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					                                Notes on Lab #1 - dc Circuits
    1. dc Circuits ................................................................................................................... 1
       1.1 Voltage .................................................................................................................. 1
          1.1.1 Voltage Difference ......................................................................................... 1
       1.1.2 Absolute Voltage ............................................................................................... 2
       1.1.3 Voltage (Power) Supply..................................................................................... 2
       1.2 Current .................................................................................................................. 3
       1.3 Basic Laws of Electrical Circuits .......................................................................... 4
          1.3.1 Ohm's Law ..................................................................................................... 4
          1.3.2 Kirchoff's Laws .............................................................................................. 5
       1.4 Power in Electrical Circuits .................................................................................. 6
       1.5 Equivalent Resistance ........................................................................................... 7
          1.5.1 Resistors in Series .......................................................................................... 7
          1.5.2 Resistors in Parallel........................................................................................ 8
       1.6 Voltage Divider ..................................................................................................... 9
       1.7 Loading, "output impedance", and the Thevenin Model .................................. 10
          1.7.1 Adding a "load resistor" ............................................................................... 10
          1.7.2 Thevenin Model ........................................................................................... 12
          1.7.3 A prescription for determining Vth and Rth for an arbitrary circuit ............ 13


1. dc Circuits1
There are two fundamental quantities that you need to keep track of in electrical circuits,
voltage and current.

In short: A current is the rate at which electrical charge is flowing by. A voltage, or
more strictly, a voltage difference, is the “agent” which causes a current to flow.

Let’s state this a bit more formally.


         1.1 Voltage

          1.1.1 Voltage Difference
The voltage difference (or potential difference) between two given points, V , is
defined as the amount of work required to move a positive unit test charge from one point
to the other.



1
  You may have seen much of the material in this first lab during an introductory physics
course or somewhere else along the line. But these basic ideas are so central to what
follows for much of the rest of this course, so it will be helpful to revisit these topics. It is
probably a good idea to reread the relevant sections of an introductory physics text (e.g.
Wolfson and Pasachoff Chapters 28 and 29, or Serway, Chapters 28 and 29.)


                                                               page 1
For example suppose that V is defined as Vb  Va in the drawing below. Note that
in this case V > 0 since positive work is required to move a positive unit test charge
from its initial position at a to its final position at b.

To gain intuition about potential differences it may be helpful to think about the
analogous case of gravitational potential: In the same way that a ball will roll downhill
towards a point of lower gravitational potential, a positive charge will tend to move
toward regions of lower electrical potential.

The SI unit for voltage difference is the volt (abbreviation: V). By definition, a voltage
difference of 1 volt exists between two points if 1 joule of work must be done in moving
1 coulomb of charge between the two points. This implies that:


               1 volt = 1 joule / 1 coulomb = 1 "joule per coulomb"


      1.1.2 Absolute Voltage
By convention, we choose one point to serve as a reference point, which is defined to
have an (absolute) voltage of 0 V. Usually the "earth" or "ground" is selected to be the
reference point. (Since the earth is a pretty good conductor it is usually a good
approximation to treat all points on the earth as being at the same voltage.) Furthermore,
any point connected directly to the earth through a good conductor is usually also at or
very near 0 V.

The absolute voltage Va at any point a is then defined to be equal to the amount of work
that must be done to move one coulomb of charge from the earth to point a.


      1.1.3 Voltage (Power) Supply
An ideal voltage (or power) supply is defined to be a device that maintains a constant
voltage difference V between its two terminals independent of the "load" that is


                                          page 2
connected to these terminals.

For example:

       A battery is a type of dc voltage supply. ("dc" usually means that the size of the
       potential difference between the battery’s terminals V does not change with
       time, although sometimes it means only that the direction of V doesn't change
       with time.)

The schematic symbol for battery is or other kinds of dc voltage supply is:




A function generator is an example of an ac voltage supply. In this case the potential
difference between the output terminals of the function generator V changes over time
in a user-controlled manner. Because of it’s ability to transmit information time varying
voltage V t  is often referred to as a signal.

