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What is AC through Pure Resistance?

ac through pure resistance

In this article, we shall discuss the response of a purely resistive circuit to alternating current supply, i.e. AC through a pure resistance.

Consider an alternating voltage source v is applied across a resistor of resistance R as shown in figure-1.

ac through pure resistance

Let the alternating voltage is given by the following expression,

`\v=V_m  sin⁡ωt…(1)`

According to Ohm’s law, the electric current flowing through the resistor is given by,

`\i=v/R`

`\⇒i=(V_m  sin⁡ωt)/R`

`\∴i=I_m  sin⁡ωt…(2)`

Where Im = Vm/R is the maximum value of the current through the resistor R.

From equations (1) and (2), it is clear that in pure resistance, the voltage and current are in the same phase. The phasor diagram for the purely resistive AC circuit is shown in figure-2.

ac through pure resistance

The phasor diagram shows that the phase angle between the voltage across and current through a resistance is zero.

Power Relation for Purely Resistive AC Circuit:

The instantaneous power consumed by the resistance is given by,

`\p=v.i`

`\⇒p=(V_m  sin⁡ωt )(I_m  sin⁡ωt )`

`\⇒p=V_m I_m  sin^2⁡ωt`

`\∵sin^2⁡θ=(1-cos⁡2θ)/2`

`\∴p=(V_m I_m)/2 (1-cos⁡2ωt )…(3)`

Therefore, the total average power consumed by the resistor in one cycle is given by,

`\P=1/(2Ï€) ∫_0^(2Ï€) (p.dωt)`

`\⇒P=1/(2Ï€) ∫_0^(2Ï€) (V_m I_m)/2 (1-cos⁡2ωt ).dωt`

`\⇒P=(V_m I_m)/(4Ï€) ∫_0^(2Ï€) 1.dωt-(V_m I_m)/(4Ï€) ∫_0^(2Ï€) cos⁡(2ωt).dωt`

`\⇒P=(V_m I_m)/(4Ï€) [(1)-(sin⁡2ωt )]_0^(2Ï€)`

`\⇒P=(V_m I_m)/(4Ï€) (2Ï€)`

`\∴P=(V_m I_m)/2=V_m/\sqrt{2}  I_m/\sqrt{2}`

`\⇒P=VI…(4)`

Where V and I be the RMS value of applied voltage and resulting current respectively. From equation (3), it can be noted that the instantaneous power cannot be negative, and it has two components, i.e. a constant component and a pulsating component having a frequency double the supply voltage.

ac through pure resistance

The double frequency alternating component of the instantaneous power is equal to zero at ωt = 0° and 180°, and it is maximum at ωt = 90° and 270° as shown in figure-3.

Hence, this is all about alternating current through pure resistance.

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