SUPER CONDUCTIVITY:
The electrical resistivity of many metals and alloys drops suddenly to zero when the specimen is cooled to a sufficiently low temperature often a temperature in the liquid helium range. This phenomenon is called superconductivity. At a critical temperature Tc the specimen undergoes a phase transition from a state of normal electrical resistivity to a superconducting state Fig 17. This phenomenon was observed by Heike Kamerlingh Onnes in 1911.
The superconducting state is on ordered state of the conduction electrons of the metal. The order is in the formation of loosely associated pairs of electrons. The electrons are ordered at temperatures below the transition temperature and they are disordered above the transition temperature Tc.
Occurrence of superconductivity
Superconductivity occurs in many metallic elements of the periodic system and also in alloys, intermetallic compounds and doped semiconductors.
Tc values of some elements, alloys and compounds are given in table (2.2)
Electrical Resistivity in Alloys
The resistivity of a pure metal increases at a particular temperature when we add more and more impurities or when we make alloy. So the resistivity of alloys depends upon the percentage of impurity material in the base material at a particular temperature.
According to Nordheim's rule, at a particular temperature, the residual or ideal resistivity of an alloy can be written as
High resistivity alloys
High resistivity alloys is made by mixing of various metals with appropriate percentage and temperature.
1. Nichrome
79.80% Ni, 19.20% Cr, 1.15% Mn and small amount of iron, used as heating elements in heaters and furnaces.
2. Manganin
86% Cu, 12% Mn and 2% Ni used as standard resistances.
3. Constantan
60% Cu and 40% Ni used as thermocouples, rheostats and starters for electric motors.
4. Kanthal
69% Fe, 23% Cr, 6% Al and 2% Co, used as heating elements in heaters and furnaces.
Conducting Materials
The choice of a conductor for a given application depends not only on its electrical conductivity, but also on various other parameters, such as corrosion resistance, mechanical strength and workability. Copper and aluminium are the most widely used high conductivity materials, of the two, copper is more widely used. Only recently is aluminium being used as a substitute, particularly in view of the increasing price of copper. Hard copper is used in bus bars, commutators segments, contact functions like switches and relays while soft copper is used for magnet wires, cable strands etc.
Copper is the most widely used material as it has (i) the highest conductivity next only to silver, (ii) can be drawn into wires, strips or rolled into sheets (iii) fairly resistant to corrosion and (iv) has good mechanical strength. Generally in electrical power utilities copper of high purity better than 99% is required, very often purity of the order of 99.9% or better is needed. The impurities present affect both electrical and mechanical properties. 0.2% of Fe or 0.3% of As brings down the conductivity to 50% as shown in Fig. 14. Here the oxygen content must be less than 0.1%. Copper with total impurity content of less than 0.5% and of oxygen less than 0.02% has excellent mechanical properties. Such copper is used to draw thin wires. The wire produced by a cold working process has light tensile strength. Annealing produces soft copper having good ductility and low hardness.
Aluminium is progressively replacing copper in many applications. The electrical resistivity of aluminium is about 1.6 times that of copper and it has about one-third the density of copper. Also, linear coefficient of expansion, specific heat and melting point are higher. In mechanical properties it is inferior to copper. For the same resistance and length of wire, aluminium has half the weight of copper. Impurities have a similar effect on the electrical resistivity as in copper. An oxide film formed on the surface of aluminium prevents further corrosion of the material.
Classification of solids on the basis of band theory
A solid is determined as a conductor, insulator or semiconductor by the energy band structure.
Conductors
Thus if there are N atoms in a solid piece of sodium, its 3s valence band will contain N energy levels can hold 2N electrons. Thus the 3s band in sodium is only half filled by electrons and the Fermi energy EF lies in the middle of the band. When a potential difference is applied across a piece of solid sodium, 3s electrons can pick up additional energy while remaining in their original band. The additional energy is in the form of KE, and the drift of the electrons constitute an electric current. Sodium is therefore a good conductor.
Insulators
Semiconductors
Silicon has a crystal structure like diamond; a gap separates the top of its filled valence band from an empty conduction band above it. The band gap in silicon however is only about 1eV wide. At low temperature silicon is little better than diamond as a conductor, but at room temperature a small number of its valence electrons have enough thermal energy to jump the forbidden band and enter the conduction band.
