Electric Circuits

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					ELECTRICAL CIRCUITS
                 The CELL
The cell stores chemical energy and transfers it to

electrical energy when a circuit is connected.

                    When two or more cells are
                    connected together we call this
                    a Battery.

                    The cells chemical energy is
                    used up pushing a current round
                    a circuit.
What is an electric current?
An electric current is a flow of microscopic particles
called electrons flowing through wires and
components.

                              -
                          +




In which direction does the current flow?
from the Negative terminal to the Positive terminal of a
cell.
             simple circuits
Here is a simple electric circuit. It has a cell, a
lamp and a switch.


      cell                                      wires




    switch                                       lamp



To make the circuit, these components are connected
together with metal connecting wires.
             simple circuits
When the switch is closed, the lamp lights up. This is
because there is a continuous path of metal for the
electric current to flow around.




 If there were any breaks in the circuit, the current
 could not flow.
             circuit diagram
Scientists usually draw electric circuits using symbols;




    cell         lamp          switch         wires
           circuit diagrams
 In circuit diagrams components are represented by
 the following symbols;




 cell     battery     switch      lamp     buzzer




ammeter   voltmeter   motor     resistor   variable
                                           resistor
          types of circuit
There are two types of electrical circuits;


  SERIES CIRCUITS         PARALLEL CIRCUITS
        SERIES CIRCUITS




The components are connected end-to-end, one
after the other.
They make a simple loop for the current to flow
round.
If one bulb ‘blows’ it breaks the whole circuit and
all the bulbs go out.
    PARALLEL CIRCUITS




The components are connected side by side.

The current has a choice of routes.
If one bulb ‘blows’ there is still be a complete circuit to
the other bulb so it stays alight.
    measuring current
Electric current is measured in amps (A) using

an ammeter connected in series in the circuit.




                                  A
        measuring current
This is how we draw an ammeter in a circuit.




A                         A




    SERIES CIRCUIT            PARALLEL CIRCUIT
        measuring current
SERIES CIRCUIT

• current is the same    2A             2A
  at all points in the
  circuit.                    2A



PARALLEL CIRCUIT

                         2A             2A
• current is shared
  between the                      1A
  components
                                   1A
copy the following circuits and fill in the
missing ammeter readings.




   ?
  4A                        ?
                           3A                 3A

                                       ?
                                      1A
                    ?
                   4A

           4A                         1A


                                       ?
                                      1A
            measuring voltage
The ‘electrical push’ which the cell gives to the current
is called the voltage. It is measured in volts (V) on a
voltmeter




                                         V
           measuring voltage
Different cells produce different voltages. The
bigger the voltage supplied by the cell, the bigger the
current.

Unlike an ammeter a voltmeter is connected across
the components


Scientist usually use the term Potential Difference
(pd) when they talk about voltage.
         measuring voltage
This is how we draw a voltmeter in a circuit.




     V
                                      V
  SERIES CIRCUIT             PARALLEL CIRCUIT
    measuring voltage




                   V


V       V
                    V
             series circuit
• voltage is shared between the components




                          3V




                 1.5V          1.5V
             parallel circuit
• voltage is the same in all parts of the circuit.



                         3V

                         3V




                         3V
measuring current & voltage

copy the following circuits on the next two
slides.

complete the missing current and voltage
readings.

remember the rules for current and voltage
in series and parallel circuits.
measuring current & voltage

 a)
                   6V
      4A                    A


           V            V

               A
measuring current & voltage

   b)
             6V
        4A            A
             V

                  A


             V

                  A
                          answers

a)                               b)
                                           6V
                6V                    4A             4A
                                           6V
 4A                         4A

      3V             3V                         2A


           4A
                                           6V

                                                2A
             Electric Circuits
In electricity, the concept of voltage will be
  like pressure. Water flows from high
  pressure to low pressure (this is consistent with
  our previous analogy that Voltage is like height
  since DP = rgh for fluids) ; electricity flows
 from high voltage to low voltage.
But what flows in electricity? Charges!
How do we measure this flow? By Current:
     current = I = Dq / Dt
UNITS: Amp(ere) = Coulomb / second
           The rate at which electrons move
           along field lines is called drift speed,
           typically about 10-4 m/s

