### 1. In this Chapter

We begin this chapter by exploring one of the basic components in a circuit: the resistor. We learn how to apply Ohm’s law to measure the value of a resistor and then measure the equivalent resistor value in a series or parallel combination. We will use the NI Multisim circuit teaching environment to verify these calculated results with example circuits that can be used by any educator or student.

If you do not have NI Multisim installed on your computer, you can download a free 30 day evaluation at http://www.ni.com/multisim/try/

### 2. Example Courses

Listed below are example courses that teach this concept at their schools.

Course Name | School | Learn More |

Electrical Principles | Conestoga College | http://www.conestogac.on.ca/fulltime/0071.jsp |

Electronic Technology 1 | Macomb Community College | http://www.macomb.edu/noncms/Search/Courses/coursekey.asp?coursekey=ELEC-1161) |

### 3. Resistors in Series

In series, the current entering the combination is the same and not divided, however the voltage drop across each device differs based on the value of that device.

Therefore the equivalent resistance **R _{eq}** is the summation of the individual values of resistances:

**R**(n = number of resistors) [1].

_{eq}= R_{1}+ R_{2}+ ... + R_{n}Using NI Multisim we can verify the above equation by measuring the voltage and current in a resistive series circuit.

__STEP 1:__ Open circuit file “resistors_series.ms12”. You will see the circuit below [2].

__STEP 2: __Place probes at the desired points in our circuit (as shown in the figure below) by selecting the probe from the instrument’ bar on the right hand side of the screen in NI Multisim and then run the simulation by selecting “Simulation>>Run Simulation”.

You will notice that indeed the current passing through a combination of resistances in series is not divided and is the same through all the resistances and how the voltage across these devices is in fact divided and differs based on the value of each resistor.

__Answer Sub-Step 1: Calculate the equivalent resistance for the above circuit.__

**R _{eq} **=

**5K + 5K + 10K + 5K + 5K**

**=**

**30K Ω**

__STEP 3: __Open “series_eq_circuit.ms12” circuit file and place a probe as shown below then run the simulation.

You will notice that the measured current is equal to the current we measured in the original circuit and that confirms that we have calculated the correct value of the equivalent resistance.

### 4. Resistors in Parallel

In parallel, the voltage drop remains the same and is not divided (since we are measuring the voltage drop between the same two points), however the current is faced with one or more branches and is forced to split with its values dependant on each resistor in each branch.

Therefore the equivalent resistance **R _{eq}** is equal to the reciprocal of the summation of the individual resistor values:

**R**(where n = number of resistors) [3].

_{eq}= 1/((1/R_{1}) + (1/R_{2}) + ... +(1/ R_{n}))Using NI Multisim we can verify the above equation by measuring the voltage and current in a resistive parallel circuit.

__STEP 4:__ Open the circuit file “resistors_parallel.ms12”. You will see the circuit below [4].

__STEP 5: __Again, place probes at the desired points in our circuit (as shown in the figure below) and then run the simulation by selecting “Simulation>>Run Simulation”.

You will notice that indeed the current passing through a combination of resistances in parallel is divided into parallel paths only to be recombined and rejoined eventually to form the current **I**. The voltage however is the same across each resistor.

__Answer Sub-Step 2: Calculate the equivalent resistance for the above circuit.__

**R _{eq}** = 1/ ((1/80K) + (1/20K) + (1/4K)) = 1/(0.125 + 0.05 + 0.25) = 1/0.3125 =

**3.2K Ω**

__STEP 6:__ Open “parallel_eq_circuit.ms12” circuit file and place a probe as shown below then run the simulation.

You will notice that the measured current is equal to the current we measured in the original circuit and that confirms that we have calculated the correct value of the equivalent resistance.

### 5. Example Problem

Let us now examine the following circuit containing both series and parallel combinations and solve to get the equivalent resistance. You can use the attached circuit to investigate the theory as well as follow the steps below.

__STEP 7:__ Begin by opening “series_parallel.ms12” circuit file in NI Multisim: [5]

__Answer Sub-Step 3: Calculate the equivalent resistance R _{eq} for the above circuit in Ohms.__

We have a combination of series and parallel resistors. Break the combinations down and solve one combination at a time.

__Answer Sub-Step 4: Calculate the equivalent resistance R _{parallel} for R_{5} and R_{6}.__

**R _{parallel}** = 1/(1/R

_{5}) + (1/R

_{6})) =

**20 Ω**

Thus the resulting circuit will look like this:

__Answer Sub-Step 5: Calculate the equivalent resistance R _{series} for R_{3} , R_{4} and R_{parallel}.__

**R _{series}** =

**R**+

_{3}**R**+

_{4}**R**=

_{parallel}**40 Ω**

Thus the resulting circuit will look like this:

__Answer Sub-Step 6: Calculate the equivalent resistance R _{parallel} for R_{2} and R_{series}.__

**R _{parallel}** = 1/((1/

**R**) + 1/

_{2}**R**)) =

_{series}**20 Ω**

Thus the resulting circuit will look like this:

__Answer Sub-Step 7: Calculate the equivalent resistance R _{eq} for R_{1} and R_{parallel}.__

**R _{eq}** =

**R**+

_{1}**R**=

_{parallel}**40 Ω**

__STEP 8:__ Open circuit file “series_parallel_eq_circuit.ms12” in NI Multisim then place a probe as shown below and simulate the circuit.

Comparing the value of the current in the equivalent circuit to the original circuit verifies the correct reduction for **R _{eq}:**

### 6. Suggested NI Solution

National Instruments offers a number of products that combine to provide a scalable and powerful teaching platform for educators. The solution includes:

NI Multisim circuit teaching environment: Combining an intuitive circuit definition environment, with powerful SPICE simulation technology, educators can use NI Multisim to easily teach the ins-and-outs of circuits in a safe environment.

NI ELVIS teaching and measurement platform allows educators to provide students with a compact, all-in-one unit for their measurement and analysis needs. Combining an oscilloscope, function generator, DMM, bode analyzer and 8 other instruments into a small platform; it simplifies the laboratory experience for students and lab instructors.

### 7. References

[1] University of Guelph, Department of Physics. “DC Circuits”. __Resistors in Series.__

[http://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.intro.series.html]. (16/01/2013)

[2] University of Guelph, Department of Physics. “DC Circuits”. __Resistors in Series.__

[http://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.example.series.html]. (16/01/2013)

[3] University of Guelph, Department of Physics. “DC Circuits”. __Resistors in Parallel.__

[http://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.intro.parallel.html]. (16/01/2013)

[4] University of Guelph, Department of Physics. “DC Circuits”. __Resistors in Parallel.__

[http://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.example.parallel.html]. (16/01/2013)