These electrons, though few, are still enough to allow a small amount of current to flow when an electric field is applied. Thus silicon has a resistivity intermediate between those of conductors and those of insulators and is called as semiconductors. The energy band diagram of semiconductor is given in Fig. 13.
Band Theory Of Solids
The energy band structure of a solid determines whether it is a conductor, an insulator or a semiconductor.
A solid contains an enormous number of atoms packed closely together. Each atom, when isolated, has a discrete set of electron energy levels 1s,2s,2p,....... If we imagine all the N atoms of the solid to be isolated from one another, they would have completely coinciding schemes of their energy levels.
Let us study what happens to the energy levels of an isolated atom, as they are brought closer and closer together to form a solid. If the atoms are brought in close proximity, the valence electrons of adjacent atoms interact. Hence the valence electrons constitute a single system of electrons common to the entire crystal with overlapping of their outermost electronic orbits. Therefore, the N electrons will now have to occupy different energy levels. This is brought about by the electric forces exerted in each electron by all the N nuclei. As a result of these forces, each atomic energy level is split up into a large number of closely spaced energy levels. A set of such closely spaced energy levels is called an energy band.
For example, the 11 electrons in a neutral sodium atom, each occupy a specific energy level as indicated in fig.10.
In the energy band, the allowed energies are almost continuous. These energy bands are in general, separated by regions, which have no allowed energy levels. These regions are known as "forbidden bands" or 'energy gaps'.
The amount of splitting is not the same for different levels. The levels filled by the valence electrons in an atom are disturbed to a greater extant, while those filled by inner electrons are disturbed only slightly fig.10.
If there are N atoms in a solid, there will be N allowed quantum states in each band. Each quantum state can be occupied by a maximum of two electrons with opposite spins. Thus each energy band can be occupied by 2N electrons. The band formed from the atomic energy levels containing valance electrons is called valance band. These electrons have the highest energy. Above the valance band, there is the band of next higher permitted energies called the 'conduction band'. The conduction band corresponds to the first excited states; electrons can move freely in this band and are called 'conduction electrons'. The interval between conduction band and valence band in which electrons cannot occupy is called 'Forbidden gap'.
A solid contains an enormous number of atoms packed closely together. Each atom, when isolated, has a discrete set of electron energy levels 1s,2s,2p,....... If we imagine all the N atoms of the solid to be isolated from one another, they would have completely coinciding schemes of their energy levels.
Let us study what happens to the energy levels of an isolated atom, as they are brought closer and closer together to form a solid. If the atoms are brought in close proximity, the valence electrons of adjacent atoms interact. Hence the valence electrons constitute a single system of electrons common to the entire crystal with overlapping of their outermost electronic orbits. Therefore, the N electrons will now have to occupy different energy levels. This is brought about by the electric forces exerted in each electron by all the N nuclei. As a result of these forces, each atomic energy level is split up into a large number of closely spaced energy levels. A set of such closely spaced energy levels is called an energy band.
For example, the 11 electrons in a neutral sodium atom, each occupy a specific energy level as indicated in fig.10.
The energy levels of sodium become bands when the atoms are close together. In figure r0 is the distance between the atoms in solid sodium. When the atoms are in solid, they interact with each other and the electrons have slightly different energies.
In the energy band, the allowed energies are almost continuous. These energy bands are in general, separated by regions, which have no allowed energy levels. These regions are known as "forbidden bands" or 'energy gaps'.
The amount of splitting is not the same for different levels. The levels filled by the valence electrons in an atom are disturbed to a greater extant, while those filled by inner electrons are disturbed only slightly fig.10.
If there are N atoms in a solid, there will be N allowed quantum states in each band. Each quantum state can be occupied by a maximum of two electrons with opposite spins. Thus each energy band can be occupied by 2N electrons. The band formed from the atomic energy levels containing valance electrons is called valance band. These electrons have the highest energy. Above the valance band, there is the band of next higher permitted energies called the 'conduction band'. The conduction band corresponds to the first excited states; electrons can move freely in this band and are called 'conduction electrons'. The interval between conduction band and valence band in which electrons cannot occupy is called 'Forbidden gap'.
