Question

In a single-phase ac circuit, for a general load composed of RLC elements under sinusoidal-steady-state excitation, the average reactive power is given by

(a) ${\mathbf{V}}_{\mathbf{rms}}\mathbf{}{\mathbf{I}}_{\mathbf{rms}}\mathbf{}\mathbf{cos\varphi }$

(b) ${\mathbf{V}}_{\mathbf{rms}}\mathbf{}{\mathbf{I}}_{\mathbf{rms}}\mathbf{sin\varphi }$

(c) Zero

[Note: $\mathbf{\varphi }$ is the power-factor angle]

Short answer

Answer

The correct option is (b); that is,${\mathrm{V}}_{\mathrm{rms}}{\mathrm{I}}_{\mathrm{rms}}\mathrm{sin\varphi }$

Step-by-step solution

Define average reactive power

The average reactive power absorbed by a load is the product of the rms load voltage${\mathbf{V}}_{\mathbf{rms}}$ , the rms load current ${\mathbf{I}}_{\mathbf{rms}}$, and localid="1652496348445" $\mathbf{sin}\mathbf{(}\mathbf{\delta }\mathbf{‐}\mathbf{\beta }\mathbf{)}$ where $\mathbf{\delta }$ is the phase of the voltage waveform and $\mathbf{\beta }$ is the phase of the current waveform.

Use the theory of reactive power for RLC load

For a$0°$phase of the voltage waveform, $\delta =0°$. Then, the power factor becomesrole="math" localid="1652495753247" $\sin \left(0°-\beta \right)=\sin \varphi$, where $\varphi$ is the relative phase of the current waveform. Thus, the average reactive power absorbed by an RLC load is defined as follows:

$Q={\mathrm{V}}_{\mathrm{rms}}{\mathrm{I}}_{\mathrm{rms}}\mathrm{sin\varphi }$ …… (1)

We know from the above equation that the average power absorbed by an RLC load is ${\mathrm{V}}_{\mathrm{rms}}{\mathrm{I}}_{\mathrm{rms}}\mathrm{sin\varphi }$. Thus, Option (b) is correct.

Explanation for Option (a)

We know from Equation (1) that the average power absorbed by an RLC load is ${\mathrm{V}}_{\mathrm{rms}}{\mathrm{I}}_{\mathrm{rms}}\mathrm{sin\varphi }$ . However, the average power provided by Option (a) is a function of the power factor $\cos \varphi$ which is not equal to${\mathrm{V}}_{\mathrm{rms}}{\mathrm{I}}_{\mathrm{rms}}\mathrm{sin\varphi }$ Thus, Option (a) is incorrect.

Explanation for Option (c)

We know from Equation (1) that the average power absorbed by an RLC load is ${\mathrm{V}}_{\mathrm{rms}}{\mathrm{I}}_{\mathrm{rms}}\mathrm{sin\varphi }$. However, Option (c) has zero for the average power. Therefore, it is an incorrect option.