Electric current defined in
terms of the flow of positive
charge opposite the electrons is
called conventional current
Current will always be in the
same direction as the local
electric field
         Voltage Sources:
    batteries and power supplies
A battery or power supply supplies voltage. This is
  analogous to what a pump does in a water system.
Question: Does a water pump supply water? If
  you bought a water pump, and then plugged it in
  (without any other connections), would water
  come out of the pump?
Question: Does the battery or power supply
  actually supply the charges that will flow
  through the circuit?
            • Charges move from
              higher to lower potential
            • For the process to
              continue, charges that
              have moved from a
              higher to lower potential
              must be raised back to a
              higher potential again
            • A battery is able to add
              charges and raise the
              charges to higher electric
              potential


Symbol for a battery
          Voltage Sources:
    batteries and power supplies
Just like a water pump only pushes water (gives
  energy to the water by raising the pressure of the
  water), so the voltage source only pushes the
  charges (gives energy to the charges by raising
  the voltage of the charges).
Just like a pump needs water coming into it in order
  to pump water out, so the voltage source needs
  charges coming into it (into the negative terminal)
  in order to “pump” them out (of the positive
  terminal).
          Voltage Sources:
    batteries and power supplies
Because of the “pumping” nature of voltage sources,
  we need to have a complete circuit before
  we have a current.
           Circuit Elements
In this first part of the course we will consider
  two of the common circuit elements:
      capacitor
      resistor
The capacitor is an element that stores
  charge for use later (like a water tower).
The resistor is an element that “resists” the
  flow of electricity.
             Electrical Resistance
                       &
                 Ohms’ Law
The current established is directly proportional to
 the voltage difference
Ohm’s Law: ΔV ∝ I

In a plot of ΔV vs I,
the slope is called the electrical resistance
               Resistance
Current is somewhat like fluid flow. Recall
 that it took a pressure difference to make
 the fluid flow due to the viscosity of the
 fluid and the size (area and length) of the
 pipe. So to in electricity, it takes a voltage
 difference to make electric current flow due
 to the resistance in the circuit.
                Resistance
By experiment we find that if we increase the
 voltage, we increase the current: V is
 proportional to I. The constant of
 proportionality we call the resistance, R:
     V = I*R         Ohm’s Law

UNITS: R = V/I so Ohm = Volt / Amp.

