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Electric current Previously we dealt with electrostatic problems, where field problems are associated with electric charges at rest. We now consider the charges in motion that constitute current flow. There two types of electric current are caused by the motion of free charges. Conduction current, which are governed by Ohm’s law, in conductors and semiconductors are caused by drift motion of conduction electrons and / or holes. Convection current results from motion of electrons and / or ions in a vacuum or rarefied – such as in a cathode ray tube. Basically, what we are interested in electronics, are the possibilities for controlled motion and transport of electric charge, especially within conductors – where the free electrons as the carriers of charge are already present in the medium – that means we are interested in conduction current. In the case of conduction currents there may be more than one kind of charge carrier (electrons, holes, and ions) drifting, with different velocities – all of which contribute to what we measure as current.

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Electric current Since electric charge is quantized (in discrete multiples of the electron charge), it is instructive to look at electric current as the movement of multiple microscopic charge carriers as discrete point particles – although this may not directly correspond to the actual physical reality. The physical quantity that deals with the description of the motion of electric charge is called electric current: - defined physically as the continuous [and directed ] flow of electric charge through a conducting material - defined mathematically as the rate of charge flow past a given point in an electric circuit, or in other words, the quantity of charge passing through some region of space (specific point in a circuit) in a given time period Electric current is measured in Amperes [A] and 1 Ampere corresponds to a rate of 1 Coulomb of charge per 1 second, where 1 Coulomb is the charge of 6.24x1018 (6.24 quintillion ) electrons. Note that the definition for current can be applied to other systems of flowing discrete particles – which will enable us to relate electric current to flow of water. Also, the electric current definition demands only mobile charged particles – this does not necesarilly mean only electrons.

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**Current, conductors and field**

We are already aware of the fact that These findings are confirmed through the fact, that when a conductor is placed in electrostatic field, its constituent free electrons move and redistribute in such manner that they cancel the field within the conductor – and there is no further electron movement. This also means that the potential is constant throughout the conductor. - Charge can move under the influence of an electrostatic field (=potential difference). The direction of electron movement is from a region of negative potential to a region of positive potential. - Charge can freely move in a conductor (material medium), just as well as in vacuum. - Conductors which are in electric contact have the same potential – they share excess charge between themselves (charge can move between as well as in them), and equalize their originally different internal potentials.

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**Electromotive force - emf**

Eventually an internal field due to the repositioned charge cancels an eventual external electrostatic field resulting in zero current flow. The conclusion is that we cannot induce directed motion of charge carried by the free electrons in a conductor, by simply placing the conductor in an external electrostatic field – the free electrons cancel the effect of the external electrostatic field within the conductor. To move the charge within the conductor, in a directed fashion, we must set up a field within the conductor, or in other words, we need to set up potential difference at the ends of the conductor – in that direction that we want the free electrons to move. Such a potential difference at the ends of a conductor can be set up by means of what is known as electromotive force or emf. So, charge can flow in a material under the influence of an external electric field. To maintain a potential drop (and flow of charge) requires an external energy source known as electromotive force – EMF, such as battery, power supply, signal generator, etc. Another definition could be that emf is a physical quantity which describes the ability of an electrical source to deliver energy - or the property of the source which creates current in a circuit. The emf of a battery is responsible for producing a potential difference between the battery's terminals. In order to understand the concept of electromotive force, we should take a closer look at the meaning of potential in a conductor medium.

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**Potential and charge density**

In review: - An electrostatic field is due to an electrically charged particle. - A potential gradient indicates an electrostatic field different from zero – and vice versa. In order to have directed motion partucles in the conductor, we need a field directed in the conductor, indicated by a corresponding potential gradient. Locally in space, electrostatic potential is related to the strength of the electrostatic field at that point. However, the strength of field is related to quantity of charge (or number of electrons) on a charged body that causes the field, and the distance from it – and thus the potential, locally, is related to both of these.

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**Potential and charge density**

This means that if the distance to a charged body is not changed, we can use electrostatic potential locally as an indication of the quantity of charge (or the concentration of charge carriers) that produces the field. Considering that the metal conductors already posess a significant number of free electrons as charge carriers, the electrostatic potential within a conductor can be related to the measure of the repulsive electrostatic forces acting between the free carriers themselves – whose strength depends on the concentration of carriers in the conductor.

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**Potential and charge density**

Considering that the metal conductors already posess a significant number of free electrons as charge carriers, the electrostatic potential within a conductor can be related to the measure of the repulsive electrostatic forces acting between the free carriers themselves – whose strength depends on the concentration of carriers in the conductor. The image displays a rendering of the potential (and thus local field) due to the mutual repulsive forces from a concentration of charges. Note that in normal conditions in a conductor, there is a given concentration of free carriers, for which these repulsive forces are compensated by the molecular forces that hold the metal crystal together. If that normal concentration is related to a ”zero” potential, then a negative potential means rarefaction, and positive means densification in relation to the normal density of free charge. Remember from electrostatics that we can deposit charge on a conductor (charging), which means effectively that the concentration (density) of charge changes in a conductor!

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**Potential and charge density – relation to pressure**

Here we can observe one of the correspondances between fluids and free electrons. Namely molecules in fluids (gases, liquids) have a certain freedom of movement, executed due to thermal energy being able to cope with molecular forces (Thermal Physics and Properties of Matter). Due to this, fluids assume the shape of a container. In addition, due to the chaotic thermal movement, they hit each other and the walls of the container, causing on average a measure of a pressure – force acting on given area. The pressure is thus an indication of how much repulsion is there between the (free) molecules due to collisions during their thermal motion. Similarly, the potential is an indication of how much repulsion is there between the free electrons due to the repulsive electrostatic field. This relation helps with the understanding that the movement direction for free carriers is from a state of high repulsion (potential / pressure) towards low repulsion (possibly equilibrium). Electrons free to move (metal) Molecules free to move (fluid) Less dense – less electrostatic repulsion – less force per area (or per num. charges ) More dense – more electrostatic repulsion – more force Less dense – less collisions due to thermal motion – less force per area (or per num. molecules ) More dense – more thermal collisions – more force

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**Potential and charge density – electron gas**

