The Flow of Current

Before we cover AC Voltage, for a few lines let’s go back to the basics . A piece of electrical gadget runs on electricity, which really means electrons run through it. Voltage is a force that makes these electrons flow. Current is another name for the flow of electrons. But there is a twist.

Based on how current flows in a circuit we distinguish between:

  • Direct Current (DC)
  • Alternating Current (AC)

Electrons flow from the negative to the positive terminal. However, they got this wrong the first time and thought current flows the other way. But it was too late to change.

Conventional Current flows in the opposite direction to the electrons, from the positive to the negative terminal.

It is said that current is the flow of holes, which are essentially the empty spaces electrons leave behind.

Difference Between AC and DC

Direct current is simple.

  • We have a battery that has a fixed positive and negative terminal. Connect up a light bulb and now we have a closed circuit, electrons start to flow. The light will come up. This is DC as electrons only flow in one direction.

Alternating Current is a bit more complex.

  • We don’t have a simple battery here. We have a generator instead that converts rotational energy to electrical energy. (water turbines for example)
  • At the end of the prime mover (rotating shaft) there is a magnet. As the magnet turns, it induces an alternating Electro Magnetic Force (voltage) in the surrounding coils.
  • This simply means, that the positive and negative terminals will constantly swap places. This is why voltage alternates 50 times a second. Since voltage is what makes electrons move, current will alternate too.
3-phase Electricity Generator

Figure 1: A three-phase generator. A turbine rotates the prime mover. At the end of the prime mover there is a salient-pole DC electromagnet with 4 poles. This means this magnet has 2 North and 2 South poles. Since this is an electromagnet, it is powered via slip rings (uses brushes). The rotating magnet is the rotor. The rotor is placed inside a hollow metal cylinder – the stator – that has 3 separate coil windings since we have 3-phases. In the picture X marks the ‘in the paper’ direction of the copper wiring and the dot the ‘coming out from the paper’ direction. As the rotor turns, an Electro Motive Force will be generated in all 3 individual coils simultaneously. These coils are120° apart in phase.

Working Principle of an AC voltage Generators

A magnet has a North and a South pole. The only way to induce voltage in the surrounding coils is based on Faraday’s Law of Electromagnetic Induction. For this, we need a constantly rotating magnetic field which we already have as the rotating shaft rotates the magnet.

So, when the magnet turns from North to South, a positive voltage will be induced that moves the electrons to one direction. But when the magnet next turns from South to North, a negative voltage will be induced that pushes the electrons in the other direction.

Essentially, we’ll end up having an Electro Magnetic Force (Voltage) that constantly changes direction, so electrons fluctuate between forward and backward directions.

For more info on how Synchronous & Asynchronous Generators work check out my previous post: Induction vs Synchronous Machines.

Time-Varying Sinusoidal Signal

An Alternating Current (AC) is caused by an alternating voltage. Due to the continuous change of direction, we refer to such electronic signals as time-varying.

The varying voltage (or current) is represented with a sinusoidal signal.

Comparison between DC, AC and AC with DC offset singals

Figure 2: Comparison between DC (a), AC (b) and AC signal with a DC offset (c).

The above figure represents 3-type of signals:

  • DC signal: Voltage does not change over time. At any point of time we have the same voltage. Like a battery’s terminals.
  • AC signal: Voltage changes over time. At t=0 we have different voltage value than example t=1. Like the above mentioned generator.
  • DC/AC signal: This is a kind of signal that is AC but has a DC offset too. Electrons fluctuate back and forth in the cable but overall they also move constantly forward.

Equation of a Time-Varying Sinusoidal Signal (AC Voltage)

For a DC signal we don’t really need an equation. We know that a voltage has a particular constant value and that is about it. For example V = 5V.

AC signals change with time.

When it comes to AC, we need a way to tell the voltage value at a given time. So, if we were to substitute all the relevant values (properties of the wave) and also a chosen time t, we would get the instantaneous voltage value at that very moment.

Alternating Voltage General Eqaution

where:

  • v: the instantaneous AC voltage
  • Vm: the max value of the voltage
  • ω: the angular speed
  • Φ: phase shift

While we write DC signals with capital letter, AC signals are written with small letters.

