The rms value of the resultant current in a wire which carries a dc current of 20 A and a sinusoidal alternating current of peak value 20 A is

Question

Prepp Team · Accepted Answer

When a wire carries both a direct current (DC) and an alternating current (AC) simultaneously, the resultant current's root mean square (RMS) value is calculated by considering the effective power dissipation of both components. The RMS value is a crucial concept in electrical engineering as it represents the equivalent DC current that would produce the same heating effect as the varying current.

Resultant Current Overview

We are given two distinct components of current flowing through the wire:

DC Current: This current has a constant magnitude.
Sinusoidal Alternating Current: This current varies sinusoidally with time, and its peak value is provided.

DC Current RMS Value

For a direct current (DC), its RMS value is identical to its constant magnitude. In this problem, the DC current ($I_{dc}$) is given as 20 A.

Therefore, the RMS value of the DC current, $I_{rms, dc} = 20 \text{ A}$.

Sinusoidal AC Current RMS Value

For a purely sinusoidal alternating current, the relationship between its RMS value ($I_{rms, ac}$) and its peak value ($I_{peak, ac}$) is defined by the formula:

$I_{rms, ac} = \frac{I_{peak, ac}}{\sqrt{2}}$

The peak value of the sinusoidal AC current is given as 20 A.

Substituting this value into the formula:

$I_{rms, ac} = \frac{20}{\sqrt{2}}$

Since $\sqrt{2} \approx 1.414$, we calculate:

$I_{rms, ac} \approx \frac{20}{1.414}$

$I_{rms, ac} \approx 14.14 \text{ A}$

Resultant Current Calculation

To find the RMS value of the total (resultant) current when both DC and AC components are present, we use the principle of superposition based on power. The total average power dissipated in the wire is the sum of the average powers dissipated by the DC and AC currents independently. This leads to the formula for the resultant RMS current ($I_{rms, resultant}$):

$I_{rms, resultant} = \sqrt{I_{rms, dc}^2 + I_{rms, ac}^2}$

Now, let's substitute the calculated RMS values of the DC and AC currents into this formula:

$I_{rms, resultant} = \sqrt{(20)^2 + (14.14)^2}$

First, square the individual RMS values:

$20^2 = 400$

$14.14^2 \approx 199.9396 \approx 200$

Next, sum these squared values:

$I_{rms, resultant} = \sqrt{400 + 200}$

$I_{rms, resultant} = \sqrt{600}$

Finally, calculate the square root:

$I_{rms, resultant} \approx 24.4948 \text{ A}$

Rounding this value to one decimal place, we get approximately 24.5 A.

Current Values Summary

Current Type	Given Value	Calculated RMS Value
DC Current	$I_{dc} = 20 \text{ A}$	$I_{rms, dc} = 20 \text{ A}$
Sinusoidal AC Current	$I_{peak, ac} = 20 \text{ A}$	$I_{rms, ac} \approx 14.14 \text{ A}$
Resultant Current	Not applicable	$I_{rms, resultant} \approx 24.5 \text{ A}$