In reality, free electrons in a conductor behave a lot like gas molecules in container. They not only move and redistribute throughout the conductor – effectively assuming the shape of the container, but they also exhibit chaotic thermal movement as well. During such movement they can collide with other free electrons, or other particles within the conductor – which is why the free electrons in a conductor are referred to as electron gas. It is important to realise that a free electron does not remain in free chaotic movement as a free molecule in gas does. The situation is usually that an electron gains energy to become free, moves for a short while due to thermal energy and/or external field, and then experiences a collision where it gives its energy, and becomes bound to an atom again. The sitation we have is that on average during a given time period, we always have a given number of free electrons moving, which relates to the gas (fluid) aspect. The fact that the free electrons fill out the conductor (the shape of the container) as fluid (gas), helps us understand a conductor as a fluid container, and relate it macroscopically to a pipe filled with fluid (for instance water). motion of free electrons in a conductor collisions Note that in this analogy, a gas is a compressible fluid, while water is not compressible – however, due to the electrostatic repulsive force, the “pressure” applied at one part gets transferred to another, like in liquids.

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Conduction path The fact that the free electrons fill out the conductor (the shape of the container) as fluid (gas), helps us understand a conductor as a fluid container, and relate it macroscopically to a pipe filled with fluid (for instance water). That means that a conductor’s shape, as a container, also represents a path for its constituent free electrons – a path where they could freely move – in similar manner a pipe as a container represents a path where water can freely flow. If we want electrons to flow in a certain direction to a certain place, we must provide the proper path for them to move, just as a plumber must install piping to get water to flow. Remember that electrons can flow only when they have the opportunity to move in the space between the atoms of a material – and that opportunity is present in a conductive material. This means that there can be electric current only where there exists a continuous path of conductive material providing a conduit for electrons to travel through. A single path is thus related to a single piece of conductor which we know as a conducting wire.

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**Conduction path connection**

As mentioned before, whent two (or more) conducting bodies are in electrical contact, they can share excess charge – which means charge can move between the two. In addition, they attempt to equalise their potential, so eventually, the system of connected conductors behaves like one conductor, with one potential shared for all participating conducting bodies. It is in this way we can connect conductors, and thereby establish a continuous path for the electrons to move. This can of course be related to a system of connected water pipes, which establish a path for water flow. In both cases, at start we have two conductors (pipes) containing different ammounts of particles, and thus dufferent levels of internal potential (pressure). When a counductor (pipe) is connected between the two, the particles can freely flow between the two, and aim to redistribute so that the potential (pressure) is equalized (see Pascal’s Principle), thus current begins to flow – a mass transport of particles. Since now the two original bodies, and the connecting conductor represent one conducting body (pipe), a disturbance in potential (pressure) will be gradually transferred to all the particles in the container – which is the essence of the conduction mechanism. Note that when the potential (pressure) is equalised, the current stops. Water flow Electron flow

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Transient current Note that we could relate this example to a system of two charged conducting bodies, which become connected with a conductor (say, a wire). Since after the connection they must redistribute their excess charge between themselves, so the field inside is zero, current must flow through the conductor wire. As soon as the charge is evenly distributed, there is no more flow of particles – current. Such current persists for a very short time and is known as a transient current. In the more general applications, currents more persistant in time are needed – and these are known as continuous currents. If the rate of the number of free electrons crossing a given point is constant in a given time interval, it is known as a steady current. If the direction of the free electron flow does not change in relation to some referently chosen direction during time, then it is known as direct current (DC). The direction of the current can be easily determined through the normal of our ”test surface” – which help us count the number of electrons past a certain point. Note that in the most basic case, in order to have current, as flow of particles, we need to have a difference of potential[pressure], a conducting path and free particles that will move.

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**Transient current – capacitor discharge**

The given system of two charged conducting bodies separated by a dielectric is by definition a capacitor, and in a capacitor, deposited charge means voltage: That means that since we have deposited excess charge, it has redistributed (on the surface) so the field inside is 0, which means there is a constant potential inside, which we can relate to the average electrostatic repulsion a free electron might feel from the other free electrons. This repulsion, is of course dependant on how many free electrons there are –excess charge included. In that sense, potential in metals with free carriers is related to concentration of carriers. And hence for two conducting bodies (separated by a dielectric – capacitor), a difference in potential, or voltage, means also a difference of concentration of charge. (Finally, the difference in potential between two points means existence of an electric field.) And it is this difference, regardless of how we see is (as difference of potential = voltage, as difference of concentration or as difference in the repulsive pressure the electron ”sees”) that will cause the particles to move from more negative (higher pressure) to more positive potential (lower pressure) – given a conducting path. These are the minimum requirements for current flow – presence of free carriers, a conducting path, and potential difference (voltage) at the ends of the conducting path.

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**Transient current – capacitor discharge**

In the example, in the start, we have two separate conducting bodies, separated by non conductng medium (dielectric) – which constitutes a capacitor. There is a difference in concentration, which implies that one of the bodies carries excess charge. Thus, there exists a difference of potential or voltage, between the two bodies, due to excess charge. When the dielectric is replaced with conductive material, we establish a path between the two bodies, on which ends now there is potential difference, and the free carriers can move. By moving, we effectively are taking away excess electrons from one body and placing it on the other, so the current flow reduces the excess charge – and thus the potential difference between the two. When the excess charge is levelled out, there is no more voltage, and thus there is no more current. This is the example of a charged capacitor, having its terminals connected with a conductor – and the transient current that appears is known as capacitor discharge current. Due to the excess charge there is voltage on its terminals. When the conductor is connected to the terminals, the excess charge from one body is taken away – discharge – which lowers the potential difference. Although this illustrates a minimal situation for current, it results only in a transient one - we are interested in continuous currents – that persist over time, so we should take a closer look at the conduction mechanism.

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Conduction mechanism The requirements for a path and free carriers in order to have current are both satisfied by metals as conductors – since they already contain a number of free electrons. The mechanism of movement is thus not as straightforward as thinking that one free electron crosses the entire distance of the conductor. As each electron moves uniformly through a conductor, it pushes on the one ahead of it, such that all the electrons move together as a group. The starting and stopping of electron flow through the length of a conductive path is virtually instantaneous from one end of a conductor to the other, even though the motion of each electron may be very slow. An approximate analogy is that of a tube filled end-to-end with marbles.’ The tube is full of marbles, just as a conductor is full of free electrons ready to be moved by an outside influence. If a single marble is suddenly inserted into this full tube on the left-hand side, another marble will immediately try to exit the tube on the right. Even though each marble only traveled a short distance, the transfer of motion through the tube is virtually instantaneous from the left end to the right end, no matter how long the tube is. With electricity, the overall effect from one end of a conductor to the other happens at the speed of light: 300 million meters per second!!! Each individual electron, though, travels through the conductor at a much slower pace.