Breaking Down the AC Voltage Equation

Let’s have a simple wave.

This wave below represents a sinusoid voltage as it varies with time. Sinusoid means it completes a cycle, and next it repeats.

The Sine wave is the most basic sinusoidal wave. At time 0 the voltage = 0V.

This is a Cosine wave below, which is a Sine Wave with a +90° phase shift. So at time 0 the voltage is at its max.

Cosine Wave showing Period

Figure 3: A cosine function. Cosine functions have maximum value at time o. One cycle is referred to as 2pi radians. More on this below.

In the above example the voltage starts off at 3V, drops down to -3V and raises back up to 3V again. Next, this repeats. This signal is said to be periodic.

– one cycle – is one change from 3V to -3V and back to 3V.

– period (T) – the amount of time it takes to complete one cycle. Measured in seconds (s).

– frequency (f) – how many cycles in one second. Measured in Hertz (Hz)

Here is the general equation again:

Alternating Voltage General Equation

1. v – the instantaneous voltage

Small v represents the instantaneous voltage.

We substitute a particular Vm, ω and Φ at a given time t, and we get the value of the voltage at that very moment.

2. Vm – the max value of the waveform

In the example above this is 3V.

3. Understanding the Cosine function

We know that the basic sine function starts at 0 (from the origin). We also know that contrary to this, the cosine function starts at its max (as above at 3V).

These waves are functions.

It simply means that the value of the y axis will depend on the x axis. Above, we have voltage on the X-axis and time on the Y-axis. At different times (different X values) we have different voltages (different Y values).

Okay, Sine and Cosine are functions. Functions mean one thing depends on the other, right? So what is depending on what to get the Sine and Cosine functions?

Let’s represent our AC voltage with this circle below. The Y-axis shows the Voltage as before, but the time now becomes the circumference of the circle. As the time goes, we go around the circle anticlockwise.

So, the X-axis is not the time in the case of the circle anymore. As the Y-axis, it also represents the momentary voltage. When we have a Sine wave we use the Y value, for Cosine we use the X value.

Picture the X-axis as the starting point. When point P is on the X-axis it is time 0. X is at its max (Cosine function), Y is at 0 (Sine function).

Circular motion of Sine and Cosine Waves

Figure 4: The image shows the circular motion of Sine and Cosine waves. The angle Φ represents the angle between the radius r and the X-axis. The radius is a connection between the Origin and point P. At any point around the circle Point P has an X and Y coordinate. If you look at the OPX triangle, sine of angle Φ simply is the opposite side over the hypotenuse. Cosine is the adjacent side over the hypotenuse.

Going Around The Circle

  • Let point P indicate the instantaneous moment in time.
  • We start from P being on the X-axis and we move around the circle anticlockwise.
  • The values on the Y and X axis represent the value of the voltage at a given moment. They will be different as the Y value will draw the Sine function, while the X value draws the Cos function.
  • The radius of the circle is always Vm – the max value of the voltage. In our case 3V.
  • As we arrive back to the X-axis, we completed a cycle, just as before.

At any point around the circle point P has an X and Y value. If we take the above example, we see that the radius r and the X-axis encompass an angle, which we denote as Φ.

The Sine of angle Φ is simply the ratio of the momentary Y value to the maximum Y value (the radius).

The cosine of angle Φ is simply the ratio of the momentary X value to the maximum X value (the radius).

So, in figure 3, as it is a Cosine function, any given point represents the ratio of the monetary value of X on the circle to 3V.

4. Angular Speed (ω)

Angular speed is essentially the speed of point P around the circle measured in radians/sec.

The whole circle is 360º or 2π radians around. They are essentially the same. 1 Radian is a portion of the circle, namely 57.2958°. We’ll look at this later on with a bit more detail.

Angular Speed Formula

The circle is 360° around because 2π *57.2958° = 360°. For now let’s look at the relationship how the angular frequency is related to frequency (f) and period (T).

Angular Speed Formula

Figure 5: Angular Speed and Period equations respectively. Think of angular speed this way: Under what time (T) does the point P go around the circle (2pi radians or 360°). The ω formula has 2pi in the numerator which has unit radians, and T in the denominator which as seconds as units. Therefore, ω will have units of rad/sec. Pi is a ratio, and it represents the number 3.14, but in the formula what we mean in the numerator, is the circle around, 360º, which is 2pi*radians.

