If a direct current of a ampere is superimpos | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The resulting current is $I = a + b \sin \omega t$.The effective $(RMS)$ value of the current is defined as $I_{rms} = \sqrt{\frac{1}{T} \int_{0}^

Vedclass · Accepted Answer

$[a^2 + \frac{1}{2}b^2]^{1/2}$

Vedclass · Answer

(B) The resulting current is $I = a + b \sin \omega t$.The effective $(RMS)$ value of the current is defined as $I_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} I^2 dt}$.Squaring both sides,we get $I_{rms}^2 = \frac{1}{T} \int_{0}^{T} (a + b \sin \omega t)^2 dt$.Expanding the integrand: $I_{rms}^2 = \frac{1}{T} \int_{0}^{T} (a^2 + b^2 \sin^2 \omega t + 2ab \sin \omega t) dt$.Integrating over one full time period $T$: $\int_{0}^{T} a^2 dt = a^2 T$$\int_{0}^{T} b^2 \sin^2 \omega t dt = b^2 \int_{0}^{T} \frac{1 - \cos 2\omega t}{2} dt = \frac{b^2 T}{2}$$\int_{0}^{T} 2ab \sin \omega t dt = 0$ (since the average of a sine wave over a full cycle is zero).Substituting these values: $I_{rms}^2 = \frac{1}{T} (a^2 T + \frac{b^2 T}{2} + 0) = a^2 + \frac{b^2}{2}$.Therefore,$I_{rms} = \sqrt{a^2 + \frac{b^2}{2}}$.

If a direct current of $a$ ampere is superimposed with an alternating current $I = b \sin \omega t$,then the effective value of the resulting current is

Similar Questions

The r.m.s. voltage of domestic electricity supply is $220 \ V$. Electrical appliances should be designed to withstand an instantaneous voltage of......$V$.

For a domestic $AC$ supply of $220 \ V$ at $50 \ \text{cycles per sec}$, the potential difference between the terminals of a two-pin electric outlet in a room is given by

What is the approximate percentage value of the ratio of maximum voltage to its rms value in an $LCR$ $AC$ circuit (in $\%$)?

Define the root mean square (rms) value of an alternating current and provide its formula. Also,show the relationship between the rms current and the peak current on a graph of current versus $\omega t$.

$A$ direct current of $5\,A$ is superposed on an alternating current $I = 10 \sin \omega t$ flowing through a wire. The effective value of the resulting current will be

Vedclass Test Series

Exam Paper Generator

Online Exam Module