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Conduction mechanism Remember that electrons can flow only when they have the opportunity to move in the space between the atoms of a material. This means that there can be electric current only where there exists a continuous path of conductive material providing a conduit for electrons to travel through. In the marble analogy, marbles can flow into the left-hand side of the tube (and, consequently, through the tube) if and only if the tube is open on the right-hand side for marbles to flow out. If the tube is blocked on the right-hand side, the marbles will just "pile up" inside the tube, and marble "flow" will not occur. [from here] However, that also means that when we connect such ”blocked” conductors (in our case the copper conductor would be ”closed” with air – which is why we actually call this situation an ”open” circuit) any eventual ”push” on one side of the conductor (resulting from say an electric contact with a charged body), will be transferred to the other side of the conductor.

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**Conduction mechanism – wave aspect**

If we think about what happens from an electrostatic perspective, upon an insertion of an extra electron (marble): the local density of charge carriers increases, increasing the repulsive force at that point, and correspondingly, the overall electrostatic field is changed due to this electron. Since we know that the electrostatic field travels with the finite speed of light, the information of the change represented through the field will also travel at the same speed. Note that if the conductor was in equillibrium previously (and charges did not move due to the field), the charges further on in the conductor (say 1 cm apart) will start moving only after the information of the changed field has reached them (for 1 cm, and speed of lights, this is 333 nanoseconds!). That means that the changed field due to changed density in a conductor travels as a distrubance through the medium of the conductor, which is by definition a wave.

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**Conduction mechanism – wave aspect**

This basically means that the starting and stopping of electron flow through the length of a conductive path is propagates from one end of a conductor to the other with the speed of light as a wave – the electrons start their actual movement right aftwreards. For the wave aspect of conduction, read Understanding electrical conductivity in wires The wave propagation if a cause of reflections in wires (transmission lines), and the need to account for characteristic impedance of cables. Applets from the site: Reflection – mathematically – see tri line applet here Interaction of electrons – see five el applet here Physically, via speed of particles – see terminated applet here for terminated wire, see open applet here for open wire

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**Emf potential as power source**

We can now consider the two terminals of a source of emf as two conducting bodies – and say that the negative terminal has more free electrons than the usual number for that particular metal – which causes a more negative potential in that piece of material, in relation to the positive terminal – and hence the potential difference, or voltage, between the terminals of the source. Normal concentration for the terminals conductor The electromotive force tries to maintain this potential difference – and the difference of concentration of free charge at the terminals - in time. A useful metaphor is that in order to maintain the difference in concentration of charge, the battery must take away charge from the positive terminal conductor and move it to the negative terminal – performing work while doing so – which is how we can provide continouous and steady current. Since it must maintain this difference over time (otherwise the excess charge will redistribute and equalise the potential at the terminals) – it must continually perform work as time goes. By definition, rate of work is power, and it is thus a main characteristic of sources of emf.

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**Electromotive force - emf**

We could rephrase: The electromotive force EMF of a source of electric potential energy is defined as the amount of electric energy per Coulomb of positive charge as the charge passes through the source from low potential to high potential. A difference in charge between two points in a material can be created by an external energy source such as a battery. This causes electrons to move so that there is an excess of electrons at one point and a deficiency of electrons at a second point. This difference in charge is stored as electrical potential energy known as emf. It is the emf that causes a current to flow through a circuit. A circuit consists of electrons flowing from the negative terminal of a battery to the positive terminal of the battery. However, these electrons must then return to the negative terminal, or the current will stop flowing. The electrons are forced into this higher potential by an electromotive force – and this is intuitively very similar to the notion of the energy of a water pump. Emf is not a ”force” - the “electromotive force” represents an electric potential or voltage and is measured in Volts [symbol V].

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Electric circuit? We mentioned presence of free carriers, a conducting path, and potential difference (voltage) at the ends of the conducting path as minimum requirements for electric current. Since we were interested in continuous current, an emf generator can be used as source of potential difference, and a conductor can be used to provide the free carriers and the conducting path. Due to the definition of work of the power source – that it takes away charge from one terminal and moves it to the other, it is obvious that a closed loop conducting path (toward the source) is needed if energy is to be utilized outside the source; this conducting loop is known as an electric circuit. This is the basic meaning behind electric circuit. However, because of the same definition, we need to consider that if work is put in, that energy must be used up somewhere – so we need to look at the power relations. electric circuit - complete (unbroken) conducting path along which an electric current exists or is intended or able to flow. A circuit of this type is termed a closed circuit, and a circuit in which the current path is not continuous is called an open circuit.

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Power and resistance Since we defined our source via power, we must recognize the law of conservation of energy. That is, if the generator put some work in a system, that energy must be somehow used in the system. This can be visualised on a system made of frictionless track, a ball that can move on the track, and a power generator which can perform work on the ball pull it toward itself and accelerate it on the other direction. If we cover a part of the track with sand or other material that causes friction, then the acceleration given by the source is used up on compensating the friction forces – which effectively turns the energy of the source into heat. The ball arrives ”tired” at the source, with reduced velocity, and the cycle will then continue in a predictable fashion.

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**Power and resistance – short circuit**

However, if we disregard the resistance, then the ball arrives at the source with the same speed it has reached during the past acceleration, and as the cycle repeats, it gains more and more energy, and theoretically, it should reach infinite velocity, which is impossible. (see applet) Consider that in reality, we do not deal with one particle, but with a number of them – and according to the conduction mechanism - the energy, without resistance and via progressive transfer of the push from particle to particle, will simply return to the power source and force it to accept energy instead of give it – which is a situation that is destructive to the generator. This corresponds to the situation of a short circuit – where the terminals of the power source are connected only through a conductive wire, which has small resistance – and we must always be careful to avoid this!

