Experiment 3

Series and Parallel Resistors

 

Objectives:

 

The objective of this experiment is to verify Ohm's law applied to

 

1.      series resistors,

2.      parallel resistors, and

3.      a module of parallel resistors collectively in series with another resistor.

 

Equipment:

 

A few ceramic resistors (100 to 700 ohms), a dc-power source, 2 multi-meters, a calculator, and a few connecting wires with alligator clips

 

Theory:

 

Ohm's law simply states that the ratio of voltage across an electric device to the current through that device is a constant called the electric resistance of that device.

If V is in volts and I is in amperes, R will be in ohms.  A more often used version of this formula is

 

V = R I

 

i)          Series Resistors:

 

Theory and Procedure:

 

Series resistors experience the same current, but (possibly) different voltages.  A typical pure series circuit is shown below.  On one hand, the current I has to be the same everywhere.

 

Fig. 1

 

On the other hand, the voltage will be different across each resistor (if their resistances are different).  The battery voltage Vbat will be divided between the three resistors R1, R2, and R3, proportional to their resistances, such that

    Vbat = Vab + Vbc + Vcd, or                                                       (1)

 

    Vad = Vab + Vbc + Vcd.                                                            (2)

 

Also, for series resistors:                   

   Rad = Rab + Rbc + Rcd                                                                                  (3)

  1. Measure the four ceramic resistors you are provided with a multi-meter (set on ohm setting) and name them R1, R2, R3, and R4.  Name the smallest R4. 

  2. Arrange a circuit as shown in Fig.1.  To measure the voltages, there is no need for three voltmeters.  One voltmeter is sufficient.  The voltages across R1, R2, and R3 can be measured one at a time by placing the voltmeter terminals across each resistor.

  3. To measure the current, the ammeter must be placed in the circuit in series.  Note that the circuit must be opened at one of the connections and the ammeter be placed in between the open ends. Note that the approximate current must be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will be damaged.

  4. Read the current, I, through the circuit and the voltage across each resistor and record them.

  5. Turn the circuit off or disconnect the battery.

 

Data:

Given:

Use an ohmmeter to measure more-accurate values for the following resistances.

R1 = (100 - 700) Ω 

            R2 = (100 - 700) Ω

            R3 = (100 - 700) Ω

            R4 = (100 - 700) Ω

Vbat = 9 V (When the circuit is on, use the voltmeter to measure the exact voltage supplied by the battery (Vbat ).  This is the voltage that must be used in your calculations).            

 

Measured:

When the circuit is closed, measure the following values:

V1 = ?        V2 = ?        V3 = ?        I = ?    (These are measured values.  Do not use these in calculations).

Calculations:

 

Knowing the given values for Vbat, R1, R2, and R3, solve for I, V1, V2, and V3, and use these calculated values as accepted values.  Note that the exact values of R1, R2, and R3 measured by the ohmmeter must be used in your calculations.

 

Comparison of the Results:

Corresponding to every measured value, there is an accepted value.  Calculate a percent error for each, using the percent error formula you used in Experiment 2.

 

ii)      Parallel Resistors

 Theory and Procedure:

 

Parallel resistors experience the same voltage, but (possibly) different currents.  This is in contrast to series resistors.  A typical pure parallel circuit is shown below.  The voltage ( Vab ) has to be the same for the three resistors, because they are all between the same two points of the circuit (a and b).   The current in each resistor is determined by the equation I=V/R.  Since the voltage V is the same for each resistor (V1=V2=V3=Vab), the larger the resistance R of a particular resistor, the smaller the current through that resistor.   Charge or flow conservation requires that

I = I1 + I2 + I3.

 

Point a is a dividing point.  Current I divides at point a into I1, I2, and I3.  At point b,  I1, I2, and I3 add up to form I again.

Fig. 2

Note that the voltage across each of the resistors is Vab.  If we neglect the small voltage-drop across the ammeter, the voltage Vab is equal to the battery voltage Vbat.  This is because there is no circuit element (other than the ammeter) between point a and the battery, or between point b and the battery.  The ammeter (A) does not contribute to any significant voltage drop; the voltage drop across an ammeter can usually be neglected.

 

For a different case, where there is a fourth resistor between point a and the battery, or between point b and the battery, Vab is not equal to Vbat.  This is going to be case (iii) in this experiment.  Here, for parallel resistors:

  1. Use the same three resistors R1, R2, and R3.

  2. Arrange a circuit as shown in Fig.2.  To measure the voltages, place the voltmeter across the junction points a and b.  The voltage you read is the voltage across each of R1, R2, and R3.  Parallel resistors share the same voltage.

