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Phasors Explained
Learning Goal: To understand the concept of phasor diagrams and be able to use them to analyze AC circuits (those with sinusoidally varying current and voltage).
Phasor diagrams provide a convenient graphical way of representing the quantities that change with time along with \cos(\omega t+\phi), which makes such diagrams useful for analyzing AC circuits with their inherent phase shifts between voltage and current. You have studied the behavior of an isolated resistor, inductor, and capacitor connected to an AC source. However, when a circuit contains more than one element (for instance, a resistor and a capacitor or a resistor and an inductor or all three elements), phasors become a useful tool that allows us to calculate currents and voltages rather easily and also to visualize some important processes taking place in the AC circuit, such as resonance.

Let us assume that a certain quantity I(t) changes over time as I(t)=I_0\cos(\omega t). A phasor is a vector whose length represents the amplitude I_0 (see the diagram ).This vector is assumed to rotate counterclockwise with angular frequency omega; that way, the horizontal component of the vector represents the actual value I(t) at any given moment.

In this problem, you will answer some basic questions about phasors and prepare to use them in the analysis of various AC circuits.
In parts A - C consider the four phasors shown in the diagram . Assume that all four phasors have the same angular frequency omega.
Part A
At the moment t depicted in the diagram, which of the following statements is true?
ANSWER:
I_2 leads I_1 by \pi.
I_1 leads I_2 by \pi.
I_2 leads I_1 by \frac{\pi}{2}.
I_1 leads I_2 by \frac{\pi}{2}.
Part B
At the moment shown in the diagram, which of the following statements is true?
ANSWER:
I_2 lags I_3 by \pi.
I_3 lags I_2 by 2\pi.
I_2 lags I_3 by \frac{\pi}{2}.
I_3 lags I_2 by \frac{\pi}{2}.
Part C
At the moment shown in the diagram, which of the following statements is true?
ANSWER:
I_2 leads I_4 by \pi.
I_2 lags I_4 by 2\pi.
I_2 leads I_4 by \frac{\pi}{2}.
I_2 lags I_4 by \frac{\pi}{2}.
Let us now consider some basic applications of phasors to AC circuits.
  • For a resistor, the current and the voltage are always in phase.
  • For an inductor, the current lags the voltage by \frac{\pi}{2}.
  • For a capacitor, the current leads the voltage by \frac{\pi}{2}.
Part D
Consider this diagram. Let us assume that it describes a series circuit containing a resistor, a capacitor, and an inductor. The current in the circuit has amplitude I, as indicated in the figure.

Which of the following choices gives the correct respective labels of the voltages across the resistor, the capacitor, and the inductor?

ANSWER:
V_1; V_2; V_3
V_1; V_2; V_4
V_1; V_4; V_2
V_3; V_2; V_4
V_3; V_4; V_2
Part E

Now consider a diagram describing a parallel AC circuit containing a resistor, a capacitor, and an inductor. This time, the voltage across each of these elements of the circuit is the same; on the diagram, it is represented by the vector labeled V_0.

The currents in the resistor, the capacitor, and the inductor are represented respectively by which vectors?
ANSWER:
I_1; I_2; I_3
I_1; I_3; I_4
I_1; I_4; I_2
I_1; I_2; I_4
I_1; I_3; I_2
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