### 1. In this Chapter

We begin this chapter by exploring AC circuits and the different relationships current and voltage have with respect to passive circuit elements like resistors, capacitors, and inductors. We will use the NI Multisim circuit teaching environment to verify these relationships 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. AC Circuits

In AC circuits the AC source provides a stimulus with a magnitude and a frequency.

In a resistive circuit with an AC source, there is no change to the magnetic field as current passes through the resistors [1].

In an inductive circuit with an AC source, the inductor opposes a change in the current flow, thereby causing current to lag the voltage by 90°. In an RL circuit the current will lag somewhere between 0 and 90° from the input signal [1].

In a capacitive circuit with an AC source, the capacitor opposes a change in the voltage thereby causing current to lead the voltage by 90°. In an RC circuit the current will lead somewhere between 0 and 90° from the input signal [1].

Using NI Multisim we can begin investigating this relationship between the voltage and current with the oscilloscope instrument.

__STEP 1: __Open the “ac_circuit_resistor.ms12” circuit file. You will see the circuit below [2].

__STEP 2: __Run a simulation in NI Multisim by selecting “Simulation>>Run Simulation”

__STEP 3:__ Double-click on the XSC1 oscilloscope instrument in the circuit.

Since the resistor has the same response in either DC or AC circuits, we notice that the input and output signals are in phase (no phase shift). The output signal (in blue) however is at a smaller voltage value due to the resistive network dividing the input voltage.

__STEP 4:__ Open the “ac_circuit_capacitor.ms12” circuit file in NI Multisim. You will notice the circuit below [2].

__STEP 5: __Again run the simulation and open the oscilloscope instrument by double-clicking. Running the circuit and examining the oscilloscope results in the following graph you will notice that the input signal (blue) is leading the output signal (red).

__STEP 6: __Finally, open circuit file “ac_circuit_inductor.ms12” and once more run the simulation. Open the oscilloscope instrument [2].

__STEP 7:__ Run the simulation and notice that in this inductor circuit the input signal (blue) lags the output signal (red wire) as shown below.

### 4. Example Problem

Let us now examine the below circuit with an AC source and calculate the maximum voltage drops across each device and find out the average power dissipated. You can use the attached circuit to investigate the theory as well as follow the steps below.

__STEP 8:__ Begin by opening the “ac_example.ms12” circuit file in NI Multisim [3].

__Answer Sub-Step 1: Calculate the angular frequency of the source in the above circuit in radians/second.__

Since our source is 120V with frequency of 60 Hz therefore: ω = 2∏f = 2 x ∏ x 60 = 120 ∏ rad/sec

__Answer Sub-Step 2: What is the V _{max} for resistor R_{1}?__

R_{1} = 50 Ω

ΔV_{max} = I_{max} R_{1}

__What is I _{max} ?__

__Answer Sub-Step 2: What is the V _{max} for capacitor C_{1}?__

C_{1} = 20 μF

ΔV_{max} = I_{max} X_{C}

X_{C} = 1/(ωC) = 1/((120∏ rad/sec)(20μF)) = 133 Ω

__What is I _{max} ?__

__Answer Sub-Step 3: What is the V _{max} for inductor L_{1}?__

L_{1} = 10 mH

ΔV_{max} = I_{max} X_{L}

X_{L} = ωL = (120∏ rad/sec)(10mH) = 3.77 Ω

__What is I _{max} ?__

__Answer Sub-Step 4:__ Since X_{C} is much bigger than X_{L} we now know that the current will be leading the voltage for this circuit.

__Answer Sub-Step 5: Calculate I _{RMS} which allows us to calculate Imax for our ΔV_{max} formulas above.__

V_{RMS} = 120V = I_{RMS} Z (where Z is the overall impedance of the circuit)

Z = √ [R^{2} + (X_{L} – X_{C})^{2}] = √[50)^{2} + (133 – 3.77)^{2}] = 139 Ω

Therefore rearranging for I_{RMS}, we calculate I_{RMS} = 120/139 = 0.86 A

__Answer Sub-Step 6: Calculate I _{max}__

I_{RMS} = I_{max} / √2

Therefore I_{max} = I_{RMS} √2 = 0.86 x √2 = 1.22 A

__Answer Sub-Step 7: Calculate ΔV _{max}__

Substituting this value back in the above equations to find the individual maximum voltage drop across each device:

ΔV_{max} = I_{max} x R = 1.22 x 50 = 61V

ΔV_{max} = I_{max} x X_{C} = 1.22 x 133 = 162V

ΔV_{max} = I_{max} x X_{L} = 1.22 x 3.77 = 4.6V

__Answer Sub-Step 8: Calculate average power__

P_{avg} = I_{RMS}^{2} x R = (0.86)^{2} x 50 = 37W

### 5. 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.

### 6. References

[1] Boston University, Department of Physics. “AC Circuits”. __AC Circuits.__

[http://physics.bu.edu/~duffy/PY106/ACcircuits.html]. (05/02/2013)

[2] Missouri University of Science and Technology, Electrical and Computer Engineering Department. “Circuits Lab”. __Experiment Number 8: Capacitor Current –Voltage Relationship.__

[http://ece.mst.edu/media/academic/ece/documents/c/1/circuitslab8.pdf]. (05/02/2013)

[3] Georgia State University, Department of Physics and Astrology. “AC Circuits”. __RLC Series Circuit.__

[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcser2.html]. (05/02/2013)