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Electric circuit We can now put up a more complete basic definition for an electric circuit – one that can sustain continuous currents due to a power source. Namely, it is obvious that some sort of friction of the conducting path – resistance to the flow of current, or user of the provided energy of the source - must be included in the model. Locally, as property of conductive materials this property is called resistivity (r), and as a property of conductive objects with finite dimensions – resistance [R]. All conductive materials (such as metals) are resistive – some more, some less. This brings about the distinction in conductive materials as: - conductors – good conductive properties, small resistivity (copper, silver, ... ) - resistors – big resistivity (graphite). Additionally, theoretically we can define an ideal conductor, which has resistivity zero. .

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Electric circuit electric circuit - complete (unbroken) conducting path along which an electric current exists or is intended or able to flow. The term is usually taken to mean a continuous path composed of conductors and conducting devices and including a source of electromotive force that drives the current around the circuit. A closed loop conducting path is needed if energy is to be utilized outside the source; this conducting loop is known as an electric circuit. The device or circuit component utilizing the electric energy is called the load. Note that we now have a situation where free electrons circulate in a conductive medium, which was not possible solely with an electrostatic field. .

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**Electric circuit – fluid mechanics**

The basic notion of an electric circuit can be demonstrated through a fluid example. Most practical applications of electricity involve the flow of electric current in a closed path under the influence of a driving voltage, analogous to the flow in a water circuit under the influence of a driving pressure. A fluid is either a gas (like air) or liquid (like water) that does not hold its own shape. .

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Ohm’s law This analogy also gives way for intuitive understanding of Ohm’s law, as formulating the necessity for the conductive loop, free particles, potential difference and resistance, in order to have a possible circuit that will sustain steady current – where the energy exchange demanded by power genator is possible: U I R Said in other words, all that the power source needs to “see” on its terminals is a finite resistance (not zero and not infinite), in order for continuous current to flow. .

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Ohm’s law U I R In the water analogy we have that greater pressure results with greater speed – which results with more particles crossed through a hole per given time interval. That can be directly applied to electron flow, where increased voltage results with more free electrons crossed through a point in the conductor per given time interval. .

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Ohm’s law and sensors. Ohm’s law, even on this intuitive level is of essential meaning for sensors. To transmit information costs energy: Every message is associated with a material object or radiation, and must thus be accompanied by some energy. So, every signal must convey some energy/power — except the trivial case of the signal “0 Volts”. Thus, when you apply an input voltage to, say, an oscilloscope, it must also draw a small current to make it recognise that a signal has arrived. This can be related to the fact that when current flows, that means that energy exchange is happening between the generator and the load. Any piece of equipment which accepts input signals will require both a voltage and a current to make it work..All instrumentation modules, which actively gather data, must extract energy from sensors in order to measure information. A transduction element will commonly provide a relationship between some physical parameter, and its own resistance or generated emf. Basically, what we need to do in order to read this information – (or to interface with the sensor), is to bring the transduction element in a ciruit where current flows – where there is energy exchange – and transfer this energy to the load – the user circuit.

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**Ideal power source – current and voltage**

We have already mentioned that the power source produces voltage by moving electrons from the positive to the negative terminal – this being the energy input in the circuit. Since work is defined as charge times potential difference, the rate of work which is power will be potential difference times rate of charge (current), which is: On the other hand, we saw that a resistor is necessary to utilize this energy in the circuit – and the very parameter of a resistance sets up a relationship between voltage and current. That means, that a voltage applied to a resistor determines the current through it – so we say that when voltage is applied to a resistor, it draws current – or, when current is forced into a resistor, voltage appears on its terminals – related to the resistance. That means that when the power source starts its work, and realizes the resistance, it has two options on how to proceed – either to keep the potential difference on its ends, and dose the ammount of charge as according to what the resistor demands, or to keep a constant rate of charge flowing into the resistor, and develop a potential difference based on what the resistor produces for that particular current. These represent idealized electrical sources of power: Ideal voltage source – that keeps a constant potential difference on its ends, no matter what the current through it – EMF refers to the pot. difference of an ideal voltage source Ideal current source – that keeps a constant rate of charges flowing, no matter what the voltage across it. .

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**Ideal power source – current and voltage**

These represent idealized electrical sources of power: Ideal voltage source – that keeps a constant potential difference on its ends, no matter what the current through it – EMF refers to the pot. difference of an ideal voltage source An ideal voltage source is an element in which the voltage across its terminals is an independently specified function of time Vs(t). It is capable of supplying, or absorbing, infinite current in order to maintain the specified voltage. Ideal current source – that keeps a constant rate of charges flowing, no matter what the voltage across it. The ideal current source is an element in which the current supplied to the system is an independently specified function of time Is(t). The terminal voltage of a current source is defined by the system to which it is connected. These two ideal sources may continuously supply or absorb energy since in each, one power variable is independently specified while the complementary power variable is determined by the system to which the source is coupled. Ideal sources are capable of supplying (or absorbing) infinite power and are idealizations of real sources, which have finite power and energy capability - and thus only approximate real power-limited sources.

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**Power sources and energy exchange**

Since we have defined a relation for the power of the source, and since we used resistance to account for the energy expenditure (heat dissipation) in the circuit – we can now calculate that for the elementary circuit, the power produced by the source will be the same as the one used by the load: V i P=Vi Produced by Source Or Used by Load Load + - Since the resistance was introduced to account for the energy expenditure, we can rewrite the relation for the power expenditure in the load through the property of resistance or:

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Real power sources The idealised model for the power sources, does not take into account the fact that any power source has some internal resistance. This internal resistance can be modelled as a resistor in series with an ideal voltage source, and a resistor in parallel with an ideal current source.

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**Resistivity and resistance**

However, in order to understand Ohms law better, we need to take a look a closer look at resistivity – on the molecular scale (more: Ohms Law and Materials Properties). This is because we have introduced resistance in order to model some sort of friction, to account for the impossibility of infinite velocity; however there is no friction on the molecular scale – only effects of fields. In addition, Ohm’s law, as it is given, is limited to only a class of conductive materials called ohmic materials – where the resistivity is generally not dependant on the electric field (and thus the voltage). The best way to model resistivity is through free electrons, as particles, being able to experience a hit or collision with another particle. During this hit, the momentum that the particle has can be given to the other particle, and transferred as heat, and the particle may continue in a different direction. This behaviour is very similar to gas in a container, which is why free electrons in conductors are referred to as the “electron gas”. Even without influence of a field, the free electrons are in a state of chaotic movement and collision, due to the thermal energy of the crystal. During these an electron may lose energy and even fall back into a shell of the crystal ion – possibly giving energy to another bound electron so it becomes free. .