The schematic symbol for a function generator: or other signal source is




      1.2 Current
The electrical current I is defined as the rate of flow of electrical charge. The SI unit for
current is the ampere. A current of 1 ampere means that a coulomb of charge is passing
through a fixed point in 1 second.

A very useful analogy (if you don't push it too far) exists between electrical currents
flowing in a circuit and water flowing though pipes:

Electrical Circuit                                    Water Flowing in Pipes

       voltage         <=============> water pressure

       current         <=============> rate of water flow (volume/unit time)




                                           page 3
      1.3 Basic Laws of Electrical Circuits

         1.3.1 Ohm's Law
The electrical resistance R of an object is defined to be the ratio of the voltage difference
across the object to the current flow through the object:


                      V
                R
                       I
Common point of confusion: The above equation is not Ohm's Law, it is the definition of
resistance. It turns out that for many (but certainly not all) objects that we might wish to
place in an electrical circuit the value that one obtains for this ratio is to a good
approximation independent of the value of V . Such objects are called resistors and
are said to obey Ohm's Law. (From this point of view a piece of copper wire is for
example a resistor with a very small value for R.) Many other objects (e.g., a light bulb
filament, a diode) do not obey Ohm's Law, as you shall see in lab.

Note: The use of the word "Law" is somewhat misleadingly, since it is disobeyed so
commonly. However, by definition, Ohm's Law is ruthlessly enforced for resistors.

       Ohm's Law restated: If a current I flows through a resistor whose resistance has
       a value R, then the voltage difference across the resistor is given by V  IR .

This simple statement is tremendously useful. It should be kept constantly in mind and
applied fearlessly as you think about electrical circuits.

IV Curves


                                           page 4
An electrical device can be characterized by plotting its IV curve, which is a graph of
amount of current that flows through the device vs. the voltage difference across the
device. A resistor has a linear I-V curve:




A note on sign conventions: The quantity I is thought of as a flow of positive charges.
This means that there is a voltage drop as current flows across a resistor (because positive
charges will flow towards decreasing voltages). Of course in reality mostly electrical
currents are associated with the movement of negative charges (electrons). These
electrons are in fact moving in the opposite direction of the current I. But (in this course
at least) one almost never has worry about this. We can think of electrical current as
being due to moving positive charges and never run into trouble.


         1.3.2 Kirchoff's Laws
Though we will not mention them by name very often in this course, Kirchoff's Laws
are used implicitly all the time.

Kirchoff's current law - This "law" states something that should be pretty obvious from
common sense and the principle of conservation of charge: The sum of all the currents
into a point in a circuit equals the sum of the currents out. Engineers like to refer to such
a point as a node.




                                 "amount in = amount out"




                                           page 5
Thinking about the water flow analogy helps makes this law obvious.

Kirchoff's voltage law – There are three different ways of saying the same thing:

       1) The sum of the voltage drops around any closed circuit is zero.

       2) Things wired in parallel have the same voltage across them.

       3) The sum of the voltage drops in moving from point A to point B in a circuit is
       independent of the path taken.




The Kirchoff voltage law is necessary if the voltage is have a single, well-defined value
at a given point in the circuit. (It’s very much analogous the keeping track of altitude
changes as you go on a hike. The net change in altitude between your starting and ending
points doesn’t depend on the route you take. And if you end up back where you started
out, your net altitude change over the course of the hike must be zero.)


      1.4 Power in Electrical Circuits
The electrical power, P, is delivered to a given electrical device is defined as the rate
which electrical energy is delivered to the device, or, equivalently, the rate at which work
is done on the device. The SI unit for is the watt. One watt corresponds to energy being
delivered at a rate of one joule per second.




                                          page 6
                P  I V
This follows from the of definitions of power, voltage and current. (The above equation
simply states that (work/time) = (charge/time) x (work/charge).) For V in volts and I in
amps, P is in watts.

