we’ll look at the [latex]y[/latex]-coordinate first. The only good reason for that choice is to keep the diagram horizontal so that it fits on the screen easier—the [latex]x[/latex]-coordinate is just as easy to deal with conceptually.
In effect, in concentrating on the [latex]y[/latex]-coordinate, we are looking at just one of the dimensions (the vertical one) of the 2-dimensional circular motion. This ‘projection’ of the circular motion can be visualised by imagining shining a light from the left-hand side (parallel to the plane of the circle), and following the shadow of the blue dot on a wall on the right. The shadow will move up and down in one dimension as the blue dot moves in its circular path.
This means that the mathematics of circular motion can be used to describe oscillating quantities like, say, an alternating voltage. That idea is one of the central points of this page.
The diagram below shows the rotating blue dot again, but added to it is a graph, showing the [latex]y[/latex]-coordinate of the blue dot’s position at time [latex]t[/latex]. The projection on the [latex]y[/latex]-axis (green dot) oscillates up and down as the blue dot moves in circles.
There’s a lot going on in that picture. So the video below builds up the diagram, in case you need a more step-by-step explanation.
This idea of oscillations as a 1-dimensional projection of circular motion (2-dimensional) is very powerful, as we will see later on this page, and in Section 14: Oscillations and Waves.
Dealing with an initial phase
In the discussion above, the blue dot started on the [latex]x[/latex]-axis. In other words, at [latex]t=0[/latex], the angle [latex]\theta[/latex] is also [latex]0[/latex]. What if that isn’t the case? What if we ‘start the stopwatch’ when the dot has already covered an angle [latex]\phi[/latex], at the position of the red dot, as in the diagram below?
The [latex]y[/latex]-displacement is still given by the black arrow, which in turn is equal to the length of the grey dotted line, down from the blue dot to the [latex]x[/latex]-axis. And that grey dotted line is one side of a right-angled triangle, the other two sides being the [latex]x[/latex]-axis and the blue radius vector. The angle between the radius vector and the [latex]x[/latex]-axis is [latex](\theta+\phi)[/latex]. The angle [latex]\theta[/latex] is time-dependent, and increases as the blue dot goes about its journey. The angle [latex]\phi[/latex] is not time-dependent; it is just the initial angle, and as such has a constant value. By extension from our previous equations, the value of [latex]y[/latex] is given by
[latex]y=r \sin {(\theta+\phi)}[/latex]
We also know that [latex]\theta[/latex], the time-dependent part, is given by [latex]\theta=\omega t=2 \pi ft[/latex], and this means that
[latex]y=r \sin {(\omega t+\phi)}[/latex]
Or
[latex]y=r \sin {(2 \pi ft+\phi)}[/latex]
The only extra addition from our previous expressions for [latex]y[/latex] is the extra angle [latex]\phi[/latex] in the sine function. You can clearly see the time-dependent part (it has [latex]t[/latex] in it) and the constant part in the trigonometric function.
Both dimensions at once
So far, we have concentrated on the [latex]y[/latex]-coordinate of the rotating blue dot. And we have seen that this [latex]y[/latex]-coordinate is also a representation of an oscillation in one dimension. Now let’s consider the [latex]x[/latex]-coordinate at the same time. The [latex]x[/latex]-coordinate (red dot in the diagram below) behaves similarly to the [latex]y[/latex]-coordinate in that it oscillates in a sinusoidal manner. The only difference is that at the beginning (at time [latex]t=0[/latex]) the [latex]x[/latex]-displacement is at its maximum value, whereas the [latex]y[/latex]-displacement is zero. You can see that graphically in the diagram. Mathematically it is taken care of using the cosine function instead of the sine function. And indeed, looking at the right-angled triangle containing the blue dot and the angle [latex]\theta[/latex], the [latex]x[/latex]-displacement is indeed given by [latex]x=r \cos {\theta}[/latex].
Compiling the results so far, we can say that the position [latex](x,y)[/latex] of the blue dot at any time [latex]t[/latex] is
[latex](x,y)=(r\cos{\omega t},r\sin{\omega t})[/latex]
That result is going to be extremely important below.
