To find the RMS (Root Mean Square) value of the resultant current $i_R$ for two parallel-connected electric devices, we first need to find the resultant current equation and then compute its RMS value.

Step 1: Write the current equations

You are given the two currents:

1. $i_1 = 11.3 \sin (377t + \frac{\pi}{4})$
2. $i_2 = 8.5 \sin (377t - \frac{\pi}{4})$

We can find the resultant current $i_R$ by adding the two sinusoidal currents. However, first, we'll express both currents in a way that allows for easy addition.

Step 2: Use trigonometric identities

We use the following identity for the sum of two sinusoids:

$A \sin(x + \phi_1) + B \sin(x + \phi_2) = \sqrt{A^2 + B^2 + 2AB \cos(\phi_1 - \phi_2)} \sin\left(x + \phi_{\text{res}}\right)$

Where:

• $A = 11.3$, $\phi_1 = \frac{\pi}{4}$
• $B = 8.5$, $\phi_2 = -\frac{\pi}{4}$

So, the phase difference $\phi_1 - \phi_2 = \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{2}$.

Step 3: Calculate the amplitude of the resultant current

Now, we can calculate the amplitude of the resultant current:

$I_{\text{res}} = \sqrt{(11.3)^2 + (8.5)^2 + 2(11.3)(8.5) \cos\left(\frac{\pi}{2}\right)}$

Since $\cos\left(\frac{\pi}{2}\right) = 0$, the formula simplifies to:

$I_{\text{res}} = \sqrt{(11.3)^2 + (8.5)^2}$

$I_{\text{res}} = \sqrt{127.69 + 72.25} = \sqrt{199.94} \approx 14.14$

Thus, the amplitude of the resultant current is approximately $I_{\text{res}} = 14.14$ A.

Step 4: Compute the RMS value of the resultant current

The RMS value of a sinusoidal current is given by:

$I_{\text{rms}} = \frac{I_{\text{res}}}{\sqrt{2}}$

Now, calculating the RMS value:

$I_{\text{rms}} = \frac{14.14}{\sqrt{2}} \approx \frac{14.14}{1.414} \approx 10.0 \, \text{A}$

Final Answer:

The RMS value of the resultant current is approximately 10.0 A.

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Would you like to dive deeper into the derivation, or have any more questions about this problem?

Related questions to explore:

1. How do phase differences affect the amplitude of resultant currents?
2. What happens if the two currents have the same phase angle?
3. How do you calculate the RMS value for non-sinusoidal waveforms?
4. What is the relationship between peak current and RMS current in AC circuits?
5. Can this method be applied to voltages as well?

Tip: For sinusoidal signals, always remember that the RMS value is $\frac{\text{Amplitude}}{\sqrt{2}}$, a useful formula for power calculations in AC circuits.