The energy delivered to a device often ends in the form of heat (e.g., if the device is a
resistor), but it can also end up as mechanical (kinetic) energy (e.g., if the device is a
motor), electromagnetic radiation (lamps, transmitters) or stored (potential) energy
(capacitors, batteries).


      1.5 Equivalent Resistance

         1.5.1 Resistors in Series




                I1  I2  I            (from conservation of charge)


                V  V1  V2  IR1  IR2                     (from Ohm's Law)

Together, these equations imply:


                V  IR1  R2 


                                           page 7
Thus the equivalent resistance for resistors in series is given by


                           V
               Rseries        R1  R2 
                            I
Bottom line: This means that the series resistance is "dominated by" the largest resistor
in the combination. (Think of it as a "Bottleneck Effect." Again the water flowing
through pipes analogy helps here.)


         1.5.2 Resistors in Parallel




For the circuit shown above, Kirchoff's current law implies


               I  I1  I2

while Kirchoff's voltage law along with Ohm’s law implies


               V  I1 R  I2 R2.
                        1

By definition, the equivalent resistance between the terminals is given by


                              V
               Rparallel 
                               I
Combining the above results and making use of Ohm's Law we conclude


                                 V        V
               Rparallel              V V
                              I1  I2        R
                                        R1     2




                                          page 8
                              1
               Rparallel  1    1
                           R  R2
                            1


                                          1
                Rparallel  R  R 
                               1  1
                             1
                                 2 
                                     
Bottom line: For a parallel combination of resistors, the equivalent resistance is
dominated by the smallest resistor ("Bypass Effect").

Example: Consider the circuit below that contains a battery and two identical light bulbs
A and B and a simple mechanical switch. What is the relative brightness of the bulbs a)
before the switch is closed and b) after the switch is closed?




      1.6 Voltage Divider
This is possibly the most important circuit you will see all semester:




Note convention of placing “inputs” on the left side of the schematic diagram and
“outputs” on the right, so that the diagram is “read left to right”.

Using the above result for a series combination of resistors implies:


                                          page 9
                        Vin
                I 
                      R1  R2

Using Ohm's Law to calculate the voltage drop across R2 :


               Vout  IR2

Combining these two results:


                              Vin             R2
               Vout                R2           V
                        R1  R2            R1  R2 in

Note:

         If R2  R1 then Vout  Vin .

         If R2  R1 then Vout  Vin

Application: Light Sensor.

A photocell is a material whose electrical resistance varies depending on how much light
is incident on it. You can build a simple light sensor simply by using a photocell as one
of the resistors in the divider circuit. When you are doing lab 1-4, try using one of the
“LEGOized” photocells. By measuring the output voltage can you deduce whether the
resistance of a photocell increases or decreases when the amount of light hitting the cell
increases.


        1.7 Loading, "output impedance", and the Thevenin Model

         1.7.1 Adding a "load resistor"
Suppose R2 in the divider circuit shown above is a "variable resistor" (often called a
potentiometer, or "pot" for short) whose resistance can vary from 0 ohms (a short circuit)
all the way the a value much greater than R1. This means that Vout can be made to vary
from 0 V all the way up to Vin , just by adjusting the potentiometer ( "twiddling the
pot"). It looks like we may have stumbled upon a simple "variable voltage supply".
Such a variable voltage supply might be useful, for example, for "driving" the dc motors
we will use in a robot. (The speed at which a dc motor turns can be varied by varying the
voltage that is applied to the motor. Do you remember the motors you built in the
Physics 108 lab?) Sounds easy, huh? But how “good” and useful a variable voltage
supply have we really built?

