An alternating current is given by the equation i = i1 sin(wt) + i2 cos(wt). The rms current will be

Q: Why can't we just add the RMS values of the two currents directly?

Because the currents are not in phase. They are orthogonal (phase difference of $90^\circ$), so their RMS values add in quadrature, much like perpendicular vectors.

Q: Does this method work if the frequencies are different?

No, the phasor method and this specific RMS addition only work if the two superimposed waves have the exact same angular frequency $\omega$.

Q: Why is the cosine term drawn at $90^\circ$ to the sine term?

Because mathematically, $\cos\omega t = \sin(\omega t + 90^\circ)$. In a phasor diagram, a positive phase shift corresponds to a counter-clockwise rotation.

Q: What is the physical meaning of the RMS value?

The RMS (Root Mean Square) value is the equivalent DC current that would produce the same average heating effect in a resistor over a complete cycle.

Q: Can we solve this using integration instead of phasors?

Absolutely! You can square the entire expression, integrate it over one time period, and divide by the time period. The cross term $\sin\omega t \cos\omega t$ will integrate to zero, leaving the sum of the squares of the individual RMS values.

Question

An alternating current is given by the equation $i = i_1 \sin\omega t + i_2 \cos\omega t$. The rms current will be

Options:

(A) $\frac{1}{\sqrt{2}} (i_1^2 + i_2^2)^{1/2}$

(B) $\frac{1}{\sqrt{2}} (i_1 + i_2)^2$

(C) $\frac{1}{2} (i_1^2 + i_2^2)^{1/2}$

(D) $\frac{1}{\sqrt{2}} (i_1 + i_2)$

Sigma Percentile · Accepted Answer

The problem asks us to find the RMS (Root Mean Square) value of an alternating current that is given as a sum of a sine and a cosine function. At first glance, it might seem like we need to dive into complex integration, but there is a much more elegant and visual way to solve this: **The Phasor Method**.

### The Superposition of Currents

Imagine the given current equation:
$$i = i_1 \sin\omega t + i_2 \cos\omega t$$

This equation tells us that the total current is the superposition of two separate alternating currents. The first current, $i_1 \sin\omega t$, oscillates with a peak value of $i_1$. The second current, $i_2 \cos\omega t$, oscillates with a peak value of $i_2$.

But here is the crucial detail: they are not oscillating in sync! Because one is a sine function and the other is a cosine function, there is a phase difference of exactly $90^\circ$ (or $\frac{\pi}{2}$ radians) between them.

### The Power of Phasors

To visualize this, we can use a phasor diagram. A phasor is simply a rotating vector that represents a sinusoidal quantity.

Let's take the sine term as our reference. We draw a vector of length $i_1$ along the positive x-axis. Now, where does the cosine term go? Since $\cos\omega t = \sin(\omega t + 90^\circ)$, the cosine term leads the sine term by $90^\circ$. Therefore, we draw a vector of length $i_2$ along the positive y-axis.

### Calculating the Peak Current

The total current is the vector sum of these two phasors. Because they are at a right angle to each other, they form a right-angled triangle. The resultant phasor, which represents the peak value of the total current ($I_0$), is the hypotenuse of this triangle.

Using the Pythagorean theorem, we can easily find the length of this resultant phasor:
$$I_0 = \sqrt{i_1^2 + i_2^2}$$

This $I_0$ is the maximum amplitude that the combined alternating current will ever reach.

### The RMS Connection

Now that we have the peak current, finding the RMS current is a breeze. For any purely sinusoidal alternating current, the RMS value is simply the peak value divided by $\sqrt{2}$.

$$I_{rms} = \frac{I_0}{\sqrt{2}}$$

Substituting the value of $I_0$ we just found, we get:
$$I_{rms} = \frac{\sqrt{i_1^2 + i_2^2}}{\sqrt{2}}$$

We can rewrite this expression to match the options provided in the question:
$$I_{rms} = \frac{1}{\sqrt{2}} (i_1^2 + i_2^2)^{1/2}$$

And there we have it! By visualizing the currents as vectors, we bypassed the tedious integration and arrived at the correct answer elegantly.

Animated Solution for Physics - Electromagnetic Induction: An alternating current is given by the equation $i = i_{1} sin ω t + i_{2} cos ω t$ . The rms current will be

Select Answer:

Visualized Solution

The Alternating Current Equation

Phasor Representation of $i_{1}$ and $i_{2}$

Resultant Peak Current $I_{0}$

RMS Value Formula

Final RMS Current $I_{r m s}$

General Superposition for Phase $ϕ$

The Sigma Insight: Alternating Current (AC) and Voltage

The Superposition of Currents

The Power of Phasors

Calculating the Peak Current

The RMS Connection

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Animated Solution for Physics - Electromagnetic Induction: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } An alternating current is given by the equation i=i1​sinωt+i2​cosωt. The rms current will be