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					                                           Sensors Technology MED4: Electron Flow – Current
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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                                            Sensors Technology MED4: Electron Flow – Current
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 Q passing through some region of space (specific point in a
    circuit) in a given time period t
                                          Q
                                       I
                                          t

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.
                                                                                                               2
                                          Sensors Technology MED4: Electron Flow – Current
Current, conductors and field

We are already aware of the fact that
 - 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.
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.




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                                          Sensors Technology MED4: Electron Flow – Current
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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                                             Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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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                                              Sensors Technology MED4: Electron Flow – Current
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!                            7
                                                    Sensors Technology MED4: Electron Flow – Current
  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   Less dense – less collisions due to thermal motion – less
area (or per num. charges )                                  force per area (or per num. molecules )
More dense – more electrostatic repulsion – more force       More dense – more thermal collisions – more force           8
                                          Sensors Technology MED4: Electron Flow – Current
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         motion of free electrons in
movement as a free molecule in gas does. The situation is usually that an               a conductor
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.
                                                                                 collisions
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).

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.                                                                                          9
                                          Sensors Technology MED4: Electron Flow – Current
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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                                         Sensors Technology MED4: Electron Flow – Current
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
Water flow               Electron flow            (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.
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                                            Sensors Technology MED4: Electron Flow – Current
  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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                                         Sensors Technology MED4: Electron Flow – Current
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:
                                                           Q
                                                        C
                                                           U
                            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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                                           Sensors Technology MED4: Electron Flow – Current
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.                                                                                    14
                                               Sensors Technology MED4: Electron Flow – Current
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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                                         Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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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                                         Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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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                                           Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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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                                           Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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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                                            Sensors Technology MED4: Electron Flow – Current
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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                                           Sensors Technology MED4: Electron Flow – Current
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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                                           Sensors Technology MED4: Electron Flow – Current
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.




 .




                                                                                                            26
                                            Sensors Technology MED4: Electron Flow – Current
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                         U

                    I                           I

                       R                         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.




.




                                                                                                     27
               Sensors Technology MED4: Electron Flow – Current
Ohm’s law

                U
       U        I
    I          R

       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.


.




                                                                           28
                                           Sensors Technology MED4: Electron Flow – Current
Ohm’s law and sensors.

Ohm’s law, even on this intuitive level is of essential meaning for sensors.
                                                                                                  U
                                                                                             I
                                                                                                  R
 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.


                                                                                                         29
                                            Sensors Technology MED4: Electron Flow – Current
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:
                                                 P UI
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:
1. 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
2. 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:

1.   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.




2.   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:


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


                                        P UI  RI 2



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                                          Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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                               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
      r throughout                        R as “box”                            reapplied throughout the
                                                                                dimensions of the material.
                                                                                Therefore, we basically
                                                                                “lump” physical processes
                                l                                               involved with resistivity inside
                            Rr                                                 a single package – ”black
                                                                                box” - called a resistor,
                                S                                               described with only one
                                                                                parameter - resistance (R).
                                                                                The resistance is the overall
                                                                                effect of resistivity (effect of
          S                                                                     all the “sieves”) within the
                                                                                resistor’s entire volume on
                                                                                the flow of current.
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Resistivity and resistance
                                                    l
                                                Rr
                                                    S
            S




     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                         U
                                                                              I
                                                                                 R                     44
                                          Sensors Technology MED4: Electron Flow – Current
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               Consider the Venturi Tube simulation applet for
demonstration of the gradient of potential
.                                                    demonstration that the local pressure, during
and concentration, as well as the change             current flow, acts in all directions – and given a
in speed according to density of particles,          conduit at that position, that “push” will be
and wave propagation of the field – and              propagated through the conduit. The same
power relations between source and load.             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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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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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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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.                                                                                        49
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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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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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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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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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                                         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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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.

. the energy storage elements the constraint is expressed as a differential or integral relationship, that
In
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.                                                                                                                    55
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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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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.
                                                                              U
 If the element is a resistor, its elemental equation algebraically is   I
                                                                              R
 and geometrically relates to a linear UI characteristic.

 .
 We have to solve this equation – graphically or
 numerically - in order to get the current through                                                  1
 the element.
                                                                                               I     U
                                                                                                    R

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                                                                                                       1
                                                                                                  I     U
                                                                                                       R


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




.




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                                      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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                                +                 P=Vi
                                                                           +
                           V          i     Produced by Source      i   Load   V
                                                    Or
                                -              Used by 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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                                          Sensors Technology MED4: Electron Flow – Current
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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                                       Sensors Technology MED4: Electron Flow – Current
Circuit theory – schematic symbols

circuit schematic symbols        Resistor                        Conductive wires




                        Power source




.




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                                         Sensors Technology MED4: Electron Flow – Current
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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                                           Sensors Technology MED4: Electron Flow – Current
 Circuit theory – basic laws


                                                                               
                                                                                   
                      
                                                                                       




                                                                                    
                                                                               
                                                                                
              A circuit with 5 branches                        A circuit with 3 nodes (2 principal)
               and 2 (principal) nodes                                    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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                                           Sensors Technology MED4: Electron Flow – Current
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:
                       which basically governs the voltage across and current through a
Ohms law:       U      branch that is modelled by a resistor. For non ohmic elements, we
             I
                R      need a corresponding relation between the current and voltage!

Kirchoff laws:
 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.                                                                                69
                                       Sensors Technology MED4: Electron Flow – Current
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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                                         Sensors Technology MED4: Electron Flow – Current
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.                         71
                              Sensors Technology MED4: Electron Flow – Current
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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                              Sensors Technology MED4: Electron Flow – Current
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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                                  Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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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                                         Sensors Technology MED4: Electron Flow – Current
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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                                          Sensors Technology MED4: Electron Flow – Current
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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