Where:

  • ω – angular speed (rad/s)
  • f – frequency (Hz)
  • π – pi (rad) – pi is 3.14 rad or180° in angle. 1 rad is 57.2958° so, 57.2958° * 3.14 = 180°
  • T – period (s) – the time P completes one cycle.

What are Radians?

Let’s recap quickly where the circumference formula comes from. If we take any size of circle, the ratio of the circumference and the diameter (which is 2*radius) is approximately 3.14. This number is called pi (π) and it has no units.

Visual demonstration of Pi = circumference/diameter

Figure 6: Breakdown of pi and the Circumference formula. Pi is 3.14 and does not have units. It is a ratio.

Now, let’s measure out a sector (slice) from the circle where each side’s length is r. One side is curved, but nevertheless it has length r as well.

The angle the two blue radii encompass is 1 Rad and it is 57.2958°.

Visual Demonstration of What 1 Radiian is

Figure 7: Visual representation of 1 Radian. From the Circumference formula we know that is equal to the radius’s length 2pi times. So if we multiply the red curve above 2*3.14 times we go around the circle once. From the diagram it is obvious that similarly if we take the 1 Radian 2pi times around, we complete the circle once again. In other words we complete 360°. This means 360° = 2pi Radians.

4. Phase (Φ)

Here is the general equation for a time varying wave again:

Alternating Voltage General Equation

The phase shifts the entire wave either to the left or right.

  • Negative phase values shift the wave to the right
  • Positive phase values shift the wave to the left

Now, let’s expand ω in the general equation:

Alternating Voltage General Equation with Angular Frequency Expanded
  • If we chose T=2π this means that 2pi radians are completed under 2pi seconds.
  • So 1 radian is completed in 1 sec.
  • Since the whole circle is 2pi radians around, it is completed under 2pi seconds, which is 2*3.14 = 6.28 seconds.
  • Considering there is no phase shift, this is the most basic Cosine formula: v=cos(t). Let’s plot it:

v = cos(t)

Plot: v = cos(t)

Figure 8: T=2pi sec. This means 1 cycle is completed in 2pi seconds which is 6.28 seconds. This is clear on the graph. From the function v=cos(t) it is also clear that Vm=1V and Φ=0°. ω=1rad/sec is T=2pi sec. Vm is the max amplitude of the wave, therefore, the max Y value is 1V. The X-axis represents time. So, 6.28 sec on the X-axis is also 2π rad. Graphical calculators also give us the option to use degrees instead of radians. What they do, is they multiply 2π by 57.2958. That way the X-axis is in degrees instead of radians.

Increasing the angular frequency

Point P can go around the circle more than one time.

With 2 cycles it completes 2×320 = 720°. Here is where it gets a bit confusing. As we increase the frequency, we complete more cycles within 1 sec. Although, 2pi radians (or 360°) is always at the same position on the X-Axis.

How can Point P complete 4 cycles in 360°? Well. the numbering on the X-axis is a reference to v=cos(t). When we use the general equation, we care about the value of the voltage on the Y-axis at a certain time on the X-axis. It is not really relevant whether the point P completed 360° or 720°. Point P will be at the same position around the circle regardless.

v = cos(2πt)

Plot: v = cos(2πt)

Figure 9: T = 1 sec. Since f=1/T the frequency = 1Hz. This is also clear from the graph as one cycle is completed under 1 sec.

v= 3cos(4πt)

Plot: v= 3cos(4πt)

Figure 9: T = 1/2 sec and f=2 Hz. 2 cycles are completed in 1 sec.

v = cos(t-1rad)

v = cos(2πt-1rad)

Plot: v = cos(t-1rad) and v = cos(2πt-1rad)

Figure 10: Comparing phase shifts (in radians) between two waves with different ω values. If ω=1rad/sec one cycle is 6.28 seconds (or radians). Having a -1rad/sec shift simply offsets the wave towards the right by 1 sec (or rad/sec). However, having ω=2π rad/sec with -1rad/sec shift the wave shifts to the right by 1/2π seconds (or radians).