Calculating Power Factor in an AC circuit, given voltage and current

[bardia sepehrnia]

Homework Statement

Calculate the power factor for each case:
v(t) = 540 cos(ωt + 15◦) V, i(t) = 2 cos(ωt + 47◦) A
v(t) = 155 cos(ωt − 15◦) V, i(t) = 2 cos(ωt − 22◦) A

Relevant Equations

Vrms=V/sqrt(2)
Irms=I/sqrt(2)

My attempt at solving this question:

I realized my attempt is wrong however I just don't know how to proceed in the first step. How can I calculate the phase shift? and find Voltage and Current in phasor form??If I know that, then I can use power factor formula: pf=Pav/V*I

 

[Babadag]

The plane in which voltages and currents are represented rotates with ω radians per seconds. The angles between voltages and currents are constant and rotate all the time with the same velocity. Power factor it is the cos(ϕ).

 

[bardia sepehrnia]

Yeah, I realized I just had to subtract the voltage angle from the current one and take cosine of that. Thank you.

 

[bardia sepehrnia]

The plane in which voltages and currents are represented rotates with ω radians per seconds. The angles between voltages and currents are constant and rotate all the time with the same velocity. Power factor it is the cos(ϕ).

can you also help me with this one pls?

I understand that the first 2 are capacitive as power factors are leading. In C, I think I understand why it's inductive because first if we change current equation to cosine, then: i(t)=4.2 sin(ωt-(pi/2)), and the phase angle is 0-(-pi/2)=pi/2, and positive phase angle indicates that the load is inductive. If that reasoning is true, then d, should i also be inductive, but it is resistive. I thought the phase angle has to be for the load to be resistive.

 

[Babadag]

You are right.

 

[bardia sepehrnia]

Thank you very much!