First slide

Alternating Current and voltage

Question

If a direct current of “a” ampere is super imposed on an alternating current $I = b \sin ω t$ flowing through a wire, what is the effective value of resulting current in the circuit

Moderate

A

${(a^{2} - \frac{b^{2}}{2})}^{1 / 2}$
B

${(a^{2} + b^{2})}^{1 / 2}$
C

${(\frac{a^{2}}{2} + b^{2})}^{1 / 2}$
D

${(a^{2} + \frac{b^{2}}{2})}^{1 / 2}$

Solution

Given
Direct current $I_{d c} = a$
Alternative current $I_{a c} = b \sin ω t$
Total current
$\begin{array}{l} I = I_{d c} + I_{a c} \\ \Rightarrow I = a + b \sin ω t \\ \Rightarrow I_{rms}^{2} = \frac{\int i^{2} d t}{\int d t} = \frac{1}{T} \int_{0}^{T} (a + b \sin ω t)^{2} d t \\ \Rightarrow I_{rms}^{2} = \frac{1}{T} [\int_{0}^{T} a^{2} d t + \int_{o}^{T} b^{2} \sin^{2} ω t + \int_{0}^{T} 2 b a \sin ω t] \\ W e k n o w, \int_{0}^{T} \sin ω t d t = 0 a n d \\ \int_{0}^{T} \sin^{2} ω t d t = \frac{T}{2} \end{array}$
$\begin{array}{l} \Rightarrow I_{r m s}^{2} = \frac{1}{T} [a^{2} T + b^{2} \frac{T}{2} + 0] \\ ∴ I_{rms} = {[a^{2} + \frac{b^{2}}{2}]}^{1 / 2} \end{array}$

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