Question

An electric current both A.C and D.C components. The value of the Dc. Components are equal to 12A. While the A.C component is given as $I = \;9\sin \omega tA$ . Determine the formula for the resultant current and also calculate the value of ${I_{rms}}$

Answer 1

Hint: Here, we are going to apply the concept of alternating current as well as direct current and determine the resultant current by adding the AC and DC component of the current. Use the formula for the rms value of the current to determine the rms current.

Complete step by step answer:
We will now write the formula for ${I_{rms}}$ i.e. the root mean square value of current.
${\left( {{I_{rms}}} \right)^2} = \dfrac{{\int\limits_0^T {{I^2}dt} }}{T}$
$ \Rightarrow {\left( {{I_{rms}}} \right)^2} = \dfrac{{\int\limits_0^T {{{\left( {12 + 9\sin \omega t} \right)}^2}dt} }}{T}\\$
$ \Rightarrow {\left( {{I_{rms}}} \right)^2} = \dfrac{1}{T}\left[ {144t + \dfrac{{216}}{\omega }\cos \omega t + \dfrac{{81t}}{2} + \dfrac{{81\sin \omega t}}{{4\omega }}} \right]_0^T\\$
$ \Rightarrow {\left( {{I_{rms}}} \right)^2} = \dfrac{{369}}{2}\\$
$ \Rightarrow {I_{rms}} = \sqrt {\dfrac{{369}}{2}} \;A$

Therefore, the RMS value of current is \[\sqrt {\dfrac{{369}}{2}} \;A\].

Additional information:
There are following advantages of alternating current over direct current
1. Long distance transmission can be done at high voltages, thereby reducing the current, and consequently losses in transmission.
2. Generation of alternating current is easier, and so is transmission.
3. Generation and transmission can be at high voltages, so the fact that it can be stepped up and down from level to other easily and efficiently. (we conveniently shift voltage by transformers ,in D.C. transformers are not used)
4. Cable sizes for AC are quite lighter in the comparison of D.C.
5. Power loss in AC is less as compared to DC during transmission.
6. AC can be easily converted into DC.
7. Voltage at which power can be sent can be higher in AC. These levels can be dangerous in DC.

Note: Also remember that alternating current can be defined as a current that changes its magnitude and polarity at regular intervals of time. It can also be defined as an electrical current which repeatedly changes or reverses its direction opposite to that of Direct Current or DC which always flows in a single direction.