Question 21.5: Average Power in an RLC Series Circuit GOAL Understand power...
Average Power in an RLC Series Circuit
GOAL Understand power in RLC series circuits.
PROBLEM Calculate the average power delivered to the series RLC circuit described in Example 21.4.
STRATEGY After finding the rms current and rms voltage with Equations 21.2 and 21.3,
I_{\mathrm{rms}}={\frac{I_{\mathrm{max}}}{\sqrt{2}}}=0.707I_{\mathrm{max}} [21.2]
\Delta V_{\mathrm{max}}=\frac{\Delta V_{\mathrm{max}}}{\sqrt{2}\mathrm{}}=\ 0.707\ \Delta V_{\mathrm{max}} [21.3]
substitute into Equation 21.17,
P_{\mathrm{av}}\equiv I_{\mathrm{rms}}\Delta V_{\mathrm{rms}}\cos\phi [21.17]
using the phase angle found in Example 21.4.
First, use Equations 21.2 and 21.3 to calculate the rms current and rms voltage:
I_{\mathrm{rms}}={\frac{I_{\mathrm{max}}}{\sqrt{2}}}={\frac{0.255\,\mathrm{A}}{\sqrt{2}}}=0.180\,\mathrm{A}
\Delta V_{\mathrm{rms}}={\frac{\Delta V_{\mathrm{max}}}{\sqrt{2}}}={\frac{1.50~\times~10^{2}\,\mathrm{V}}{\sqrt{2}}}=106\,\mathrm{V}
Substitute these results and the phase angle \phi=-64.8^{\circ} into Equation 21.17 to find the average power:
P_{a \text{v}}=I_{\mathrm{rms}}\,\Delta V_{\mathrm{rms}}\cos\phi=(0.180\,\mathrm{A})(106\,\mathrm{V})\,\cos\,(-64.8^{\circ})
= 8.12 W
REMARKS The same result can be obtained from Equation 21.16, { P}_{\mathrm{av}}=I_{\mathrm{rms}}^{2}R.
P_{\mathrm{av}}=I_{\mathrm{rms}}^{2}R [21.16]