Electrical Circuits: Parallel vs Series
          Prof. Julia Thompson, University of Pittsburgh and SIUE 


I.  What are voltage, current, and resistance?

First, we need to define, briefly, some basic ideas.


A. Voltage.
Macroscopic matter is made up of atoms and molecules which
are bound together by electrical forces.  The atoms are made up of
electrically neutral neutrons, and positively charged protons.
In batteries, or other sources of electrical power, positive 
and negative charges are separated. Since unlike charges attract
each other, work must be done to separate them.  You have already
studied conservation of (mechanical) work and energy.  Electrical
energy is just another form of energy, and what we have found
is that all kinds of energy can change form, from one kind to another,
but the total energy is conserved. Under the right conditions,
we can get back the work we put into separating the charges.
A source of electricity is rated by the "voltage", or work
per unit charge which could be recovered if a unit positive 
charge moved from the "positive" side of the battery or electricity
source to the "negative" side. (or, which is what  usually happens,
a unit negative charge moved from the "negative" side of the 
battery to its "positive" side.

B. Current.  The amount of charge which passes an observer's
station per unit time is called the electrical current.
Since Voltage is work/unit charge and current is charge/unit time,
we see that Voltage * Current is Work/time, which is power,  or 
the rate at which work is done.  The utility company charges
by the total work (or power * time), but you need to know how
much power an electrical device can use, and how much power your
electrical power source provides, before
connecting your device  
to your electrical power source.

C. Resistance.  When charges have been separated onto 
positive and  negative "terminals", and suddenly the two 
terminals are touched together, the charges will move very 
quickly to equalize the charge... and there will be a  large
spark, which corresponds to a huge instantaneous current 
(transfer of charge over a very short time).  On the other
hand if the terminals are in a vacuum, no current will flow
between them.  Any intermediate situation will give 
some intermediate current.  The "resistance" of a given
piece of material placed between two terminals with difference
in voltage V is defined as: V = i * R, where i is the 
current which flows between the two terminals when 
the resistance R is connected.  R can always be defined
in this way.  For some materials, R depends on the voltage
across the two ends of the material, or it may depend on the 
temperature of the material.
For other materials, R is nearly independent of temperature,
the voltage across it, and the current through it.
Materials with variable resistance are fascinating both for
their function and their construction.  If you go on to 
design electrical circuits you will work with such variable 
resistor components as diodes and transistors, or you might 
worry about the breakdown voltages of gases at which current
starts to flow.  But in our lab today, we will start at the 
beginning with materials called "resistors", for which the 
"resistance" R is independent of current and voltage. That is,
a graph of voltage  vs current would be a straight line,
with slope R.

II. Circuits

There are two kinds of circuits:

A. Series Circuits

First,  series, in which the electrical
current flows through the components one after the other "in series".
 In a series circuit, the current is the same in each element.


   -----XXXXXX-----XXXXX------XXXXXX......XXXXXX------
     i-> R1         R2          R3          R4      
  V0            V1        V2         V3              V4


               Figure 1


Figure 1 shows a series circuit.  If V0 = 0, then 
the work done in moving a charge from the left hand side
of R1 to the right hand side of R4 will  be:

    V4 = V1 + (V2 - V1)  + (V3 - V2)  + (V4 - V3)  
(since the work done to move a charge through all the resistors
is just the sum of the work to move the charge through each of
the individual resistors in turn.

Therefore:
    
    V4 = i * R1 +  i * R2 + i * R3 + i * R4 

or:   V4 = i*(R1 + R2 + R3 + R4) 

and we see that 
      R_{eff, series}  = R1 + R2 + R3 + R4. 


B. Parallell Circuits

 Next, parallel, in which there are different "parallel" paths which 
current can take, shown in Figure 2.


