Electricity takes a long way to our homes. Power plants are located at remote areas mainly due to their polluting and potentially unsafe nature. This is where nature’s energy is converted to 3-Phase electric power and being transmitted via transmission lines at high voltages.

This is a follow up on a previous topic called Transmission Lines & Electricity Distribution. Check it out!

What is 3-Phase Electricity?

Basics of AC electricity generation

Electricity is generated very simply: Have a coil of wire and rotate a magnet around it. The constant change of North and South poles of the magnet will induce an Alternating EMF (voltage) in the coil.

Diagram: Single Phase Electricity Generation bases on Faraday's Law.

Figure 1: Single phase electricity generation based on Faraday’s Law of Electromagnetic Induction. If a coil is present in a varying magnetic field, an EMF will be induced in it. As the magnet rotates, it changes polarity, thus, changes the direction of the EMF. The momentary direction is represented by the red arrow.

EMF is a force that wants to move electrons. For this, we have to have a closed circuit. If this coil is part of a closed circuit, electrons start to flow – which is essentially the flow of current.

Note, all the the coil sees is once a North pole, next a South pole. The North pole has the EMF appear in one direction inside the coil, the South pole the other. Direction matters, so, the electrons will fluctuate. The North pole moves them one way, next they stop and the South pole moves them backwards. This is called Alternating Current (AC).

3-Phase Electricity

In a 3-Phase Electricity Generator there are 3 coils instead of one, and they are 120° apart. This is the most basic setup. In real life, each coil is split up to two coils (same wire) that are on opposite sides of the stator.

  • Called a 6-pole 3-phase electricity generator.
  • The magnet is called a Rotor in a 3-phase machine as it rotates.
    • It is not a permanent magnet, but an electromagnet.
  • The coils are attached to the Stator, which does not move.
Diagram: 3-Phase Electriity Generation

Figure 2: Layout for a 3-Phase Electricity generator. The are 3 coil pairs. Each coil pair is connected in series and are opposing each other. This means that for a particular coil pair the wiring for coil 1 starts from top to bottom, for the other one bottom to top. This is done as they are facing different poles of the magnet at a time. This way the EMF generated will point to the same direction for both coils, reinforcing one another. This is a more efficient design that having only 3 coils.

Time Domain Representation of 3-Phase Voltage

So, as the magnet turns, the North pole initially will start to increase the voltage in the first (red) coil in the positive direction (upward).

The three coils are 120° apart, so some time later, when the North pole arrives at the green coil, it will induce a voltage there too. Same happens with the yellow coil.

The South pole will change the voltages to the negative direction.

Time Domain Representation of 3-Phase.

Figure 3: Time Domain representation of 3-phase electricity. The differences in time between the phases are due to the 120° position offset of the coil pairs.

As we see above, the voltages raise and fall at different times, as the magnet reaches each coil at different times.

In Europe, we have 50Hz AC in the wall, which means the electromagnet turns 50 times around. If you notice, for each phase the period is 20ms. This is the time the wave completes one whole cycle.

Calculation: Period of a 50Hz Waveform.

Benefits of 3-phase

Picture a light bulb. If it is only powered by a single phase what happens when the phase is at 0V momentarily? There will be no current at that second, thus, no light. This would happen every time the waveform crosses the x axis. With 3 phases, we are never at 0V.

Okay, we are not powering a bulb with 3 phases in real life, but some engines we do. Also note, transmitting a single phase would require 2 lines: live and neutral. In 3-phase, all phases share the same, single neutral wire. For long distances we are saving up on wire materials!

Star & Delta 3-Phase Configurations

The 3-phase electricity that is generated at the power plants can be transmitted in two configurations:

  • Star Configuration
  • Delta Configuration

Diagram: Star vs Delta Configurations

Figure 4: Star vs Delta Configuration wiring.

The idea lies in how the Load is wired.

For Delta config, we don’t have a neutral line. To save on materials, it is used for long distances transmissions.

For electricity distribution (between houses), both configuration can be used. Some 3-phase machines operate on Delta, others on Star configurations. Note, the everyday socket has a single phase in it, which is 240V.

Key Terms:

  • Line Voltage: Voltage between phase and neutral.
  • Phase Voltage: Voltage between any two phases.
  • Line Current: Current flowing in a phase in transmission lines.
  • Phase Current: Current flowing through a load (resistor).

Star Config

  • Line Voltage ≠ Phase Voltage
  • Line Current = Phase Current
  • Requires neutral line
  • If well balanced, there is no current in neutral, thus, the neutral conductor is smaller.

Delta Config

  • Line Voltage = Phase Voltage
  • Line Current ≠ Phase Current
  • There is no neutral line

3-Phase Phasor Representation

A phasor is a way to represent the phase relationship between each voltage induced in the coils of the generator. We know now that that the voltages are 120° apart as the poles of the magnet reaches them at different times.

Phasor for Star Configuration

In the phaser diagram, the arrows represent the magnitude of the voltage in each phase. This is 240V in Europe.

Diagram + Equation for Star Configuration

Figure 5: Phasor representation of the voltages in a 3-Phase Star Configuration.

For a Star Configuration, the phase and the line-line voltages are not equal. If we draw the line voltage between two phases (red arrow), it gives us a triangle.

To find the relationship between the line and the phase voltages we have to use trigonometry. For this we need a right angled triangle:

  • Half the triangle to get a right-angled triangle.
  • The 120° angle will be halved too.
  • We know that the sine of an angle is simply the opposite side’s length over the hypotenuse (the longest side in the triangle).
Derivation: Relationship between Line and Phase voltage in Star Configuration

Pop in the calculator ‘sin60‘ and you’ll see it is sqrt(3/2). Rearrange the equation and you will get the relationship between Line and Phase Voltage.