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**Resistivity and resistance**

The free electron might collide with (and scatter from): Another free electron A defect in the crystal lattice A vibration of the crystal latice called a phonon which essentially represents the Drude model of conduction. .

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**Resistivity and resistance**

Under the influnce of electric field, the chaotic movement does not stop – instead, the overall movement gains a direction – as in a swarm of bees. This net directional velocity is called drift velocity. drift .

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**Resistivity and resistance**

The influence of the chaotic movement can be averaged out, and then we can work only with the drift velocity, which is due to the field: .

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**Resistivity and resistance**

This gives us the possibility to use a test surface, normal to the direction of flow of current, and have a concept of number of electrons that crossed the surface amongst all that random movement – and a numerical value for current. This is also a good place to note that generally, it doesn’t have to be only free electrons that move – those could be positive particles, such as lacks of shell electrons in a crystal due to impurities! In that case, we add the effects of the positive and the negative particles. . Actually, it is much easier to think in terms of current of free electrons, as a flow of positive particles from the positive to the negative terminal of the battery! Although that is not the case, this direction is widely used and is known as technical direction of current.

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**Resistivity and resistance**

We needed the previous steps because we need to see what happens on average, in between the collisions. As the source performs work and begins to insert electrons, electrons further on in the conductor sense the field and begin to move, repelled by it. However, some of those will experience collisions, and they might end up in the opposite direction of where they are supposed to go. These on average, make the drift velocity lower. That at the same time means that locally, those that have gone back, are contributing to the greater density of free charges at that point – and thus greater electrostatic repulsion, and greater potential. So, because of collisions and scattering with other particles – which is our cause of resistivity – during current, there appears a accumulation of charge locally at the conductor, which on average move slower – have a lower drift velocity. However, greater density means also stronger potential – and stronger repulsion for those that actually did “cross the surface” and contributed to the current. That means that they will be locally less dense, and faster! .

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**Resistivity and resistance**

The situation that we have now in the conductor is that due to the resistance due to collisions - a difference in concentration of free charge is established, due to deposited charge that went back from the collision, along the line of flow of current – and the corresponding potential gradient, due to which, there is also a difference in the speed of particles. This concept of hits and deposition can be directly related to the macroscopic multilayered sieve (the one pictured is on the right for materials analysis). It can also be related to flow of water, in the sense of a sieve(strainer) intercepting the flow of water. The sieve can be related to the static crystal lattice of the conductor, and the water molecules to the free electrons. Basically, the effect of the crystall latice on the electron flow as a sieve would relate to our measure for resistivity, locally in the conductor.

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**Resistivity and resistance**

If water was compressible, it would not be difficult to relate a “deposition” of water molecules and increase of water pressure because the molecules on their way hit the walls of the sieve and try to turn back. Thus, effectively the sieve can be seen as a nozzle – which effectively lowers the area through which water can flow. Due to the principle of conservation of mass – whatever flows in must flow out, regardless if the cross section area is smaller – and that is possible only if the molecules move faster (being too close together, they will start repelling and increase the pressure – which on microscopic level is close to the logic of the electrons repelling). Thus a smaller cross sectional area in direction of the flow represents resistance to flow, and causes particles to move faster. In spite of the fact that there is a gradient of concentration of carriers – it will still result in the same current – as average number of electrons that crossed a certain point - throughout the conductor.

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**Resistivity and resistance**

Another visual rendering of the interaction of free electrons with the crystal lattice that results with hits and slowdown, can done be through the following macroscopic model – which might represent for instance a “top” view at the “sieve” structure of the lattice, while the electrons are moving through it. The analogy is also that the “sieve” effect of the lattice is distributed throughout the conductor material:

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**Resistivity and resistance**

The most important finding is that for ohmic materials, resistivity – as parameter of expected hits due to the specific build up of the conductor – does not depend on the electric field. In that case, we can treat a given conductive object as a whole, and express ts resistivity and its geometry (which we have seen also influences the flow of current) through one parameter called resistance: Note that this is the resistance which is called real resistance, which attributes for the development of heat and dissipation of the energy of the source. The resistivity r is a parameter which would describe the specific “sieve” of a conductor – however, it is reapplied throughout the dimensions of the material. Therefore, we basically “lump” physical processes involved with resistivity inside a single package – ”black box” - called a resistor, described with only one parameter - resistance (R). The resistance is the overall effect of resistivity (effect of all the “sieves”) within the resistor’s entire volume on the flow of current. r throughout R as “box” S

44
**Resistivity and resistance**

For ohmic materials, both resistivity and R are independent of the el. field (and voltage), and means that the resistive effect upon the flow of current is constant in relation to voltage. So, for these materials, Ohms law will become a linear equation – and the relationship between the voltage across and the current through a resistor will be a linear one – linear UI characteristic. Ohm’s Law now provides a numerical relationship between current, voltage and resistance of a resistor. R

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**Current flow simulation**

Applets: Resistance at molecular level – demostration of collisions Electrons in metal simulation – demonstration of how the crystal lattice (the ”sieve” geometry) and thus collisions, depend on temperature. Light Bulb applet Resistance applet Consider the simple circuit applet for demonstration of the gradient of potential and concentration, as well as the change in speed according to density of particles, and wave propagation of the field – and power relations between source and load. Consider the Venturi Tube simulation applet for demonstration that the local pressure, during current flow, acts in all directions – and given a conduit at that position, that “push” will be propagated through the conduit. The same concept applies for potential in electric circuits – and allows us to measure voltage (as potential diff. on the ends of a piece of cond. material) .