Importance Of Fermi Energy:
Fermi energy of a metal separates the filled states and empty states at 0 K.
Fermi energy acts as a reference energy level. Based on its value, the number of free electrons per unit volume in a metal is determined
Therefore the electrical conductivity of a metal depends on its Fermi energy.
In explaining any physical property like electrical conductivity, thermal conductivity, specific heat, susceptibility or optical absorption only Fermi level electrons are used.
In the case of semiconductors or insulators Fermi energy level is in the band gap situated between the conduction band and valence band. Here the position of Fermi level varies with temperature and carrier concentration. Thus the value of the Fermi energy determines the electrical conductivity of semiconductors and insulators also.
Effective mass of electrons
When an electron in a periodic potential is accelerated relative to the lattice in an electric field or magnetic field, then the mass of that electron is called effective mass. Consider that an external electric field, is applied to an electron of change, q and mass, m inside the crystal.
Fermi energy acts as a reference energy level. Based on its value, the number of free electrons per unit volume in a metal is determined
Therefore the electrical conductivity of a metal depends on its Fermi energy.
In explaining any physical property like electrical conductivity, thermal conductivity, specific heat, susceptibility or optical absorption only Fermi level electrons are used.
In the case of semiconductors or insulators Fermi energy level is in the band gap situated between the conduction band and valence band. Here the position of Fermi level varies with temperature and carrier concentration. Thus the value of the Fermi energy determines the electrical conductivity of semiconductors and insulators also.
Effective mass of electrons
When an electron in a periodic potential is accelerated relative to the lattice in an electric field or magnetic field, then the mass of that electron is called effective mass. Consider that an external electric field, is applied to an electron of change, q and mass, m inside the crystal.
DENSITY OF STATES:
The number of states per unit volume of the crystal which are contained in the energy interval between E and E+dE defines a density of states Z(E).
The Fermi function does not gives us the number of electrons which have certain energy. It gives us only the probability of occupation of an energy state by a single electron. In a small energy interval dE there are many discrete energy levels. So the concept of density of state is introduced to calculate the number of electrons with a given energy. By multiplying the number of states by probability occupation we get the actual number of electrons.
If N(E) is the number of electrons in a system that have energy E and Z(E) is the number of states at that energy, then
N(E) dE = Z(E) p(E) (48)
The number of energy states, with a particular value of E, depends on how many combinations of the quantum number results in the same value of n. Since we are dealing with almost a continuum of energy levels, we may construct a space of points represented by the values of nx, ny and nz and let each point with integer values of the coordinate represent an energy state.
Electron energy in metal and Fermi energy:
If we consider the three dimensional metal, electrons will move in all directions so that three quantum numbers n x, n y, n z are needed. Let us assume the potential energy inside the cubic crystal with side 'a' is zero and infinity at outside.
Therefore the permitted energy levels can be written as from equation (42).
Fig. 6 The Fermi distribution curve
Application of Schrodinger wave equation: Electron in an infinitely deep potential well (one dimension):
Consider an electron placed in an infinitely deep potential well with width 'a'. It is assumed that the movement of the electron is restricted by the walls and the electron is moving only in the x-direction. The collision of electrons with walls is perfectly elastic. Since the electron is moving freely, inside the well its potential energy V = 0 and V = outside the well so electron cannot escape from the well through the sides.
Electrical Conductivity in CONDUCTING MATERIALS
When an electric field is applied to a conductor, the free electrons are accelerated and give rise to velocity called drift velocity vd. The drift velocity is defined as the average velocity acquired by an electron in the presence of an electric field and the drift velocity direction is opposite to the field direction.
Let us consider E, the electric field intensity applied to a conductor, -e, charge of electron, m, mass of the electron v, the velocity of electron and A, area of cross section. The Force acquired by the electrons can be written as
Complex Circuits :Solving for unknown time
Sometimes it is necessary to determine the length of time that a reactive circuit will take to reach a predetermined value. This is especially true in cases where we're designing an RC or L/R circuit to perform a precise timing function. To calculate this, we need to modify our "Universal time constant formula." The original formula looks like this:
If ex = a, then ln a = x.
If ex = a, then the natural logarithm of a will give you x: the power that e must be was raised to in order to produce a.
