## Objectives

- To understand and calculate the power factor of a passive circuit.
- To verify that resistive components dissipate power while reactive components do not.

## Equipment

- Breadboard
- Function generator
- Oscilloscope
- Digital multimeter (DMM)

## Background

The steady-state response is the response that exists after the initial conditions and transient or natural response die out. AC steady state analysis determines the steady state response of a circuit when the inputs are sinusoidal functions. The steady state voltages and currents in the circuit will also be sinusoidal, with the same frequency as the input signal. The maximum amplitude and phase angle of the steady state response will, in general, differ from that of the source.

The angle referred to as the power factor angle, is involved in the calculation of the average and reactive power. The power factor is the cosine of this angle

A lagging power factor implies that the currents lags the voltage, hence an inductive load. A leading power factor implies current leads the voltage, hence a capacitive load. The power factor (PF) is the ratio of the average power to the apparent power. The average power absorbed by the element is calculated by,

Eq 6-1 |

and the apparent power is given by,

Eq 6-2 |

where V_{m} and I_{m} are the magnitudes of the voltage and current respectively, V_{RMS} and IRMS are the rms RMS values voltage and current respectively, θ_{V} and θ_{I} are the phase angles of the voltage and current respectively.

The average power (often simply called ‘power’) dissipated in a circuit with a periodic input signal of period T is defined as

Eq 6-4 |

The average power dissipated by a resistor is given by

Eq 6-5 |

and the average power dissipated by reactive components, such as inductors and capacitors, is equal to zero. In other words, reactive components are storage elements and do not dissipate power.

# Preparation

Consider the circuit in Figure 6 – 1 with V_{in} magnitude of 5 V and frequency of 3 kHz.

- Calculate the total impedance of the circuit.
- Calculate the power factor of the circuit.
- Calculate the voltage and current (they are both complex numbers) in each element in the circuit.
- Calculate the average power dissipated in each element in the circuit.
- Calculate the total average power provided by the source.
- Verify that the power generated by the source equals the total power dissipated in all the components in the circuit.
- Assume that and (V
_{x }and V_{xm }are the magnitudes of V_{in}and V_{x}respectively, is the phase angle between V_{in}and V_{x}, and they can be read on the oscilloscope). Prove that the following equation is valid,

Eq 6-6 |

where is the power factor angle of the source. Hence the power factor can be found.

**Figure 6 – 1 Circuit**

## Simulation

- In Multisim, build and simulate the circuits in Figure 6 – 1. Set the input voltage to and frequency to 3 kHz.
- Determine the voltage and current in each element using a DMM. Remember, the numbers you obtained are in RMS.
- Determine the average power dissipated in each element.
- Determine the total power provided by the voltage source (or the total power consumed by the circuit).
- Compare all the results from step 2 to 4 with those obtained in PREPARATION.
- Verify the law of conservation of energy.
- Determine V
_{m}, V_{xm }and (as in ) using oscilloscope and cursors. Calculate the power factor using the equation in PREPARATION step 7. Compare this result with that from PREPARATION step 2.

# Experiment

- Build the circuit on breadboard, and use the same input settings as in SIMULATION. Place the Ch1 probe at V
_{in }and the Ch2 probe at V_{x}. Display both V_{in }and V_{x }on the screen. These two curves should have the same frequency but with a phase shift between each other. - Measure V
_{m }and V_{xm }using the oscilloscope. Note that is measured indirectly. First measure the time difference between the two peaks of V_{in }and V_{x}. Expand the time scale in order to get a better reading with the cursors. Then use the equation (rad) to calculate . Pay attention to the unit. Convert to degrees if needed. - Calculate the power factor using the equation in PREPARATION step 7.
- Measure the voltage and current in each element using a DMM. How does this result compare with that from PREPARATION and SIMULATION?
- Calculate the average power dissipated in each element using the results (voltage and/or current) from the last step. Compare the result with that from PREPARATION and SIMULATION.
- Measure the total voltage and current provided by the voltage source.
- Calculate the total power provided by the voltage source. Compare this result with that from PREPARATION and SIMULATION.
- Compare the total power delivered by the voltage source with the total power dissipated by all the elements in the circuit. Explain your result (e.g. what element contributes to the power dissipation and what element doesn’t).

# REPORT

Prepare your report as per the guidelines given in APPENDIX III.