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**Circuit theory – graphs and network topologies**

We have already realised that the flow of current, first and foremost means that there is energy exchange of some sort, and in the simplest possible setup, this is an exchange between source --- a provider of energy in the circuit, such as a generator, and load – a user of energy in the circuit, such as a resistor. Systems for transport of free electrons (current flow), can of course be more complicated than the basic case. Nonetheless, the above findings must still be valid. As the complexity of the transport system grows, it can be difficult to keep track of all the parameters which influence the flow of current – especially since the geometry of the connections influences possibilities for flow. In general: The internal flow of fluids through pipes, vessels, and pumps, and the external flow around vehicles, aircraft, spacecraft, and ships are complex phenomena involving flow variables that are continuous functions of both space and time. As such they generally cannot be represented in terms of pure lumped elements. With some simplifying assumptions, however, a number of significant characteristics of the dynamic behavior of fluid systems, particularly for one dimensional pipe flows, can be adequately modeled with lumped parameter elements. .

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**Circuit theory – graphs and network topologies**

With some simplifying assumptions, however, a number of significant characteristics, can be adequately modeled with lumped parameter elements. : In circuit analysis, important characteristics are grouped together in “lumps” (separate circuit elements) connected by perfect conductors (“wires”) An electrical system can be modeled by an electric circuit (combination of paths, each containing 1 or more circuit elements) if l = c/f >> physical dimensions of system l = Distance travelled by a particle travelling at the speed of light in one period Example: f = 100 Hz l = 3 x 108 m/s / 100 = 3,000 km We define a set of lumped parameter elements that store and dissipate energy in network-like fluid systems, that is systems which consist primarily of conduits (pipes) and vessels filled with incompressible fluid. These definitions are analogous to those for mechanical and electrical system networks. A transportation network enables flows of people, freight or information, which are occurring along links. A graph is a symbolic representation of a network. It implies an abstraction of the reality so it can be simplified as a set of linked nodes. Graph theory is a branch of mathematics concerned about how networks can be encoded and their properties measured. .

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**Circuit theory – graphs and network topologies**

A transportation network enables flows of people, freight, information – or free electrons, which are occurring along links. A graph is a symbolic representation of a network. It implies an abstraction of the reality so it can be simplified as a set of linked nodes. Graph theory must thus offer the possibility of representing movements as linkages. The goal of a graph is representing the structure, not the appearance of a network. The conversion of a real network in a planar graph is a straightforward process which follows some basic rules: The most important rule is that every terminal and intersection point becomes a node. Each connected nodes is then linked by a straight segment. The outcome of this abstraction, as portrayed in the above figure, is the actual structure of the network. The real network, depending on its complexity, may be confusing in terms of revealing its connectivity (what is linked with what). However, the graph representation reveals the connectivity of a network in the best possible way. .

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**Circuit theory – graphs and network topologies**

As the complexity of the transport system grows, it can be difficult to keep track of all the parameters which influence the flow of current – especially since the geometry of the connections influences possibilities for flow. Since it is only the geometry of connections that matter - the structure, not the appearance of a network - what we have obtained with the graph is the topology of a network. And we can redraw it until it reveals the connectivity of a network in the best possible way. The ideas involved in network topology come from a branch of geometry (topology), which is concerned with the properties of a geometrical figure that do not change when the figure is drawn in alternate forms, where those alternate forms do not involve taking apart or joining together any parts of the figure. In the case of a given electric circuit, it can be redrawn in many ways, and it still is the same circuit. So regardless of the way the circuit is drawn, it can be analyzed in the same fashion.

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**Circuit theory – graphs and network topologies**

We have already realised that the flow of current, first and foremost means that there is energy exchange of some sort, and in the simplest possible setup, this is an exchange between source and load. As the complexity of the transport system grows, it can be difficult to keep track of all the parameters which influence the flow of current – especially since the geometry of the connections influences possibilities for flow. That is why it is useful to go back to the elementary circuit of energy exchange between source and load – and realise the basic aspects of representation of topologies of electric circuits. The figure displays three representations - superimposed upon one another - of the topology of an elementary circuit between a source and a load: branches & nodes (graph representation) (red) Connection of elements with ports (schematic representation) (gray) Connection of networks with ports (black) all of which are based on, and related to, the graph representation of this circuit. .

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**Circuit theory – graphs and network topologies**

Generally A branch may consist of more than one element A network may encompass a more complex connection than just a single branch (In the elementary circuit, one branch consists of one element each, and each network consists of one element) The graph representation is the basics for the remaining two. It basically represents the path of transport – which in our case is conductive path for transport of free electrons or current – without any regard to the special conditions on that path. Each path segment is modelled with a branch (wire / pipe), each end (termination) of a path is modelled with a node (end of wire / pipe - terminal). The nodes of both branches are connected (that is why we are focused on topology of connections), so we form a circular (closed) conducting path for current flow – a circuit. So, we need at least two branches to close a circuit. Since our definition for current – elementary circuit – demands at least two types of elements (source and load) – neither of them can alone represent a circuit – so they must each have (at least) two terminals. The two terminals then correspond to the two nodes each branch has. + .

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**Circuit theory – graphs and network topologies**

Terms from graph theory: Graph. A transportation network, like any network, can be represented as a graph. Vertex (Node). A node v is a terminal point or an intersection point of a graph. It is the abstraction of a location such as a city, an administrative division, a road intersection or a transport terminal (stations, terminuses, harbors and airports). Edge (Link - Branch). An edge e is a link between two nodes. Path. A sequence of links that are traveled in the same direction. Circuit. A path in where the initial and terminal node corresponds. It is a cycle where all the links are traveled in the same direction. + .

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**Circuit theory – graphs and network topologies**

The schematic representation of a circuit, adds the possibility of modelling the transport conditions (such as resistivity) of a path (branch) to the graph representation. This is done by assigning a lump model (such as resistance) to the branch in the form of an element. The element must retain its two terminals, and these are modelled by ideal conducting paths – in our terms, ideal conductors. The schematic diagram consists of idealized circuit elements each of which represents some property of the actual circuit. It thus has its own symbolic language for representation of these elements on the graph: circuit schematic symbols .

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**Circuit theory – graphs and network topologies**

The flow of current through the elements can be modelled with arrows. On the other hand, we are aware that current is related to potential difference between the terminals of an element. Since we must retain the difference in potentials, we must index the nodes - number (name) them, so that we can write that “node A is on higher potential that node B”. This allows us to interpret the value of nodes as potentials – and assign current (flow) as through variable, and voltage (difference in potential) as across variable to any element with the two end terminals (and in the general sense - any branch with the two end nodes). This is again related to energy exchange and brings about the concept of causality. .

