### Experiment 3

#### Series and Parallel Resistors

Objectives:

The objective of this experiment is to verify Ohm's law applied to (a) series resistors, (b) parallel resistors, and (c) a module of parallel resistors in series with another resistor

Equipment:

A few ceramic resistors (100 to 500 ohms), a dc-power source, 2 multi-meters, a calculator, and a few connecting wires with alligator clips

I) Series Resistors:

Theory (I):

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

If V is in volts and I in amps, R will be in ohms.   The preferred version of this formula is V = R I.

Series resistors experience the same current, but generally different voltages.  A typical series circuit is shown below.   Current I is the same everywhere.

Figure 1

Voltage across different resistors are different.  The battery voltage Vbat  divides amongst the three resistors R1, R2, and R3  proportional to their resistances such that

Vbat = Vab + Vbc + Vcd ,   (1)

Also, for series resistors:

Rad = Rab + Rbc + Rcd      (2)

Procedure (I):

1. Select 4 ceramic resistors between 100Ω to 500Ω.  Measure their resistances with a multi-meter (set on ohm setting) and name them R1 , R2 , R3 , and R4 in increasing order.

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 V1 , V2 , and V3 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 opened ends.  Also, the approximate current may be calculated first and the correct and safe mA range be selected; otherwise, the ammeter will get damaged.

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

5. Turn the circuit off.

Data: (I):

Given:  The measured resistances of the selected resistors are to be treated as "given."

R1 = ........ Ω,    R2 = ........ Ω,    R3 = ........ Ω,   and R4 = ........ Ω

Vbat = ........... =  whatever you read for the voltage between Points a and d when the circuit is on and stabilized.  Doing this, you eliminate the small voltage drop by the ammeter.  This is like bypassing the ammeter.

Measured:

When the circuit is on and stabilized, measure the following values:

V1 = ..........   V2 = .........   V3 = ........., and    I ........... .

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

Calculations (I):

Knowing the given values for Vbat , R1 , R2 , and R3, solve for I , V1 , V2 , and V3 .    Use these calculated values as accepted values.

Comparison of the Results (I):

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

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II) Parallel Resistors:

Theory (II):

Parallel resistors experience the same voltage, but generally different currents.  A typical parallel module is Fig. 2.

We want to develop a formula that calculates the equivalent resistance between Points a and b.  See Fig. 2.  Let's name this equivalent resistance Rab.  To find the imaginary Rab that can do the job of the three parallel resistors R1 , R2 , and R3 , it is important to emphasize that current I at Point a divides into sub-currents I1 , I2 , and I3 .  At Point b, the sub-currents I1 , I2 , and I3 merge to form I again.  We may writeI  = I1 + I2 + I3   (3)  in which

I = Vab /Rab ,   I1 = Vab /R1 ,    I2 = Vab /R2 ,  and     I3 = Vab /R3 .

Substitution in (3) results in:

Dividing through by Vab , we get:

Rab = 1 / (1/R1 + 1/R2 + 1/R3 ).   (4)

R1 , R2 , and R3 all together draw the same current from the battery that Rab would draw if replaced them.  Rab is the "equivalent resistance" between a & b.

Figure 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 any other circuit element between Point a and the battery, or between Point b and the battery to cause a voltage drop.  The ammeter A is just a measuring device and does not contribute to significant voltage drop.  The voltage drop across an ammeter can usually be neglectedIn this part, it is better to measure Vab and use it as Vbat in your calculations.  This is like bypassing the ammeter.

Equation (4) calculates the equivalent resistance R of the three parallel resistors R1 , R2 , and R3 .

Procedure (II):

1. Use the same three resistors R1 , R2 , and R3 as in Part I.

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 opened ends.   Also, the approximate current may be calculated first and the correct and safe mA range be selected; otherwise, the ammeter can get 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.

Data (II):

Given:  The measured resistances of the selected resistors are to be treated as "given."

R1 = ........ Ω,    R2 = ........ Ω,    R3 = ........ Ω,   and R4 = ........ Ω

Vbat = ........... =  whatever you read for the voltage between Points a and b when the circuit is on and stabilized.  Doing this, you eliminate the small voltage drop by the ammeter.  This is like bypassing the ammeter.

Measured:

When the circuit is on and stabilized, measure the following values:

I1 = ..........   I2 = .........   I3 = ........., and   I ........... .

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

Calculations (II):

Knowing the given values for Vbat , R1 , R2 , and R3, solve for Rab , I , I1 , I2 , and I3 .    Use these calculated values as accepted values.

Comparison of the Results (II):

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

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III) A Parallel Module in Series with Another Resistor:

Theory (III):

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

Vbat = Vab + Vbc .       (5)

The parallel module R1 , R2 , and R3 may be replaced by Rab  given by

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      or     Rac =  Rab + R4 .

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 I , the main current, the battery can push through.  Ammeter A is a low-resistance device, and its resistance may be ignored.

Figure 3

When I is determined, it can be multiplied by Rab to calculate Vab.  It can also be multiplied by Rbc to calculate Vbc .

Vab = Rab      and      Vbc = Rbc I.

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

I1 = Vab/R1 .

I2 = Vab/R2 .

I3 = Vab/R3 .

Procedure (III):

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 opened ends.   Also, the approximate current may be calculated first and the correct and safe mA range be selected;  otherwise, the ammeter may get damaged.  Do this for each branch in the parallel section and record its value in mA.

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.

Data (III):

Given:  The measured resistances of the selected resistors are to be treated as "given."

R1 = ........ Ω,    R2 = ........ Ω,    R3 = ........ Ω,   and R4 = ........ Ω

Vbat = ........... =  whatever you read for the voltage between Points a and c when the circuit is on and stabilized.  Doing this, you eliminate the small voltage drop by the ammeter.  This is like bypassing the ammeter.

Measured:

When the circuit is on and stabilized, measure the following values:

V1 = ........  V2 = .......   V3 = ......., V4 = ........  and   I ......... .

I1 = ........  I2 = ........  I3 = .......... .

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

Calculations (III):

Knowing the given values for Vbat , R1 , R2 , R3 and R4 , solve for I , I1 , I2 , I3 , V1 , V2 , V3 and V4 , and use these calculated values as accepted values.

Comparison of the Results (III):

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.