a) Suppose that between the two terminals of an element there is a voltage v (t) and that a current i (t) enters through the terminal marked with a positive sign. Show that if v (t) and i (t) are sinusoidal and are out of phase by an angle ϕ, then the active power P that is consumed by said element is proportional to cos ϕ. b) How much is P for a pure inductor and how much for a pure capacitor? Does this make physical sense? c) When specifying the power factor seen between two terminals, why is it necessary to indicate whether it is lagging or leading?
Answer the following points in detail and rigorously. Instead of resorting to formulas, develop from basic principles (laws of voltages and currents, definition of electrical power, properties of phasors, among others).
a) Suppose that between the two terminals of an element there is a voltage v (t) and that a current i (t) enters through the terminal marked with a positive sign. Show that if v (t) and i (t) are sinusoidal and are out of phase by an angle ϕ, then the active power P that is consumed by said element is proportional to cos ϕ.
b) How much is P for a pure inductor and how much for a pure capacitor? Does this make physical sense?
c) When specifying the power factor seen between two terminals, why is it necessary to indicate whether it is lagging or leading?
d) Show that the complex power consumed by two elements in parallel is the sum of the individual complex powers. Then repeat the procedure but for two elements in series.
e) What meaning would you give to the P consumed by an element? Which to Q?
f) Suppose you know P and Q for a two-terminal element. What is the power factor of this element? When would you be ahead and when would you be behind?
g) Suppose you know the equivalent resistance R and the equivalent reactance X of a two-terminal passive network. What is the power factor of this network? When would you be ahead and when would you be behind?
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