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**Circuit theory – graphs and network topologies**

Causality - Each of the primitive elements is defined by an elemental equation that relates its through and across-variables. This equation represents a constraint between the across-variable and the through-variable that must be satisfied at any instant. An immediate consequence is that the across-variable and the through-variable cannot both be independently specified at the same time. One variable must be considered to be defined by the system or an external input, the other variable is defined by the elemental equation. This is known as causality. . In the energy storage elements the constraint is expressed as a differential or integral relationship, that defines the element as having integral or derivative causality. Dissipative elements always operate in algebraic causality because the through and across-variables are related by algebraic equations. Causality leads to the possibility of using an equation (or a UI characteristic [comes from U for voltage and I for current – graphical representation of the equation]) to find out the current if the voltage is known – or vice versa. It also leads to acknowledgment that power can either be specified to voltage or current supplied, but not both.

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**Circuit theory – graphs and network topologies**

Causality leads to the possibility of using an equation (or a UI characteristic [a graphical representation of the equation - comes from U for voltage and I for current]) to find out the current if the voltage is known – or vice versa. It also leads to acknowledgment that power can either be specified to voltage or current supplied, but not both. For each element: There are two terminals, and there is a potential assigned to each (Va1 and Va2; or VA and VB). By reference we can assign one terminal potential to be higher (+) (above Va1 or VA ) than the other which determines a reference voltage across the element and VAB = VA – VB or Va = Va1 – Va2 the reference current flowing through the element (from + to -) .

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**Circuit theory – graphs and network topologies**

For each element: There are two terminals, and there is a potential assigned to each (Va1 and Va2; or VA and VB). By reference we can assign one terminal potential to be higher (+) than the other The difference in between the terminal potentials is the voltage across the element: V = V+ – V- or Va = Va1 – Va2 - The current through the element, can be determined from the equation / UI characteristic. If the element is a resistor, its elemental equation algebraically is and geometrically relates to a linear UI characteristic. We have to solve this equation – graphically or numerically - in order to get the current through the element. .

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**Circuit theory – graphs and network topologies**

The definition of voltage as “across” variable, and current as “through” variable, indicates the way measuring instruments should be interfaced: - Ammeter (ampermeter) measures current : so we must break a circuit at a given branch, and insert the ampermeter as an additional branch that completes the given branch (and the circuit) in order to measure current through the given branch - Voltmeter and Oscilloscope measure voltage – or the difference in potentials at the terminals : so we must position the voltmeter probes so they are in electric contact with the terminals of the element. .

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**Circuit theory – graphs and network topologies**

The network representation is meant to provide means for simplified representation of more complex network systems – focused on the simple fact that energy must be exchanged if current is to flow, no matter how complex the system. So, it geometrically recognizes the fact that we need at least two terminals for an element that can participate in energy exchange in current flow. This recognition comes in the form of a port, which is nothing but a pair of terminals through which the system can exchange energy with another system. Note that also othe energy systems than electric can be modelled with this approach – mechanical, fluid etc. So, in the elementary circuit we will have two networks exchanging energy, one representing the generator element, and the other – the resistor. Since both of these networks sport only two terminals each, they are both one-port networks. Note that again, we need at least two one-port networks to close a circuit – so a network in this sense does not represent a circuit – although internally – since they can represent complex circuits – there could be current flowing and energy exhange going on. The port just represents the energy that has been exchanged with the “outside”. .

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**Circuit theory – graphs and network topologies**

The benefit of network representation is more obvious when we consider that we can represent more complex connections through one or more port networks. If we remember the intuitive interpretation of Ohms Law, all that a source needs to see on its terminals is some finite resistance for current to start flowing – no matter how complex the network may be. Network representation makes this perception of a source ”seeing” resistance on its terminals easier – which is the core of the method known as Thevenin/Norton theorem, which simplifies complex networks down to a one-port network. .

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**Circuit theory – graphs and network topologies**

We can compare the different representations – in the case of the elementary circuit – on the figure below: .

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**Circuit theory – graphs and network topologies**

On the figure above, the shaded areas represent the branches in the same elementary circuit, drawn when the generator and resistor switch places. Here we can see the topological aspect – that the connection structure doesn’t depend on how we draw. . Namely, the fact that for the generator branch, current flows from – to + node, means that it is giving energy – and the same is valid for the resistor branch - current flowing from + to - node, means that it is using/receiving energy. This is a valid conclusion for a models of branch (& its nodes), as well as for elements and their terminals, and networks and their ports – and is the basic method for recognizing energy exchange.

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**Circuit theory – graphs and network topologies**

V i P=Vi Produced by Source Or Used by Load Load + - .

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Circuit theory These theories bring about circuit theory as a way of solving electric circuits – analysis of more complex topologies than the elementary circuit. As the complexity of a circuit grows, we can expect a great number of branches to be interconnected in different ways, resulting in a number of currents through and voltages across hose branches. To solve a circuit means to determine all currents and voltages in a circuit in a given instant. Circuit theory applies graph theory – for the graph representation of the conductive transport network which constitutes the electric circuit. The graph is supplied in the form of a schematic diagram, with the corresponding schematic symbols. Then, a correspondence between the graph and a system of equations can be developed, and the system solved, basically using only three laws: First and second Kirchoff law (laws of conservation of charge and energy) Ohms law (for a resistor, or any UI relation that exists for a given element – represented on the graph with a schematic symbol) (Additionaly Thevenin/Norton theorem may be used) The algebraic system of equations can determine each current and voltage in the circuit. For strictly ohmic elements, these equations are also linear, and enable us to quickly solve circuits. .

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Circuit theory Circuit theory means that now we can work approximatively, and instead of taking into account local properties (such as concentration of charge, potential distribution and resistivity), we can solve a set of linear equations connected to a graph, and quickly gain integral, general measures – such as current and voltage. In addition, we do not need to care how the circuit physically appears – all we have to be careful about is that we have correctly represented the connection between the elements (connection topology) in the schematic as in the physical circuit. .

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**Circuit theory – schematic symbols**

circuit schematic symbols Conductive wires Resistor Power source .

