Series And Parallel Resistors

• Electrical components can be connected together in two basic ways: parallel and series. When resistors are connected in series (figure 2.3) a fraction of the supply voltage is dropped across each resistor and each resistor dissipates some of the total energy from each coulomb of charge. 
• The total resistance (RT) of the circuit (ignoring the resistance of the wires) is the sum of the resistances and the same current flows through each resistor.

Temperature And Resistance

• When a 2V supply is connected to a 60W, 240V bulb it draws a current of 25mA, its resistance is

                                              Thus R=U/I = 2/0.025 =80ohms

• When the same bulb is connected to the correct 240V supply it glows white hot and draws a current of 250mA and its resistance is:

                                             R=U/I = 240/0.250 = 9600ohms

• Thus we can see that a larger current has caused the bulb to get very hot and its resistance has increased by 12 times. To take account of the change in resistance with temperature the temperature coefficient of resistance (α) is used. 
• α for a material at 0°C is the change in resistance of a 1Ω sample of that material when the temperature increases from 0°C to 1°C. Matters are complicated furtherer because it is not easy to measure the resistance of a conductor at 0°C hence, a value for α is often quoted for a temperature increase of 20°C to 21°C. 
• Table 2.2 contains the temperature coefficient of some metals.

•  We can see from the above table that when the temperature increases from 20°C to 21°C the resistance of a 1Ω copper resistance will increase to 1.00396Ω. Thus: 

                                                          Rt=Ro(1+αT)
where:
Rt = total conductor resistance at T (Ω)
R0 = resistance of a conductor at 0°C (Ω)
α = temperature coefficient of resistance
T = temperature (°C)
And: Rt=R20(1+αT)
where:
R20 = resistance of a conductor at 20°C (Ω)
If the resistance of a conductor is not known at the temperature for which α is known the following method can be used:
R1 = the conductor resistance at temperature T1
R2 = the conductor resistance at temperature T2
R1=Ro(1+αT1) and R2=Ro(1+αT2)
Dividing: R1/R2 = (Ro(1+αT1))/(Ro(1+αT2)) = (1+αT1)/(1+αT2)

Hence:
                                   R2=(R11+αT2)/(1+αT1)
Therefore the value of R0 has been eliminated from the equation.

Resistivity (also known as specific resistance)

• When electrons flow through a wire they experience resistance and lose energy, the furtherer along the wire they flow the more energy they lose, therefore, we can say that the total resistance of a wire is proportional to its length.
• Since the electrons are evenly distributed throughout the wire and since the current is the rate at which a charge passes any point on that wire, we can see that to provide any specific current the electrons in a wider wire will have to flow a shorter distance than electrons in a narrower wire (figure 2.2). 
• We can therefore say that resistance is inversely proportional to the conductor's cross-sectional area. 
• Therefore thicker wires have less resistance per meter and will cause less energy to be lost as heat.

• Putting the previous two concepts together given us:

                                           R~1/a (*proportional)

• where l is the length of wire, a is the cross-sectional area and α means proportional to. The resistivity (ρ) of a material is defined as the resistance between opposite faces of a cube of that material with a given side length. 
• ρ is very small for most conductors and is usually quoted in micro-ohms (μΩ) for a 1 meter cube expressed as μΩm. For example aluminium has a resistivity of 0.0285μΩm. 
• Thus the resistivity of a wire is proportional to the resistivity of the material from which it is made. We can combine resistivity with the previous equation to give:

                                                                    R=ρl / a

• Note that the resistance calculated from this equation will be given in the same units of resistance used for the resistivity (i.e. μΩ). Also l and a must be in the same length units as ρ so that if ρ is in μΩm l must be in m and a must be in m2. Table 2.1 contains the resistivities of some metals.

Example:
Calculate the resistance of 1000m of 16mm2 single annealed copper wire.
From the table: ρ = 17.2μΩmm (since the cross sectional area is given in mm2)
l = 1000m = 10^6m
R=ρl / a = (17.2*10^6)/16 =1075000μΩ=1.075Ω

• All wires and cables have some resistance, therefore there will always be some energy lost and a voltage drop within. Thin wires will get very hot and may burn out if they carry too much current, they may also cause such a large voltage drop that equipment may not function properly. 
• Thick wires will reduce these phenomena however, because they contain more copper they will be considerably more expensive than thin wires. Voltage drops in cables are normally recommended to be no more than 4% of the supply voltage.
• That means a 9.6V drop for a 240V supply or a 4V drop for a 110V supply. Note that you can only measure a voltage drop across a length of cable when a current is flowing in it.

