Question

The instantaneous power absorbed by the load in a single-phase ac circuit, for a general RLC load under sinusoidal-steady-state excitation, is

(a) Nonzero constant

(b) Zero

(c) Containing double-frequency components

Short answer

Answer

Therefore the correct option is (c) containing double frequency components.

Step-by-step solution

Define the voltage and current

Consider that the instantaneous voltage is $\mathbf{\text{V}}\left(t\right)\mathbf{=}{\mathbf{V}}_{\mathbf{max}}\mathbf{}\mathbf{cos}\left(\mathrm{\omega t}+\delta \right)\mathbf{}$and the instantaneous current is $\mathbf{I}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{I}}_{\mathbf{max}}\mathbf{}\mathbf{cos}\mathbf{(}\mathbf{\omega t}\mathbf{+}\mathbf{\delta }\mathbf{)}\mathbf{}$ , at constant frequency $\omega$.

Use the definition of instantaneous power

The instanetaneous power is the product of the instantaneous voltage and current, defined as,

$$\begin{gathered} P\left(t\right)=v\left(t\right)I\left(t\right) \\ ={\mathrm{V}}_{\max }\cos (\mathrm{\omega t}+\mathrm{\delta }){\mathrm{I}}_{\max }\cos (\mathrm{\omega t}+\mathrm{\delta }) \\ ={\mathrm{V}}_{\max }{\mathrm{I}}_{\max }{\cos }^2(\mathrm{\omega t}+\mathrm{\delta }) \\ ={\mathrm{V}}_{\max }{\mathrm{I}}_{\max }\left(\frac{1+\cos (2\mathrm{\omega t}+\mathrm{\delta })}{2}\right) \end{gathered}$$ …….. (1)

Rewrite equation (1) as,

$\mathrm{P}\left(\mathrm{t}\right)=\frac{{\mathrm{V}}_{\max }{\mathrm{I}}_{\max }}{2}+\frac{{\mathrm{V}}_{\max }{\mathrm{I}}_{\max }\cos \left(2\mathrm{\omega t}+\mathrm{\delta }\right)}{2}$ …… (2)

From equation (2), the instantaneous power is a function of double frequency components. Thus option (c) is correct.

Explanation for option (a)

From equation (2), the instantaneous power is a function of double frequency components.

Since, the instantaneous power is not a function of nonzero constant. Thus the option (a) is incorrect.

Explanation for option (b)

From equation (2), the instantaneous power is a function of double frequency components.

Since, the instantaneous power is not zero. Thus the option (b) is incorrect.