AC Voltage Applied to a Resistor: AC Voltage and Equations

When voltage changes its direction after every half cycle is known as alternating voltage. The current flows in the circuit at that time are known as alternating current. The alternating current(AC) follows the sine function which changes its polarity concerning time. Most of the electrical devices are operating on the ac voltage.

Resistance is a substance that produces obstructions in the flow of electric current. It is usually denoted by R and its SI unit is Ohm. In DC voltage the resistance and current are related to each other using Ohm’s Law. In this article, we will understand how to apply Ohm’s law when the voltage applied is in the form of AC.

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AC Voltage Applied to a Resistor

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Consider an alternating current source that produces potential difference across both terminals and responsible for the flow of electric current. Its representation is given in the figures below:

Let the voltage be given by:

V = V₀ sin ωt ----(1)

Here V₀ = maximum amplitude of voltage 

ω = angular frequency

On applying Kirchhoff’s law to find the value of current in the resistance, we get:

∑ V(t) = 0

Equation (1) hence becomes:

V₀sin ωt = I. R

Or

I = V₀sin ωt / R 

According to Ohm’s Law:

V = I. R Or V/R = I

Then,

V₀/R = I₀

Hence, we obtain the following relationship between current, resistance and voltage in an AC circuit:

I = I₀sin ωt

The above equation shows that Ohm’s law is equally working on the AC and DC voltages. The values of AC and DC reach their maximum and minimum simultaneously. The current and voltage in the ac circuit with resistor follow sinusoidal wave function having in the same phase.

The video below explains this:

AC voltage applied to a resistor Detailed Video Explanation:

Also Read:

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Average Value of Current in R – Circuit

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The average or mean value of an alternating current is the total current flown in one complete cycle in a given time dividing by the time required to complete one cycle.

In other words, the average alternating current value is the average of every instantaneous value of AC from origin to its peak value.

i_avg = 0∫^Ti (t) dt/ ₀∫^Tdt

i_avg = 1/T ₀∫^T i (t) dt

i_avg = 1/T ₀∫^T / 2 i₀sin ωtdt +1/T  _T/2∫^T i₀sin ωtdt

i_avg = i₀/T (- cos ωt/ω)₀/2 + i₀/T (-cos ωt/ω)₀/2^T

i_avg = i₀/Tω (cosπ – cos0 – cosπ + cos²π)

Hence

i_avg = 0

It is clear from the above equation that the average value of alternating current in one complete cycle is zero because it is positive in the first half cycle and negative in a second-half cycle with equal magnitude.

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Root Mean Square (RMS) Value of Current in R – Circuit

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Root mean square or RMS value is defined as the amount of direct current passing through the resistance in the circuit for a given time that will produce the same amount of heat as the alternating current does in the same resistance at the same time. Mathematical derivations of I_rms is given below:

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Average value of power in AC circuit having a resistor

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p = i². R = (i₀)².R sin² ωt

here i₀ and R are constants

using the trigonometric equation we have

sin ²ωt = ½ (1 – cos ωt)

since cos 2ωt = 0 then we can write

sin ² ωt = ½

p = ½ i₀². R

The presentation below highlights these basic concepts:

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