### Table of contents

1. Direct Current

2. Ohm's Law

3. Alternative Current

4. Effective Value

5. Phase Difference

6. Impedance

7. Ohm's Law for AC

8. Power

9. Three-Phase

### Direct Current

Analogy between the potential difference or the voltage and difference of water level of which static pressure results can be made. The pressure will enable a water flow. In the same way, the voltage will enable a flow of electrons called current.

If the liquid is more viscous than water, it will be harder for it to flow. Morever, its circulation will as more difficult as pipes’ cross sectional area is low. The circuit will have a certain resistance to the flow. The same phenomenon occurs in **electricity**: an **electrical resistor** opposes the flow of **current**.

When the current circulates only in one direction, it is called **direct current**

The basic law is Ohm’s Law: V=RI

Voltage V is expressed in Volts (V)

Current I is expressed in Amps (A)

Resistance R is expressed in Ohms

### Ohm’s Law

It can be imagined that the flow of electrons that constitute the current will have more difficulty to pass through a long conductor with a low cross sectional area than a small conductor with a larger cross sectional area. Opposition to the current flow is the resistance.

The flow of the current through a resistor releases heat (Joule effect) with a power such as P = RI².

**Energy and power**

Time takes part in energy calculation which is expressed in Joules, but also in kWh for electricity bills. 1 kWh corresponds to 3,6 MJ.

Time does not take part in power calculation.

The thiner a conductor is, the larger its resistance will be. According to the formula, for a same current, a thin conductor will dissipate more power in the form of heat (Joule effect) and thus will get warmer .

The current that can pass through a given cross sectional area is thus limited

### Alternative Current

The current circulates alternatively in a direction then in the other.

Used in the energy distribution of 50hz or 60hz in sinusoidal AC Current.

One complete cycle at 50hz lasts 0.02s = one period.

**50Hz sinusoidal current**

The angular frequency is linked to the frequency with a simple relation. At a frequency of 50 Hz, angular frequency is around 314 rad/s

### Effective Value

The effective value of an alternative quantity (voltage or current) corresponds to the direct quantity which would dissipate the same power.

When the value of an alternative quantity is expressed, it is always its effective value. For example, the voltage at domestic electrical plugs is 230V (effective value).

### Phase Difference

Example of some receivers : A coil is a winding of wire which will create a magnetic field.

This coil will oppose the AC current flow and will have an inductive effect , i.e. the current that passes through it will be in phase lag toward the coil ’s voltage

The reverse occurs in a capacitor: the current will be in phase lead toward the voltage

A pure resistor does not phase shift the current

In AC, vectorial sums of the electrical quantities have to done in order to take into account their phase differences.

### Impedance

**Basic Receivers**

is the argument of the impedance. If the voltage is set at the phase origin ( = 0). The current has a phase shift of ° to the voltage. For a capacitor, the current will have a positive phase shift with the voltage and therefore be in phase lead. For an inductor, the phase difference will be negative, the current therefore being in phase lag.

Pure resistor: the impedance is the value of the resistance.

In the case of capacitors and inductors, it can be noticed that the impedance varies with frequency, because the term (OMEGA) appears.

The value of a capacitor’s capacitance is expressed in farad (F).

For a capacitor, the impedance will be all the lower as frequency is higher and the value of the capacitance is high. its effect is purely reactive.

The value of an inductance is expressed in Henri (H).

The Impedance of an inductor is all the larger as frequency is higher and inductance is higher

Perfect capacitor and inductor have only one reactive impedance and the current is 90° phase shifted.

### Ohm’s law of AC

The formula as it is presented, does not show the phase differences. It supplies information only about the effective values.

The impedance Z can be a coil, a capacitor, a resistor or a combination of the three.

**Example circuit**

The current I is in-phase with Ur, in lag quadrature with UL and in °- phase lag toward U

### Power

**Active Power**

It can be noticed that for a given voltage and current, the active power is maximum when cos = 1 or = 0.

By re-working on the previous RL circuit, It can be noticed that the inductive component does not consume any active power

It is the power produced by the part of the current in-phase with the voltage

**Reactive Power**

The reactive power is consumed by the circuits that have an inductive character, for the magnetization of these circuits.

On the other hand, it is provided by the receivers including a capacitive component.

It is not transformed into output power (mechanical, heat) and causes the circulation of an additional current in the lines. These lines must be able to carry this current.

So, if the reactive power becomes too important toward the active power, it will be necessary to oversize the lines, the transformers for a same active power, which is useless

This power in the lines have to be composated to the maximum, i.e. it is necessary to have the best cos . A common value for cos is at least 0,03

In this circuit, there is only an inductive component which is involved in reactive energy.

Q is positive because a coil consume reactive energy unlike the capacitor which provides

it.

It can be imagined the possibility to compensate the energy by including a a capacitor

With a capacacitor, the power is calculated in the same way.

As the capacitor condensateur provides reactive reactive power, a (–) minus sign is given to the reactive power.

That ’s why there is a (–) minus sign in the formulas.

**Representation and relation**

**Power conservation principle**

When the active power is consumed , a positive sign is assigned to it.

If there are several receptors connected to the network, the active powers can be simply added

**Power Factor**

### Three phase

**Line to Neutral and Line to Line Voltage**

U1, U2 and U3 are the line-to-line voltages or phase-to-phase voltages or line voltages

V1, V2, V3 are the line-to-neutral voltages or phase voltages

The example is a value of 230 V for V and of 400 V for U

**Wye Coupling**

In Wye coupling, the receptor is subject to line-to-neutral voltages.

The value of the line current is the same as the value of the phase current.

There is one common point to the three receivers

If the network is balanced ( 3 same voltages with a 120° – phase shift) and the receptor is balanced ( 3 same loads), there is no current in the neutral and the voltages are perfectly balanced. There is no need for a neutral cable.

If case of imbalance, over-currents and over-voltages can appear if the neutral is not connected.

If the neutral is connected, the voltages an the currents will remain the same and a current will circulate by the neutral to compensate for imbalances.

**Delta Coupling**

In Delta coupling, the receptor is subject to line-to-line voltagethe value of the line current is root of 3 times the phase current.

There is no more common point , therefore there is no possibility of connecting the neutral

These formulas are valid only in balanced state ( case of motors)

Compared to the single phase, the only difference is the factor 3 or root of 3, it depends if it ’s the line-to-neutral voltages or the line-to-line voltages that are used

**Unbalanced Three Phase**

The balanced network I1, I2, I3 is a direct-succession system, because the order of succession is 1 then 2 then 3.

The system Id1, Id2, Id3 constitutes a balanced system with direct succession. In the case of motors, these are the components which create the engine torque.

The inverse system will create a torque on the opposite side which will slow down the motor and will cause an additional overheating.

The effect of the homopolar current is to generate an overheating of the machine..

The system of homopolar currents only exists if the neutral is distributed. In this case, each homopolar current ’s value is the third of the value of the current in the neutral.

When a machine has a fault current at the ground which ends at the neutral, it can be detected by measuring the homopolar current