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Animating AC Phasors

In this post, we try to get an intuitive feeling about special quantities called phasors, used to represent current and voltage in AC circuits. We will see the theory at work, learning about complex numbers, sinors, and how these are used to represent voltage and current. At the end, we'll be able to input our custom values in the animation!

Sinusoids:

Any quantity that varies in the form of a sine or a cosine function is called a sinusoid. When a sinusoidally time-varying voltage is applied across an electrical component we get a corresponding sinusoidally time-varying current which we call an Alternating Current (AC). The voltage applied is called the alternating voltage. In the general form if ` v(t) = V_{0} sin(ωt) `, then the current is in the form ` i(t) = I_{0} sin(ωt + φ) `, where ` φ ` is called the phase difference, created by the properties of the electrical component.

Alternatively, if the current is taken as ` i(t) = I_{0} sin(ωt) ` then voltage is of the form ` v(t) = V_{0} sin( ωt + φ) `, where again voltage is said to be ` φ` phase ahead of current. Indeed when `φ` is negative, we say that voltage is `φ` phase behind the current.  

Transient State and Steady State:

In typical systems, when a voltage is applied, the redistribution of the charges causes the current.

As a consequence, for an idle electrical component, our first task is to create a potential difference between the terminals of the component. To do this we apply, say, a sinusoid ` v(t) = V_{0}sin(wt + φ) `. This will gradually build up current in the system as determined by the properties of the components. This state is called the transient state, where the current is building up.  

Fig: The transient state of a typical RL circuit

After some time, the build-up smoothens out and we get a sinusoidally time-varying current `i(t) = I_{0} sin(wt + φ)`. This state is called the steady state. For an observer beginning her observation in this steady state, it is equivalent whether the current is `φ` phase ahead of the voltage, or the voltage is `φ ` phase behind the current. 

In this post, we will be dealing with the steady state condition. Since both conclusions are equivalent, henceforth, for convenience, we'll take current at zero phase, and voltage at a phase `φ` ahead of current. i.e. 

` i(t) = I_{0}sin(wt) `

` v(t) = V_{0} sin(wt + φ) `

Here `φ` can of course be negative.

A Complex Number:

A complex number is represented in the Argand Plane as ` x + jy ` where `x`  is called the real part and `y` is called the imaginary part. In the polar form, it is represented as `r angle φ`, where `r` is the magnitude and `φ` is the angle that it makes with the real axis. In the exponential form, it is represented as `r e^{iφ} `, where again `r` is the magnitude and `φ` is the angle. 

In particular, ` e^{jφ} ` is represented by a sinor with unit magnitude. When, `r angle φ = re^{jφ} ` is multiplied ` e^{j θ}`, we have,

`re^{jφ} \cdot e^{jθ} = re^{j(φ+θ)} = r angle(φ + θ) `

i.e. the sinor just rotates by an angle `θ`. We shall use this property of complex numbers to create rotating sinors.

Fig: Rotation of the complex no. `r angle φ` by an angle `θ`

AC Phasors:

It is difficult to keep track of a sinusoidal time-varying quantity, which is indeed difficult to imagine and even harder to work with. To simplify the reckoning, a sinusoid is often represented by a quantity called phasor. A phasor is simply a complex number that is used to represent a sinusoidally varying quantity. 

From Euler's formula, a complex number ` e^{j φ}` can be written as, 

` e^{j φ} = cos(φ) + j sin( φ)`

By which we can consider `cos φ` and `sin φ ` as the real and imaginary part of `e^{j φ} ` respectively, and thus we can write, 

` cos(φ) = Re(e^{jφ}) `

` sin(φ) = Im(e^{jφ}) `

Since ` e^{j φ} ` represents a complex number with unit magnitude. We can represent ` r angle φ` as ` r e^{j φ} `. Given a sinusoid ` v(t) = V_{0} sin (wt + φ) `, it can represented in the phasor form as,  

` v(t) = V_{0} sin (wt + φ) = Im(V_{0} e^{j(wt + φ)}) `

` v(t) = Im(V_{0} e^{j(wt + φ)}) ` 

Thus, 

$$ \boxed{v(t) = Im(\textbf{V} e^{jwt})} $$

where, $ \textbf{V} = V_{0}e^{jφ}$ `= V_{0} angle φ ` is called the phasor of the sinusoid `v(t)`. 

Fig: Projection of the imaginary part of the phasor

The factor `e^{jwt}` rotates the phasor in the argand plane. As `t` varies, at any time the imaginary part of the rotating phasor gives us the value of the sinusoid at that time.  

Impedance of a Circuit:

The impedance $ \textbf{Z} $ of a Circuit is defined as the ratio between the phasor voltage and the phasor current. 