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**Circuit theory – basic laws**

We can see that all the elements are modelled as two terminal or one port elements. This is related to the graph concept of a branch, and analogous to a conducting path – for each branch we can have an across and through variable, related to voltage and current, and then the actual behavior can be attributed to a particular idealized model – say a resistor. With this in mind, in circuit theory we work in essence with branches and nodes on schematic diagrams – graphs that represent electric circuits, which use schematic symbols for depicting elements. Thus we will rephrase the terms from graph theory in circuit theory context. . A circuit with 5 branches. A circuit with 3 nodes

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**Circuit theory – basic laws**

A circuit with 5 branches and 2 (principal) nodes A circuit with 3 nodes (2 principal) and 3 branches We redefine an electric circuit as a connection of electrical devices that form one or more closed paths. A branch: A branch is a single electrical element or device. A node: A node can be defined as a connection point between two or more branches. (principal node or junction – at least three branches.) If we start at any point in a circuit (node), proceed through connected electric devices back to the point (node) from which we started, without crossing a node more than one time, we form a closed-path. A loop is a closed-path. An independent loop is one that contains at least one element not contained in another loop. Ground is a reference node located at an arbitrary point in the circuit to which for convenience is assigned a potential of zero volts. .

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**Circuit theory – basic laws**

To solve a circuit, means to determine currents through and voltages across all branches in the circuit. Three laws govern the distribution of currents and voltages in a circuit with resistors: Ohms law: Kirchoff laws: which basically governs the voltage across and current through a branch that is modelled by a resistor. For non ohmic elements, we need a corresponding relation between the current and voltage! the 1st or current law (KCL) or the junction rule: for a given junction or node in a circuit, the sum of the currents entering equals the sum of the currents leaving. This law is a statement of charge conservation – what goes in must come out. . the 2nd or voltage law (KVL) or the loop rule: around any closed loop in a circuit, the sum of the potential differences across all elements is zero. This law is a statement of energy conservation, in that any charge that starts and ends up at the same point with the same velocity must have gained as much energy as it lost.

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**Circuit theory – basic laws**

An important corollary of 1st KL: A current through a branch remans the same, no matter where in the branch it is measured. .

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AC Current So far, we have drawn our conclusions based on a steady current – one that does not change its magnitude or direction in time. Such a current is known as DC (Direct Current) to point at the fact that the current does not change direction. However, a direct current might also be changing in magnitude and not be steady. To realise the behaviour of current or voltage in time, we use time diagrams. A voltage source that creates a constant potential difference on its ends (which cases steady current in a resistor) is a DC voltage source. A voltage source that creates a sinusoidal potential difference on its ends (which is changing in magnitude and direction, but is mathematically predictable) is a AC voltage source. A voltage source that creates a potential difference on its ends, which is changing in magnitude and direction, and is random (only statistically predictable) is a random voltage source. Information signals (speech) also appear random in time, so if information is encoded in the change of voltage in time, generally we have a signal voltage source.

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**AC Current – Interpreting sinusoidal time graphs**

The image displays a sine function that represents a sinusoidally changing current. Here the function graph gives information on how many particles crossed a given point in circuit at a given point in time.

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**AC Current – Interpreting sinusoidal time graphs**

The image displays a sine function that represents a sinusoidally changing voltage. Here the function graph gives information on the potential difference between two points in a circuit (such as element terminals) at a given point in time. Note that we could obtain such a sinusoid from for example as a result from a measurement with an oscilloscope. By that we obtain information only about the potential difference on the terminals – not their exact potentials! They depend on the point that we use as reference (ground – 0 Volts) and on the circuit topology itself. Thus, when we obtain voltage for a given moment, it could mean any kind of potential distribution – until we connect that voltage to the rest of the circuit and obtain node potentials.

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**AC Current – Using the UI characteristic.**

Once we start working with voltages which are changing in time, it can become difficult to relate the equations to what is happening in the circuit. However, since the UI characteristic is a geometric rendering of the equation of the element, it can be used to graphically determine the look of the signals. The image displays a UI characteristic of a resistor (linear UI characteristic). We know the resistance doesn’t change with voltage, so the resistor will “resist” all the same, no matter what voltage is applied. So the look of the sine will remain – only its amplitude will reflect Ohms law. Graphically, we can align the graphs for change of voltage (and current) in time, with the corresponding sides of the UI characteristic. Then, at each moment in time we have definite values for U and I called a working point. As the voltage changes, the working point moves on the UI char, and traces out the changes of current in time. So, we now have a graphical method for obtaining voltages and currents in time.

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**Soldering Basic Soldering photo guide**

Electric contact - a physical contact that permits current flow between conducting parts. A. Solder is used to hold two (or more) conductors in electrical contact with each other. B. Solder is not used to make the electrical contact. C. Solder is not used to provide the main mechanical support for a joint. D. Solder is used to encapsulate a joint, prevent oxidation of the joint, and provide minor mechanical support for a connection.

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Final Remarks Electricity CAN be dangerous – read the following document on electrical safety: Electrical Safety and How can electric current *NOT* cause a shock? The magnetic field is an effect that appears around charges in movement – and is per defintion an ffect that follows electric currents – as phenomena of transport (movement) of free electrons. The magnetic field will however not be looked at during this part of the course. A conductor which is not a part of a loop in a circuit, yet is connected to some source of (AC) current, will experience that a charge will be deposited on it – according to some equivalent capacitance it might have in relation with some other conductor. If this deposited charge changes due to a changing current, then it becomes a source of a changing electrostatic (and thus also electromagnetic) field – which we are aware propagates through space like a wave with the speed of light. The the conductor becomes an antenna.

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**Resources Conductors in Electric Field**

All About Circuits - basic concepts of electricity Fluid Dynamics Hydrostatics Direct-Current Circuits Direct current circuits Understanding electric flow on the quantum level Graphs, matrices and circuit theory Circuit models Lecture - Circuit Theory Circuit Variables Network Models

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1 Fall 2004 Physics 3 Tu-Th Section Claudio Campagnari Lecture 15: 18 Nov. 2004 Web page:

1 Fall 2004 Physics 3 Tu-Th Section Claudio Campagnari Lecture 15: 18 Nov. 2004 Web page:

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