To answer this question we must consider what happens when a "load" such as a motor is


                                           page 10
attached to the output terminals of our voltage supply: We can model such a load as a
simple resistor:




The "brute force" way to answer this question is to simply replace the parallel
combination of Rload and R2 by the equivalent resistance for the parallel combination
and use the above result for a voltage divider:


                            Rp
               Vout            V
                         R1  Rp in


                         Rload R2
               Rp 
                        Rload  R2


                             Rload R2 
                                       
                            Rload  R2 
                                       
               Vout                          V
                               Rload R2  in
                         R1             
                              Rload  R2 
                                         
Typically when we set out to design a good voltage supply, an important criterion is that
we would like the value of Vout to not change appreciably when the load resistor is
connected. It is not so easy to tell from looking at the above, rather complicated
expressions exactly when this criterion will be met. However, it's pretty clear that if
Rload is small compared to R1 and R2 then Vout will drop when the load is
connected. For example, consider what's wrong with the following attempt to design a
variable voltage supply for one of the 4.5 volt LEGO motors, which have an internal
resistance of about 20 .


                                          page 11
The only way to get the voltage divider to supply the desired voltages to the motor is to
make R1 and R2 much smaller. But do you see how wasteful this is?


         1.7.2 Thevenin Model
There is an easier way to think about circuits such as the one above that makes use of
something called the Thevenin Model. More importantly, as the circuits we build
become more complicated, the advantages of the Thevenin Model will become
increasingly apparent.

The essence of a good model is that it provides easier way to think about a more
complicated situation. (E. g., “consider a spherical cow”)

Consider an arbitrary arrangement of batteries and resistors inside a "two-terminal black
box":




Thevenin's Theorem states that there exists an equivalent circuit of the form:


                                         page 12
The two circuits are equivalent in the sense that no electrical measurement made outside
the black box at the terminals can distinguish which of the two circuits is inside the black
box. Thus the Thevenin Model is useful because it allows us to think about the behavior
of a complicated circuit in terms of a simpler one.




         1.7.3 A prescription for determining Vth and Rth for an arbitrary circuit
The Thevenin equivalent voltage Vth is simply equal to the "open circuit voltage", Voc .
By "open circuit voltage" we simply mean that the voltage between the terminals is
measured without allowing any appreciable current to flow through the terminals. If no
current flows through Rth then there is no voltage drop across Rth (Ohm's Law!) and
therefore Voc  Vth .


               Vth  Voc

 (The open circuit voltage can be measured with an "ideal voltmeter", which would have
an infinitely large input resistance. The digital voltmeters that we use in this lab have
input resistances of 10 M, which is sufficiently large so that in most cases they can be
effectively regarded as measuring an open circuit voltage.)

The Thevenin equivalent resistance Rth can be determined from the following
procedure:

       1) Measure the "short circuit current" I sc by connecting an ammeter (internal
       resistance ~ 0) between the terminals.

       2) Use Ohm's Law to calculate




                                          page 13
                    V          Voc
               Rth  th 
                     Isc        Isc


                    V
               Rth  oc
                     Isc

 (Often we can perform this procedure mentally, rather than experimentally. In fact
actually shorting the terminals in order to measure I sc could damage the circuit or the
ammeter.)

Example: Find the Thevenin equivalent circuit for the voltage divider shown below:




From our discussion of the voltage divider above we have


                                  R2
               Vth  Voc  Vin
                               R1  R2

The Thevenin equivalent resistance Rth is given by


                    V
               Rth  oc
                     Isc

Since, from Ohm's Law the short circuit current is


                    V
               Isc  in
                     R1


                                         page 14
we have


                                        R2
                                Vin
                     V                R1  R2
                Rth  oc             Vin
                      Isc
                                      R1

                          R2 R1
                Rth 
                        R1  R2

That is, the Thevenin equivalent resistance, Rth , of the voltage divider is equal to the
equivalent resistance of the parallel combination of R1 and R2 .

With a Thevenin equivalent circuit for the voltage divider in hand, it is now easier to
quickly see what the effect of attaching a particular load will be:


                           Rload      
               Vout  Voc             
                          Rload  Rth 
                                      
Thus, for a voltage divider to be a "good" power supply we generally require that


                Rth  Rload .

In a case such as this where the terminals involved are "outputs" the Thevenin equivalent
resistance is also referred to as the output resistance or, more commonly, the output
impedance of the device.




                                           page 15

				
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