Question

The alternating current is given by \(i = \{\sqrt{42} \sin(\frac{2\pi}{T}t) + 10\} A\). The r.m.s. value of this current is _________ A.

Hint:

Remember that RMS values do not add directly. When dealing with a mix of AC and DC, you must use the formula \(I_{rms} = \sqrt{I_{dc}^2 + I_{ac,rms}^2}\). This is because power is proportional to the square of the current, and for non-interacting components (like DC and AC), the average powers add up.

Answer 1

Correct Answer: 11

Step 1: Understanding the Question:
The given current is a superposition of a sinusoidal alternating current (AC) and a direct current (DC). We need to find the root-mean-square (r.m.s.) value of this total current.
Step 2: Key Formula or Approach:
The r.m.s. value of a function \(i(t)\) over one period T is defined as \(I_{rms} = \sqrt{\frac{1}{T}\int_0^T [i(t)]^2 dt}\). For a current that is a sum of a DC component (\(I_{dc}\)) and an AC component (\(i_{ac}(t)\)), the total r.m.s. value is given by: \[ I_{rms} = \sqrt{I_{dc}^2 + (I_{ac,rms})^2} \] where \(I_{ac,rms}\) is the r.m.s. value of the AC component alone. For a sinusoidal current \(i_{ac}(t) = I_0 \sin(\omega t)\), its r.m.s. value is \(I_{ac,rms} = \frac{I_0}{\sqrt{2}}\), where \(I_0\) is the peak current.
Step 3: Detailed Explanation:
The given current is \(i(t) = 10 + \sqrt{42} \sin(\frac{2\pi}{T}t)\).
By comparing this with \(i(t) = I_{dc} + i_{ac}(t)\), we can identify the components:
The DC component is \(I_{dc} = 10\) A.
The AC component is \(i_{ac}(t) = \sqrt{42} \sin(\frac{2\pi}{T}t)\).
The peak value of the AC component is \(I_0 = \sqrt{42}\) A.
Now, we find the r.m.s. value of the AC component: \[ I_{ac,rms} = \frac{I_0}{\sqrt{2}} = \frac{\sqrt{42}}{\sqrt{2}} = \sqrt{\frac{42}{2}} = \sqrt{21} \text{ A} \] Finally, we calculate the total r.m.s. value of the current: \[ I_{rms} = \sqrt{I_{dc}^2 + (I_{ac,rms})^2} \] \[ I_{rms} = \sqrt{(10)^2 + (\sqrt{21})^2} \] \[ I_{rms} = \sqrt{100 + 21} = \sqrt{121} \] \[ I_{rms} = 11 \text{ A} \] Step 4: Final Answer:
The r.m.s. value of this current is 11 A.