Let's see how this all works on a real example circuit. Taking the same resistor-capacitor circuit from the beginning of the chapter, we can work "backwards" from previously determined values of voltage to find how long it took to get there.
Complex Circuits
What do we do if we come across a circuit more complex than the simple series configurations we've seen so far? Take this circuit as an example:
The simple time constant formula (Ï„=RC) is based on a simple series resistance connected to the capacitor. For that matter, the time constant formula for an inductive circuit (Ï„=L/R) is also based on the assumption of a simple series resistance. So, what can we do in a situation like this, where resistors are connected in a series-parallel fashion with the capacitor (or inductor)?
The answer comes from our studies in network analysis. Thevenin's Theorem tells us that we can reduce any linear circuit to an equivalent of one voltage source, one series resistance, and a load component through a couple of simple steps. To apply Thevenin's Theorem to our scenario here, we'll regard the reactive component (in the above example circuit, the capacitor) as the load and remove it temporarily from the circuit to find the Thevenin voltage and Thevenin resistance. Then, once we've determined the Thevenin equivalent circuit values, we'll re-connect the capacitor and solve for values of voltage or current over time as we've been doing so far.
After identifying the capacitor as the "load," we remove it from the circuit and solve for voltage across the load terminals (assuming, of course, that the switch is closed):
Again, because our starting value for capacitor voltage was assumed to be zero, the actual voltage across the capacitor at 60 milliseconds is equal to the amount of voltage change from zero, or 1.3325 volts.
• REVIEW:
• To analyze an RC or L/R circuit more complex than simple series, convert the circuit into a Thevenin equivalent by treating the reactive component (capacitor or inductor) as the "load" and reducing everything else to an equivalent circuit of one voltage source and one series resistor. Then, analyze what happens over time with the universal time constant formula.
The simple time constant formula (Ï„=RC) is based on a simple series resistance connected to the capacitor. For that matter, the time constant formula for an inductive circuit (Ï„=L/R) is also based on the assumption of a simple series resistance. So, what can we do in a situation like this, where resistors are connected in a series-parallel fashion with the capacitor (or inductor)?
The answer comes from our studies in network analysis. Thevenin's Theorem tells us that we can reduce any linear circuit to an equivalent of one voltage source, one series resistance, and a load component through a couple of simple steps. To apply Thevenin's Theorem to our scenario here, we'll regard the reactive component (in the above example circuit, the capacitor) as the load and remove it temporarily from the circuit to find the Thevenin voltage and Thevenin resistance. Then, once we've determined the Thevenin equivalent circuit values, we'll re-connect the capacitor and solve for values of voltage or current over time as we've been doing so far.
After identifying the capacitor as the "load," we remove it from the circuit and solve for voltage across the load terminals (assuming, of course, that the switch is closed):
Again, because our starting value for capacitor voltage was assumed to be zero, the actual voltage across the capacitor at 60 milliseconds is equal to the amount of voltage change from zero, or 1.3325 volts.
• REVIEW:
• To analyze an RC or L/R circuit more complex than simple series, convert the circuit into a Thevenin equivalent by treating the reactive component (capacitor or inductor) as the "load" and reducing everything else to an equivalent circuit of one voltage source and one series resistor. Then, analyze what happens over time with the universal time constant formula.
Kinetic Energy Storage and Release
Now let's consider a mechanical analogy for an inductor, showing its stored energy in kinetic form:
This time the cart is on level ground, already moving. Its energy is kinetic (motion), not potential (height). Once again if we consider the cart's braking system to be analogous to circuit resistance and the cart itself to be the inductor, what resistance value would facilitate rapid release of that kinetic energy? Maximum resistance (maximum braking action) would slow it down quickest, of course!
With maximum braking action, the cart will quickly grind to a halt, thus expending its kinetic energy as it slows down. Without any braking action, the cart will be free to roll on indefinitely (barring any other sources of friction like aerodynamic drag and rolling resistance), and it will hold its kinetic energy for a long period of time. Likewise, an inductive circuit will discharge rapidly if its resistance is high and discharge slowly if its resistance is low.
Hopefully this explanation sheds more light on the subject of time constants and resistance, and why the relationship between the two is opposite for capacitive and inductive circuits.
