Problem 51-2    Source:  Problem 30 - page 403 -   BOYLESTAD, Robert - Book: Introdução à Análise de Circuitos - 10ª edição - Ed. Pearson Education do Brasil - 2004.

Calculate the phase difference between the waveforms of each pair below:

a)  v = 2 cos (ωt - 30°)   and   i = 5 sin (ωt + 60°)

b)  v = - sin (ωt + 20°)   and   i = 10 sin (ωt - 70°)

c)  v = -4 cos (ωt + 90°)   and   i = -2 sin (ωt + 10°)

Solution of the Problem 51-2   

Item a   

Note that in this item v and i are written to different functions. Therefore, you should write them as sine or cosine function. It was chosen to write the i function as a cosine function. Since the cosine function is always advanced by 90° with respect to the sine function, then i = 5 cos (ωt + 60° - 90°) = 5 cos (ωt - 30°). Soon, knowing that:

Making the substitution by the values of the angles we have:

Conclusion: v e i are in phase.

Observation -    The function v could have been written as a sine function. Since sine is delayed by 90° in relation to cosine, then v = 2 sin (ωt - 30° + 90°) = 2 sin (ωt + 60°). Comparing the two functions we conclude that they are in phase, or θ = 0°.

Item b   

In this case, the negative sign of the function v can only be removed by adding an angle of 180 °, or v = sin (ωt + 20° + 180°) = sin (ωt + 200°). Now there are two sine functions, so the angle of lag can be calculated, or:

Thus, we can say that voltage is advanced by 270° in relation to current or, what is the same thing, voltage is delayed by 90° in relation to the current .

Another way to solve the problem is to subtract 180° from the function v, that is, v = sin (ωt + 20° - 180°) = sin (ωt - 160°) . Therefore, the angle between v and i will be of:

The same conclusion is similar from the previous solution: the voltage is delayed by 90° in relation to current i.

Item c   

Writing v as a sine function, you must subtract 90°, that is, v = - 4 sin (ωt + 90° - 90°) = - 4 sin (ωt + 0°) . As v and i are written with the same function, we can calculate the angle of lag, or:

Thus, it can be stated that the voltage is delayed by 10° in relation to the current or, what is the same thing, the current is advanced by 10° in relation to the voltage.

Observation -    The function i could have been written as a cosine function. Because cosine is advanced by 90° in relation to sine, then i = - 2 cos (ωt + 10° + 90°) = - 2 cos (ωt + 100°). Performing the calculation, θ = 90° - 100° = - 10°. That is, the same conclusion is reached.