Phasor for Delta Configuration

At the Delta Config, the Phase and Line voltages are equal but the Phase and Line currents are not. The equation is similar as earlier.

Background Theory:
Trigonometry – Breaking down the concept of ‘Sine’ using The Unit Circle

Let’s have a circle with radius of 1 unit.

Any point on the circle will have an X and a Y coordinate as well have a distance of r=1 from the Origin.

Let’s take the X coordinate of the point. We get this by drawing a perpendicular line from the point to the X axis. This line closes a 90° angle with the X axis. This gives us a right angled triangle.

Θ is the angle between r and the x axis.

SinΘ represents the ratio of the Y coordinate and the hypotenuse of the point drawn on the unit circle.

Since hypotenuse = r =1 -> SinΘ = the Y coordinate. We use a circle with r=1 just to make the relationship more visible:

  • If Θ = 0° -> SinΘ = 0.
  • If Θ = 90° -> SinΘ = 1.
The Unit Circle: Demonstrating what the sine of an angle is.

Figure 6: Representation of the Unit Circle where the radius = 1. A point is taken on the circle with X and Y coordinates. This example demonstrates that sine of Θ represents the opposite line (value of Y) over the hypotenuse (r).

The unit circle has 4 quadrants. We can easily have our chosen point in Quadrant 2 for example, although the Θ angle will be obtuse. That is greater than 90°.

What we do in this case is we draw a line once again perpendicular onto the X axis. Instead of Θ, we calculate with 180 – Θ. The magnitude of the green sine wave at each phase above represents the value of the Y coordinate.

Quadrant 3 is the same magnitude as Q1 but negative, and the same is true for Q4.

Power in 3-Phase Systems

We already know that power is the product of the current and voltage: P=IV. At 3-Phase we have 3 phases, thus:

Equation: Power Formula for Purely Resistive Load

But here comes the trick. Up till now, when we talked about ‘load’ we were thinking about a fully resistive load such as a light bulb. When electrons enter a fully resistive load, they get a harder time to pass through and some of the power will dissipate as heat.

This is not the case with induction motors that uses coils.

Reactive Power

There are 3 types of load in AC circuits:

  • Resistive Load
  • Capacitive Load
  • Inductive Load

While power is fully dissipated in a Resistive Load, in Capacitive and Inductive Loads the power will bounce back and froth from generator to the load not doing any active work. We refer to this as Reactive Power.

Voltage & Current Phase relationship in Resistive, Capacitive & Inductive Loads

Figure 7: Voltage & Current phase relationships in Resistive, Capacitive and Inductive Loads.

  • Resistive Load: Current is in phase with voltage.
    • As voltage rises, current starts to flow straightaway.
  • Capacitive Load: Current leads voltage by 90°.
    • A capacitor is essentially 2 plates parallel to each other. It charges and discharges. One plate collects the electrons and let them go. By charging, less and less current will flow across the plates. But this means the potential will grow between the plates.
  • Inductive Load: Current lags voltage by 90°.

The power at any given point is the product of the momentary voltage and current.

  • Active Power (Resistive): Always is in the positive. – useful work as circuit is delivering power.
  • Reactive Power (Inductive & Capacitive): Fluctuates back and forth – delivers power to load, next takes away power from load. – not useful work.
  • Apparent Power: The combination of Active & Reactive Power.
Power in Resistive, Capacitive & Inductive Loads

Figure 8: Power in Resistive, Capacitive and Inductive Load. Since the latter two has 50% of the power in the negative, Purely Capacitive and Inductive Loads do not do active work. What they do is take power away from the source to deliver it to the load, next take the same amount from the load and deliver it back to the source.

Complex Impedance

If a load is not fully resistive, it has an imaginary part called reactance. This reactance is either capacitive or inductive.

A resistor is purely resistive. It has a resistance, which is a real number. If a component is not fully resistive but has this imaginary reactive part to it, instead of a resistance we say it has an impedance.

Z = R + jX

Where:

  • Z: Impedance (a complex number)
  • R: Resistance (a real number)
  • X: Reactance

j is a complex number where j = √-1.

While resistance can only be positive, reactance can be either negative or positive. We can represent this in the complex plane:

Complex Impedance in polar form

Figure 9: Complex Impedance representation in the complex plane. The X axis represents resistance while the Y axis the reactive component.

Inductance has a positive reactance while capacitance has negative. They are both 90° from the real axis for the above mentioned reason where the current is either leading or lagging by 90°.

The load is usually inductive. This means we can represent this the following way, where we have a real component and an inductive component. Using the Pythagoras Theorem we can calculate the magnitude of the impedance which is the hypotenuse of the triangle.

Illustration of the Power Factor

Figure 10: Impedance representation in the complex plane

Power Factor = CosΦ

CosΦ = the ratio of the active & apparent power.

The Power Factor is the ratio of the active power and the apparent power.

The Power in a 3-Phase system:
Formula: Power in a 3-Phase System
  • The power factor in an inductive load is lagging.
  • cosΦ=1 gives us the most power.
  • example: cosΦ =0.8 means the real power is 80% and the reactive is 20%.
  • Inductive devices: example induction motors. Capacitors are placed in parallel to increase the power factor.

Efficiency of a 3-Phase system

Formula: Efficiency of 3-Phase Systems