This time the cart is on level ground, already moving. Its energy is kinetic (motion), not potential (height). Once again if we consider the cart's braking system to be analogous to circuit resistance and the cart itself to be the inductor, what resistance value would facilitate rapid release of that kinetic energy? Maximum resistance (maximum braking action) would slow it down quickest, of course!
With maximum braking action, the cart will quickly grind to a halt, thus expending its kinetic energy as it slows down. Without any braking action, the cart will be free to roll on indefinitely (barring any other sources of friction like aerodynamic drag and rolling resistance), and it will hold its kinetic energy for a long period of time. Likewise, an inductive circuit will discharge rapidly if its resistance is high and discharge slowly if its resistance is low.
Hopefully this explanation sheds more light on the subject of time constants and resistance, and why the relationship between the two is opposite for capacitive and inductive circuits.
Potential Energy Storage and Release
This may be analogously understood by considering capacitive and inductive energy storage in mechanical terms. Capacitors, storing energy electrostatically, are reservoirs of potential energy. Inductors, storing energy electromagnetically (electrodynamically), are reservoirs of kinetic energy. In mechanical terms, potential energy can be illustrated by a suspended mass, while kinetic energy can be illustrated by a moving mass. Consider the following illustration as an analogy of a capacitor:
The cart, sitting at the top of a slope, possesses potential energy due to the influence of gravity and its elevated position on the hill. If we consider the cart's braking system to be analogous to the resistance of the system and the cart itself to be the capacitor, what resistance value would facilitate rapid release of that potential energy? Minimum resistance (no brakes) would diminish the cart's altitude quickest, of course!
Without any braking action, the cart will freely roll downhill, thus expending that potential energy as it loses height. With maximum braking action (brakes firmly set), the cart will refuse to roll (or it will roll very slowly) and it will hold its potential energy for a long period of time. Likewise, a capacitive circuit will discharge rapidly if its resistance is low and discharge slowly if its resistance is high.
Capacitor and Inductor Discharge
In either case, heat dissipated by the resistor constitutes energy leaving the circuit, and as a consequence the reactive component loses its store of energy over time, resulting in a measurable decrease of either voltage (capacitor) or current (inductor) expressed on the graph. The more power dissipated by the resistor, the faster this discharging action will occur, because power is by definition the rate of energy transfer over time.
Therefore, a transient circuit's time constant will be dependent upon the resistance of the circuit. Of course, it is also dependent upon the size (storage capacity) of the reactive component, but since the relationship of resistance to time constant is the issue of this section, we'll focus on the effects of resistance alone. A circuit's time constant will be less (faster discharging rate) if the resistance value is such that it maximizes power dissipation (rate of energy transfer into heat). For a capacitive circuit where stored energy manifests itself in the form of a voltage, this means the resistor must have a low resistance value so as to maximize current for any given amount of voltage (given voltage times high current equals high power). For an inductive circuit where stored energy manifests itself in the form of a current, this means the resistor must have a high resistance value so as to maximize voltage drop for any given amount of current (given current times high voltage equals high power).
Universal Time Constant Formula For Transient Circuits
• To analyze an RC or L/R circuit, follow these steps:
• (1): Determine the time constant for the circuit (RC or L/R).
• (2): Identify the quantity to be calculated (whatever quantity whose change is directly opposed by the reactive component. For capacitors this is voltage; for inductors this is current).
• (3): Determine the starting and final values for that quantity.
• (4): Plug all these values (Final, Start, time, time constant) into the universal time constant formula and solve for change in quantity.
• (5): If the starting value was zero, then the actual value at the specified time is equal to the calculated change given by the universal formula. If not, add the change to the starting value to find out where you're at.
Inductors have the exact opposite characteristics of capacitors. Whereas capacitors store energy in an electric field (produced by the voltage between two plates), inductors store energy in a magnetic field (produced by the current through wire). Thus, while the stored energy in a capacitor tries to maintain a constant voltage across its terminals, the stored energy in an inductor tries to maintain a constant current through its windings. Because of this, inductors oppose changes in current, and act precisely the opposite of capacitors, which oppose changes in voltage. A fully discharged inductor (no magnetic field), having zero current through it, will initially act as an open-circuit when attached to a source of voltage (as it tries to maintain zero current), dropping maximum voltage across its leads. Over time, the inductor's current rises to the maximum value allowed by the circuit, and the terminal voltage decreases correspondingly. Once the inductor's terminal voltage has decreased to a minimum (zero for a "perfect" inductor), the current will stay at a maximum level, and it will behave essentially as a short-circuit.