Resistance And Ohm's Law

• Electrical resistance is a little like friction and electrons flowing in wires lose energy overcoming it. 
• All conductors have some resistance and the larger the resistance, the larger the amount of energy dissipated within it as electrons flow, and therefore, the greater the energy needed to move electrons around a circuit. 
• Hence, if a fixed voltage supply is connected to a circuit with a low overall resistance more current (i.e. coulombs of charge per second) will flow than if the same supply is connected to a higher resistance circuit. 
• Some electrical components, such as water heaters, are designed to have a large resistance so that a lot of electrical energy is converted into heat as current flows through them. 
• Wires and cables on the other hand should have a low resistance so that not much energy is lost when electricity is supplied from one place to the other. 
• The unit of resistance (R) is the ohm (Ω) and one ohm is the resistance that causes a drop of one volt when a current of one ampere flows. 
• For a metallic conductor which remains at a Constance temperature Ohm's law applies:

                                              R=U/I          I=U/R         U=I*R
Components in circuits that are designed to have a fixed resistance are usually shown as a simple
box, as depicted in figure 2.1.

Example
An electrical heater is used on a 240V supply draws a current of 12A. Its resistance is:
R=U/I=240/12=20ohms.

Electrical Circuits, Voltmeters And Ammeters

                    Figure 1.2: (a) a closed circuit where current (I) can flow; (b) an open circuit where no current can flow.

•  The path around a circuit must be unbroken for an EMF to push electrons around it, such a circuit is said to be closed. If a switch is included in the circuit it can be used to break the path and produce an open circuit where no electrons can flow (figure 1.2a and 1.2b).
• In a high voltage circuit, current may arc across a small gap causing considerable damage. Most switches will arc briefly when opened however, as long as the current doesn't jump the gap for too long, it is not a problem. 
• Voltmeters measure a potential difference between two points and so must be connected in parallel. If you take a voltage reading across a supply in a open circuit you will be measuring the supply's EMF, you can only measure voltage drops across a load (e.g. a piece of equipment) if a current is flowing. Figure 1.3 shows the placement of voltmeters in a circuit. 
• Ammeters measure the current flowing through a point in a circuit, therefore they are connected in series. Figure 1.4 shows the placement of a voltmeter in series.

Electromotive Force, Potential Difference And Voltage

• Moving electrons through a conductor as an electrical current requires energy because, as the electrons move they constantly dissipates energy (as heat). 
• This energy can be supplied by many means including batteries or generators. In the case of a battery chemical energy is converted into electrical energy, whereas a generator converts chemical energy to mechanical energy (through combustion) before turning it into electrical energy. 
• For historical reasons the amount of energy needed to move one coulomb of charge around a circuit is called the electromotive force or EMF (E). 
• The unit of EMF is the volt (V) and one volt is one joule per coulomb, thus:

                                              EMF (V) = Energy (J)/Charge (C)
                                                         E = W/Q
Example:

• A battery with an EMF of 6V can supply a current of 5A round a circuit for five minutes. How much energy is provided?

                    The total charge transferred in 5 minutes: Q = It = 5 * 5 * 60 = 1500C
                    The total energy supplied to do this: energy = E * Q = 6 * 1500 = 9000J

• When you lift a ball to a height of two meters above the ground you give it potential energy. When the ball is dropped this potential energy is converted to kinetic energy until, at the point when it hits the ground, there is no potential energy left. 
• A coulomb of charge travelling around a circuit behaves in a similar way. Each coulomb of charge leaving from a battery with an EMF of 6V has six joules of potential energy. 
• By the time the charge reaches the battery again, having travelled around the circuit, it will have dissipated all six joules and posses no potential energy. 
• The amount of energy expended by one coulomb of charge when travelling between any two points in a circuit is known as the potential difference (PD) between those points. Since the potential difference is a number of joules per coulomb the unit of PD is the volt (i.e. the same unit used for EMF). 
• The PD between two points is thus called the voltage drop and has the symbol U (the symbol V used to be used). Therefore, from the previous equation we can say:

                                                                               U = W/Q
Example:

• How much electrical energy is converted into heat each minute by an immersion heater which takes 13A from a 240V supply?

                       Energy = lost by each coulomb = 24J
                      Quantity of charge flowing per minute: Q = It = 13 * 60 = 780C
                      Energy converted in one minute = U * Q = 240 * 780 = 187 200 = 187.5kJ

Energy, Work and Power.