The symbol for resistance is
                 Resistance
Just as with fluid flow, the amount of
  resistance does not depend on the voltage
  (pressure) or the current (volume flow).
  The formula V=IR relates voltage to
  current. If you double the voltage, you will
  double the current, not change the
  resistance. The same applied to capacitance:
  the capacitance did not depend on the charge and
  voltage - the capacitance related the two.
As was the case in fluid flow and capacitance, the
  amount of resistance depends on the
  materials and shapes of the wires.
                Resistance
The resistance depends on material and
 geometry (shape). For a wire, we have:
                  R=rL/A
where r is called the resistivity (in Ohm-m)
 and measures how hard it is for current to
 flow through the material, L is the length of
 the wire, and A is the cross-sectional area of
 the wire. The second lab experiment deals with
  Ohm’s Law and the above equation.
           Electrical Power
The electrical potential energy of a charge is:
           U = q*V .
Power is the change in energy with respect to
  time:    Power = DU / Dt .
Putting these two concepts together we have:
  Power = D(qV) / Dt = V(Dq) / Dt = I*V.
           Electrical Power
Besides this basic equation for power:
     P = I*V
remember we also have Ohm’s Law:
     V = I*R .
Thus we can write the following equations for
  power: P = I2*R = V2/R = I*V .
To see which one gives the most insight, we
  need to understand what is being held
  constant.
                 Example
When using batteries, the battery keeps the
 voltage constant. Each D cell battery
 supplies 1.5 volts, so four D cell batteries in
 series (one after the other) will supply a
 constant 6 volts.
When used with four D cell batteries, a light
 bulb is designed to use 5 Watts of power.
 What is the resistance of the light bulb?
               Example
We know V = 6 volts, and P = 5 Watts; we’re
 looking for R.
We have two equations:
     P = I*V and V = I*R
which together have 4 quantities:
     P, I, V & R..
We know two of these (P & V), so we should
 be able to solve for the other two.
                Example
Using the power equation we can solve for I:
 P = I*V, so 5 Watts = I * (6 volts), or
 I = 5 Watts / 6 volts = 0.833 amps.
Now we can use Ohm’s Law to solve for R:
 V = I*R, so
R = V/I = 6 volts / 0.833 amps = 7.2 W .
            Example extended
If we wanted a higher power light bulb,
  should we have a bigger resistance or a
  smaller resistance for the light bulb?
We have two relations for power that involve
  resistance:
P=I*V; V=I*R; eliminating V gives: P = I2*R and
P=I*V; I=V/R; eliminating I gives:   P = V2 / R .
In the first case, Power goes up as R goes up; in the
   second case, Power goes down as R goes up.
Which one do we use to answer the above question?
            Example extended
Answer: In this case, the voltage is being held
  constant due to the nature of the batteries. This
  means that the current will change as we change
  the resistance. Thus, the P = V2 / R would be the
  most straight-forward equation to use. This means
  that as R goes down, P goes up. (If we had used the
  P = I2*R formula, as R goes up, I would decrease – so it
  would not be clear what happened to power.)
The answer: for more power, lower the
 resistance. This will allow more current to
 flow at the same voltage, and hence allow
 more power!
              Kirchhoff’s Laws
• Junction Law: at a junction in a circuit, the
  sum of the current entering the junction will
  equal the sum of the current leaving.
              ΣI= ΣI
        in     out


• Loop Law: the sum of the potential drops
  around any closed loop must add to
              ΣV= 0
       loop
         Connecting Resistors
There are two basic ways of connecting two
  resistors: series and parallel.
In series, we connect resistors together like
  railroad cars; this is just like we have for
  capacitors:
                 +   -     +   -
high V                                   low V
               R1        R2
          Formula for Series:
To see how resistors combine to give an
  effective resistance when in series, we can
  look either at                 R1
                           I
V = I*R,
or at                    +      V1 V     R2
                    Vbat            2

R = rL/A .               -
              Formula for Series
Using V = I*R, we see that in series the
 current must move through both resistors.
(Think of water flowing down two water falls in series.)
 Thus Itotal = I1 = I2 .
Also, the voltage drop across the two resistors
 add to give the total voltage drop:
(The total height that the water fell is the addition of the two
  heights of the falls.)
Vtotal = (V1 + V2). Thus, Reff = Vtotal / Itotal =
(V1 + V2)/Itotal = V1/I1 + V2/I2 = R1 + R2.
          Formula for Series
Using R = rL/A , we see that we have to go
 over both lengths, so the lengths should add.
 The distances are in the numerator, and so
 the values should add.
This is just like in R = V/I (from V = IR)
 where the V’s add and are in the numerator!
Note: this is the opposite of capacitors
 when connected in series! Recall that
 C = Q/V, where V is in the denominator!
   Formula for Parallel Resistors
The result for the effective resistance for a
 parallel connection is different, but we can
 start from the same two places:
(Think of water in a river that splits with some water
  flowing over one fall and the rest falling over the
  other but all the water ending up joining back
  together again.) V=I*R, or R = rL/A .
                 Itotal
         +
  Vbat                    I1   R1 I2   R2
             -
   Formula for Parallel Resistors
                V=I*R, or R = rL/A
For parallel, both resistors are across the same
  voltage, so Vtotal = V1 = V2 . The current can go
  through either resistor, so: Itotal = (I1 + I2 ) .
Since the I’s are in the denominator, we have:
         R = Vtotal/Itotal = Vtotal/(I1+I2); or
1/Reff =     (I1+I2)/Vtotal = I1/V1 + I2/V2   = 1/R1 +
                          1/R2.
  Formula for Parallel Resistors
If we start from R = rL/A , we can see that
   parallel resistors are equivalent to one
   resistor with more Area. But A is in the
   denominator (just like I was in the previous
   slide), so we need to add the inverses:
       1/Reff = 1/R1 + 1/R2 .
             Terminal Voltage
Terminal voltage, VT , is the potential difference
  between the terminals of a battery.
Ideal voltage, VB , is determined by the chemistry
  of the battery.
Internal resistance, ir : some charge will be los due
  to the random thermal motion of the battery