When the switch is first closed, the voltage across the inductor will immediately jump to battery voltage (acting as though it were an open-circuit) and decay down to zero over time (eventually acting as though it were a short-circuit). Voltage across the inductor is determined by calculating how much voltage is being dropped across R, given the current through the inductor, and subtracting that voltage value from the battery to see what's left. When the switch is first closed, the current is zero, then it increases over time until it is equal to the battery voltage divided by the series resistance of 1 Ω. This behavior is precisely opposite that of the series resistor-capacitor circuit, where current started at a maximum and capacitor voltage at zero. Let's see how this works using real values:
Just as with the RC circuit, the inductor voltage's approach to 0 volts and the current's approach to 15 amps over time is asymptotic. For all practical purposes, though, we can say that the inductor voltage will eventually reach 0 volts and that the current will eventually equal the maximum of 15 amps.
When the switch is first closed, the voltage across the inductor will immediately jump to battery voltage (acting as though it were an open-circuit) and decay down to zero over time (eventually acting as though it were a short-circuit). Voltage across the inductor is determined by calculating how much voltage is being dropped across R, given the current through the inductor, and subtracting that voltage value from the battery to see what's left. When the switch is first closed, the current is zero, then it increases over time until it is equal to the battery voltage divided by the series resistance of 1 Ω. This behavior is precisely opposite that of the series resistor-capacitor circuit, where current started at a maximum and capacitor voltage at zero. Let's see how this works using real values:
Just as with the RC circuit, the inductor voltage's approach to 0 volts and the current's approach to 15 amps over time is asymptotic. For all practical purposes, though, we can say that the inductor voltage will eventually reach 0 volts and that the current will eventually equal the maximum of 15 amps.
Capacitor Transient Response
Because capacitors store energy in the form of an electric field, they tend to act like small secondary-cell batteries, being able to store and release electrical energy. A fully discharged capacitor maintains zero volts across its terminals, and a charged capacitor maintains a steady quantity of voltage across its terminals, just like a battery. When capacitors are placed in a circuit with other sources of voltage, they will absorb energy from those sources, just as a secondary-cell battery will become charged as a result of being connected to a generator.
A fully discharged capacitor, having a terminal voltage of zero, will initially act as a short-circuit when attached to a source of voltage, drawing maximum current as it begins to build a charge. Over time, the capacitor's terminal voltage rises to meet the applied voltage from the source, and the current through the capacitor decreases correspondingly. Once the capacitor has reached the full voltage of the source, it will stop drawing current from it, and behave essentially as an open-circuit.
When the switch is first closed, the voltage across the capacitor (which we were told was fully discharged) is zero volts; thus, it first behaves as though it were a short-circuit. Over time, the capacitor voltage will rise to equal battery voltage, ending in a condition where the capacitor behaves as an open-circuit. Current through the circuit is determined by the difference in voltage between the battery and the capacitor, divided by the resistance of 10 kΩ. As the capacitor voltage approaches the battery voltage, the current approaches zero.
Once the capacitor voltage has reached 15 volts, the current will be exactly zero. Let's see how this works using real values:
The capacitor voltage's approach to 15 volts and the current's approach to zero over time is what a mathematician would call asymptotic: that is, they both approach their final values, getting closer and closer over time, but never exactly reaches their destinations. For all practical purposes, though, we can say that the capacitor voltage will eventually reach 15 volts and that the current will eventually equal zero.
Electrical Transients
This chapter explores the response of capacitors and inductors to sudden changes in DC voltage (called a transient voltage), when wired in series with a resistor. Unlike resistors, which respond instantaneously to applied voltage, capacitors and inductors react over time as they absorb and release energy.
Definitions About Three-Phase Voltages:
If you look at the wave and phasor diagrams in figure 14.1 b and c it should be clear that if the three phases are added together the result is that the voltages cancel each other out and VR + VY + VB = 0V. Therefore, if one end of each coil (R1, Y2 & B3 from figure 4.1a) are connected together a common neutral line at zero potential is created and the result is called a 4-wire 3-phase supply.