  3. To measure the current, the ammeter must be placed in the line of each resistor once to read the current in that branch.   Note that each branch must be opened at one of the connections and the ammeter be placed in between the open ends. Note that the approximate current must be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will be damaged.  Do this for each branch and record the values in (mA).

  4. Read the current, I, through the main branch (between the battery and a or between b and the battery).  That gives the current in the main branch.  It is the current that leaves the battery from one terminal and finally goes into the battery at its other terminal.  Record its value.

  5. Turn the circuit off or disconnect the battery.

Data:

 Given:

 

R1 = (100 - 700) Ω 

            R2 = (100 - 700) Ω

            R3 = (100 - 700) Ω

Measured:

When the circuit is closed, measure the following values:

V1 = V2 = V3 = ?       I = ?       I1 = ?       I2 = ?      I3 = ?   

(These are measured values.  Do not use these in calculations).

Calculations:

 

Knowing the given values for Vbat, R1, R2, and R3, solve for R, I, I1, I2, I3, V1, V2, and V3, and use these calculated values as accepted values.

 

Comparison of the Results:

Corresponding to every measured value, there is an accepted value.  Calculate a percent error for each, using the appropriate formula as in Experiment 2.

 

iii)     Parallel Resistors in Series with Another Resistor:

 

Theory and Procedure:

 

Fig. 3 shows a parallel portion between points a and b that is in series with another resistor between points b and c.  Here, the battery voltage partially drops across the ab portion and partially across the bc portion.  It is logical to write:

 

Vbat = Vab + Vbc

 

The voltage across resistors 1, 2, and 3 is the same because they are in parallel.  (The parallelism is not because they are drawn parallel to each other on paper; it is because the current between points a and b must divide into three branches such that I = I1 + I2 + I3.  The same voltage Vab drives currents I1, I2, and I3 into resistors 1, 2, and 3.)

 

This balance-of-currents equation may be written in terms of voltages and resistances as:

 

I = I1 + I2 + I3, or

Dividing through by Vab yields:    1 / Rab = 1 /R1  +  1 /R2   +  1 /R3

 

Rab = 1 / (1 / R1 + 1/ R2 + 1/ R3 )

This equation is good only for the parallel portion.  To find the total resistance of the circuit, one may write:

Rac = Rab + Rbc      where Rbc = R4         (Rab and Rbc are in series)

 

Once the total resistance Rac is determined, the main current I can be calculated by using Ohm's law:

 

Vbat = Rac I.

 

The total resistance Rac determines how much current the battery can deliver.  (The ammeter (A) is a low-resistance device, and its resistance may be ignored.)  The reason that Rac is the determining factor is that there are no other significant resistors between point a and the battery or point c and the battery.

Fig. 3

When I is determined, it can be multiplied by Rab to calculate Vab.  It can also be multiplied by Rac to calculate Vac as shown:

 

Vab = Rab I      and        Vac = Rac I

Now that Vab is known, Ohm's law may be used again to calculate I1, I2, and I3, as follows:

I1 = Vab / R1

I2 = Vab / R2

I3 = Vab / R3

 

  1. Use the four resistors R1, R2, R3 and R4.

  2. Arrange a circuit as shown in Fig. 3.  To measure the voltages, place the voltmeter across  points a and b to read Vab and then across points b and c to read Vbc.  Of course, Vab is the voltage across each of R1, R2, and R3, and  Vbc is the voltage across R4 .

  3. To measure the current, the ammeter must be placed in the line of each resistor once to read the current in that branch.   Note that each branch must be opened at one of the connections and the ammeter be placed in between the open ends. Note that the approximate current must be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will be damaged.  Do this for each branch in the parallel section and record its value in (mA).  Record the

  4. Read the current, I, through the main branch (between the battery and a or between c and the battery).  That gives the current in the main branch.  It is the current that leaves the battery from one terminal and finally goes into the battery at its other terminal.  Record its value.

  5. Turn the circuit off or disconnect the battery.

 

Data:

Given:

R1 = (100 - 700) Ω 

            R2 = (100 - 700) Ω

            R3 = (100 - 700) Ω

            R4 = (100 - 700) Ω

 Measured:

When the circuit is closed, measure the following values:

            V1 = V2 = V3 = ?         V4 = ?         I = ?       I1 = ?       I2 = ?      I3 = ?   

            (These are measured values.  Do not use these in calculations).

Calculations:

 

Knowing the given values for Vbat, R1, R2, R3and R4, solve for I, I1, I2, I3, V1, V2, V3and V4, and use these calculated values as accepted values.

 

Comparison of the Results:

Corresponding to every measured value, there is an accepted value.  Calculate a percent error for each.

Conclusion:                        To be explained by students

 

Discussion:             To be explained by students