Question

In column I, variation of current $i$ with time $t$ is given in the figure. In column II, root mean square current $i_{\text{rms }}$ and average current are given. Match column I with corresponding quantities given in column II

Column I		Column II
i		a	$i_{rms}=\frac{i_0}{\sqrt{3}}$
ii		b	Average current for positive half cycle is $i_0$
iii		c	Average current for positive half cycle is $\frac{i_0}{2}$
iv		d	Full cycle average current is zero

• A. i. $\to$ a.; ii. $\to$ b., d.; iii. $\to$ b., c.; iv. $\to$ b.
• B. i. $\to$ d.; ii. $\to$ a., d.; iii. $\to$ b., d.; iv. $\to$ b.
• C. i. $\to$ b.; ii. $\to$ a., c.; iii. $\to$ c., d.; iv. $\to$ d.
• D. i. $\to$ c.; ii. $\to$ c., d.; iii. $\to$ b., d.; iv. $\to$ b.

Answer 1

Answer: (B)

i. For sinusoidal curve $i_{rms}=i_0/\sqrt{2}$ and average current for positive half cycle is $2i_0/\pi$.
ii. Let us find the average current (or rms current) for $t=0$ to $T/4$. Then it will also be average current (or rms current) for half cycle, i.e., from $t=0$ to $T/2$.
For $t=0$ to $T/4:i=\frac{4i_0}{T}t$
then $i_{av}=\frac{\underset{0}{\overset{T/4}{\int }}idt}{T/4}=\frac{\underset{0}{\overset{T/4}{\int }}\frac{4i_0}{T}tdt}{T/4}=\frac{i_0}{2}$
and $i_{ms}=\sqrt{\frac{\underset{0}{\overset{T/4}{\int }}{i}^{2}dt}{T/4}}=\sqrt{\frac{64i_0^2}{T^3}\underset{0}{\overset{T/4}{\int }}{t}^{2}dt}=\frac{i_0}{\sqrt{3}}$
Full cycle average current is zero.
iii. For positive half cycle, average current
$=\frac{\int idt}{\int dt}=\frac{i_0(T/2)}{T/2}=i_0$
Full cycle average current is zero.
iv. Average current for positive half cycle is $i_0$.
For full cycle, average current
$=\frac{\int idt}{\int dt}=\frac{i_0(T/2)+0}{T}=\frac{i_0}{2}$
and $i_{rms}=\sqrt{\frac{\underset{0}{\overset{T/2}{\int }}{i}_0^{2}dt}{T}}=\frac{i_0}{\sqrt{2}}$