             R1,i1
       ---XXXXXXX------      
       |              |
       |     R2,i2    |  
       ----XXXXXXX-----
  i-->-|              |-->i
       |     R3,i3    |
       ---XXXXXXX------      
       |              |
       |     R4,i4    |  
       ----XXXXXXX-----

 
             Figure 2

For parallel circuits, since all the
resistors
are connected at the beginning and the end of the current paths
it takes the same work for a charge to travel through any 
resistor.  Since charge must be conserved, the total current
must split, giving:

 i = i1 + i2 + i3 + i4 

But for each  resistor, V=iR, so,

  i = V/R1 + V/R2 + V/R3 + V/R4 ,

which gives:

   1/Reff, parallel = 1/R1 + 1/R2 + 1/R3 + 1/R4 

You can show that there is a simple form for two resistors:
  Reff, parallel = R1*R2/(R1+R2).  

Note above that the series effective  resistance is always
larger than the resistance of any one resisitor,
and  the Reff, parallel is always less than the resistance of 
any one resistor. 

III. What will we actually do?

Your equipment will be a battery, an electrical meter which 
can measure voltage and current, a clipboard for electrical 
connections, and some (nearly) identical resistors.


Some useful information: The resistor colorcode.
 black:   0
 brown:   1
 red:     2
 orange:  3
 yellow:  4
 green:   5
 blue:    6
 violet:  7
 gray:    8
 white:   9

 tolerances: gold 5%, silver 10%, nothing 20%, otherwise labelled.

Resistors are labelled with two color bands for significant
figures, one for a power of 10, and one for tolerance.  eg,
red, violet,yellow gold, would be 27 *$10^{4}$, with a 5% 
tolerance.  Gold, as a power of 10, can also mean power -1.

First, find the nominal  value of your resistors using the color code.



Second, use Ohm's law, V=iR, for each resistor in turn, to 
calculate the "measured" resistance.  [Teacher's note: important 
to match batteries, resistors and meters so measurements are in a 
reasonable region of the meter scale if using an analog meter.]
How different are they? Are your measurements within the 
expected tolerances?  Students' notes:  Check the meter
zero and adjust if necessary. For an analog meter, the 
number to which the switch is turned indicates the maximum
value of the scale to be used; look at the meter directly from above,
to avoid parallax.

Below are several other possible explorations.  Do the ones of
most interest to you, but you must have completed at least two
before taking the "mystery" resistor to test.
For  all the steps below, a simple circuit diagram should accompany each
measurement.



Third, connect your resistors in series, and in series with 
the battery. By connecting the meter connections to opposite
ends of each resistor, one at a time, measure the voltages across 
each resisitor, and check that the total voltage  is the sum of 
the individual voltages.  Check the current at different points
in the circuit by connecting your meter (set on the curent scale)
between two resistors (in series with the rest of the circuit).
Is the current the same throughout the circuit? 

Fourth, measure the current in the series circuit for different numbers of
resistors (in series). Qualitatively do you see what you expect, 
that as the resistance rises, the current drops?
Think of a way to display your results in a graph, to demonstrate
Ohm's law (V=iR) for a series circuit. (i vs n? i vs 1/n? 1/i vs
n? 1/i vs calculated Reff?)

Fifth, connect your resistors in parallel, and in parallel with 
the battery.  Check that the voltage across each resistor
is the same.  Check the currents flowing through each resistor
(by putting one end of the meter connections to the 
common voltage point, and the other to the tip of the resistor..
that is, the meter will be in series with the resistor whose
current you are measuring, and the meter and the resistor being
measured 
will be in parallel with the rest of the resistors.

Sixth, measure the current provided by the batter  for different
numbers of resistors. Are the results qualitatively as you 
expect, that is, as the number of current paths increases, with the
voltage held constant for each current path, the total current increases? 
Think of a way to display your results in a graph, to demonstrate
Ohm's law for a parallel circuit.

Seventh, if there is time, once you are comfortable with your
circuit, borrow one of the "mystery" resistors, and find
its resistance.