Figure 4.2a shows how the generator (or transformer) coils can be connected to give a 4-wire supply, this configuration is called a star, wye or Y connection and the point x is called the neutral point, star point or zero point. Figure 4.2b shows an alternative connection for the generator coils, known as a delta or mesh connection, this configuration has no neutral wire and provides a 3-phase 3-wire supply.
4-wire supplies are normally used to distribute domestic supplies since they can provide an earthed neutral. 3-wire systems are more commonly used for the transmission of high voltage supplies between substations because money is saved by not providing a neutral wire. If we consider a 4-wire supply we can see that there are 4 possible voltage supplies available, these are shown in figure 4.3.
(b) Delta connection;
The power supplies shown in figure 4.3 are:
- The three phase wires together give a 3-phase 3-wire supply suitable for heavy machinery such as
- 3-phase motors. With a 240V supply in each line the 3-phase supply has an RMS voltage of 240√3 = 415V. (The mathematical derivation of why √3 is used is too complex for this text)
- Three single-phase 240V supplies are available between each phase line and the neutral wire.
The three phase single-phase 415V supplies are available between any of the three phases.
The wave diagram for the potential difference between two 240V lines, 120° out of phase is shown in figure 13.11b. (Note that the PD when a load is connected between phase lines is one voltage minus the other voltage, the resultant voltage when two phase lines are connected together is the line voltages added together (figure 4.4a).)
The three phase lines and the neutral together give a 3-phase 4-wire supply with a RMS voltage of 240√3 = 415V.
Loads can be connected to a 3-phase supply using the same two methods used for generator coils; that is star and delta. Figure 4.5 shows these connections. The current that flows in the lines is called the line current (IL) and the PD between any two phase lines is called the line voltage (UL).
The current in each load is called the phase current (IP) and the PD across each load is called the phase voltage (UP).
For a balanced system:
• Each load has the same magnitude of impedance;
• Each load impedance has the same phase angle;
• Each phase voltage (and current) is equal in magnitude;
• Each phase voltage is displaced by 120° from each other
The example in section 2.5 shows a analogous system for a DC supply where the branch containing R3 is the equivalent to the neutral wire. When the system is balanced no current flows through R3, only when the system becomes unbalanced does any current flow through this resistor.
Figure 4.4: The wave diagrams for: (a) adding together two 240V lines that are 120° out of phase to find the voltage supply when two lines are connected to the same point; (b) subtracting two 240V lines that are 120° out of phase to find the PD between the lines. The resultant is shown as a bolder line. It is interesting to note that the potential difference between any two phase lines at U volts is √3U volts whereas the same two phases added together result in a voltage of U.
Figure 4.5a shows that for a star connected loads the PD across any load is the PD between the phase line connected to it and the neutral, however the line voltage will be greater than this because it is the PD between two phase lines. The current through the load will be the same as the line current. Thus for balanced star connected loads:
Figure 4.5: 3-phase load connections: (a) a 4-wire star connection (note that a 3-phase wire connection is also possible when the neutral wire is left out); (b) a 3-wire delta connection. The line voltages are shown in black and the phase voltages are shown in grey.
Thus for a balanced system with 240V lines UL = 415V, and UP = 240V. For a balanced system no neutral wire is needed and even in a star supply the neutral wire can be left out. Only when the star connected loads are unbalanced will any current flow down the neutral, in a balanced system one of the other lines will always be at a lower potential than the neutral and the current will 'return' down this type of load connection is usually used for domestic distributions, where houses connect to one phase line and the neutral. If more houses are connected to one line than the others the loads become unbalanced and the neutral wire carries current.
Figure 14.5b shows that for delta connected systems of loads the phase voltage is equal to the difference between two lines. The phase current for each load is coming from two lines (because each load is connected to two lines) and will be more than the line current. Thus for delta connected loads:
UL = UP
IL = √3IP
Thus for a balances system with 240V lines UL = UP = 415V. This method of load connection is usually only used for machines with windings such as 3-phase motors which have three identical windings, since the absence of a neutral can means that unbalanced loads can cause problems.
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