$\boxed{\textbf{Z} = \frac{\textbf{V}}{\textbf{I}}} $

Mark that though impedance is the ratio of two phasors, it is not a phasor itself since it does not represent a sinusoidally varying quantity (see note 1). Impedance is a constant complex number as it does not contain the rotating factor. More accurately,

` \text{Z} = \frac{\text{V}e^{jωt}}{\text{I}e^{jωt}} `

` \text{Z} = \frac{\text{V}}{\text{I}} `

Clearly, the rotating factor cancels out.

In another form, we can write, 

$ \boxed{\textbf{V} = \textbf{IZ}} $

This suggests that if the current is at `0` phase and voltage is `φ` phase ahead of current, `\text{Z}` just rotates the sinor of `\text{I}` by the angle `φ` and makes its magnitude equal to that of`\text{V}`.

The same analogy can easily be extended when we write the equation in the form, `\frac{\text{V}}{\text{Z}} = \text{I}`.

Animating The Phasors:

So far, we have covered the fundamentals of phasors and how they represent alternating current and voltage. Now let us explore their behavior visually.

The below, is a graph created in a graphing software called Desmos. First, let us understand what the different parts of the graph mean. In the graph, the red curve represents voltage, and the blue curve represents current. On the left side of the vertical axis, you’ll find phasor circles in the complex plane. These circles are the locus of the end-points of the rotating phasors. The imaginary parts of these phasors, projected onto the vertical axis, give the instantaneous values of voltage and current. 

In this graph, time is an intrinsic parameter. The time `t` running at the origin represents the real time, and the values at the origin are the real instantaneous values (see note 2). The wave moving towards the right is but just a visual extension of the values that occurred at previous times. For example if `t` at the origin is `25 s`, then the wave height at a point right to the origin represents the instantaneous value of the voltage at some earlier time like `20 s`. It's crucial to understand this.

The different parameters in the graph are as follows:

• `V_{0}`→ Peak Voltage (default: 3 V)
• `I_{0}` → Peak current (default: 1.5 A)
• `k` → Phase difference. (default: 0.523 rad, equivalent to `30^{\circ}`). `k` is in radians, with each unit on the slider representing a `15^{\circ}` increment. Here, as we had discussed earlier, voltage is `k` phase ahead of current. Symbol `k` has been used instead of `φ`. A table of conversion for deg to rad is given below.
• `w`→ Angular frequency (default: 0.8 rad/s)

Now, try adjusting the sliders! For example:

• Set `I_{0}` = 4 A and `V_{0} = 5 V`. Did you see how the circles and the peak values moved?
• Set `k = 1.570`, equivalent to `90^{\circ}`. This models a purely inductive circuit.
• Increase `w` to 2 rad/s. Did the phasors become faster?

You can also hide the input board by clicking on the double arrow next to the settings icon. Use your fingers or mouse to zoom in/out or move the graph. Try continuous sliding to experiment with different values. The graph is yours to enjoy!

(To scroll the web page, swipe outside the graph area)

(If you are using a smartphone or tablet, turn on the 'desktop site' mode on your browser, and rotate your your device to get a better experience of the graph)

You can also access this graph here.

Table for `k`:

Value in degree	Value in radian
`k = 15^{\circ}`	`\frac{π}{12} = 0.261`
`30^{\circ}`	`frac{π}{6} = 0.523`
`45^{\circ}`	`frac{π}{4} = 0.785`
`60^{\circ}`	`frac{π}{3} = 1.047`
`75^{\circ}`	`frac{5π}{12} = 1.308`
`90^{\circ}`	`frac{π}{2} = 1.570`

In India, the standard transmission voltage is 50 Hz 230 V RMS, giving a peak value of 325 V. In the US, the peak value of voltage is about 170 V at 60 Hz. Thus, we see that the practical values are much larger. However, if we plot these values, the waves become very thin and the phasors become extremely fast. This makes them difficult to observe.

For this reason, the upper limits have been to 10 V and 10 A. You can still, of course, try the practical values for fun by typing them manually.

The same understanding stated here can easily be extended to all kinds of phasors.

____________________________________________________

Notes:

1. It is a common misconception to treat impedance as a phasor. `\text{Z}` is not a phasor, since it does not represent a sinusoidally varying quantity. It is also wrong to take `\text{Z} = frac{v}{i}`. To understand this, see that in the general case, for two complex numbers `z_{1}` and `z_{2}`, `frac{z_{1}}{z_{2}} \ne frac{Im(z_{1})}{Im(z_{2})}`.

2. By real time, I mean the time that will be measured by an observer, when she begins her observation across an electrical component. Similarly, by real instantaneous values, I mean the values that are measured by the observer in the real time.

References:

A major inspiration has been taken from 'Fundamentals of Electric Circuits' by Charles Alexander & Matthew Sadiku (5th Ed.).