Terminal voltage will be:    VT = VB - ir
For recharging a battery:    VT = VB + ir
             Current Division
When current enters a junction, Kirchhoff’s first
 law tells you the sum of the current entering
 must equal the sum of the current leaving.
Example:         8 = I + 4I + 5I  I = 0.8 A

                 1/RP = 1/20 +1/5 +1/4 = ½  RP = 2Ω

                 V = IRP = 8 • 2 = 16V

                 16 = I1(20)  I1 = 0.8A
                 16 = I2(5)  I2 = 3.2A
                 16 = I3(4)  I3 = 4A
             Simple Circuits
• A simple circuit is a connection of batteries
  and resistors that meets 2 criteria

  1. All batteries are in series
  2. The equivalent resistance of the entire circuit
     can be obtained by repeated use of just the
     series and parallel equivalent resistance
     formulas
VB = 60 – 18 = 42V
                               1/RP = 1/12 + 1/6 = ¼  RP = 4Ω

Requiv = 4 + 8 + 6 + 3 = 21Ω
                               42 – I(21) = 0  I = 2A
VT60 = 60 – 2 • 1 = 58V
VT18 = 18 – 2 • 2 = 22V
                               2 = I + 2I  I = 0.67A
              Capacitance
We define capacitance as the amount of
  charge stored per volt: C = Qstored / DV.
UNITS: Farad = Coulomb / Volt
Just as the capacity of a water tower depends
  on the size and shape, so the capacitance of
  a capacitor depends on its size and shape.
  Just as a big water tower can contain more
  water per foot (or per unit pressure), so a
  big capacitor can store more charge per
  volt.
      Parallel Plate Capacitor
For a parallel plate capacitor, we can pull
  charge from one plate (leaving a -Q on that
  plate) and deposit it on the other plate
  (leaving a +Q on that plate). Because of the
  charge separation, we have a voltage
  difference between the plates, DV. The
  harder we pull (the more voltage across the two
  plates), the more charge we pull: C = Q /DV.
  Note that C is NOT CHANGED by either
  Q or DV; C relates Q and DV!
              Capacitance
• What happens when a water tower is over-
  filled? It can break due to the pressure of
  the water pushing on the walls.
• What happens when an electric capacitor is
  “over-filled” or equivalently a higher
  voltage is placed across the capacitor than
  the listed maximum voltage? It will
  “break” by having the charge “escape”.
  This escaping charge is like lightning - a
  spark that usually destroys the capacitor.
               V or DV ?
When we deal with height, h, we usually refer
 to the change in height, Dh, between the
 base and the top. Sometimes we do refer to
 the height as measured from some reference
 point. It is usually clear from the context
 whether h refers to an actual h or a Dh.
With voltage, the same thing applies. We
 often just use V to really mean DV. You
 should be able to determine whether we
 really mean V or DV when we say V.
       Parallel Plate Capacitor
For this parallel plate capacitor, the
  capacitance is related to charge and voltage
  (C = Q/V), but the actual capacitance
  depends on the size and shape:
   C plate ∝ A / d
A is the area of each plate, d is the distance
  between the plates
            Energy Storage
If a capacitor stores charge and carries
   voltage, it also stores the energy it took to
   separate the charge. The formula for this is:
       Estored = (1/2)QV = (1/2)CV2 ,
where in the second equation we have used
   the relation: C = Q/V .
          Energy Storage
Note that previously we had:
     U = q*V ,
and now for a capacitor we have:
     Ucap = ½ QV = ½ CV2 = ½ Q2/C
             Energy Storage
The reason is that in charging a capacitor, the
 first bit of charge is transferred while there
 is very little voltage on the capacitor (recall
 that the charge separation creates the
 voltage!). Only the last bit of charge is
 moved across the full voltage. Thus, on
 average, the full charge moves across
 only half the voltage!
 Hooking Capacitors Together
Instead of making and storing all sizes of
  capacitors, we can make and store just
  certain values of capacitors. When we need
  a non-standard size capacitor, we can make
  it by hooking two or more standard size
  capacitors together to make an effective
  capacitor of the value we need.
            Two basic ways
There are two basic ways of connecting two
  capacitors: series and parallel.
In series, we connect capacitors together like
  railroad cars; using parallel plate capacitors
  it would look like this:
                 +   -     +   -
high V                                   low V
               C1        C2
                   Series
If we include a battery as the voltage source,
   the series circuit would look like this:
                   C1
  +
 Vbat
                               C2
Note that there is only one way around the
   circuit, and you have to jump BOTH
   capacitors in making the circuit - no choice!
                 Parallel
In a parallel hook-up, there is a branch point
  that allows you to complete the circuit by
  jumping over either one capacitor or the
  other: you have a choice!