Using the complex plane to describe circular motion
So far, we have drawn the blue dot rotating around the origin of an [latex](x,y)[/latex] Cartesian coordinate frame. But that choice of frame is totally arbitrary. Now, we’ll replace the Cartesian frame with the complex plane (see Section 06: Complex Numbers if that makes no sense to you). We know it’s worthwhile to do that because we’ve seen this before and we know where we are heading. We admit it seems like a weird thing to do if you are new to this…
As we know, the blue dot has coordinates [latex](r\cos{\omega t},r\sin{\omega t})[/latex]. On the complex plane, those coordinates represent the complex number
[latex]r\cos{\omega t}+i\times r\sin{\omega t}[/latex]
Simplifying and factorising gives
[latex]r(\cos{\omega t}+i\sin{\omega t})[/latex]
Euler’s formula tells us that [latex]\cos{\theta}+i\sin{\theta}[/latex] can be written as the complex exponential [latex]e^{i\theta}[/latex] (if that doesn’t make sense to you, you might want to go to Section 12 before continuing). When we put our blue dot on the complex plane, its coordinates are a time-dependent version of this:
[latex]\cos{\omega t}+i\sin{\omega t}=e^{i\omega t}[/latex]
So that
[latex]r(\cos{\omega t}+i\sin{\omega t})=re^{i\omega t}[/latex]
This is a really important result. The position vector of our rotating blue dot, as expressed in the complex plane, is given by
[latex]\mathbf{r}=re^{i\omega t}[/latex]
[latex]re^{i\omega t}[/latex]represents rotation at angular velocity [latex]\omega[/latex] with radius [latex]r[/latex]. And [latex]re^{-i\omega t}[/latex] represents the same rotation but in the opposite direction.
As we saw before, the projection of 2D circular motion on each axis is a one-dimensional sinusoidal oscillation, so that:
- The [latex]x[/latex]-component of the position vector [latex]\mathbf{r}[/latex] is the real part of the complex number [latex]re^{i\omega t}[/latex]
- The [latex]y[/latex]-component of the position vector [latex]\mathbf{r}[/latex] is the imaginary part of the complex number [latex]re^{i\omega t}[/latex]
In other words:
[latex]x=r\cos{\omega t}=\mathrm{Re}[re^{i\omega t}][/latex]
And
[latex]y=r\sin{\omega t}=\mathrm{Im}[re^{i\omega t}][/latex]
So whenever you see a time-dependent complex exponential, in the form [latex]e^{i\omega t}[/latex], think immediately “rotation or oscillation”…
Complex amplitude
Earlier on the page, we asked “what if we ‘start the stopwatch’ when the dot has already covered an angle [latex]\theta[/latex]?”, so that there is an initial non-zero phase at time [latex]t=0[/latex]. Well, we’ll ask the same question again, now we have our new complex exponential representation of rotation.
Remember, we dealt with an initial phase [latex]\phi[/latex] by replacing [latex]\omega t[/latex] with [latex](\omega t+\phi)[/latex], so that [latex]y=r\sin{\omega t}[/latex] became [latex]y=r\sin{(\omega t+\phi)}[/latex].
Well, similarly, we can write:
[latex]\mathbf{r}=re^{i(\omega t+\phi)}[/latex]
Now, the laws of indices (and in particular, [latex]x^a\times x^b=x^{a+b}[/latex]) allow us to write the expression for [latex]\mathbf{r}[/latex] as
[latex]\mathbf{r}=re^{i \omega t}e^{i\phi}[/latex]
And for convenience, we can write this in a different order:
[latex]\mathbf{r}=(re^{i\phi})\times e^{i \omega t}[/latex]
In this way of writing the rotation, the [latex]e^{i \omega t}[/latex] part represents the (time-dependent) rotation. The [latex]re^{i\phi}[/latex] part is called a complex amplitude. It encodes information about both the amplitude of the motion and the initial phase in one complex number.
Sometimes, the representation [latex]\mathbf{r}=(re^{i\phi})\times e^{i \omega t}[/latex] is called a ‘phasor’, although this term is also sometimes reserved for just the complex amplitude part of the expression. Phasors are useful, for example, when dealing with alternating currents, where the voltages are sinusoidal, and where the voltages across different components (e.g. the resistors, capacitors and inductors of an LCR circuit) have different phases relative to eachother. All the relevant information can be encoded within the complex amplitude representation we have discussed.
Link to Section 06: Complex numbers
If you need a reminder of the complex plane before tackling this page
Link to Section 12: Euler’s formula
In case you need a reminder of Euler’s formula and complex exponentials before tackling this page
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