• Generally speaking energy can either be useful or wasted. For example energy can be used to lift a weight but energy can be lost as heat because of friction. 
• The useful bit of energy is called work and if we lift a weight we are said to have 'done work'. Work is linked to force by the following equation:

                   work done (J) = distance moved (m) * force for movement (N) 

• In this equation force is measured in newtons (N), distance is measured in meters (m) and the work done is measured in joules (J), which is the unit normally used for any energy. 
• To lift an object you need to over come the force of gravity with an equal force acting is in the opposite direction. For this reason the 'force for movement' will be equal but opposite to the force opposing the movement.
• It is important to note the difference between mass and weight. An object's mass is due to the amount of matter it contains and has the units of kilograms (kg). Weight is the force that an object exerts downwards towards the ground and is measured in newtons (N). 
• The weight of an object can be calculated by multiplying its mass by the acceleration due to gravity which is 9.8 meters per second per second (often rounded up to 10 m/s2).

Example:

• How much energy (or how much work) is needed to lift a 70kg man off the floor to a height of 3m (using the acceleration due to gravity = 10 m/s2).

                             mass weight = 70 * 10 = 700N                   work done = distance moved * opposing force = 3 * 700 = 2100J

• Power is the rate of using energy or doing work. Lifting the man in the previous example to 3m will always require the same amount of work but the quicker you do it, the more power is needed.

                           power = work or energy /time = joules/seconds
                           P = W/t

• The unit of power is the watt (W, not to be confused with the symbol for work). 
• One watt is the power used supplying one joule per second. The symbol W is also used for energy used or work done, so be careful not to get confused with W used as the units of power.

Unit Of Charge And Current.

• Each electron carries the same amount of negative charge and we could use the charge on a single electron as the unit of charge. 
• However, the charge on an electron is tiny and electric currents contain many electrons flowing through a wire. 
• The unit of charge is thus the coulomb (C) and one coulomb is approximately equal to the total charge of six million million million electrons (to be more exact the charge on one electron is 1.6x10-19, therefore 1C comprises the charge of 6.25x1018 electrons). 
• If electrons flow past a point on a wire so that 1 coulomb of charge passes every second, a current of one ampere (A) is said to flow. 
• Therefore, electrical current is the rate at which charge passes through a wire. Mathematically we can say:

Where:
Q = the charge transferred in coulombs (C)
I = the current in amperes (A)
t = time during which the current flows in seconds (s)

The Atom And Electrical Current.

• All matter is made of atoms. Atoms are very small and when placed in a straight line about 8 million will fit into 1mm. 
• Atoms themselves are made up from three particles: electron, proton and neutrons. Electrons and protons have equal but opposite charges; the electron has a negative charge and the proton has a positive charge. 
• The neutron has no charge. Charges behave like magnetic poles in that alike charges repel and unalike charges attract. 
• The protons and neutrons make up the nucleus which is at the centre of the atom and the electrons reside around the outside (figure 1.1). 
• Normally an atom will have equal numbers of electrons and protons so that as a whole the atom has no overall charge. 

•  In solid materials that are conductors the electrons are free to move between atoms. The movement may be random in all directions throughout the conductor however, if the electrons move predominantly in one direction an electric current flows. 
• Since the protons are at the centre of the atom they can not break free, therefore electricity is the flow of negative charge and not the flow of positive charge. 
• Because of these phenomena any region of negative charge is formed by an excess of electrons gathering and a region of positive charge is due to electrons moving away, leaving a deficit. 
• Solid materials, where the electrons are not able to move are insulators and such materials do not conduct electricity. 
• Materials that conduct electricity under certain conditions are called semiconductors. In a piece of metal, a wire for instance, electrons are free to move in any direction. 
• However, as soon as a battery is connected to both ends of the wire the electrons will move towards the positive battery terminal. For every electron that leaves the wire through this terminal, another enters the other end of the wire from the negative terminal; thus the total number of electrons in the wire remains the constant. 
• This flow of electrons from a negative terminal, through the wire, to the positive terminal is called an electric current. For historic reasons, in most circuits electricity is considered to flow from the positive terminal to the negative terminal; in fact for most circuits it does not matter which direction you consider the electricity to flow. 
• When a current flows in a conductor two important things happen:

1. heat is generated,
2. a magnetic field is set up around the conductor.