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# Vectors and Fields

The use of vectors and vector fields greatly simplifies the analysis of many electromagnetic (and indeed other) systems. Due to their usefulness, these concepts will be used extensively in this book. For this reason it will be useful to begin with a general treatment of the subject.

### Vectors[edit | edit source]

A vector is a quantitative which has both magnitude, direction and sense. The magnitude represents the vector's size or physical quantity. The direction represents the vector's position with respect to a reference axis. The sense represents the vector's orientation and it is represented by its arrow head. This contrasts with the definition of a scalar which has only magnitude. Examples of scalar quantities include temperature, resistivity, voltage and mass. In comparison, examples of vector quantities would include velocity, force, acceleration and position.

The most familiar and intuitive use of vectors is in the two-dimensional (x, y coordinates) or three-dimensional (x, y and z coordinates) Cartesian coordinate system.

### Fields[edit | edit source]

The term *field* has a general meaning in mathematics and physics, but here we will be referring only to the special cases of scalar and vector fields. Generally, a field is a region in space where the quantity in question, exist and its influence is being felt. A scalar field is a region of space in which each point is associated with a scalar value. A classic example of a scalar field is a temperature field in a heated block of material.

If some heat source is applied to a cube of a conductive material, such as a metal, the temperature in the block will be highest where the heat source is applied, dropping off as we move away from the source in any direction. At every position inside the block a value could be assigned which is the temperature at this point. These temperature values make up the scalar temperature field in the block. It might be that it is possible to model accurately these values with some mathematical function, but the field itself is simply the variation in the scalar quantity in the space occupied by the block.

A vector field differs from a scalar field in that it has not only a magnitude at every position, but also *direction*. A good example of a vector field is the velocity of the fluid flow in a winding river of changing width. Clearly at every point in the river, the velocity of the fluid will have a magnitude (the speed) which will be lower where the river is wide, and higher where the river is narrow. However, the flow will also have a direction which changes as the water is forced around the river bends. If we noted the fluid speed and direction everywhere in the river, the result would be a vector field of the fluid flow.

In addition to varying in space, fields can also vary in time. In the first example, if we started with a cold block and then applied the heat source, mapping the temperature field at set time intervals, it would be seen that the values of the temperature at every point would change as the heat conducted throughout the block over time. The result therefore is a scalar field that varies in the three dimensions of space and one of time.

# Magnetic Field Concepts

## Magnetic Field Concepts[edit | edit source]

An electromagnetic field is a region of space in which electrical charges experience forces. The classical definition of the electromagnetic field is given by the following equation

What this equation states, is that an electrical charge of size q experiences a force when in the presence of an electric field denoted by **E**. Furthermore if the charge is moving, with velocity **v**, it will experience a further force if in the presence of a magnetic field denoted by **B**. Therefore the force on a electric charge *defines* what the electric and magnetic fields are in a given region of space. Without the presence of a charged particle it could never be known what their magnitudes or directions were.

#### Further Reading on Magnetism[edit | edit source]

- See Magnetic field and Magnetic phenomena and Physics Study Guide/Magnetism and Permanent magnets and Current's magnetic effects and Electromagnetism and Magnetism's units.
- Wikipedia: Magnet and Wikipedia: Magnetic forces and Wikipedia: Magnetism and Wikipedia: Magnetism's origin

#### Further Reading on Electric Current[edit | edit source]

- See Current and Kirchoff's Current Law (KCL) and Current (Physics) and Relationships: voltage, current and resistance.
- Wikipedia: Current and Wikipedia: DC and Wikipedia: AC.

# AC Motors and Generators

- See Single-phase motors and generators, and 3-phase motors and generators, and AC basics, and also this.

## AC generators[edit | edit source]

- A very simple AC generator consists of a permanent magnet that rotates inside a coil in such a way that the N-pole and S-pole alternate as seen from the coil. An analog voltmeter (or rather a millivoltmeter?) that has its zero at the middle of the scale is connected to the ends of the coil. As the magnet is rotated the voltmeter moves first one way, then the other way. The speed of rotation determines the number of "cycles per second", called Hertz(Hz). A rotation speed of 3000 revolutions per minute(RPM) produces 50 Hz, and 3600 RPM produce 60 Hz.
- The rotating permanent magnet can be replaced by another coil that is fed by DC and acts as an electromagnet. Doubling the number of coils will double the number of, what is called "the poles", and then only half the rotation speed is required for a given output frequency.
- See also Wikipedia: Alternator

AC generator works on the principle of Faraday's laws electro-magnetics.

## AC motors[edit | edit source]

AC motors are generally divided into two categories, induction and synchronous motors. The most common AC motor is the "Squirrel cage motor", a type of induction motor. These have only one or more coils within which a special kind of mechanical rotor is free to rotate. There is no electrical connection to the rotor from the outside. The general formula to determine the synchronous speed of an induction motor is

For induction motors, this is a theoretical speed, even though it will never be obtained. The motor will always run slower than synchronous speed with a slip of S. If a motor were to be operated at full synchronous speed, the relative speed of the rotor to the stator would be 0, making it impossible to induce a voltage (Faraday's law) in the rotor windings. This in turn would make the flow of current impossible. Without current no magnetic field can be generated.

Most AC motors require a starter, or method of limiting the inrush current to a reasonable level. Types of motor starting include reactive (capacitor start and inductive start), and electronic (frequency drives and soft start drives). The reactive start method is usually used on fractional horsepower motors, and the electronic method is usually reserved for larger motors (cost of the drives is the main reason for this). Connecting these motors to computers, PLC's (programmable logic controllers), and interfacing with automation systems, is becoming more prevalent.

## DC Generators[edit | edit source]

DC generators are basically AC generators whose output voltage is switched the other way round at the proper moment, so that the direction of the voltage is always in a single direction. But the magnitude of the voltage keeps changing, just as it does in an AC generator, and it can be said that the output of a DC generator is DC plus a "superimposed" AC voltage, called "ripple". Connecting a capacitor across the output terminals reduces that ripple. See also Wikipedia: "Testatika" Electrostatic generators

## DC Motors[edit | edit source]

Direct Current (DC) motors have a "Commutator" that switches the part of the coil that is closest to the poles at the time, more or less similar to the legendary "donkey" that tries to catch the carrots, but never succeeds. See the very simplified commutator shown in blue.

Usually a commutator has many "segments", as many as there are taps on the coil. Starting a DC motor requires often an external resistor or rheostat to limit the current. The value, in Ohms, of that resistor is reduced in steps as the speed of the motor increases, until finally that resistor is removed from the circuit as the motor reaches close to its final speed. See also Wikipedia: Car starter

## Other electric motors[edit | edit source]

#### Universal[edit | edit source]

See Wikipedia: Universal motors

### Stepper[edit | edit source]

A stepper motor is a brushless, synchronous electric motor that can divide a full rotation into a large number of steps, for example, 200 steps.

See Robotics: Stepper Motors and Wikipedia: Stepper motor

## Exotic motors[edit | edit source]

### Electrostatic[edit | edit source]

### Superconducting[edit | edit source]