High V          C1                 Low V



                C2
             Parallel Circuit
If we include a battery, the parallel circuit
   would look like this:

   +              +                +
  Vbat           C1              C2
        Formula for Series:
To see how capacitors combine to give an
 effective capacitance when in series, we
 can look either at C = Q/V, or at
 C plate = A /d
           Formula for Series
Using C = Q/V, we see that in series the
 charge moved from capacitor 2’s negative
 plate must be moved through the battery to
 capacitor 1’s positive plate.
                    C1
       +        +Q
Vbat                            C2
       -                   -Q
            (  +Qtotal)
           Formula for Series
But the positive charge on the left plate of C1 will
  attract a negative charge on the right plate, and the
  negative charge on the bottom plate of C2 will
  attract a positive charge on the top plate - just
  what is needed to give the negative charge on the
  right plate of C1. Thus Qtotal = Q1 = Q2 .
                   C1        (+Q1  )

       +         +Q1     -Q 1       +Q2
Vbat                                      C2
       -                           -Q2
              (  +Qtotal)
           Formula for Series
Also, the voltage drop across the two
  capacitors add to give the total voltage drop:
Vtotal = (V1 + V2).
Thus, Ceff = Qtotal / Vtotal = Qtotal / (V1 + V2),
  or (with Qtotal = Q1 = Q2)
 [1/Ceff] = (V1 + V2) / Qtotal = V1/Q1 + V2/Q2 =

            1/C1 + 1/C2 = 1/Ceffective
               Parallel Circuit
For parallel, both plates are across the same voltage,
  so Vtotal = V1 = V2 . The charge can accumulate
  on either plate, so: Qtotal = (Q1 + Q2).
Since the Q’s are in the numerator, we have:
Ceff = C1 + C2.
   +                           +Q1              +Q2
  Vbat              C1         -Q1     C2       -Q2
                    +Q1 
          +Qtotal = (Q1+Q2)          +Q2
              Review of Formulas
Electric Current         I= ΔQ/Δt
Resistor Voltage Drop    V = IR
Resistivity              R = rL/A
Electric Power           P = IV
Resistance Power P = I2R = V2/R
Junction Law             SI = SI
                        in    out

Loop Law                SΔV = 0
                       loop

Series Resistors           Rs = R1 + R2 +…
Parallel Resistors 1/RP = 1/R1 + 1/R2 +…
Terminal Voltage VT = VB ± IR
Capacitance                C = Q/V
Parallel Plate Capacitor C plate ∝ A / d
Capacitors in Series       1/Cs = 1/C1 + 1/C2 +…
Capacitors in Parallel     CP = C1 + C2 +…
Energy in Capacitor        U = ½ Q2 /C = ½ CV2 = ½ QV

				
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